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The Luminescent Solar Concentrator
Citation
Batchelder, John Samuel
(1982)
The Luminescent Solar Concentrator.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/W5W7-9660.
Abstract
The Luminescent Solar Concentrator (LSC) allows sunlight to be concentrated through the use of light pipe trapping of luminescence. Such concentrators do not require tracking, and they can reduce the cost of solar energy conversion by reducing the required area of photovoltaic cells. We have conducted the following experimental and theoretical investigations in order to optimize the LSC's performance.
The spectral characteristics of 18 organic laser dyes are studied for their applicability as luminescing centers. The spectral homogeneity and self-absorption characteristics of representative dyes are examined in detail. The relative spectral homogeneity of such dyes is shown to depend upon the surrounding material using narrow band laser excitation. We develop three independent techniques for measuring self-absorption rates; these are time-resolved emission, steady state polarization anisotropy, and spectral convolution. Prototype devices are tested for performance, and the componant dyes are tested for stability to solar exposure.
A model is developed which predicts the efficiency and gain of and LSC from the spectroscopic characteristics of its components. A critical optical density (CODE) is assigned to the dyes surveyed which predicts the self-absorption limited performance for a particular dye. The maximum efficiency of an LSC is found using a simple model and the experimentally measured Stokes shift required to minimize self-absorption.
We find that the performance of LSCs which achieve high light concentration is primarily limited by self-absorption and by photodegradation. The maximum efficiency possible is about 9% in such systems, and present devices can achieve about 3%. A typical lifetime for an LSC using organic laser dyes due to photodegradation is on the order of a month.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Applied Physics)
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Applied Physics
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Zewail, Ahmed H.
Thesis Committee:
Zewail, Ahmed H. (chair)
Bridges, William B.
McCaldin, James Oeland
Murray, Bruce C.
Rutledge, David B.
Defense Date:
4 August 1981
Funders:
Funding Agency
Grant Number
Caltech
UNSPECIFIED
Department of Energy (DOE)
UNSPECIFIED
Record Number:
CaltechETD:etd-11152004-162115
Persistent URL:
DOI:
10.7907/W5W7-9660
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4566
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THE LUMINESCENT SOLAR CONCENTRATOR

Thesis by
John Samuel Batchelder

In Partial Fulfillment of the Requirements
for the degree
of
Doctor of Philosophy

California Institute of Technology
Pasadena, California
1982

(Submitted August 4, 1981)

Dedicat ed to my parents ,
Joyce and John Batchel der

iii

ACKNOWLEDGMENTS
I wish to express my gratitude to Professor Ahmed H. Zewail
for his invaluable support and guidance in this work.
I have also greatly profited from useful discussions with
Dr. Terry Cole, Dr. James McCaldin, Dr. John Lambe, and Dr.
Amitava Gupta. I wish to thank Dr. Krishna Koliwad, Dr. James Liu,
Bob Mueller, Taher Daud, Sandy Hyland, and co-workers at JPL for
their generous assistance in prototype testing. I am thankful to
Stuart Vincent for his help in casting plates and in the degradation
experiments, and to Prakash Kasiraj for his meticulously written
software. I am grateful to my compatriots Bill Lambert, Dave
Millar, Dean Neikirk, Joe Perry, and Duane Smith for their help and
comradeship. I have appreciated the assistance of many of the staff
at the Institute, particularly Tom Dunn, Linda Dozsa, Fran Bennett,
Charlie Beebe, Delmer. Dill, and Jim Olson. I especially thank
Tina Wood for her care and effort in typing this thesis.
I appreciate the financial support that I have received from the
California Institute of Technology and from the Department of Energy
Solar Energy Research Institute.

ABSTRACT

The Luminescent Solar Concentrator (LSC) allows sunlight
to be concentrated through the use of light pipe trapping of luminescence.
Such concentrators do not require tracking, and they can reduce the cost
of solar energy conversion by reducing the required area of photovoltaic
cells. We have conducted the following experimental and theoretical
investigations in order to optimize the LSC 's performance.
The spectral characteristics of 18 organic laser dyes are
studied for their applicability as luminescing centers.

The spectral

homogeneity and self-absorption characteristics of representative dyes
are examined in detail.

The relative spectral homogeneity of such dyes

is shown to depend upon the surrounding material using narrow band laser
excitation. We develop three independent techniques for measuring selfabsorption rates; these are time-resolved emission, steady state polarization anisotropy, and spectral convolution.

Prototype devices are

tested for performance, and the componant dyes are tested for stability
to solar exposure.
A model is developed which predicts the efficiency and gain of

and LSC from the spectroscopic characteristics of its components.

critical optical density (CODE) is assigned to the dyes surveyed which
predicts the self-absorption limited performance for a particular dye.
The maximum efficiency of an LSC is found using a simple model and
the experimentally measured Stokes shift required to minimize self-

absorption.
We find that the performance of LSCs which achieve high
light concentration is primarily limited by self-absorption and by
photodegradation.

The maximum efficiency possible is about 9% in

such systems, and present devices can achieve about 3%.

A typical

lifetime for an LSC using organic laser dyes due to photodegradation
is on the order of a month.

TABLE OF CONTENTS
Page
Chapter 1.

LSC Concept, History, and Thesis Outline

IN TROD UC TION

II.

LSC CONCEPTS

III.

IDSTORICAL DEVELOPMENT

IV.

OUTLINE OF THE THESIS

SUMMARY

Chapter 2.

Experimental Procedures and Data

SURVEY OF COMPONENT MATERIALS

II.

PHOTOBLEACIDNG MEASUREMENTS

III.

SELF-ABSORPTION IN RHODAMINE-575

IV.

PROTOTYPE TESTING

124

Chapter 3.

Single and Multiple Dye .Performance Model

SINGLE DYE MODEL FOR THE PERFORMANCE OF
AN LSC

132

II.

MULTIPLE DYE SYSTEMS

163

III.

SUMMARY

173

Chapter 4.

Self-absorption Modeling and CODEs

INTRODUCTION

175

II.

THE SCATTERING PLATE

175

III.

SELF-ABSORPTION IN A SEMI-INFINITE ROD

180

IV.

SELF-ABSORPTION AND EMISSION POLARIZATION

184

vii

Page
V.

SELF-ABSORPTION AND TRANSIENT EMISSION

190

VI.

CHARACTERISTIC LENGTH APPROXIMATION

193

VII.

CRITICAL OPTICAL DENSITY(CODE)

195

VIII.

SUMMARY

202

Chapter 5. Data Analysis
I.

ANALYSIS OF EXPERIMENTAL RESULTS

205

II.

SUMMARY

221

Chapter 6.

Thermodynamic Considerations

INTRODUCTION

224

II.

DETAILED BALANCE CALCULATION

225

III.

GENERALIZED BRIGHTNESS THEOREM OF
YABLONOVITCH

230

IV.

OPTIMAL EFFICIENCY MODEL

236

SUMMARY

242

Chapter 7. Discussion of LSC Performance
I.

INTRODUCTION

244

APPENDIX I.

ECONOMICS OF LSC APPLICATIONS

247

APPENDIX II.

MOLECULAR ABSORPTION AND
LUMINESCENCE

253

APPENDIX Ill. TOTAL INTERNAL REFLECTION AND
WAVEGUIDES

261

APPENDIX IV. BLACK BODY RADIATION

264

APPENDIX V.

268

SOLAR CELLS FOR USE WITH LSCs

viii

Page

APPENDIX VI.
REFERENCES

THE BRIGHTNESS THEOREM

272
274

-1-

CHAP TER 1

-2-

INTRODUCTION
Electricity generated from solar energy is an appealing alter-

native energy source.

It has the acclaimed advantages of being non-

polluting, renewable, widely distributed, and of delivering peak power
at the times of peak load.

The use of photovoltaic energy today is re-

stricted primarily to satellites and other specialized applications due
to tis high cost.

The motivation for this study of Luminescent Solar

Concentrators (LSCs) is to explore a potential technique for reducing
the cost of photovoltaic power generation.

Since the cells which per-

form the conversion from light to electricity are typically the most
expensive componant of such systems, it is advantageous to reduce
the required acreage of cells for a given power output by concentrating
the sunlight prior to illuminating the cells (Hovel, 1978).

LSCs offer a

unique advantage with respect to other types of concentrators such as
mirrors and lenses.

We have demonstrated LSCs which concentrate

sunlight by factors of five, and which should be capable of factors
greater than twenty with some modification, without requiring hourly
or seasonal tracking.

We also show that the penalty paid for obtaining

such light concentration without tracking is that the upper limit on the
overall efficiency of the converter is about 9%.
II.

LSC CONCEPTS
There are three princip.al processes which are combined in an

LSC.

These processes are light concentration, light pipe trapping,

and luminescence.

Since the solar cells are in general the most

-3-

expensive part of the converter, we wish to span the area directed at
the sun with a concentrator which directs the incident light towards a
smaller area of cells.

This idea imitates nature's use of chlorophyll

as an antenna for absorbing sunlight and transmitting the resulting excitation to a center for chemical reaction.

The typical LSC is a plate

of transparent material, such as glass or plastic, which contains
luminescing centers that adsorb and then emit light, such as phosphorus
or organic laser dyes. Sunlight enters the upper face of the plate and
is partially absorbed by these centers. A fraction of the resulting
luminescence is trapped by total internal reflection. Successive reflections transport the luminescence to edge-mounted solar cells, which
in turn generate electricity.

Figure 1 depicts an example of the opera-

tion of an LSC.
Suppose that the surface of the LSC plate which faces the sun has
an area Aface' and that the edge on which the solar cells are mounted
has an area Aedge· We will refer to the ratio of the area of the face
to the area of the edge as the geometric gain of the plates: Ggeom =
Aface/Aedge·

This geometric gain is roughly analogous to the con-

centrating power of a mirror or lens.

For example, if an LSC is

completely efficient, such that all of the sunlight which can be con..,
verted by the edge-mounted cells is absorbed by the plate and transmitted
without any losses to the cells, then the cells would
absorb a flux that was denser than the incident sunlight by a factor
equal to the geometric gain. Our task to determine the performance
of an LSC is, therefore, to quantify these loss mechanisms.

-4-

Figure 1. Operation of an LSC. Sunlight enters from above, passes
through the plate, through an air gap to a mirror, and back through
the plate.

Part of this light is absorbed by luminescing material,

which then emits into the plate.

About 7cP/o of this emission is

trapped by total internal reflection.

This light propagates in the

plate until it is absorbed by the edge-mounted cells.

MIRRORED
EDGE

MIRRORED
BACK

xLY
PHOTOVOLTAIC
CELLS

OF R LUMINESCENT
SOLRR CONCENTRATOR CLSC)

~PERRTION

CJ1

-6-

ID.

lllSTORICAL DEVELOPMENT
A number of precursors to the LSC developed independently

over the last twenty years.

Lerner at :MIT had built a solar collecting

device embodying this principle, consisting of a solution of laser dye
contained between two sheets of glass.

This was described in a 1973

NSF grant proposal. In 1975 D. P. Weilmenster, working under
Lerner's direction at MIT, submitted a senior thesis on this device.
A similar device for radiance amplification in scintillation counters

was proposed by Shurcliff (Shurcliff and Jones, 1949, and Shurcliff,
1951). Garwin discussed some thermodynamic aspects of trapped
radiation converters (Garwin, 1960). Kiel published results of experiments on plastic radiation converters for scintillation counters
(Keil, 1969).

More recently, quantum counters for scintillation de-

tection analogous to LSCs have been explored (Manda!, Pearson, and
Demas, 1980).
The first actual use of an LSC was probably not for solar energy
at all.

The LSC concept was employed for (unpublished) astronomical

observations by Weekes. She used sheets of dye-doped plastic, which
were edge-coupled to photomultiplier tubes, in an attempt to observe
Cerenkov radiation from cosmic ray showers in the upper atmosphere
(Lerner, 1979).
Weber and Lambe, working at Ford Research, initially suggested
both the concept of luminescence trapping for solar concentrators, and
the name of luminescent greenhouse collector (Weber and Lambe, 1976).
The term greenhouse was suggested because of the similarity between

-7the LSC 's absorption of sunlight in the visible followed by em~ssion in
the red, and the well-known atmospheric greenhouse effect.

The

first prototype devices were made at Ford by Levitt and Weber from
ED2 neodymium glass and from rhodamine-590 doped polymethyl
methacrylate (PMMA) (Levitt and Weber, 1977). (The PMMA plate
was a commonly available red-colored drafting triangle).
Photon Sorting
Since the luminescing material selectively absorbs radiation in
a certain energy window, then a number of different luminescing
materials, each in separate LSC plates, can separate light into its
spectral components. It is thus possible to un-mix broadband sunlight in a manner similar to a dispersive prism.

Goetzberger,

Greubel and Baur in Germany proposed this use of a multi-layered
LSC structure to further increase the overall efficiency (Goetzberger
and Greubel, 1977, and Baur and Greubel, 1977).

This technique is

analogous to the splitting of concentrated sunlight by a dichroic mirror
so that higher energy flux is converted by gallium arsenide cells and
lower energy flux is converted by silicon cells.
Dye Cascade
The selective absorption of luminescing material is advantageous
for sorting light by its energy, but it is problematic when one LSC
plate is suppose to utilize the largest possible portion of the solar
spectrum.

A possible solution is to include a variety of properly

chosen luminescing materials in a single plate (Swartz, Cole and
Zewail, 1977).

The materials are chosen such that their absorption

bands form a sequence from highest to lowest energy, such that the

-8-

absorption of one material overlaps strongly with the emission band
of the next material.

Light absorbed by one material will be emitted

and then absorbed by the next material until it is emitted by the last
material.

The result is an LSC which appears to have a very broad

absorption band across the solar spectrum.
Dye Orientation
Organic laser dye molecules usually have absorption and emission radiation patterns similar to that of a simple dipole antenna. We
observed that if these molecules could be oriented in the plate such
that these emission dipoles were principally normal to the plane of the
plate, then the emission from these dipoles would be more likely to be
trapped by total internal reflection (Batchelder, Zewail, and Cole,
1979).

For example, an isotropically absorbing and emitting source

in a PMMA plate will have about 74% of its emission trapped in the
plate, while a dipole emission source in the plate oriented perpendicular to the plate will have about 91 % of its emission trapped in the plate.
Glass Systems
Organic dyes tend to undergo photodegradation in sunlight. Some
materials are considerably hardier, however, and these are being explored for applicability to LSCs. Work on LSCs based on inorganic
ion luminescence in glasses, such as uranium glass, has been done
by Andrews and Lempicki at GTE and by Offenhartz and Micheels at
EIC Corporation (Lempicki et al. , 1980, and Offenhartz and Micheels,
1980).

-9-

Thin Films
Owens-Illinois is developing LSC plates which incorporate organic laser dyes which reside in thin plastic films, which are then
attached to clear substrates (Friedman, 1980).

Relatively exotic

plastic hosts can be used, and dyes which would otherwise react with
each other can be isolated in separate film layers.

Finally, if the

dyes degrade due to solar exposure, the films can be removed and
replaced.
Geometry
Hexagonal LSCs appear to give the best compromise between
uniform illumination of the cells, close packing of the concentrator
plates, and moderate light pathlengths in the concentrators.

The

amount of light trapped in the plate can be increased by curving the
upper surface of the plate (Batchelder, Zewail, and Cole, 1979).

The

geometric gain of a plate can be somewhat increased by forming the
edge of the plate into a mirrored surface that focusses the output
light onto the edge (Goetzburger and Schirmer, 1979).

For example,

this technique can increase the concentration by about 30% in a PMMA
plate.
IV. OUTLINE OF THE THESIS
Chapter 2 contains experimental procedures and data. We surveyed the spectroscopic characteristics of component LSC materials.
Preliminary degradation studies were made on organic laser dyes
under solar exposure in a variety of conditions. The relative homogeneity of the absorption and emission spectra of rhodamine-575 was

-10-

measured in three different hosts, and its self-absorption rates were
measured using spectral convolution, emission polarization, and
transient lifetime techniques.

Finally, prototype devices were .made

and tested.
In Chapter 3 we present a model for predicting the performance
of an LSC. We start with a flow chart for the various energy flow
channels.

The necessary device parameters, such as solar absorp-

tion, trapping probability, and self-absorption probability, are defined or derived.

These are computed for a specific device geometry.

Chapter 4 further elaborates on self-absorption models. We
derive relations for the performance of a purely scattering plate,
which is a limiting case of a highly self-absorbing system.

A tech-

nique is developed which calculates the output emission spectrum
from an LSC rod, with the contribution of higher order generations of
self-absorbed excitations taken into account. We develop models to
distill self-absorption rates from emission polarization and transient
lifetime measurements.

Two approximate techniques are developed

to include self-absorption effects.

The first is the characteristic

pathlength approximation, in which the self-absorption probability is
found by assuming an average pathlength for emission propagation in
the plate.

The second is the critical optical density calculation,

(CODE), which assigns to each dye a number which specifies the
largest plate in which the dye can be efficiently incorporated without
significant self-absorption losses.
In Chapter 5 we analyze the data from Chapter 2 using the techniques developed in Chapters 3 and 4. We find that the literature

-11-

values for the quantum efficiency for photodegradation (molecules
destroyed per absorbed photon) for typical laser dyes in general
are comparable to those that we have measured. We find that
dye spectra are relatively homogeneous in solution and in diffused
plastic, and are predominantly inhomogeneously broadened in cast
plastic.

The self-absorption model for the emission spectrum from

an LSC rod compares favorably with experimental spectra. We find
very good agreement between the three techniques for measuring the
self-absorption rate of rhodamine-575. We perform a calculation of
the efficiency of a single dye LSC as a function of size and concentration, and find that it compares well with prototype results.

Finally,

the CODE s for the dyes surveyed are calculated and tabulated.
In Chapter 6 we analyze the performance of an LSC from a thermodynamic point of view. We show that the light amplifying characteristics can be calculated by substituting black body absorbers for
the edge-mounted solar cells, and by balancing the input and output
energy to this absorber. We compare the generalized brightness
theorem result of Yablonovitch as a function of the positions of the
absorption and emission bands of the dyes to the CODE calculations
for the gain of an LSC, and we find that the Yablonovitch result is useful only for dyes with the smallest energy difference between their
absorption and emission bands.

Finally, a simple model similar to

that used to calculate the optimal efficiency of an ideal solar cell as a
function of its bandgap energy shows that the optimal efficiency of a
high gain LSC is about 9%.

-12-

Chapter 7 summarizes the important results.

We find that

dye degradation rates are typically two orders of magnitude higher than
what would be required for a 20 year useful life. We have experimentally determined that peak position Stokes shifts of about 0. 7 ev
(6, 000 cm -i) are necessary to reduce self-absorption to a negligible

level. We have demonstrated performance of concentration ratios
(flux gains) of 5. 1 and efficiencies of 1. 9% (not by the same device).

The highest efficiency reported to date is 3. 2%.

The maximum per-

formance predicted by the thermodynamic model is an efficiency of
about 9% for a flux gain of about 100.
A series of appendices are included which give background

material to topics relevant to this thesis.
economic model for LSCs.

Appendix II describes the process of ab-

sorption and emission in molecules.
as a waveguide.

Appendix I develops a simple

Appendix III describes the LSC

Appendix IV describes some characteristics of a

photon gas which are used in Chapter 6.

Appendix V is a discussion

of solar cell efficiency approximations used in the text.

Appendix VI

gives a description of the brightness theorem from geometric optics.
V.

SUMMARY
1.

The conceptual operation of an LSC is as follows: Sunlight

enters the face of a transparent plate of material containing a luminescent specie.

Part of this light is absorbed and re-emitted.

Part of

this emission is trapped by total internal reflection, so that it propagates in the plate until it reaches the edge-mounted solar cells, where
it is converted into electricity.

-13-

The primary motivation for developing LSCs is economic.

we wish to reduce the area of the most expensive component, which
is usually the solar cells, by concentrating the sunlight onto a smaller
area of cells.

LSCs have an advantage over other types of concentra-

tors in that they can achieve moderately high concentrations without
hourly or seasonal tracking.

Their disadvantage is that they are less

efficient than most lens and mirror systems.
3. We define the geometric gain, Ggeom' of an LSC to be the
area of the face exposed to the sun divided by the area of the edge
which is covered by solar cells. If an LSC were completely efficient,
such that all of the sunlight which can be converted by the solar cells
is absorbed by the plate and transmitted losslessly to the cells at an
energy which can be converted by the cells, then the light output would
be concentrated by a factor equal to the geometric gain.

-14-

CHAPTER 2

-15J.

SURVEY OF COMPONENT MATERIALS
There are three functionally distinct components in an LSC:

these are the plate or matrix material, the luminescing specie(s), and
the photovoltaic converter.

Our aim was to develop an optimal com-

bination of these by cataloging their relevant individual characteristics,
and by using these data to make a priori prototype designs.
All possible candidate systems could not be examined. We
omitted inorganic ion-glass systems because of their usually unacceptably low quantum efficiency of luminescence at room temperature
(typically less than 50%).

Color- or F-center materials were omitted

for the same reason. Nonlinear optical techniques usually require
electric field strengths of about four orders of magnitude greater than
that available from unconcentrated sunlight (100, 000 volts/cm vs
10 volts/cm; see Bloembergm, 1965).
The luminescent material of choice was the organic laser dyes.
A variety of such dyes have been developed in response to demand

from the tunable dye laser industry.

They typically have high quantum

efficiencies, and offer a choice of absorption peaks from the ultraviolet to the near infrared.

They can be utilized in liquids or in suit-

able solid matrices in which the dyes are soluable.
Matrix Materials
The important characteristics of the matrix or solvent material
is its index of refraction and its absorption spectrum. We needed to
know the absorption and emission spectra for the dyes in their appropriate media, as well as their quantum efficiency of luminescence.

-16-

Finally, we needed to know the spectral response of the photovoltaic
cells used, their AMl efficiency, and their change in efficiency with
light concentration. We measured only some of these quantities.
We did not measure the index of refraction or the absorption
coefficients for the various matrix materials. Our standard matrix
material was polymethyl methacrylate (PMMA, trade name Plexiglass),
which has an index of refraction of 1. 49.

The absorption spectrum

for optical fiber made from PMMA by du Pont was measured by
du Pont, and this spectrum is shown in Figure 2.

Typical attenuation

in PMMA is less than 10% per meter from 5, oooA to 7, oooA, which
is the most likely spectral band for emission from an LSC.

The

primary alternative plastic is polycarbonate (Lytle, Wilkerson, and
Jaramillo, 1979).
Dye Sample Preparation
The source of the organic laser dyes used was Exciton Chemical.
Dyes were used as received without further purification

Liquid

samples were made at known concentrations by dissolving quantities
of dye weighed on a Cahn-25 electrobalance in reagent grade methanol
These samples were stored in soda-lime glass bottles and kept in
darkness at room temperature.

The concentrations of the samples for

absorption measurements were chosen such that the peak optical density of a one centimeter pathlength of the solution was between 0. 5 and
1. 5 in the range between 10, 000 and 30, 000 cm- 1 •
Dye spectra were also taken in a variety of other hosts.

The

principal solid matrix material was PMMA, which usually contained

-17-

Figure 2.

Optical density per centimeter of polymethylmet hacrylate

(PMMA) optical fiber.

This has been adopted from data published by

du Pont on transmission of commercial fiber (Friedman, 1980).

-18-

-3
o 2xl0
Q)

Cl.

+(/)

"'O

2x1 o;-0-00_6_0_0_0__7_0......0_0__8_0
.....0_0_9_0_00
__10-000

Angstroms
Absorption of DuPont PMMA Optical Fiber

-19-

5% hydroxy ethyl methacrylate (HEMMA) by weight to increase the
solubility of the more polar dyes.

Large PMMA plates were made to

our specifications by Acrilex, Inc. Smaller test samples were fabricated in the following manner. Aldrich monomer, containing hydroquinone monomethyl ether as an inhibitor, was purified by fractional
distillation in a nitrogen atmosphere using a vacuum-jacketed vigreaux
column.

This distilled monomer was combined with technical grade

HEMMA, and the desired dyes were dissolved therein.

Small amounts

(less than 2% by volume) of methanol and/or acetic acid were added to
the solution to increase the dye solubility if rhodamine dyes were to
be used.

Some samples were prepared by adding the concentrated

dye-monomer solution to prepolymerized PMMA and continuing the
polymerization.

However, best results were usually obtained by poly-

merizing just the monomer-dye solution.

Two percent by weight of

azobisisobutyronitrile was added as initiator, and the mixture was
poured into a mold form·ed by two glass plates.

The plates were

separated by polyethylene tubing, and by aluminum spacers around the
periphery to maintain a constant plate thickness.

A very thin coat of

silicon vacuum grease on the glass plates acted as mold release agent.
These molds were then immersed in a water bath and placed in a convection oven.

Polymerization was initiated at about 85° C, when a

noticeable increase in viscosity occured, at which time the temperature was lowered to 55° C for 48 hours.

The molds were removed from

the bath for a final one hour 95° C cure.

Typically, significant frac-

tions of the dye did not go into solution, so that the dye concentration

-20-

in the final plates were assayed by measuring the peak optical density
of the plate and assuming that the peak extinction coefficient was that
of the methanol solutions.

(A possible problem with this technique is

that the peak extinction coefficient of the dyes in PMMA might be different from that of the dyes in methanol. We have observed factors of
two differences in the peak extinction coefficient of DCM in various
solvents.) We found that dye concentrations in excess of 10 micromolar caused significant amounts of monomer to remain unpolymerized
in the cured plates in the case of the rhodamine and oxazine dyes, and
that this monomer could be slowly driven out by vacuum degassing at
5Q° C.

After curing, the plates were removed from the molds and were

scribed and broken to size.
of grits.

The edges were polished with a sequence

The final buffing compound was a cerium oxide rouge.

We developed an alternative technique for making PMMA samples.
If commercial transparent PMMA plate or rod material is immersed in

a methanol solution containing the dye of interest, the dye will infuse
into the plate along with the methanol.

This technique has the great

advantage of not requiring distillation, casting, or curing.

A solution

of 9% dichloromethane by volume in methanol was found to be the best
compromise between speed of infusion and maintaining a good surface
finish on the plate.

The time required to achieve useful dye concentra-

tions in the plastic for an eigth-inch rod was about 15 minutes, and for
a sixteenth-inch thick plate the time was about 12 hours.
was at room temperature in both cases.

The infusion

Coumarin-540 infused faster

than rhodamine-640, possibly due to the difference in molecular weight.

-21Jt appeared to the eye that the dyes typically resided in a film between

o. 1 and 1. 0 mm from the surface of the plate, depending on the temperature and the soak time, so that the infusion technique does not
yield a uniform dye concentration across the thickness of the plate. It
is unlikely that the dye concentration in the film would be greater than
that of the soaking solution, so that the solution concentration yields
an upper bound on the local dye concentration.

Measuring the peak

optical density and assuming a uniform dye dispersion in the plate gives
the average concentration in the plate.
Dye Spectra
Absorption spectra of all samples were made using either a
Cary-14 or Cary-17 dual beam spectrophotometer.

These produced

strip-chart paper output, which was digitized using a Houston Instruments Hipad digitizer.
12, 500 cm

-1

Absorption spectra were usually taken from

(8, OOOA) to 30, 000 cm

-1

(3, 333A).

Baseline corrections

were always required over such a broad range, so that a base line spectrum was recorded, digitized, and subtracted from each absorption
spectrum.

The spectra were manipulated and stored on diskettes using

a Terak-packaged PDP 11-03/2.

The digitization accuracy was five-

hundreths of an inch, which is finer than the pen line of the chart recorder output.

The base line noise in the spectra was typically 1, 000

liters per mole per cm, or in other words, the signal to noise was on
the order of 100.
Emission and excitation spectra of the dyes were made at micromolar concentrations using the computer-controlled apparatus as shown

-22-

in Figure 3.

The excitation source was either a 200 watt Oriel 6323

tungsten lamp or a 200 watt Oriel 6137 high pressure mercury lamp.
The light was collimated with quartz optics, chopped by a PAR 191
chopper, and monochromated by a Jarrell-Ash (model 82-410, f-3. 5,

o. 25 meter) monochrometer with either a 6, oooA biased grating
(1180 grooves/mm), or with a 3, ooo.A blazed grating, (2365 grooves/
mm).

In order that phase sensitive detection produce accurate exci-

tation and emission spectra when the excitation source is chopped, it
is important that the lifetimes of the dyes must be short compared to
the chopping period.

The chopping pericx:l in these experiments was

2. 5 msec, which is several orders of magnitude longer than the lifetimes of the dyes measured.

The excitation linewidth was fixed at 90.A

across the tuning range.
The emission 90 degrees from the excitation was analyzed by a
similar Jarrell-Ash monochromator.

Adjustable slits allowed the

resolution to vary between 2 and 90.A.

The output light was detected by

a 928 Hamamatsu photomultiplier tube (PMT) biased at 900 volts.

The

PMT output was terminated into lOOk ohms in parallel with a PAR-HRB
lock-in amplifier.

The analog output of the lock-in was digitized and

recorded by the PDP 11-03/2. Care was taken to keep the PMT current
at a level less than 20% below its rated output current of 100 microamps.
The RC time constant of the lock-in was kept at least as short as the
time between digital sampling of the emission.

These samples were

taken typically every 10.A. Both monochromators were driven by a
Slo-Syn SPl 51 Driver such that one or both monochromators could scan
under the control of the PDP 11-03/2.

This allowed spectra to be

-23-

Figure 3.

Apparatus for emission and excitation measurements.

regulated mercury or tungsten source was focussed, chopped, and
monochromated prior to illuminating the sample.

The resulting

emission was monochromated and detected by a PMT, and the resulting signal was amplified by phase-sensitive detection. A remote
computer controlled both monochromators and recorded the spectra.

HQ OR W
LAMP

SAMPLE
MONOCHROMETER

SUPPLY

H. V.

PMT

LOCK-IN AMP.

STEPPER MOTOR
CONTROLLER

STEPPERIMOTORS

G--0--

<::::::>

MONOCHROMETER

LAMP
SUPPLY

CHART
RECORDER

A/D

I NTERFACE
1 41
DRIVER

PLOTTER

GRAPHICS

FLOPPY
DISK

1 PDP 11-03/2

t..:>
M::>.

-25acquired in cm-1even though the monochromators had wavelength
drives. Computer control also provided backlash correction.
Frequency calibration of the monochromators was done with the
5, 451.A line from a mercury germicidal lamp.

The system response

was measured using an Eppley Laboratory calibrated E PI-1669 quartz
halogen (tungsten) lamp in a custom housing.

Calibration data pro-

vided with the lamp showed that the spectral response of the lamp
could be accurately approximated by that of a 3, 074°K black booy for
a lamp current of 7. 9 amps. We used the black body spectrum to extrapolate the lamp response between the commercially calibrated data
points.

The system response function is given in Figure 4.

The

roll-off in the blue is due to the grating response, and the roll-off in
the red is primarily due to the PMT.
Emission spectra were corrected by dividing out this system
response, and were normalized so that the luminescence integrated
over all wavenumbers equals one.

The peak position of the emission

for the dyes tested is given in Table 1. Correcting the excitation
spectra for the response of the system requires a calibrated detector
as well as a calibrated source.

Using both it is possible to determine

the response of the monochromator that is scanning the excitation.
We were not successful in obtaining such a response using a Laser
Precision Rk-3440 pyrometer detector.

Instead we made the approxi-

mation that the PMT response was flat over the region of interest.
In this approximation the variation of the scanning excitation intensity
is assumed to be the measured response to the tungsten lamp.

Exci-

tation spectra were corrected by dividing out this measured response.

-26-

Figure 4. System response of the detection monochromator
and PMT.

-270 :::tt

(Y). l.J.J

a:
_J

.,_.

a:
a::

t-1

(f)

CCl

t-1
a:

zE)

(f)

_.•

I(f)

:::J

3SNfJdS38 W31SAS

-28-

Table 1. Spectroscopic parameters for 18 organic laser dyes.
The name of the dye given is the name used by Exciton.

The

number following the name is the approximate wavelength in
nanometers of the dye's peak lasing power. All extinction
coefficients were measured in methanol.
were similarly methanol solutions.

The emission spectra

Literature values are given

for the quantum efficieIJ.CY of luminescence in various solvents.
The critical optical density (CODE) is described on page 195
and following.

-29-

Table 1.

Stoke Shift

•max

IJ
Solvent

CODE
"'30%

4,560

0. 58a
ethanol

110.

4.950

5,440

0. 53&
ethanol

140.

20,400

4,900

2, 720

4,580

19, '100

5,070

2, 130

0. 78a
ethanol

80.

21, 500

4,650

15, 700

6,360

5, 760

0. 71b
DMSO

240.

20, 120

4,970

19,000

5,250

1,080

0. 85c
ethanol

25.

Exciton Dye
(Kodak name)

(mole/cm/liter
ln methanol)

Coumarin-480
(Coumarin-102)

22, 000

25, 730

3,890

21,200

4, '120

coumarin-500

19, 900

25, 660

3,900

20,200

Coumarin-535
(Coumarin-7)

52, 200

23, 120

4,320

Coumarin-540
(Coumarin-6)

52, 200

21, 860

DCM

28,900

Rhodamine- 560
(Rhodamine-110)

82,000

cm- 1

90.

35.

Rhodamlne-575

93, 800

19, 330

5, 170

18,300

5,460

1,010

Rhodamine-590
(Rbodamine-6G)

107, 000

18, 940

5,280

18,000

5,550

900

Rhodamine-610
(Rhodamine-B)

114, 000

18, 380

5,440

1'1, 500

5, 710

870

0. 5•
ethanol

36.

Ki!on red-620
(Sulforhodamine-B)

111, 000

17,990

5,560

17, 300

5,800

740

0. 83g
ethanol

16.

Rhodamine-640
(Rhodamine-101)

106, 000

17, 670

5,660

16,800

5,940

830

1. c
ethanol

17.

Rulforhodamine-640 120, 000
(Sulforhodamlne-101)

17,360

5, 760

16, 700

6,000

690

.1. h
ethanol

17.

Cresyl Violet-670
(Oxazine-9)

57, 900

16, 880

5,920

16, 100

6,220

810

o. 54f
methanol

17.

Oxazine- 720
(Oxazine-170)

81, 800

16, 170

6,190

15, 600

6,420

600

Oxazine-750

90, 600

15, 140

6,600

14, 500

6,920

680

25.

DODCI

238,000

17, 160

5,830

16, 500

8,050

630

11.

DOTCI

236,000

14, '1'10

8, 770

14,000

'1,140

'170

IR-144

153, 000

13, 560

'1,370

12,000

8,340

1,560

16.

0. 98d

25.

methanol

17.

aG. A. Reynolds, K. H. Drexhage, Optics Comm. , ~. No. 3, 222 (1975).

P. R. Hammond, Optics Comm. , ~. No. 3, 331 (1979).
cK. H. Drexhage, "Structure and Properties of Laser Dyes, "ed. F. P. Schafer, Topics in Dye Lasers, Applied Physics I (Springer-Verlag,
New York, 1977) ~
dA. Baczynski, T. Marszalek, H. Walerys,

B. Zietek, Acta Phys. Polo., ~. No. 6, 805 (19'13).

e T. Karstens, K. Kobs, J. Phys. Chem. , !!• No. 14, 1871 (1980).

'o. Magde, J. H. Brannon, T. L. Cremers, J. Olmsted m, J. Phys. Chem. ' E· No. 6, 896 (1979).
gJ. M. Dra.ke, R. T. Morse, R. N. Stepp!, D. Young, Chem. Phys. Lett., ~. No. 2, 181 (1975).
hc. F. Rapp et al., Final Report of Owens Illinois, Sand77-7005, p. 40.

-30The inadequacy of this correction procedure is evidenced by the mismatch between the excitation spectra corrected in this manner and the
corresponding absorption spectra for most dyes whose absorption peaM:;
were on the sloped regions of the system response.
Both the emission and excitation spectra of these dyes were taken
only at micromolar dye concentrations in order to minimize the redshifting effect of self-absorption.

The optical density of the sample in

the region where the absorption and emission overlap must be sufficiently low so that the blue tail of the emission spectrum is not artificially filtered out.

The excitation samples must be dilute for a sec-

ond reason. Away from the peak of the absorption, where the absorption per unit length is small, the exciting beam intensity does not vary
greatly along the beam path in the sample.

However, as the excitation

is scanned across the peak of the absorption, most of the exciting light
is absorbed in a thin surface layer in the sample if the concentration is
too high.

This effect changes the spatial distribution of the emission,

and thus can alter the observed excitation spectrum.
The results of these dye spectra measurements are given in
Figures 5 through 27.

The dyes are ordered by decreasing energy of

the absorption peaks.

All dyes are in methanol solutions except for

three additional DCM spectra in chloroform, DMSO, and PMMA. The
solar flux given on each spectrum is an idealized 5800K black body
spectrum.

The three titles in small letters in the upper right hand

corner of the plots are the diskette file names of the absorption,
emission, and excitation spectra.

The peak positions of the absorption

and emission spectra of these dyes, as well as the peak extinction

-31Figures 5 - 25. Survey of laser dye spectra.

The extinction

coefficient in liters per mole centimeter (solid lines); the
normalized emission spectrum in arbitrary units (long dashed line);
and the excitation spectrum in arbitrary units (short dashed line),
are presented for 18 common organic laser dyes.
the flux spectrum from a 5800° K black body.

The dotted line is

The small titles in the

upper right corner refer to diskette file names.

The figures are

organized as follows:
Figure 5
Figure 6
Figure ·7
Figure '8
Figure ·9
Figure 10
Figure 11
Figure 12
Figure 13
Figure 14
Figure 15
Figure 16
Figure 17
Figure 18
Figure 19
Figure 20
Figure 21
Figure 22
Figure 23
Figure 24
Figure 25

coumarin-4 80
coumarin- 500
coumarin-5 35
coumarin- 540
DCM
DCM
DCM
DCM
rhodamine- 560
rhodamine- 575
rhodamine - 590
rhodamine- 610
Kiton red-620
rhodamine- 640
sulforhoda mine- 640
DOD CI
cresyl violet- 670
oxazine- 720
oxazine-75 0
DOT CI
IR-144

in methanol
II

" DMSO
'' chloroform
" PMMA
" methanol
II

Page
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
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48
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-53-

Figure 26.

Three emission spectra of methanol dye solutions

resulting from 4, 500.A. excitation.

The top spectrum is from a

micromolar oxazine-720 solution, and the lower spectra are
micromolar and hundred micromolar concentrations, respectively,
of coumarin-540, rhodamine-640, and oxzaine-720.

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-55-

Figure 27.

Three excitation spectra of methanol dye solutions with

emission detection at 6, 400.A.

The three spectra correspond to the

same solutions used in Figure 26.

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-57coefficients, are summarized in Table 1.
Measurements were also made on the spectral characteristics
of multiple dye solutions. Dyes could not be combined indiscriminately, possibly due to agglomeration phenomenon between some dyes
which quenched their emission when they were in solution together.
For example, our oxazine-720 and oxazine-750 solutions showed
quenching of their emission when they were in solution with rhodamine610 or with rhodamine-590. Agglomeration has previously been observed to quench the emission of xanthene dyes (Joshi and Pant, 19 76,
and Kajiwara, Chambers, and Kearns, 1973). An example of a successful multiple dye combination is sulforhodamine-640, rhod.amine-590,
and coumarin-540.

Figures 26 and 27 show emission and excitation

spectra, respectively, for an oxazine- 720 methanol solution, a dilute
(micromolar) oxazine-720/ rhodamine- 640/coumarin- 540 methanol
solution, and of a concentrated (hundred micromolar) methanol solution
of the same three dyes .. The emission spectra of the multiple dye solution at low concentrations is dominated by rhodamine and coumarin
emission, while the oxazine dominates the emission at higher concentrations. If we detect the emission of oxazine, as shown in Figure 27,
as a function of excitation energy, the low concentration multiple dye
solution is dominated by the pure oxazine absorption peak, while the
high concentration multiple dye solution gives a nearly constant response across the visible spectrum.
Quantum Efficiency of Luminescence
We did not measure the quantum efficiency of luminescence for
any of the dyes.

The usual nontriviality of measuring quantum

-58-

efficiencies would have been further complicated by requiring an
accurately calibrated system response from 4, ooo.A to 8, ooo.A
(see Demas and Crosby, 1971). We have instead used literature values
for the measured quantum efficiencies.

These are given with the

appropriate references in Table 1.
II.

PHOTOBLEACHING MEASUREMENTS
Organic laser dyes have the good attributes of high quantum

efficiency of luminescence, well separated absorption and emission
spectra, and a wide selection of possible spectral positions.

They

have the bad characteristic that their performance deteriorates due to
exposure to light.

More specifically, the total luminescence from a

dye-containing sample under solar exposure will decrease with time.
The time required for the emission to drop to 50% of its initial value
can vary from minutes to years.

The precise rate of decline varies

with the particular dye, the surrounding matrix or solvent material,
impurities in the surrounding material, the spectral character of the
incident light, the intensity of the light, and on the temperature of the
sample. Other variables may also be important.
Studies in the Literature
Studies have been published on dye photodegradation rates in a
variety of systems.

Two groups at Bell Labs have measured xanthene

dye photodegradation quantum efficiences in liquid (Shank and Ippen,
1971) and in plastics (Kaminow, Stulz, Chandross, and Pryde, 1972).
Similar studies were done by Beer and Weber in Germany (Beer and
Weber, 1972). In all cases the excitation source for CW irradiation

-59was the 5, 145.A line from an argon ion laser. Sample temperatures
were near room temperature.

Photodegradation was determined by a

decrease in the peak absorption of the solution, not by change in the
emission.

The results of these measurements are summarized in

Table 2.
The optical density has also been seen to decay due to thermal
excitation in darkness. Dyes dissolved in glycerin were found to be
relatively stable for temperatures below 100° C (Weber, 1976).
Rhodamine-590 was found to be more sensitive to thermal degradation
in PMMA matrixes (Higuchi and Muto, 1981).

The decay rate in PMMA

increases sharply from about 10- /sec at 65° C to a plateau of about
8 x 10- 7/sec for temperatures from 70°C to 85°C.

The sudden in-

crease in the decay rate was postulated to be related to the glass transition temperature for the plastic.

The decay rate for rhodamine-590

in PMMA at 65° C corresponds to roughly a year at 9 hours per day of
exposures.
There is some concurrence in the literature that the dominant
degradation mechanism is related to triplet formation in the dyes.
ESR studies led Yamashita and Kashiwagi to propose two laser-induced
photochemical reactions (Yamashita and Kashiwagi, 1976).

Highly

excited molecules in the triplet state can transfer sufficient energy to
the surrounding solvent to rupture a C-H bond in the solvent molecule,
and the resulting radical combines with the dye. Alternatively, intersystem crossing from the first singlet to the lowest triplet state can
result in a different radical formation which depends on the particular
structure of the dye. Weber (Weber, 1973) found that degradation

-60-

Table 2. Literature values for the quantum efficiency of photobleaching
of laser dyes. In all cases the excitation source was the 5, 145A line
from an argon ion laser.

The number of photons absorbed was deter-

mined by the attenuation of the laser.

The number of molecules which

had been bleached was assumed to be given by the change in peak optical
density.

(These should be compared to our results for solar bleaching

on page 2 07).

-61-

Table 2.

Literature Values for Photobleaching Quantum Efficiencies.
(5, 145.A excitation)

Solvent

Rhodamine- 590

H2 0tTriton X-100

8 x 10- 6

(a)

H2 0+ Triton X-100

7x 10- 6

(c)

methanol

5 x 10- 7

(a)

0. 4 - 2x10- 6

(b)

PMMA

Molecules
Plioton

Dye

Reference

Rhodamine-610

methanol

17x10- 6

(c)

Cresyl Violet

methanol

9 x 10- 6

(c)

Fluorescein

ethanol

4 x 10- 6

(a)

Rhodamine-S

ethanol

1. 7 x 10- 6

(a)

(a) Shank and Ippen, 1971.
(b) Kaminow, Stulz, Chandross, and Pryde, 1972.
(c) Beer and Weber, 1972.

-62-

rates in a variety of solutions were reduced by the addition of a triplet quencher such as oxygen.
PMMA itself has proved very stable under solar exposure.

plate has been observed to retain 80-9 0% of its optical clarity across
the visible spectrum after 17 years of exposure on a test frame in
Albuquerque (Rainhart and Schimmel, 1975, and Fox, Isaacs, and
Stokes, 1963).
Degradation Measurements
We made preliminary measurements of the degradation rates of
dyes in a variety of hosts to determine the viability of various dyehost combinations as LSC prototypes.
Absolute absorption measurements are much simpler than absolute emission intensity measurements. In general, we would measure change in the peak absorption across the thickness of a plate and
assume that a decrease in this peak absorption was related to the
fraction of the dye moleeules that had chemically reacted. It is likely
that the emission intensity might not be directly related to the absorption. Increased scattering in the plate and degradation products both
might contribute to the absorption, causing the remaining dye concentration to be over-estimated.

To test the correlation between absorp-

tion and emission degradation, we made the following measurements on
a 3 mm thick PMMA plate containing 140 micromolar coumarin-540
and 97 micromolar rhod.amine-590.

A portion of the unexposed plate

was exposed to Michigan sunlight filtered by a plate glass window for
a total of about 1, 000 hours of direct sunlight. We measured the

-63emission of this exposed plate by illuminating the sample with a dispersed tungsten lamp. The spectral distribution of the tungsten lamp
closely approximated that of a 3000° K black body.

A silicon cell was

brought into optical contact with the edge using ethylene glycol as an
index matching fluid.

The output short circuit current of the cell was

measured with an HP-3466A digital multimeter.

The emission of

the plate exposed behind plate glass had dropped about 30% compare to
a similar sample that had been stored in the dark.

The peak optical

density of the rhodamine had dropped from 2. 7 to 2. 2 during exposure,
or a drop of about 20%, while the coumarin peak had been reduced
from 2. 3 to 1. 7, or a drop of about 30%.

A section of the same unex-

posed plate was exposed to direct sunlight for 243 total hours.

This

resulted in an emission decrease of about 30%, and in a decrease in
the peak optical density of the coumarin of about 20%. We find, therefore, that there is rough parity between change in the absorption
spectrum and change in the emission from the edge of a plate. If the
rates are not equal, the emission will degrade somewhat faster.
Simulated solar exposure tests were made on an LSC plate containing about 100 micromolar concentrations of rhodamine-590 and
coumarin-540. Samples of this material 0. 3 x 1 x 3 cm in size were
cut from the plate. Optical absorption spectra were measured, and
the optical density of the spectral peaks of rhodamine-590 and
coumarin-540 at 5, 300.A and 4, 600A, respectively, were noted for
the unexposed material.

The samples were then placed in a QUV

accelerated weathering tester, and were subjected to a test cycle consisting of 8 hours of exposure to UV light from a bank of four S-40 UV

-64fluorescent lamps at a temperature of 60° C followed by four hours of
darkness at 50° C and 100% relative humidity.
for a total of 2, 400 hours.

This cycle was repeated

The samples were removed from the QUV

at approximately 200 hour intervals at which times their optical absorption spectra were measured.

Figure 28 a plot of the peak optical

density for both dyes as a function of total test hours.

The decay

times from the slopes of these plots are about 1, 000 and 10, 000 hours
for rhodamine and coumarin, respectively.

The QUV apparatus has a

peak spectral intensity at 3, 130A., with an intensity of 1. 8 microwatt
per Angstrom per square centimeter at that wavelength. Acceleration
factors of 8-25 hours of environmental exposure per hour of QUV test
time have been suggested (Grossman, 1977).
Solar Exposure of PMMA Samples
The first sample tested was a 76 micromolar rhodamine- 590
PMMA plate 3 mm thick. Acetic acid was added to the monomeric
solution prior to polyme:rization to increase the solubility of the dye.
A two-week continuous exposure (336 total hours) caused a decrease in
the peak optical density from 4. 0 to 2. 8, as measured on a Cary-14
Spectrophotom eter. We estimate from the change in absorption that
the bleaching rate was about 10- 6 molecules per photon.

This estimate

is obtained by calculating the fraction of an AMO spectrum (assumed
to be a 5, 800°K black body) which would have been absorbed by the
initial concentration of rhodamine-590 .

This yields an approximate

number of photons absorbed by the sample per unit area.

The number

of dye molecules which have photochemical ly reacted is assumed to be
given by the decrease in peak optical density.

This calculated

-65-

Figure 28. Dye deterioration in a QUV test chamber.

Two LSC

samples consisting of rhodamine-590 or coumarin-540 in PMMA
were exposed for 2, 400 hours to simulated solar exposure.

(One

hour exposure in this apparatus is considered roughly equivalent to
eight hours of sunlight for paint samples. ) The decay times for the
two samples were about 1, 000 and 10, 000 QUV hours, respectively.

-66-

DYE DETERIDRRTIDN IN
R QUV TEST CHRMBERa
10 1

-------------- ----10 1
. + RHt'JDAMINE-6G
• Ct'JUMRR I N-6

....

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...

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1500

CUMULATIVE H~URS

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2500

-67-

degradation rate is roughly equivalent to that measured by argon ion
irradiation.
A PMMA plate containing 210 micromolar coumarin-540, 81

micromolar rhodamine-590, and 22 micromolar sulforhodamine-640
was exposed to direct insolation for 243 continuous hours.

The total

emission from the plate decreased 50%, as measured using the above
technique.

The peak optical density of the three dyes decreased 52%,

50%, and 51 %, respectively.

This sample showed particularly good

correlation between decline in emission and absorption.

Additional

samples of this plate were exposed over the same pericxi to determine
the spectral dependence of the degradation rate.

Three samples were

protected by either an eighth-inch thick soda-lime glass cover, a
Schott GG400 UV cut-off filter, or by OD-2 neutral density filters. All
samples were sealed against the outside atmosphere with polyvinylchloride electrical tape.

Figure 29 shows a plot of the absorption

spectra of the three exposed samples described, plus the unprotected
exposed sample and the control or dark sample.

Little change was

observed in the sample protected by neutral density filters. No other
spectral dependences are obvious from the data. Note that degradation
in the absorption peak is usually accompanied by a slight increase in
the near UV absorption, which might be related to bleaching products.
The 3 mm think PMMA plate containing rhodamine-590 was examined for changes in collection efficiency due to solar exposure.

silicon cell was mounted against a polished edge of the unexposed
plate, and the short circuit current from the cell was measured with
an HP-3466A DMM.

A focussed spot excitation from a 3, 000°K

-68-

Figure 29. Solar bleaching of a multiple dye LSC plate.

These are

the absorption spectra for a PMMA plate containing 210, 81, and 22
micromolar concentrations of coumarin-540, rhodamine-590, and
sulforhodamine-640, respectively.

The five spectra correspond to

an unexposed control plus 243 continuous hours of exposure through
either optical density (2) filters, 400 nm cut-off filters, soda-lime
glass, or no filtering.

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-70tungsten lamp was scanned from the center of the plate towaTds the
cell, and the short circuit current was recorded as a function of the
excitation-cell distance.

This procedure was repeated for the plate

which had been exposed for 336 total hours.
measurements are given in Figure 30.

The results of these

The output from the exposed

plate was actually greater than the output from the unexposed plate for
small excitation-to-cell distances.

At longer distances the unexposed

plate had a higher collection efficiency.
We made single dye plates containing coumarin-460, 500, and
540, rhodamine-590 and 610, and oxazine-725 in PMMA.

After about

250 hours of continuous exposure, the emission had dropped 13% for
the coumarin-540, which was the best case, and had dropped 66% for
the oxazine-725, which was the worst case, with the other dyes following in between these two limits. If the emission is assumed to decrease
in a roughly exponential manner, these correspond to half-lives ranging from 300 to 3, 000 hours (12 to 120 days).
Methanol solutions of the dyes coumarin-500, 535, and 540,
rhodamine-590 and 640, sulforhodamine-640, cresyl violet-670,
LD-700, and oxazine-720 were prepared in soda-lime glass vials 2. 5
cm in diameter. Oxygen was removed from the samples by bubbling
nitrogen gas through the solutions for an hour in a nitrogen atmosphere.
The screw-on caps were sealed with RTV silicon rubber sealant.

The

optical density of the samples were measured every few days without
disturbing the seals.

Figure 31 shows the measured peak optical den-

sities of the solutions as a function of total exposure time, which included hours of darkness.

The maximum exposure time was 432 hours,

-71-

Figure 30. Emission vs distance for a plate before and after exposure.
A spot excitation was scanned towards an edge-mounted cell, and the
resulting short circuit current was measured as a function of distance
between the spot and the cell.

-72-

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Cit

Lt_

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(f)

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(f)

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go•

- SdWtJillI
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-73-

Figure 31.

Peak optical densities of methanol dye solutions under

continuous solar exposure.

2. 5 cm i. d. soda-lime glass bottles

were filled with solutions of oxazine-720 perchlorate, LD-700
perchlorate, cresyl violet-670 perchlorate, rhodamine-590
perchlorate, coumarin-540, coumarin-535, and coumarin-500.
The screw-on caps were sealed with RTV.
were taken without distu'rbing the seals.

Absorption spectra

~ 0 •00

31.

>.._

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ij -

I '

10.

100.

\2~- ',:

C~NTINU~US H~URS EXP~SUAE

-J...

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8 - C-535
9 - C-500

SA-640 7 - C-540

2 - LD-700 5 - A-640
3 - CV-670 6 - A-590

1 - ~X-720

1.000.

-75(18 days).

This was insufficient time to obtain accurate decay rates

for the longer lifetime dyes.

Using an exponential as a rough approxi-

mation to the time dependence of the optical density, the lifetime for
the solutions ranged from 50 hours for rhodamine-640 to about
10, 000 hours (one year) for rhodamine-590.

Among the relatively

stable dyes were coumarin-540 (3, 000 hours), oxazine-720 (2, 000
hours), and cresyl violet (7, 000 hours).
We measured photodegradation rates for five of the most stable
dyes in a degassed environment.

Methanol solutions of coumarin-540,

rhodamine-590, sulforhodamine-640, cresyl violet-670, and oxazine750 were placed in quartz cuvettes having the outer dimensions 1. 3 x
1. 3 x 6 cm.

The concentrations of the solutions were adjusted so

that the peak optical density across the cuvettes were between 0. 5 and
1. 0. These samples were degassed by six freeze-pump-thaw cycles
on a vacuum minifold, and then flame sealed.

The resulting lifetimes

were 3. 3, 10. 0, 36. O, 0. 3, and 1. 0 direct sunlight hours, respectively.

Figures 32 through 36 show the absorption spectra of these

samples as a function of time.
DCM degradation rates were measured in methanol, dimethylformamide, dimethylsulfoxide, and chloroform solutions. Solutions
were prepared with accurate DCM concentrations and were placed in
soda-lime glass bottles.

The solvents were not degassed.

sealed on using RTV sealant.

Caps were

The samples were exposed for 210

hours (9 days) including hours of darkness.

Figures 37 and 38 show

before and after absorption spectra for DCM in methanol and

-76-

Figures 32 - 36. Solar exposure of degassed methanol dye solutions
in quartz cuvettes. Absorption spectra are presented which were
made after intervals of exposure to sunlight.

The hours of exposure,

and the resulting peak optical densities are listed on each plot.

Figure 32
Figure 33
Figure 34
Figure 35
Figure 36

coumarin- 540
rhodamine- 590
sulforhodamine- 640
cresyl violet-670
oxazine- 750

Page
77
78
79
80
81

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er:

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WRVENUMBERS

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1.
2.

EXP.

(/)

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5.00
·PK.~.D.

SR-640. QUARTZ CUVETTE
WITH VACUUM DEGASSED METHAN~L.

S~LAR EXP~SURE ~F

E ll

2.50

«>

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CV-670. QUARTZ CUVETTE
WITH VACUUM DEGASSED METHAN~L.

S~LAR EXP~SURE ~F

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QUARTZ CUVETTE

1. 75

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(f)

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WITH VACUUM DEGASSED METHAN~L.

S~LRR EXP~SURE ~F ~X-750.

CX>
!-\

-82-

Figures 37 and 38. Solar exposure of DCM.

Absorption spectra of

DCM in methanol (Figure 37) and dimethylformamide (Figure 38) are
presented before and after 210 hours of continuous solar exposure in
soda-lime glass bottles.

I--

t--i

0 0
• P.oo

2.00

3.00
E l!

WRVENUMBERS

t.:>

0 0.50

(f)

1---1

I-

E 0

1.00.--~~~~r-~~~~r-~--~~,..--~~~--

2.1E-5M DCM IN METHAN~L BEF~RE
RND AFTER 210 H~UAS ~F EXP~SUAE.

I-

t--t

00
· P.oo

2.00

3.00

WRVENUMBERS

E ll

D 0.60

t--f

>1-

1.20
E 0

2.0lE-SM DCM IN DIMETHYLF~RMAMIDE BEF~RE AND
AFTER 210 H~URS ~F EXP~SURE.

-85-

dimethylformamide. In all cases the dye lifetime was substantially
less than 210 hours.

m. SELF-ABSORPTION IN RHODAMINE-575
Emission and absorption spectra from luminescent materials
will typically have some overlap, such that there is a finite probability
of re-absorbing or self-absorbing the emission from a luminescing
specie. In particular this probability of self-absorption in an LSC will
increase as the pathlength traveled by the emission in the plate increases. Such an effect limits the size of an effective LSC plate. We
were, therefore, interested in a detailed study of the self-absorption
effect, at least for one particular luminescing system.
Rhodamine-575 was chosen to be studied for two reasons. It is
a typical xanthene organic laser dye, which is a large class of efficient
and commonly used luminescing dyes (Kuhn, 19 59). Its molecular
structure is shown in Figure 39, and is almost identical to that of
rhodamine-590.

The first reason rhodamine-575 was chosen was,

therefore, that we hoped results from this dye would be gnerally true
at least for the xanthene dyes.

The second reason was that we wished

to pump the dye in the extreme low energy tail of its absorption band
using our rhodamine-590 dye lasers.

This indicated the choice of

rhodamine-575 in the xanthene family.
Rod Emission
We measured the changes in the intensity and spatial characteristics of rhodamine-575 emission by scanning the position of a small
excitation spot with respect to the sample, and then observing the

-86-

Figure 39. Structure of rhodamine-575 .

-87-

HNC2 H5 "

~ "- /

CH /

~ /

0 "- ~ ""- ~ NHC2H 5

""- ~

~ ~ "-CH

~ "--coo-

Structure of Rhodamine- 575

-88-

resulting emission from a fixed point on the sample.
picts the apparatus used.

The 22, 940 cm

-1

Figure 40 de-

(4, 359A) line from a 200

watt Oriel 6137 high pressure mercury lamp was filtered by a Bausch
and Lomb 33-86-25 monochromator, modulated by a PAR-BZ-1
chopper, and focused by a x 40 microscope objective onto one end of
a one meter section of Math Associates OC-1200 glass optical fiber
bundle. Since we were interested in the change of intensity as well as
the spectral shape of the emission as a function of the pathlength
through the sample, it was important that the position of the excitation
source at the sample be adjustable without any variation in the intensity. Small changes in the curvature of an optical fiber cause little
change in its transmission, so that the free end of the fiber was used
as the mobile excitation. The free end was mounted on a micrometer
stage that would track parallel to the rod-shaped sample of LSC material, as shown in Figure 40.

Three different types of samples were

tested: methanol solutions, cast PMMA, and diffused PMMA. Samples
were mounted inside 7mm o. d. (5mm i. d.) borosilicate glass tubes to
maintain a uniform contact between the excitation fiber and the sample.
The end of the tube adjacent to the detecting monochromator was
sealed with a 1 cm glass plug.

The detection system was a 0. 25 meter

Jarrell Ash spectrometer, a Hamamatsu 928 PMT, and a PAR HR-8
lock-in amplifier. The detection monochromator was scanned and the
output of the lock-in amplifier was sampled by the PDP 11-03/2, as
described in Section I.
The results from one such measurement are shown in Figure 41.
In this case the borosilicate glass tube was filled with a 92 micromolar

-89-

Figure 40. Apparatus for measuring the intensity and spectral shifts
of sample emission as a function of sample pathlength.

The 4, 359.A

line from a mercury lamp was monochromated, chopped, and focused
onto the end of an optical fiber.

The position of the opposite end of

the fiber was scanned by a micrometer stage along a rod-shaped LSC
sample.

The output was detected by a scanning monochromator.

-90-

(.)

Q.)

: \I

-91-

Figure 41.

Emission spectra from a 92 micromolar methanol solu-

tion of rhodamine-575.

The sample pathlengths, in order of most to

least intense, were 0. 3, 1. 0, 3. 0, 10. 0, and 30. 0 cm.

-92-

N :ft

.....

(/')

:r:

NLLI

I(..!)

z :r:
a: I:c
..,_ a:

1--t

£)
f--1

(/)
(/)

Lt)

.......

r•

....

cc
0)
:L

::>

['I.()

Cf)

(Y)

t--1

--4

..•

.•

a: .......

.....•

:::c en
a:

NkJISSIW3

-93solution of rhodamine-575 in methanol. Spectra were taken for the
following distances between the fiber excitation and the glass plug
seal at the end of the tube: 0. 3, 1. 0, 3. 0, 10. 0, and 30. 0 cm. System response corrections were made to compensate for the detection
monochromato r and PMT, as described in Section I. Apart from this
correction, the amplitudes of each spectrum correspond to the actual
intensities emerging from the end of the sample rod.

The predomi-

nant observed effect was that the higher energy portion of the emission
band erodes away with greater pathlength through the rod, and that the
change in emission intensity varies roughly as the inverse of the logarithm of the pathlength. Similar results were obtained from the PMMA
samples.

Figure 42 shows a similar set of spectra for a cast PMMA

rod.

Spectral Homogeneity
We attempted to measure the spectral homogeneity of the absorption and emission bands for rhodamine-575 in different host materials.
Luminescence spectra were taken using the apparatus shown in Figure
43.

A spectra Physics 160 argon ion laser was used directly as the

exciting source, or instead served as a pump for a Spectra Physics
375A dye laser.

The dye laser operated with an ethylene glycol solu-

tion of rhodamine- 590. When we excited the sample at an energy consilerably below its peak absorption, it was important that very little
spontaneous emission of a higher energy than the principal lasing wavelength be present in the sample excitation beam.

The dye laser was,

therefore, followed by several sharp cut filters, a dispersive prism,
and a slit aperature. Neutral density filters were used to adjust the

-94-

Figure 42.

Emission spectra from rhodamine-575 cast in PMl\1.A.

The dye concentration in the rod was approximately 20 micromolar.
The emission pathlengths, in order of most to. least intense, were
0. 16, 0. 4, 1. 0, 4. 0, and 10. 0 cm.

1--1

Cf)
(f)

t--f

1.32

WRVENUMBERS

1. 72

2.12
E 14

IN A PMMA R~D
.16, .4, 1., 4., RND lO~CM. PRTHLENGTHS.

RH~DAMINE-575 EMISSI~N

-96-

Figure 43. Apparatus for measuring luminescence of liquid, cast
plastic, and diffused plastic samples at different temperatures and
excitation energies.

-97-

Q)
(J)

-a.
Q)

-98input power to the sample.

The samples were held in a glass dewar

with windows positioned for both the excitation beam and for the resulting right angle emission.

The luminescence spectra were taken

both at room temperature and with the samples immersed in liquid
nitrogen. Since the excitation was not chopped, stray room light had
to be eliminated.

The luminescence resulting from the laser excita-

tion was collected by a Spex 1419A Sample Illuminator, and was then
analyzed by a Spex 14018 double monochromator which utilized 2, 400
grooves/mm holographic gratings.

(The sample illuminator was a

collection optics stage originally designed for Raman Spectroscopy. It
contained the dispersive prism, slit, sample translation stage, and
matched f-number optics for the spectromator. ) A polarizing filter in
the collection optics was used to suppress scattered laser light,
followed by a depolarizing prism to compenstate for the polarization
response of the gratings.

The spectrally resolved output was measured

by a Hamamatsu 955 PMT followed by a Spex DPC-2 photon counter.
The final chart recorder output was digitized and stored by the
PDP 11-03/2. System response correction was done by digitizing the
measured spectrum of a calibrated tungsten lamp, as described in
Section I.
The liquid sample studied in this manner was 18 micromolar
rhodamine-575 in methanol.

This was mounted in a 1. 3 mm o. d.

capillary tube, oriented so as to minimize the pathlength of the output
emission. When the methanol was cooled to liquid nitrogen temperatures it formed a fractured glass.

The cast plastic sample was a

-9920 x 25 mm PMMA rod containing an effective concentration of 14
micromolar rhodamine-575.

The diffused sample was a 1. 6mm

diameter rod of Rhomm and Hass plexiglass that had been soaked for
4 hours in a 100 micromolar solution of rhodamine-575 in a mixture
of 0. 91 methanol and 0. 09 dichloromethane by volume.
The results from the liquid sample are shown in Figure 44.

The 200 cm- wide notches in the spectra are due to a shutter which
blocked the PMT as the spectrometer was scanning across the laser
frequency.

The lower three spectra correspond to room temperature

emission for laser excitations at 16, 509, 17, 164, and 20, 492 cm- 1 ,
respectively.

The peak of the emission for the lowest energy excita-

tion is at the same frequency as that of the highest energy excitation,
despite the fact that some of the emitted photons were blue-shift
(shifted to higher energy) as much as 1, 700 cm - i up from the excitation frequency.

This anti-Stokes shift energy is greater than 8 kT for

room temperature spectra, (kT = 0. 026 ev = 208. 5 cm -i for T = 300 K).
Introducing sharp-cut filters into the excitation beam centered intensity in proportion to the reduction in the excitation intensity, but
caused no shift in the observed spectral shape. We concluded that
these emission spectra did not include emission due to stray light
excitation of higher energy.

The upper plot shows emission spectra

at 77°Kdue to excitations at 16,877, 17,163, and 20,493cm- 1 •

The

luminescence spectrum produced by the highest energy excitation is a
slightly narrower version of the room temperature spectrum.

The

lower energy excitation at low temperature produced negligible emission at higher energies, which agrees with the previous result that the

-100-

Figure 44.

Emission spectra for rhodamine-575.

The lower three

plots show room temperature emission in methanol at three different
excitation energies.

The upper plots are similar spectra taken at 77°K.

1.39

EXCIT:

300 KELVIN

WRVENUMBERS

1.69

':.-.t

171 64.-

........,.,,.,~~

17163.

R-575 IN METHRN~L.

77 KELVIN
EXCIT: 16877.

EMISSI~N ~F

1.99
E 4

....,.
0....,.

-102emission at room temperature from low energy excitation was due to
anti-Stokes shifting of the excitation. Similar large anti-Stokes shifts
have been observed in vapor phase dye spectra (Pappalardo and Ahmed,
1972).
Figure 45 gives corresponding results for rhodamine- 575 cast
in PMMA.

The lower plot shows the room temperature luminescence

spectra normalized to unit area.

These spectra differ from the metha-

nol solutions in that the emissions produced by low energy excitations
at 16, 273 and 17, 060 cm

-1

were greatly skewed towards the red,

whereas the higher energy excitations at 19, 436 and 20, 492 cm -l produce emission with the same general shape and position as the methanol
room temperature emission.

The upper plot shows the low tempera1

ture emission due to excitations at 17, 423, 19, 436, and 20, 492 cm - •
As in the liquid samples, the anti-Stokes shifted emission was suppressed at 77° K, however in this case the emission shape varied greatly
with the energy of the excitation.
The diffused plastic sample showed anti-Stokes emission intermediate between that of the cast plastic and liquid solution.

Figure 46

shows the room temperature emission of a PMMA sample soaked in a
methanol solution of rhodamine-575.

The excitation positions were

16, 116, 17, 449, 19, 436, and 20, 492 cm- 1 •

The emission spectra have

been normalized to unit area. While there is some variation in the
peak position with excitation energy, we found that the spectra was
dominated by large anti-Stokes shifts for low excitation energies, as
was the case for the methanol solutions.

-103-

Figure 45.

Emission of rhodamine-575.

The lower three plots

represent emission in cast PMMA at room temperature for three
different excitation energies.

The upper plots are similar spectra

taken at liquid nitrogen temperatures. All spectra are normalized to
unit area.

1.39

300 KELVIN
EXCIT•
16273.

1.69

201492.

WRVENUMBERS

17060.

R-575 IN PMMA.

77 KELVIN
EXCIT•
171423.

EMISSI~N ~F

1.99

-201492.

/ 191436.

.....I

-105-

Figure 46. Emission of rhodamine- 575 in diffused PMMA at room
temperatur e.

The four normalized spectra correspond to four

different excitation energies.

1.39

EXC IT.•

16116 .

WRVENUMBERS

1.69

17449 . ---U

R-575 IN PMMA.
SURFACE DIFFUSED DYE
300 KELVIN

EMISSI~N ~F

1 .. 99
E q

/19436.
20492a

.....

O')

-107Preliminary measurements were made of the emission spectra
of rhodamine-575 in cast PMMA at liquid helium temperatures.

There

was no pronounced change in spectral shape between the liquid nitrogen
and liquid helium spectra.
Polarization Measurements
We suspected that self-absorptio n rates could be measured
rather simply by observing the relative polarization intensities of
emission from a sample excited by polarized light.

Light from a

tungsten lamp was monochromate d at 20, 200 cm - 1 , chopped, and
passed through an adjustable polarizer (Polaroid HN38).

This light

was focused onto an ethylene glycol solution of rhodamine-575 .

The

concentration was varied during the course of the experiment by
changing the dye solution.

The resulting emission perpendicular to

the axis of the excitation was passed through a second adjustable
polarizer, and the intensity at 17, 860 cm -l was measured with a second
monochromato r, a PMT; and a lock-in amplifier.

The lamp, mono-

chromators, PMT, and lock-in have been described in Section I.

The

polarizers were controlled by the PDP 11-03/2, which also digitized
and stored the output of the lock-in.
Dye solutions were used in the place of cast plates for two reasons. The first was that dyes in solution show far greater spectral
homogeneity, which allowed for more tractable modeling of the selfabsorption process.

The second reason was that the use of solutions

facilitated changing the dye concentrations in the sample. Ethylene
glycol was chosen because it has a viscosity of about 20 centipoises
at room temperature, and should produce rotational diffusion times

-108for the dye molecule which are long enough that emission from a polarized excitation would also be partially polarized, assuming that the
absorption and emission dipoles are aligned. If this is the case, then
emission polarized in the Z direction should be more intense than
emission polarized in the X direction for a Z polarized excitation
(see Figure 47). Y-polarized excitation should produce equal intensities for both polarizations of emission.
We varied the dye concentration of the sample and measured the
,.

intensity of the z- and X-polarized emission at each concentration.
The sample dye concentration was varied from 0. 2 micro molar to 100
micromolar by factors of 2. 0.

The pathlength in the sample traversed

by the emission en route to the detection monochromator was 0. 7 cm.
The observed intensities were corrected for background and for the
polarization response of the system.
Figure 48 shows the variation between the

z- and X-polarized

emissions for Z-polarized excitation as a function of concentration.
The plotted value is the polarization anisotropy, which is the emission
....
...
intensity of the Z- minus the X-polarized intensities, both divided by
,.

...

the sum of the z- plus two times the X-polarized emission intensities.
For very low concentrations, the anisotropy plateaus at about 0. 18,
or equivalently, the ratio of the parallel to the perpendicular emission
is about 1. 65. As the concentration increases, the apparent polarization of the emission decreases.

The error bars reflect variations of

repetitive measurements at a single concentration.

These errors

could probably be substantially reduced by the use of a more intense
excitation source.

-109-

Figure 47.
ments.

Principal directions for polarization anisotropy measure-

Emission is detected perpendicularly to the excitation axis.

The excitation and emission both have vertically and horizontally
polarized components. If the excitation is vertically polarized,
emission is parallel if it is vertically polarized and perpendicular if
it is horizontally polarized.

-110-

I 11 ( I vert)

I l. (I horz)

-111-

Figure 48.

Polarization anisotropy vs. sample concentration for

rhodamine-575 in ethylene glycol.

The reduced polarization anisotropy

is the difference between the parallel and perpendicular intensities
divided by the sum of the parallel and twice the perpendicular intensities.

The solid line serves only to connect the data points.

o.oo•
10-8

~ 0.50

z 1.00
a:

(f)

a: 1.50
..._

>CL

2.00
-1 I

111111

I t 111111

i ; 111111

IN GLYC~L.

. . . .....

C~NC.

10-7

10-s

X P~THLENGTH

10-6

•••••••••••••••••••••••••••••••

I .....

10-ij

10-3

• • • • • • • • • • • •,

.7 CM. PRTHLENGTH IN SAMPLE
20,00 0 WRVENUMBER EXCITATI~N

F~R RH~OAMINE-575

EMISSI~N P~LRRIZRTI~N RNIS~TR~PY

I.\:>

......
......

-113Transient Emission Measurements
We measured the time evolution of the emission intensity resulting from very short pulse excitation of dyes in a variety of hosts.
Time-resolved luminescence measurements were performed with a
mode-locked argon ion laser (Spectra Physics model 171/342) and a
photon counting apparatus.

The experimental apparatus was expertly

planned and assembled by David Millar and Raymond Robbins
(Robbins, Millar, and Zewail). Samples were illuminated with
vertically polarized 19, 436 cm -i light pulses at a repetition rate of
about lOOkhz, with a typical pulse half-width of less than 200 picoseconds. Emission from the sample was monitored by a Philips
XP2020Q photomultiplier. Cut-off filters at 5, OOOA and 5, 700A were
placed between the sample and the PMT to reduce scattered laser
light, plus an analyzing polarizer to sample different emission polarization intensities.

The system response, measured by laser light

scattered from a dilute coffee creamer solution sample, is shown in
the upper plot in Figure 49 and usually had a response time of less
than 250 picoseconds.
The first type of measurement we made was to find the total
lifetime of the dye as a function of concentration. In this case the dye
emission was filtered by a polarizer at 54. 7 degrees from the laser's
original vertical polarization so as to average between the emission of
the parallel and perpendicular polarization components.

The lower

plot in Figure 49 shows typical transient emission data for a 4. 6
micromolar solution of rhodamine-575 in methanol.

The dye was

contained in a 1. 0 x 0. 5 cm (i. d. ) cuvette so that, on average, the

-114-

Figure 49. System response and transient lifetime measurement.
The upper plots show a typical histogram of the system response to a
5, 145.A. pulse from a mode-locked argon ion laser, with a duration of
less than 200 picoseconds, scattered off of a dilute coffee creamer
solution.

The width and asymmetry of the response was predomi-

nantly due to the PMT.

The lower plot shows the observed emission

of a 4. 6 micromolar rhodamine-575 solution, superimposed on a best
fit exponential convolution of the system response.
fit gives a lifetime of 4. 1 nanoseconds.

The exponential

::::>

(/)

-10.

-5.

.. ..

10 •

NRN(jSEC(jNOS

15 •

20 •

UN-MASKED EMISSieJN

SYSTEM RESPeJNSE

25 •

-TRANSIENT EMISSI~N ~F A DILUTE DYE S~LUTI~N-

......

-116emission passed through 0. 5 cm of sample before arriving at the
collection optics.

The extinction coefficient for rhodamine- 575 at the

lasing wavelength was about 90, 000 moles per cm per liter, so that,
even at this low concentration, about 40% of the laser excitation was
absorbed by the sample. Superimposed on the raw data in Figure 49
is a least squares fit of a single exponential including a convolution of
the system response given in the upper plot.

The best exponential fit

gave an observed lifetime of 4. 13 nanoseconds at this concentration.
Similar measurements have been made on rhodamine-590 , rhodamine610 and Kiton red-620 (Hammond, 1979).

The measurement technique

used was incompatible with our need for measuring high rates of selfabsorption in arbitrary geometries. In general, lifetimes also increase with decreasing temperature (Theiss and Weber, 1974).
We measured excitation lifetimes for a sequence of dye concentrations from 0. 2 to 100 micromolar for rhodamine-575 in both methanol and ethylene glycol. As in the polarization measurements , we
opted for solution samples because of both the ease of changing concentrations and because of the better spectral homogeneity of the dye
in solution as opposed to the dye in PMMA.
Figure 50 shows the fitted lifetimes as a function of concentration
in methanol.

The highest concentration used was a factor of ten below

the critical concentration, as defined by the Forster model.

(The

critical concentration for transfer from rhodamine-575 to itself in
methanol is 4. 6 x 10- 3 M).

At low concentrations the lifetime of

rhodamine-575 approached an asymptotic value of 3. 7 nanoseconds,
While at high concentrations it increased to three times its low

-117-

Figure 50.

Measured lifetime of rhodamine- 575 in methanol as a

function of concentration.
the data points. )

(The solid curve serves only to connect

I I I Ii I

• • • • ••

I tiliil

I t I I

l,W

I iliiTI

10-8

0 . 00 La

1.00

a: 2.00

Cf)

C~NC.

10-7

I ' t

1o-5

I I I I

X PATHLENGTH

10-6

I I II

10-ij

10-3

I liliii

w 3.00

.3 CM. PATHLENGTH IN SAMPLE
19.440 WAVENUMBER EXCITATI~N

......
......

l!.00

Cf)

5.00

E 0

6 00 pr

LIFETIME ~F RH~OAMINE-575

IN METHAN~L.

T~TAL

-119concentration value.

The errors bars in Figure 50 indicate the un-

certainty of the lifetime in the numerical fitting of the data. As in the
previous case, the emission was filtered with a polarizer 54. 7 degrees from the vertical to sample equally the vertical and horizontal
(parallel and perpendicular) polarization components. Care was
taken not to mask any portion of the emission of the dye from illuminating the PMT.
Others have suggested that variations in excitation lifetime
might be due to molecular re-orientation times (Shapiro and Winn,
1980).

To help determine the mechanism for lifetime lengthening of

the dye with increasing concentration, we performed a measurement
of the emission intensity as a function of time, except that in this
case part of the sample cuvette was blocked so that dye in a portion
of the cuvette could not directly illuminate the detector.

Figure 51

shows a schematic of the sample geometry with respect to the exciting
laser and the output emission.

The sample used in this case was

rhodamine-575 in methanol at 460 micromolar.

At this concentration

99% of the excitation was absorbed in the first 0. 05 cm of the sample.
We were not interested in measuring the emission resulting directly
from this excitation, but in the emission that followed the selfabsorption of this first emission.
achieved this condition.

The mask shown in Figure 51

The transient emission spectrum resulting

from this configuration is shown in the upper plot in Figure 52. When
the mask was removed, we obtained the lower plot in Figure 52.

The

solid curves superimposed on these two spectra are fitted theoretical
functions, and will be described in the fourth chapter. It was evident

-120-

Figure 51.

Position of the first generation mask.

The 5, 145A laser

line is 99% absorbed in the first 0. 05 cm of the 460 micromolar
rhodamine-575 methanol solution. Blocking just the left portion of
the cell, as shown, prevents emission due to the initial excitation
from being detected directly.

-121-

Filter
and
Polarizer

C uvette

• < 200
Picosecond
Laser Pulse
5145 A

-122-

Figure 52.

Unmasked and masked transient emission spectra.

The

upper plot shows the transient emission histogram from a 460 micromolar rhodamine-575 methanol solution for the case where a mask is
placed between the region excited by the laser and the detector.
lower plot is the same measurement with the mask removed.

The

-123-

...-.
r-

::>
_.J

II)

_..

lJJ

ra:
a:
rzw

_..

II)

fJ)

LLJ

.......

I-

(/)

za:

en
a:

z::::>

........
a:
l--

LLJ

zE)

en
en

_..

SlNnQJ

-124-

that there were significant distortions of the single exponential decay
when the initially excited portion of the sample was masked.
JV.

PROTOTYPE TESTING

LSC research has a bottom line in the sense that the primary
goal is to build devices which are useful for converting sunlight into
other forms of energy. We believe that the best characterization of
an LSC plate is to measure its flux gain with a particular type of cell.
The flux gain, Gflux' is defined to be the factor by which the power
output of a cell increases when it is attached to the LSC plate, as
opposed to facing the sun directly. A second important parameter to
report is the geometric gain of a plate, which is the ratio of the area
of the LSC which is exposed to the sun (the LSC face) to the area
covered by solar cells (the LSC edge).
can be measured with a ruler.

The geometric gain, Ggeom'

These two numbers will dictate the

initial cost of the device. as well as its efficiency.

For example, if

the AMl efficiency of the cell is known, 11cell' then the efficiency of
the cell-LSC combination, 17LSC' is the cell efficiency times the flux
gain divided by the geometric gain, TJLSC = 11cell · Gnux/Ggeom·
Table 3 lists the performance parameters for a variety of
devices tested by ourselves and others.

The measured performance

of an LSC depends on the temperature, the angular distribution of the
incident light, the spectral content of the light, and probably other
factors.

Unfortunately, the testing procedure varies for the different

prototypes, so we will try and describe the circumstances for each of
the tests.

-125-

Table 3.

Prototype performances. Devices B and D were made

and tested by Owens Illinois.
here.

The rest were made and tested

The geometrical gain is defined to be the ratio of the area

of the plate exposed to the sun divided by the area of the edge
which absorbs light.

The flux gain is the factor by which the

output power of a solar cell is increased when it is mounted on
a plate.

The collector efficiency is the assumed AMl cell efficiency

times the flux gain divided by the geometrical gain, and corresponds
to the electrical power out per solar power input.

18%
21%
18%
21%
18%
18%

2. 1

1. 3
3.8

1. 7

5. 1
4.4

23
11
36
11
68
92

PMMA
thin film
ethylene glycol
thin film
PMMA
PMMA/DMSO

0. 9%

1. 3%

3.2%

1. 9%

2. 5%

1. 9%

Eff.

Matrix

Eff.

No.Dyes

Collector

Assumed
Cell

Flux
Gain

Device

Geometrical
Gain

Table 3.

.....I

-127Devices B and D in Table 3 were built and tested by OwensIllinois.

The dyes were contained in thin plastic films attached to the

surface of clear acrylic substrates. Measurements were made under
actual insolation (presumably direct sunlight), with the plate edges
roughened and blackened where cells were not mounted.

These plates

have achieved the highest efficiencies, but their small geometric gains
make them somewhat ineffective as concentrators.

For example,

cells mounted on device D had only a 70% increase in output over
similar cells which faced the sun directly.
Device C was constructed by gluing two 110 x 110 x 0. 3 cm
plexiglass plates on opposing sides of 0. 08 cm thick plexiglass spacers.
The assembly was mounted on a mirror backing.

The resulting gap

between the plates was filled with an ethylene glycol solution containing 57, 94, and 51 micromolar solutions of coumarin-540, rhodamine590, and rhodamine-640, respectively.

The edges of the sandwich

assembly were made light absorbing with black electrician's tape
except for the portion coupled to the cell.

The flux gain was measured

under actual insolation by first measuring the short circuit current on
a standard cell under direct insolation, and then measuring the short
circuit current for the same cell coupled to the plate, using ethylene
glycol as an index matching fluid. We calculate that this dye combination will absorb about 30% of a 5, 800°K black body spectrum in a two
pass geometry. The collector efficiency given in Table 3 is the cell
AMl efficiency times the flux gain divided by the geometric gain.
Device E was a 120 x 100 x 0. 4 cm PMMA plate containing 220
micromolar DCM.

The edges were taped as before except where the

-128cells were mounted.

The measurements were made using two cells

with AMl efficiencies matched to within 5%, and both with an active
area of 0. 4 x 2 cm. One cell was mounted co-planar to the LSC face
as a reference, and the other was contacted with the LSC edge using
ethylene glycol as an index matching fluid.

The tests were made

using a xenon flashlamp AMO simulator at JPL.

The flux gain was

determined to be the ratio of the peak power of the LSC-mounted cell
to the reference cell, corrected for flash to flash variations. We
calculate that this plate should absorb 30% of a 5, 800° K spectrum.
The collector efficiency was determined in the same manner as for
device C.
The performance of device E is less than would be expected
from the DCM CODE of 240 in methanol.

The reason is clear from

the absorption and emission spectra of DCM in PMMA, shown in
Figure 12.

The overlap is much greater in PMMA than in methanol,

so that the CODE of DC~ in PMMA is only 20. A similar effect had
been observed for other dyes in plastics by Sah (Sah and Baur, 1980),
who found that much of the original Stokes shift could be restored by
adding solvent to the monomer prior to polymerization. On the basis
of this suggestion, we made a PMMA plate containing 4. 5 x 10 M
DCM and 3 % dimethylsulfoxide by volume.

The device geometry

and testing procedure was the same as for device E.

The performance

of this plate is listed as device F. The D MSO did not appreciably restore
the solution Stokes shift, and caused significant loss of plate quality.

-129V. SUMMARY

1. The low concentration absorption, emission, and excitation
spectra in methanol for 18 organic laser dyes were catalogued.
2.

Photobleaching rates for a variety of plastic and solution

samples were measured under actual insolation. Degradation was
measured both by emission and absorption decay; the two rates were
found to be similar. Degradation rates increased with time. Sample
lifetimes varied between hours and years.
3.

Emission spectra as a function of self-absorption pathlength

were taken of rhodamine-575 in solution, in diffused PMMA, and in
case PMMA.
4.

Emission spectra of rhodamine-575 in methanol, in diffused

PMMA, and in cast PMMA were taken as a function of temperature
and excitation energy.

The peak position of the emission was inde-

pendent of temperature and excitation energy for the methanol and
diffused PMMA samples, and varied considerably in cast PMMA.
5.

The component polarization intensities of the emission of

rhodamine-575 resulting from plane polarized excitation was measured
as a function of concentration for a fixed self-absorption pathlength.
The emission became increasingly unpolarized with increasing concentration.
6.

The total lifetime of rhodamine-575 in solution was measured

as a function of concentration for a fixed self-absorption pathlength
using very short pulse excitation. The measured lifetime increased
with increasing concentration. The transient emission spectra showed

-130-

an anomolously slow build up time if the initially excited portion of
the cell was masked.
7.

Performance measurements were made on a variety of

prototype devices. Our highest gain collector had a flux gain of 5. 1
and an estimated overall efficiency of 1. 3%. Our highest efficiency
collector was a liquid cell with a flux gain of 3. 8 and an estimated
overall efficiency of 1. 9%.

llL

-131-

CHAPTER 3

-132J.

SINGLE DYE MODEL FOR THE PERFORMANCE OF AN LSC
There are many design variables in an LSC. Which dye should

be used and at what concentration? How large should the plate be?
What type of solar cell should be mounted on the plate? We wished to
find a reasonably accurate model that would take the measured characteristics of the components, including, for example, the absorption
spectrum of a particular dye, and from this would predict the light
output and electrical efficiency of an LSC using that dye. What follows
is such a model. Its strength is that it appears to be fairly accurate
when correlated to actual prototypes. Its weakness is that it is too
complicated to allow rapid assessment of a variety of JX>SSible systems.
Photon Flow Diagram
We can trace the flow of excitations in an LSC with the aid of the
flow chart in Figure 53. Above all is the sun.

Part of its incident

flux will be initially lost due to reflection from the I..SC surface, and
part is lost because its wavelength does not corres}X>nd to the absorption band of the dye used. What is left is the absorbed solar flux in
the dye ensemble, denoted by S.

There is a net rate of excitation, J,

of the dye ensemble, which in steady state must correspond to the
rate of de-excitation.

The photon output of the dye ensemble is the

quantum efficiency of luminescence, 11, times J.

This luminescence

is geometrically divided into the fraction JPTJ which is emitted within
any of the critical escape cones, and the fraction J(l-P)17 which is
trapped.

Light emitted into the critical cones has an average proba-

bility r that self-absorption will take place before the light can escape

-133-

Figure 53.

Photon flow diagram for a single dye LSC.

Light from

the sun is partially reflected, partially absorbed by the dye ensemble,
and partially transmitted through the LSC plate.

Emission is geo-

metrically divided between the critical escape cone and light trapped
by total internal reflection.

Light can be re-cycled into the dye en-

semble by self-absorption.

Collected light is converted to electricity

by a photovoltaic cell (PVC).

-134-

Photon Flow Diagram for a Single Dye LSC.
Critical
Cone

Loss

Unabsorbed
Flux

--------

~-•

Reflected
Flux

___

AIR
---------LSC

__._

PJ71 ,___ _..,.._ _~-....--.---<

---- -------,
Photovoltaic ..,._ _..._....,_---(
Conversion

QSS/(1-8}

Rpvc SQ

Transport

PVC I LSC

Loss

L- -

_._. ._. ._. - - - - - - - - -

-135out of the LSC, so that there is a feedback loop of magnitude

r JP1J

of emissions into the critical cones that are recovered as excitations
in the dye ensemble.
A similar feedback loop occurs for self-absorption of trapped
light. In this case the probability that a trapped photon will be selfabsorbed before it reaches the LSC-cell interface is r.

An additional

lumped parameter o describes the fraction of the trapped luminescence which is lost due to matrix absorption or imperfect reflections.
A hardy fraction, Q, of the originally absorbed solar photons, arrives

at the LSC-cell interface where a reflection can take place of magnitude QRcell· In this analysis we will assume that such reflected flux
is lost.
Solar Absorption - S
The solar spectrum is a variable quantity. We usually use a
5, 800° K black body spectrum in our calculations. A comparison of a
measured solar spectrum (Boer, 1977) and a 5, 800°K black body
spectrum is given in Figure 54.

The solar flux per wavenumber will

be denoted by I (v), and the total flux by I, where

...

1 =

1(v)dv

(1)

Clearly this could be extended to include angular variations in spectrum
and intensity, such as would be expected in a mixture of direct and
diffuse insolation.
A fraction of this incident light will be reflected at the surface.

-136-

Figure 54.

Comparison of an AMl solar spectrum and a 5, 800°K

black body spectrum.

88.6 WATTS/SQ.METER
100 WATTS/SQ.METER I

-;J

3000

5000

7000

WAVELENGTH IN ANGSTR~MS

9000

11000

13000

15000

Or.11-~~~~~~~..J-~~~~~~~-L~~~~~~~~..__~~~~~~~--L.~~~~~~~--l~~~~::-~~~.JCJ

3:1n

II-

(/")

.........

(/") 0

::E: 0

.......

....
(,)

......

a:o

in
.....

T~TAL

INCIDENT LIGHT DURING A
C~~L SUMMER DAY IN DELAWARE.

THE SOLRR SPECTRUMo

-138The reflection coefficient R can be computed by the Fresnel·equati on
for an uncoated LSC (Born and Wolf, 1975 ), or by similar relations
for LSCs with index-matchin g coatings (Hovel, 1975).

(The reflection

coefficient for vertical incidence on an uncoated plate with an index of

refraction n is (n-1) / (n+l) . )
Passing the air- LSC interface, the light will be partially absorbed by both the dye molecules and by the matrix material.

The

absorption pathlength for light incident on the plate with an angle e is

(2)

\11 - sin (8)/n

where D is the thickness of the plate.

The length is twice that ex-

pression if a backing mirror is used. If the matrix material has an
absorption coefficient a(v), such that the attenuation of light of wavenumber v over a distance ..e.s is given by 10 -£sa(i7),

then the light

absorbed by the plate in photons per square meter per second is

... 00
S =

I (ti) (1 - R (v)) [1 - 10-fs(C. E(v) + a(v))]

C · E(V)

a(v) + C · E (Z/)

for vertical incidence (Batchelder, Zewail, and Cole, 1979).

(3)

E (V) and

C are the molar extinction coefficient and the dye concentration,

-139-

respectively.
The index of refraction of the usual matrix material is quite low
(typically about 1. 5) so that reflection losses tend to also be low. As
the angle of incidence increases, the apparent area of the plate de. creases as the cosine of the angle of incidence. As the angle increases,
the reflection coefficient also increases, however this is usually more
than offset by the increased absorption pathlength for oblique angles of
incidence light in the plate. As a result the angle of incidence dependence of an LSC is very nearly that of an idealized black body absorber,
as shown in Figure 55.
Quantum Efficiency of Luminescence - Tl
We have calculated the number of excitations per second per
square meter of the dye ensemble due to the sunlight, which is the S
just calculated.

The quantum efficiency of luminescence is the proba-

bility TJ that these excitations will be emitted as photons.

There is an

assumption that only excitations which excite the molecules into one of
the electronic excited state manifolds are included both in S and in
measurements of TJ.
Escape Probability - P
Luminescence incident on the LSC faces at an angle greater than
the critical angle will be totally internally reflected.
angle in this case is given by 8c

(The critical

= sin- 1 (1/n), where n is the index

of refraction of the plate.) The critical cone originates at the point of
luminescence and forms a critical angle everywhere it intersects a
surface. If the luminescence is isotropic and the LSC is planar, the

-140-

Figure 55. Dependence of absorbed flux on angle of incidence.
The lowest curve indicates the flux absorbed by a millimolar
rhodamine- 590 plate 3 mm thick.

This is very similar to the

simple cosine law of a black body absorber with a reduced total
absorption, as shown in the middle curve.

-141-

ENERGY VS.
ANGLE ~F INCIDENCE.

RBS~RBED S~LRR

-"b

(XJ

IDEALIZED SINGLE DYE PSC
SINGLE DYE PSC
RH-6G IN PMMR

ANGLE ~F INCIDENCE

-142probability P that luminescence will escape out of the critical cones
is the sum of the solid angles of the upper and lower critical cones
divided by the total solid angle of emission

nee

p =

r·1T
sin(B) de + j Tf-B

sin (8) d8

-.11

sin (8) d8

= 1 - cos(8c) = 1- Jt-1/n 2

(4)

There is an approximation in the above definition of P that the
dye molecules show no angular dependence on their absorption and
emission.

Actually, to a good approximation, dye molecules typically

appear as electric dipoles, usually with the absorption dipole collinear
with the emission dipole.

This decreases the calculate amount of light

trapped, because the incident sunlight will be mostly absorbed by dipoles which are oriented in the plane of the LSC, and these dipoles have
a greater chance of emitting into the critical cones. Specifically, the
emission angular dependence, I (ee, ex), of the emission intensity in
an LSC plate as a function of the excitation angle of incidence, ex, is

(5)

-143where the brackets denote an average overall dipole orientations

oma and 6m e·

Figure 56 shows the coordinate system and angle de....

finitions for this calculation. E is ~unit vector in the direction of

the electric fields of the light. µ(Bma) and µ(Bme) are unit vectors
in the directions of the absorption and emission dipoles, respectively.
The angular dependence of the emission intensity in the plate is
therefore

,. ., 27T
-. 1T
I (8 ~, 8e) = 411T ~lo dma
dB ma sin (Oma)

1(8',B)
x e

= 4 - cos (B')
x - 2 cos (fJe ) + 3 cos (8 x') cos (Be)

_.!_ sin ( 9 ) •

(6)

For a planar geometry, the escape probability is again computed by
dividing the solid angle of the escape cone, this time times the intensity as a function of angle, by the solid angle of emission weighted by
the same intensity:

-144-

Figure 56. Coordinate system for the critical cone escape calculation.

The probability of escape out of a planar LSC with an index of

refraction n is actually greater than 1 - (1- 1/n )1/z due to the dipole
character of the absorption and luminescence. We assume sunlight
enters at some polar angle Os, is absorbed by a dipole with orientation (ea, 8e ), and is emitted by a dipole with the same orientation in
a direction (Be,

-145-

Incident
Sunlight

Air

LSC
Emission
Absorption
and emission
dipole µ. (8 ,cp )

(Be ,cf>e)

Coordinate System for Critica1
Cone Escape Calculation.

-146-

P(8~)

= 1 - (1 - 1/n )

Yz2

3 cos (8
(1 -

) -

10n2

) .

(7)

We finish by correcting the excitation angle in the plate for the solar
angle of incidence by Snell's law:

= 1 - (1 - 1/n )

(1 - +
10n

3 sin (Bx)

lOn4

(8)

This gives an escape probability of 0. 29 for vertical incident light on
an OSC with an index of~- 49, as opposed to 0. 26 for the same calculation ignoring the dipole nature of the dyes.
Self-Absorption Probabilities - r and r
We define the average probability that luminescence outside of
the critical cones will be self-absorbed to be r.

Similarly, r can be

experimentally measured in a variety of ways. We will presently
compute r indirectly via the collection probability Q. We will
approximate r by a Beer- Lambert absorption probability across the
thickness of the plate averaged over all wavenumbers:

-147-

., 00

r = j ·ctv 1 (ii) [1-10-T· c · €(ii) ]

(9)

This is a crude approximation to a proper spatial average.

We justify

it in this case because it will become evident that the probability of
self-absorption in the critical cones must be very small for an LSC to
be efficient. This will be discussed in more detail in the next chapter
as the characteristic length approximation.
Collection Probability - Q
We define Q to be the fraction of the original excitations S that
arrive at the edge-mounted solar cells. If we ignore self-absorption
and matrix absorption, this will be the quantum efficiency of emission
times the trapping probability (one minus the escape probability P).
Such luminescence is directly due to solar excitation; we will refer to
it as first generation em~ssion.

The first generation collection proba-

bility in this approximation is,, therefore, Q(l) = 11 (1- P).
include self-absorption.

Let us now

First generation emission can be self-

absorbed, leading to second generation emission. In general, ith
generation emission is self-absorbed to produce (i + 1 )th generation
emission.

The total collection efficiency is , therefore, the sum of the

collection efficiencies for each generation:

The photon flow diagram in Figure 53 shows that if we ignore transport

-148losses. in the form of matrix absorption, scattering centers·, and
incomplete internal reflection due to surface roughness, we see that
the fraction of the first generation emission that reaches the LSC-cell
interface is (1- r)(l - P), and that which is self-absorbed for the first
time is [r P + r (1 - P) ].

This process repeats itself in a geometric

series so that the total transmitted fraction Q can be expressed in
terms of r, r, 11 and P:

Q = (1- r)(l - P) 11

+ (l-r)(l-P)71 · TJ [FP+r(l-P))

+ (1- r)(l - P)71 · 71 [rP+r (1- P) J2

Q =

11 (1 - r)(l - P)

(10)

1 - 11 [r P + r (1 - P)]

We can also calculate Q from the spectroscopic and geometric
parameters for a particular plate.

This calculation is tractable for a

particular geometry called the planar solar concentrator, or PSC,
which is an infinite ribbon of width L bearing cells on both edges of
the ribbon. We will only calculate the collection of first generation
(1)

emission in this geometry, QPSC.

This is obtained by integrating

the probability of arrival for all paths from a given volume element,
over all volume elements across the width, L, of the PSC, and over all

-149-

wavenumbers of the emission weighted by the normalized luminescence spectrum of the dye f(ii).
(1)
QPSC

= _]___

2rrL

S dii

·"•

f(v) ~

r• rr

dY ~

d j

-, rr/2

sin(&) dB
()c

. [1 0 -(L-y)(a{V)+C ·E(v))/sin(O) sin(¢)

+ 10 -(L+y)(a(ii)+C ·£{V))/sin(8) sin(¢)]

(11)

E(ii) is the extinction coefficient of the dye, C is the dye concentration,
and a{V) is the matrix absorption.

The integral over y can be done

analytically giving

-. co
.· 1T /2
. 1T /2
(1)
= !l
dv f ("ii)
sin (B) dB • j
dcp
QPSC

'IT

()

. {l- lO- 2L(£(v) · C+a(v))/sin(B ) sin(~}

sin ( 8) sin ( )
L · (E (v) · C + a(v))

(12)

So how do we compute the collection probability of higher order
generations? One way is to keep track of where in the plate that selfabsorption takes place, and thereby keep a strict accounting of the excitation population of each generation. We will do this in the next
chapter.

Here we will utilize the experimental observation that, at

-150-

least in some matrix materials, the emission spectrum is independent
of the energy of the excitation (the spectrum is predominantly inhomogeneously broadened).

This means that there is no memory of the

exciting energy. If we make the approximation that the excitation
concentration in the plate is spatially isotropic, then we can assume
that the fraction of each generation that is collected is Q ( ).
We defined r and r so that Q(l) = 11 (1- r)(l - P). We have also
derived Q~~C in terms of the spectroscopic components for a particular ribbon geometry plate.

Therefore, we can solve for r in terms of

Q(l). Inserting this into the definition of the total collection probability Q gives

Q =
1 +Q

(1)

-17(1-P(l-r))

(13)

The remaining lumped parameter is r, the self-absorption probability
for luminescence in the critical cone. We will use the approximation
from Eq. (9) for this term, which is a simple Beers-Lambert law absorption probability over a pathlength of one-half the plate thickness.
Cell Response - Rcell' '11cell
A fraction Rcell of the collected light is reflected at the LSC-cell
interface. Despite the improved index matching between the LSC and
the cell with respect to the air-to-cell matching, the reflection at this
interface is greater for light emerging from an LSC due to the high
average angle of incidence of the light.

-151We can compute the angular distribution of light emerging from
a PSC geometry device in the following way.

There are four steps in

computing the angular dependent intensity I (e, v).

The first is to un-

fold the geometry by the method of images to straighten the zigzag
optical paths formed by the multiple reflections off of the LSC faces.
Figure 57a shows the usual trapped photon propagating towards an
edge-mounted cell. In Figure 57b we show the same photon propagating in the unfolded version, with the result that the optical path has
been straightened and the point of emission has been moved from point

a to point b. The second step is to divide the unfolded geometry into
a number of finite elements such that each element subtends a constant solid angle and normal thickness with respect to the point of
absorption on the cell.

This simplifies the integration by insuring that

each element contributes the same initial flux in the direction of the
cell.

The third step is to find the limits of integration in the polar

coordinates of the point on the cell dictated by the position of these
finite elements in the unfolded geometry.

Finally, we integrate over

the contribution from every element multiplied by the Beers-Lamber t
attenuation 10- r(a(ii)+C· e(ii))
cell.

where r is the radial distance to the

Figures 58, 59, and 60 show how this technique is applied to a
PSC geometry LSC.

The unfolded PSC geometry is a symmetric wedge

of width 2L and maximum height 4Lcot( ec).

The next step is to divide

this unfolded geometry into a series of ribbons parallel to the cell of
thickness dy.

These are then divided into circular sections of width

-152-

Figure 57a.

A typical photon trajectory for emission from a point

(a) in a planar LSC.

Figure 57b.

This trajectory can be unfolded by the method of images

to form a straight line from a point (b).

Figure 58. Unfolded geometry of an LSC.
symmetrical wedge.

An LSC unfolds to form a

Any trapped light trajectory in the original

geometry will have a straight line equivalent in the unfolded geometry.

-153-

r-----,._, ... ..,

r- - -

o::;:z:'31PVC

PVC

-154-

Figure 59.

The symmetric wedge divided into finite elements.

The

wedge is separated into plates of thickness dy, each plate being
parallel to the cells (PVC's).

Each ring is broken into concentric

rings, and each ring into subsections.

-155-

.....

plate thickness

......

volume elemen t= y2 sin8dy d8d¢/c os 2 e

-156-

Figure 60.
(PVC's).

Polar coordinate system in the frame of the solar cells
The dotted semicircles represent sections of the half-

sphere where there can be no incident light (because it has been lost
out of the critical cones).

For () < 1T /2 - ec, there are no restrictions

on the allowed values of cp.

For larger values of e,

from the dotted semicircles.

-157-

PVC

-158y dB/cos (8), where 8 is the polar angle from the point of absorption

on the cell.

Such a finite element is pictured in Figure 59 and satis-

fies the criteria of constant initial emission into a constant solid angle
from the absorber over all the finite elements.

The third step is to

find the limits of 8 and ¢ in the polar system about the absorption
point on the cell. By inspection of Figure 60, 0 s

cos- 1 [ c?s ~B})] and
Slll 8
rr - cos- [cos (Be) ] and similarly between rr + cos- 1 [cos (Be)] and
sin (8)
sin (B)

o s e s rr /2 - eC , and runs between

21f - cos- 1 (c?s ((:))] for Tr/2 - () s 8 s rr /2.
Sln

The integral form for

J(B,ii) simplifies to

,2L

I(B,iJ) =

dysin(B)·A·lo-y(a(ii )+C·e(v))/cos(B)

o ~ e ~ 1T /2 - ec

1 - 2/rr cos- 1 [ c?s (Be) ] rr /2- 8 s B ~ rr/2
Sln (8)

(14)

Figure 61 shows the result of calculating I ( 8, ii) for matrix optical
densities of 0. O, 0. 01, and 0. 15 as measured across the width of the
PSC.

The peak in the curves occurs at 47 degrees for PMMA matrix

material, which is the compliment of the critical angle.

It is striking

to note that if the matrix material is nonabsorbing, there is a substantial amount of light arriving at the cells at a completely glancing angle

-159-

Figure 61.

Angular dependence of LSC light output.

The intensity

peaks at the compliment of the critical angle.

The geometry used

was an infinite ribbon with an index of 1. 49.

The matrix absorption

optical densities are measured across the width of the LSC.

INCIDENT ANGLE ~N PVC

0 .15 .

0.01

I-

......

0.0

DPTICRL DENSITY

RNGULRR DEPENDENCE ~F C~LLECTED
EMISSI~N INTENSITY.

-161of incidence.

This effect is peculiar to the infinite geometry of the

PSC, and is a strong point against using such a geometry in a prototype device.
The reflection coefficient Reel! can now be calculated. If
I (8, v) is the angular dependence off the output light, the reflection
coefficient will be the appropriate Fresnel reflection coefficient
R (8, "ii) weighted over wavenumbers and angles of incidence:
-. Tl

= .Jo

--.

sin (8) d8 \ dv f(v) 1 (8, v) R (8, v)

(15)

The cells used in prototype testing typically had single layer SiO
antireflection coating, which has an index of 1. 9. We will assume
that the reflection loss for the cell facing the sun directly is negligible.
We use the approximation that the quantum efficiency of the
solar cell was independent of the energy of the exciting light for typical dye emission.

The primary justification for this approximation is

that the spectral response of the silicon cells we used was reasonably
flat over the range of 5, oooA to 8, oooA (1. 4 to 2. 7 eV) (Hovel, 1975).
We also made the approximation that there were no variations in cell
efficiency due to the intensity of the LSC output.

This is reasonable

because of the empirical fact that the LSCs typically only produce flux
gains on the order of two. Therefore, if the cell converts the solar
spectrum with a particular Air Mass One (AMl) efficiency 11cell' then
the light transmitted into the cell from the plate is converted with the
same total energy efficiency.

-162System Performance - 111.SC' Gflux
The fraction of the useful sunlight absorbed by the plate is S/l.
Q of that fraction is delivered to the LSC-cell interface.

The geo-

metric gain Ggeom is the area of the LSC exposed to sunlight divided
by the area of cell (or absorber) mounted on the edge of the plate. The

intensity of the light arriving at the solar cell is, therefore,
Ggeom' S·Q.

(1- Rcell) is transmitted into the cell. The flux gain

Gflux' or the ratio of the power from a cell mounted on the plate to
that from a cell facing the sun directly, is

0 uux

(16)

= Q · S/I .. Ggeom · (l - Reen>·

The efficiency of an LSC-cell combination, 77LSC' is the electrical
power out divided by the solar power in.

The AMl efficiency of the

cell is the electrical power out divided by the solar power in.

obtain the efficiency for the combination we multiply 17cell by the
fraction of the useful solar flux that is transmitted into the cell:

11 LSc = 17 cell · Gnux I Ggeom

= 11 cell· 8 /I · (l - Rcell) · Q

This completes the LSC performance model.
tion using this model is given in Chapter 5.

(l 7)

A sample calcula-

-163-

II.

MULTIPLE DYE SYSTEMS
Including a variety of dye types in an LSC, whether homogene-

ously mixed in the plate material or separated into thin films, will
allow a broader band of the solar spectrum to be absorbed than in a
single dye system.
device.

This does not necessarily lead to a more efficient

(The highest light output LSC plate to date contains only one

dye, DCM. ) We will now examine features of multiple dye LSCs.
Photon Flow Diagram
The task at hand is to develope a formalism for attacking the
multiple dye LSC problem, which we will do in a parallel manner to
the technique used in the single dye case.

Again a flow chart, shown

in Figure 62, will organize the interaction of the various processes.
What is shown is a single step in a sequence that is repeated to form
the photon cascade; dye i is some intermediate dye which can absorb
the luminescence :from other dyes higher in the cascade as well as
transferring the excitation to dyes further down the cascade.

The dyes

are ordered such that dye 1 has the highest energy luminescence
spectrum and dye N, the final dye, has the lowest.

Light from the

sun is partially reflected at the air-LSC interface. Of the light that is
transmitted, part will be unabsorbed by any dye, the part Si will be
absorbed by dye i, and the rest will be absorbed by other dyes in the
in the cascade.

Excitations of dye i-1 are transferred radiatively

to dye i with a probability r(i-1, i), or nonradiatively with a probability d(i -1, i).

The nonradiative channel is typically dipole-dipole

transfer, depicted by a double line. Similarly, excitations on dye i

-164-

Figure 62.

Flow chart for energy channels in a multiple dye LSC.

This diagram is essentially the same as the single dye case in
Figure 53.

Two principal changes are the additional absorption of

sunlight by other dyes, and both radiative and nonradiative transfer
of excitations between dyes.

-165-

Energy Flow Diagram for a Multiple Dye LSC

Unabsorbed
Flux

Critical

cone

Reflected
Flux

Losses

--------- -------

AIR
LSC

To other
dyes

.J . l"n . 1 )----~

1 1- · 1 1-

.J.

Pr11.. J .l''l
.,,.

1---__,..-----1K

J'.l ~.
(1-P)r 11
..
'I 1

J.P,,.
l ·11

Transport
Losses

d.
. 1J·
l l+.L l

'h.&
.J~(l-P)(l-r .. ) i - - - - - i
·11 1 l
11
Q.71 .J~
1'11 l

-- - ------

____________ Esc __ _
PVC
Photovoltaic
Conversion

-166-

can be transferred radiatively or nonradiatively to dye i + l (or to
other dyes, such as itself).

Excitations which are not transferred to

other dyes follow a similar flow as in the single dye case; in fact,
the emission of the last dye in a multiple dye LSC is treated identic-

ally to emission in a single dye LSC.

Emission in the critical cone is

reabsorbed with a probability E r(i, j) or is lost out of the critical

cone.

Emission outside of the critical cone is self-absorbed with a

probability 6 r(i, j), or is lost during transport with a probability

6(i), or is reflected at the LSC-cell interface, or if all else fails it
is absorbed and converted into electricity in the cells.
Solar Absorption - Si
Continuing the same approach as in the single dye system, each
dye has its own molar extinction coefficient Ei(ii) and concentration Ci'
which combine to give an absorption coefficient

= :E c. €.(V-)

(18)

a(v) is the matrix absorption coefficient. Apart from the abundance
of subscripts, the solar absorption is computed as in the single dye
case:
,··

Si = j

I(iT) dv (1- R(iJ)) [l -10

(a (V) + a(V))

-l

c. · E .(v)

aT(i7) + a(i7)

(19)

-167Direct Transfer Coefficient - d(i, j)
If dye i is in an excited singlet state (assuming that the ground
state is a singlet state), and nearby there is a different dye j whose
energy levels for absorption are resonant with the excitation energy
on dye i, then there is a finite probability that the excitation will be
transferred nonradiatively from dye i to dye j. In this case the dye
i is termed the donor, and dye j is the acceptor, and the average
probability that the event will occur is d(i, j) the probability of direct
transfer.

This would appear to be a very promising mode of operation

for an LSC. If all transfer of excitation had to be done radiatively,
there would be critical cone and quantum efficiency losses at every
stage in the photon cascade.
The predominant form of nonradiative or direct transfer in
systems where a dye is dispersed in a solid matrix is dipole-dipole
transfer. If the dye molecules are suspended in a fluid vehicle, or if
dipole radiation is not an allowed transition for one of the molecules,
then electron exchange and higher order transfer interactions must be
considered, respectively.

There is an effective critical distance for

dipole-dipole nonradiative transfer specifically in the limit of random
orientation and high viscosity (Forster, 1959):

9000 .fn(lO) · K · 1Ji

128 · 11 • n Nav

,... 00

d\ f.(v) € .(V) ~
J0 l
i74

i refers to the donor type and j to the acceptor.

fi(v) is the

(20)

-168-

normalized emission spectrum of the donor dye, and E j ( i7) is the
extinction coefficient of the acceptor. T/. is the quantum efficiency of

luminescence of the donor, and K is an orientation factor relating

the two dipole moments. K is 2/3 if the orientation is completely
random.

n is the index of fraction of the plate, and N av is

Avagadro 's number.
This critical transfer distance R0 (i, j) defines a critical molar
concentration C 0 (i, j) of dye j with respect to transfer from dye i
such that the average spacing of the donor and acceptor in random distribution is equal to R 0 (i, j):

C 0 (i,j) =

3000
2·7T 3 / 2 ·N ·R 0 (iJ')
av

(21)

There will be competition between a variety of acceptors, so that we
need a way to compare the associated transfer rates between different
pairs of donors and acceptors. Still assuming that dipole-dipole nonradiative transfer is the predominant mechanism, we define ri to be
the total critical concentration fraction for direct transfer out of dye i:

(22)

where the sum is overall possible species of acceptors j.

The

quantum efficiency of such tranfer has been shown by Forster to be
(Birks, 1970):

-1692

VI.·l = ..fir.1 exp (!:1 ) [1 - er! (I:)]

(23)

The dependence of the nonradiative transfer quantum efficiency on the
ratio of the actual acceptor concentration to the critical concentration
is illustrated in Figure 63.
Finally, we can define the probability of direct transfer from
donor i to acceptor j as the product of the total probability l/li that
Forster transfer out of the donor will occur times the fraction of the
acceptors of dye j weighted by the critical concentrations:

d (i, j)

(24)

It has been hypothesized that the direct transfer rate will have a strong

temperature dependence (Katraro, Ron, and Speiser, 1977).
Quantum Efficiency of Luminescence - 71i
The existence of direct transfer channels reduces the quantum
efficiency of luminescence of the participating dyes. If the quantum
efficiency of direct transfer out of dye i is lf,ti, and the quantum
efficiency of luminescence of an isolated sample of dye i is n0 (i),
then the quantum efficiency of luminescence in the presence of direct
transfer will be

(25)

-170-

Figure 63.

Nonradiative transfer efficiency vs acceptor concentra-

tion in the Forster model.

This shows the standard relationship

between the acceptor concentration (divided by the calculated critical
acceptor concentration) and the quantum efficiency for transfer
described in 3. II.

-171-

TRANSFER EFFICIENCY
VS. ACCEPT~R C~NCENTRATI~N IN THE
..
F~RSTER MeJDEL.

N~N-RADIATIVE

.,---,--,.-,.,..,iTtT"~,--,--.-,,,..,..,~--,--,-...,-~·

U- a:>
U- •

U-

cnto

<.D

CIC

1--

> :::::r'

t-1

1--

:::::r'

a:o

er:

z·
00

o'---'--'--'-~~~~~~"-'-'--__.--L-...1.--'-~~~

10-2

10- 1

1o0

ACCEPTOR CONC./CRITICAL C~NC.

10 1

-172-

Radiative Transfer Coefficients - r(i, j), r (i, j)
The most tractable method for computing the reabsorption
probabilities is to use the characteristic pathlength approximation
developed in the next chapter. In this approximation the probability
for emission from dye i outside of the critical cone to be absorbed by
dye j is

r(i, j)

(26)

where

1c is the characteristic length of the device. Similarly the

probability for absorption inside the critical cone is the same expression as Eq. (26) except the thickness T is substituted for fc.
Collection Probability - Q
A well chosen cascade will have stronly overlapping absorption

and emission bands.

Th~ effect of this on

Q is that only emission

from the final dye will be able to propagate freely in the plate, the
rest will be absorbed by other dyes in the cascade.

In steady state

the number of excitations in dye one, J 1, is
JI = SI+ JI [111(:r(l, 1) p + r(l, 1) (1 - P))
- ~ {d(l,i) + 1ii(r(l,i) P+ r(l,i)(l-P)))]

i>l

Similarly the number of excitations in the second dye is

(27)

-173-

J 2 = S 2 + J 1 (1h(r(1, 2) P + r(l, 2) (1- P) + d(l, 2))
+ J 2 (1 2 r(2, 2) P + r(2, 2)(1- P))

- ~ (d(2, i) + 112(r(2, i) P + r'(2, i) (1 - P)))]
i>2

and so on for each dye.

(28)

The final collection efficiency Q is the single

dye collection efficiency for emission from the final (Nth) dye times the
fraction of the initial solar excitations that are transferred to the final
dye:
1JN (1 - r(N, N)) (1 - P)

Q =

1 -71NN (r(N, N) P + r(N, N) (1- P))

N-1
SN+ L

iEl

Jl. (d(i, N) + T/· (r(i, N) p + r (i, N) (1 - P)))

(29)

The LSC efficiency and flux gain are then calculated in an identical
manner as for the single dye case.
Ill.

SUMMARY

1. A flow diagram for the energy flow channels in an LSC is
developed.
2.

Lumped parameters are assigned to the various channels.

A geneological model is proposed.

Self-absorption creates

emission from higher order generations. Performance

-173characteristics are obtained by a weighted sum over all generations.
3.

An analytical form for calculating performance is developed

for an infinite ribbon geometry.

The lumped parameters are then

derived in terms of this calculation.
4. We define the two important performance parameters for an
LSC.

These are the flux gain, Gnux' or the increase in the power

output of an edge mounted cell over that of the same cell directed at
the sun, and the collector efficiency, r/LSC' or the electrical power
out per solar power incident on the plate.
5.

The single dye plate performance calculation is contained

for a multiple dye plate.

The important new effects are an increase in

the initially absorbed sunlight and nonradiative transfer between
different dyes.

-174-

CHAPTER 4

-175I.

INTRODUCTION
This chapter pays special attention to self-absorption, or more

specifically to a dye ensemble's reabsorbtion of its emission.

First

we develop the limiting case of highly self-absorbing system, which
is a scattering plate with no Stokes' shift. Next we derive a Green's
function formalism for an analytical solution in the special case of an
LSC rod.

A formalism is developed to analyze the self-absorption

rates as measured by transient emission and polarization experiments.
A simplified self-absorption model is presented which assumes a

characteristic pathlength for self-absorption to occur which is determined by the size of the LSC.

Finally, a new engineering parameter,

the critical optical density of a particular dye (CODE), is proposed as
a rapid way of designing an LSC.
II.

THE SCATTERING PLATE
It is instructive to .calculate the collection efficiency of a purely

scattering plate. Consider the Planar Solar Concentrator (PSC) geometry of LSC made from a ribbon of clear material having a thickness
T and a width W. We hypothesize that there is a plane of isotropic
scattering centers located a distance T /2 from either exposed surface,
such that light impinging upon this scattering plane is isotropically
scattered with a scattering efficiency 77scAT'
both edges as shown in Figure 64.

Cells are mounted on

The fraction of an isotropically

scattered light flux incident on the scattering plane a distance x away
from one edge, and that is intercepted by that edge is:

-176-

Figure 64.

Collection from a scattering plate.

Suppose an infinite

ribbon of clear LSC material has cells mounted on both edges and
has a plane of isotropic scattering centers midway between the faces
of the ribbon.

The fraction of the incident light which is collected

with the scattering plane ig given as a function of the ratio of the
width to the thickness of the ribbon.

Cl 5.00

a:
cc

........

o.°8.oo

u 2.50

w E -1

0.50

WIDTH/THICKNESS

1.00

1.50

WIDTH_,

2.00
E 1

CELL~CELL

RIBB~N CR~SS-SECTI~N

LUMINESCENCE C~LLECTI~N BY
SIMPLE SCATTERING IN R CLEAR INFINITE
RIBB~N. LIGHT SCATTERED TWICE IS IGN~RED.

IMITRTI~N ~F

-:J
-:J

.....I

-178-

F(x) =
4 rr

21Tx

rr /2

, a+

S ·/2

-TT

5 sin (8)
a-

'1T /2
-1[

cos()

.fl+ (T cos(¢)/ x) 2

(30)

where a± = 11 /2 ± tan- (T cos ( 4>)/x). We have allowed only light
trajectories which do not intercept the scattering plane.
If the scattering plate has solar cells mounted on both edges,

the fraction of the scattered incident light which is collected without
intercepting the scattering plane a second time is:

'"'W

dx F (x)

= w 77scAT Jo

= -2T flscAT ( -W tan -1 (-)
+ - ln (1 + (-} 2 ) ]
1TW

(31)

where TJscA T is the scattering efficiency of the scattering plane.
We note that if the scattered light does not escape out of the
critical cones, and if it is not collected by the solar cells, then it
must intercept the scattering plane again. Assuming that the spatial
distribution of this scattered light is uniform in the plane, we find

-179-

that the higher generation collection efficiences are

(3)
(1) 2
(1)
Qs
= (l- P) (71SCAT-QS )· Qs

Therefore, the total probability of the solar cells collecting the scattered incident light is:

= .

(i)
Qs

(1)

Qs
(1)
l - ( l - p) ( 11 SC AT - Q S )

(32)

If we ignore intensity effects on the efficiency of the edge mounted
cells, and also ignore reflection losses at the scattering plate- cell
interface, then if 11cell is the AMl efficiency of the cell under direct
insolation then the efficiency of the scattering plate-cell combination is

11scAT collector

= 11cell · Q s

(33)

(1)

is plotted as a function of the width to thickness ratio in Figure 64.

-180-

Ill.

SELF-ABSORPTION IN A SEl\11-INFINITE ROD

Consider a semi-infinite rod of LSC material having a diameter
d, and containing luminescing centers at a concentration of C moles
per liter, with an extinction coefficient e:(V) and a normalized luminescence spectrum f (i7). a (v) is the matrix absorption.

Pis the

probability of escape out of the rod, P = 1/n where n is the index of
refraction of the matrix material.

The differential of the transmission

probability through a distance x of this material at a wavenumber ii
with respect to x is

~T(x,v) = .fn(lO)· c·e:(i7)· 10-x(a(i7)+C·e:(v))

We have assumed that a(v) is small.

(34)

The rod is a three-dimensional

system, which requires that we average over all possible paths. Suppose a molecule located on the symmetry axis of the rod emits a photon whose path makes an angle e with respect to the symmetry axis.
(We will assume that all emission from a particular disk element
occurs at the center of the disk.

This is a reasonable approximation

for emission pathlengths which are longer than the diameter of the
rod. ) If the emission occurs at a point y from the end of the rod, and
we are interested in the absorption probability per unit length at a
distance x from the end of the rod, then the length of a trajectory will
be a/cos (e), where a= Ix -y \. Similarly the incremental distance
da is increased by a factor of 1/cos (8).
per unit length becomes

The probability of absorption

-181-

A(x, y, v)

S sin(B)

dB tn(lO) C E~(iT) 10- lx-yl(a(i7)+C·E{v))/

C0S (8)

cos ( fJJ

1T /2 -

tan- 1

for

) for
2lx-yj

lx-yl > i. tan(8 )
1x - y I <

i.2 tan (e c )

The upper limit of integration depends on the distance Ix~ y

(35)

I· If, for

example, Ix -y < ~ tan (8 c), light from y with a polar angle greater
than 11 /2 - 8c can reach x, even though it would not be trapped by
total internal reflection.

Now let

= lx-·y\ · (a(v) + C · E:(il)Y cos(8)

= tan (8) de

Substituting in Eq. (35) gives

(36)

.-18.2-

-z

A(x,y,il) = \

dz !:Q_

l.n (10) · C · E (v)

\x - y \ · (a( v) + c · E ( v) )

lx-yl · (a(v) + C·E(v))/sin(O) for

+(x-y)

lx-yl > ~ tan(Bc)

(a(V)+C·e(iJ))
(37)

We perform a weighted average over the luminescence spectrum to
arrive at the probability A(x, y) that emission originating at a position
x will be absorbed at a position y:
-, 00

A(x, y) =

J dv ~(v) A(x, y, v)

(38)

Next we define Z

(i)

(x) to be the spatial distribution of the ith genera-

tion of excitations along the rod.

For example, if the rod is initially

excited by a focused light source at a position x, then z(l) (x) is
approximately a delta function at x:
(1)

(x) =

o (x) 10

Excitations directly due to absorption of the externally incident light
are the first generation excitations. Second generation excitations

-183arise from the self-absorptio n of the first generation emissions.

For

example,
('

= 7J j

dy A(x, y) Z

(1)
(y)

and in general

:=:

(i)

(x)

·'

= TJ j

dy A(x,y) Z

(i-1)

(y)

(39)

where 71 is the quantum efficiency of luminescence.

The luminescence

spectrum observed at the end of the rod (x = 0) is the sum of the intensities due to all of the different generations of excitations in each element attenuated by the appropriate absorption coefficient for propagation through the rod:

, 7T

Q(i7)

/2 - fJc.

dfJ tan(fJ)

-. oo

110- x(a(V) + C · •(ii))/cos ( 9)

di7 f(i7) · 1J

Z(i) (x)

(40)

In the limiting case that the Stokes' shift is large, the probability for
reabsorbing the emission is approximately zero everywhere, so that
_(i)
A(x, y) = 0. Therefore, .::. (x) = 0 for i > 1, and only the first generation emission is collected.

-.184IV.

SELF-ABSORPTION AND EMISSION POLARIZATION

Suppose we excite an LSC with vertically polarized light and
detect either the vertically or horizontally polarized emission intensities emitted in a horizontal direction, as shown in Figure 47. If the
orientation of the absorption dipole moments of the luminescing centers
is initially isotropic, then the probability that any center will absorb
an excitation as a function of its absorption dipole orientation is proportional to the square of the cosine of the angle between the absorption moment and the electric field.

Using a more concise notation

(Gordon, 1966 and Tao, 1969), if Ei is a unit vector in the direction
of the electric field of the polarized excitation, and fia is the absorption dipole moment, then the angular dependence of the excitation
...

...

probability is given by [Ei · µ.a] . If we assume that the emission
dipole moment i1 e is parallel to the absorption dipole, and that the
orientation of these moments is fixed on the time scale of the total
lifetime of the excitation~ then the intensity of the emitted light polarized in the direction Ef is given by (apart from a constant factor):

(41)

The brackets denote an average over all orientations. The emission
intensity polarized parallel to the excitation is, therefore, given by

-185-

41f

·~ 2 1T

.• , 7T

def> j

dO sin (0) cos (0)

(42)

Similarly the emission intensity polarized in a direction perpendicular
to the excitation is

I..l

87T

( [E.·
ii a ] [ii e XE.]

• 211

.7T

def> j

dO sin (0) cos (0)

(43)

The reduced anisotropy measures the degree of polarization of the
emission. It is defined to be

(44)

The initial excitation has no perpendicular (horizontal) component, so
that its reduced anisotropy is one.

From the intensities just calcula-

ted, we see that the parallel polarized emission have three times the
intensity of the perpendicular component, which gives RA (l) = O. 4 for

-186-

for the first generation emission. If the orientation of the hlminescing
dipole is different from that of the absorption dipole, or if the molecule rotates in space between the time of absorption and emission,
then in general the reduced anisotropy decreases by a factor e:

RA =

O::::; e ::::; 1

(45)

Emission from an LSC due to polarized excitation displaying a
characteristic anisotropy of 2e/5 can, therefore, be identified as
emission from the first generation of excitations. We will now compute
the reduced anisotropy of higher order emissions. In the radiation
field limit (where the average distance between reabsorptions is

» 5, ooo.A) the angular distribution of the emission from a dipole is
also proportional to the square of the projection of the emitting dipole
moment on the emitted electric field.

Thus the second generation

parallel and perpendicular emission intensities are

We use the identity that the cosine of the angle 'I/I between two vectors
with orientations (B,) and (fJ',c/>1 ) is cos(lf;) = cos(8) cos(B') +

-187sin(B) sin(B') cos(-').

Jn the far field limit, the parallel and perpendicular second generation

intensities are
--. 2 1T

(4 7T ) 3

d.1 j

J..

d81 sin(6l1 )

dB sin(O) j,,

2 1T

d¢2

(48)

153
,,...27T

I (2)

' 2 1T

., 1T

= 1 3 \
2 (411' ) J 0

,,1T

d 1 ~

,211'

d01 sin(81 ) j

,1T

d \

,21T

d8 sin(9) \

, 11

· j

d8 2 sin ( 82 )

· (1 - cos (8 2 )) [cos ( 8) cos ( 02) + sin ( B) sin ( 82 ) cos ( - ¢ 2)]

(49)

-188-

We have assumed that the absorption and emission dipoles are both
parallel and fixed in space on a time scale of the excitation lifetime.
Thus the second generation reduced isotropy is

(2)

(2)
(2)
1 11 - I.l
(2)
1 11

(50)

We have calculated the third generation result to be (2/5) . We believe that the ith generation reduced anisotropy is given by

(i)

2 2i -1

(51)

= (-)

The first generation reduced anisotropy is less than or equal to 0. 4.
The maximum for the second generation is 0. 064, and for the third is
0. 01024.

Thus it is reasonable to ignore the contributions from higher

order generations to the polarization anisotropy.
Suppose that we measure the reduced anisotropy of the emission
from an LSC due to a polarized excitation.

The measured value will

be the weighted average of the reduced anisotropy of each generation
times the fraction of the emission which is comprised of that
generation:

(52)

-189-

In the limit of very low concentrations, there is no self-absorption,
so that Q (l) = Q and RAexp = 2e/5.

For higher concentrations we use

the technique of averaging over each generation:

= [ e 17 (1 - r) (1 - P)

2e 3 2
+ (-) 17 (1-r)(l-P)(rP+r(l -P))
2e 5

+ (-) T/ (1-r)(l-P)(rP+r(l -P))

+ . . . ] /Q

1 -17 (r P + r (1 - P))
1-( 2e) 17 (rP+r(l-P))

(53)

If the excitation is near enough to the edge of the plate such that the

probability of self-absorption inside and outside of the critical cones
is about the same' (r = r), then

2e
RA exp = - •

(1 - 17 r)
1 _ ( 2e)2 T/ r

~ 2e (1 -TJ r)

(54)

-190-

Similar simplifications arise if the measurement is arrange· so that
Pis zero.
V.

SELF-ABSORPTION AND TRANSIENT EMISSION

The rates of self-absorption both inside and outside of the critical cones can be calculated directly from measured transient emission
spectra.

Suppose at some time t = 0 we have placed n0 excitations

in a dye ensemble with a light pulse whose duration is very short compared to the total lifetime of the excitations.

The rate of change of

this population with time will be proportional to the population by the
usual kinetic equation

aA1)(t) == _ n(l)(t)
Bt

(55)

where ; is the total lifetime. We assume that there is no transit time
dispersion due to emission from different parts of the sample, or in
other words, that the lifetime is long compared to the transit time for
the light in the sample.

Since typical fluorescent dyes have lifetimes

on the order of nanoseconds, this restricts the applicability of this
technique to devices about a foot across or smaller. If r is the
probability that the emission is self-absorbed outside of the critical
cones, and r is the probability of self-absorption within the critical
cones, P is the probability of trapping in the critical cones, and TJ

-191-

is the quantum efficiency of luminescence, then the kinetic ,equation
for the number of second generation excitations as a function of time,
n( )(t), resulting from first generation emissions is
n (2)(t)

= -

+ T/ n

(1)

(t) (F P+ r(l - P) y T

(56)

We will assume that the probability of self-absorption is independent
of the number of emissions previously experienced by the excitation.
This is equivalent to assuming the limit of a predominantly homogeneously broadened ensemble at temperatures sufficiently high to
allow anti-Stokes shifting of emission which is large compared to the
emission linewidth.

The third generation excitations are therefore
n(3)(t)

= -

+ 1J n

(2)

(t)(FP+r(l-P))/7

(57)

These form a system of' equations with the solutions

n 0 e -t/T
11 0 -

t 1J r:;;
\r P+ r(l - P)) e -t/7

n 0 [ - 77 (F P + r( 1 - P) ) ]

i-1

-t/7

/ (i - 1)

(58)

-192Summing the various generations gives the number of excitations in
the ensemble as a function of time:
00

N (t) =

~ n(i)(t)
i=l

= n 0 exp ( - _!_ (1 - 77

rr p + r( 1 - p) ) ) )

(59)

Thus self-absorbed emission will have an apparent lifetime which is
lengthened by a factor 1/[1 -71 (r P+ r(l - P)) J.

This factor is equal

to one if r = r = 0 . Measurements of the experimental transient
emission decay can, therefore, give 7, 7 /(1 - r71), and
T /(1- (r P+ r (1 - P)))

for the three cases of an extremely dilute

sample, edge emission due to an excitation. near to the edge, and from
general illumination of the plate, respectively.
If each generation of excitations is distributed throughout the

sample in such a way that the average probability of emission from
each generation reaching the detector is the same, then the transient
emission spectrum should have the simple exponential behavior of
Eq. (59). It is quite possible, however, to configure the exciting
source and sample in such a way that first generation emission has a
very different probability of reaching the detector.

For example, if

the exciting laser pulse is at an energy corresponding to the absorption
maximum for the emitting center, then nearly all of the laser light will
be absorbed in typically less than the first few millimeters of the
sample. A mask can then be placed between this excited region and

-193the detector, such that the excited region of the sample can illuminate
the rest of the sample without illuminating the detector (see Figure 51).
Assuming that none of the first generation emission illuminates the
detector due to scatter in the rest of the sample, and that r = r, we
find that the measured transient spectrum should be proportional to
the usual total decay of the population minus the first generation emission:

If the probability of self-absorptio n is high enough, a significant fraction of the first generation emission can be reabsorbed in the masked
region.

In this case, some fraction of the higher order generations

will have to be subtracted.

Because of the sensitivity to scattering

and inadvertent masking of higher order generations, this technique
is inferior to the simple ·lifetime measurements for determining selfabsorption probabilities.
VI.

CHARACTERI STIC LENGTH APPROXIMAT ION

While the technique developed in Section II is useful for obtaining
spectral information about the emission from an LSC rod, a simpler
calculation appears to be a reasonably good approximation for calculating self-absorptio n probabilities. In the simplest one-dimension al
case, the probability that self-absorptio n will occur is found by the
Beers-Lambe rt law:

-194-

dv f(v) [1-10-XC E(V) ]

(61}

r is the probability that emission with the normalized spectral distribution f(i/) will be self-absorbed by the luminescing centers having
an extinction coefficient E(v) and a concentration C over a pathlength
x.

The approximation that we will make in this section is that the

probability of self-absorption for emission in an LSC plate is given by
this equation, where x is the characteristic length of the plate.

This

characteristic length is basically a weighted average of the trajectory
lengths traversed by light in the plate.
We have not saved any labor by analytically computing the
weighted average of the trajectory lengths to arrive at a characteristic
pathlength. What we will show instead is, for a small class of device
geometries and dye concentrations, that the properly averaged characteristic length corresponds to a characteristic dimension of an LSC
plate.

For example, the characteristic length of a square LSC is the

length of the side.

The characteristic length for self-absorption in the

critical cone is the thickness of the LSC plate. In general we justify
this approximation by showing that the self-absorption probability has
roughly a logarithmic dependence on the pathlength (see Fig. 69), so that
precision in identifying the proper pathlength is not crucial.

However,

it is useful to explicitly calculate the weighted average pathlength
traveled by light collected in an LSC, and to compare this result to that
predicted by the characteristic length approximation.

-195-

We use the collection efficiency formalism of Eq. (12) to compute the first generation collection efficiency, Q ~C , for an infinite
ribbon geometry (PSC) LSC.

This is probably a worst case analysis,

since the infinite ribbon geometry can have extremely long pathlengths
at low dye concentrations. We assume that the width of the ribbon was
variable, that the thickness was one centimeter, that the index of refraction was 1. 5, and that cells were mounted on both edges of the
ribbon.

Since Q(l) = 11 (1- r) (1- P), computing Q(l) using Eq. (12)

gave the self-absorption probability r.

Using Eq. (61 ), we found the

average pathlength of the collected light in the plate that would result
in the computed r.

Figure 65 shows the results of this calculation,

where we have calculated the average pathlength as a fwiction of the
plate width for four different concentrations of the dye rhodamine- 575.
The assumed quantum efficiency was 0. 9.

The dashed line indicates

the characteristic length approximation result, which is that the average pathlength is assumed to be the width of the plate. At low concentrations and PSC widths, the average is less than the width.

The

cross-over point occurs when the width times the concentration divided
by the thickness equals about 0. 1 for this dye.

Thus the characteristic

length approximation would give the correct result for a. plate with a
geometric gain of 100 and a concentration of 5 x 10-4 molar, which is
typical for an optimized device.
VII.

CRITICAL OPTICAL DENSITY (CODE)
The preceding calculation of self-absorption probabilities and

average pathlengths traveled by light in the plate make clear that the

-196-

Figure 65.

Average pathlength of collected emission vs width of an

LSC. We have calculated the average distance traversed by emission
in a plate, which is collected at the edge, for four different concentrations of rhodamine-575.

The plate was assumed to be 1 cm thick,

so that 10- M corresponds to a peak optical density of one through
the plate.

The dashed line indicates the characteristic length

approximation.

-19 7-

10, 000

:r: 0

..... -

(!)C f)

1,0 00

zcn
w~

-1w

:r:
.-o
a.. f-

100

w (...)
(!)

et: _J

> (...)

1....._._..... .....u.......i--_.__._......_,..,,_.,.___

._~~~

100

PSC WIDTH

1,0 00

-198-

performance of an LSC varies in a rather complicated way with a host
of parameters such as the size, the types of dyes used, and the concentration of those dyes.

From an engineering standpoint, we do not

desire to know the performance of a plate for all possible conditions;
there is only a relatively narrow class of size and efficiency which is
interesting to look at. We are, therefore, interested in finding a
single scalar parameter which will predict LSC performance for a
particular dye in this restricted class with relative ease.
We define a new self-absorption parameter, the critical optical
density (CODE), to be the peak optical density of a sample containing
a luminescing species, such that the emission from the species has a
fifty percent chance of being self-absorbed while traversing the sample.
Mathematically, if a luminescing species has a normalized luminescence spectrum f(V), a molar concentration c, and an extinction
coefficient E(v) with a maximum value of E(v max), and this species
is contained in a sample with an optical length L, then the critical
optical density is

(62)

where the concentration and pathlength have been adjusted so that
,... 00

..!.. = \

f(v) dv 10 -L c E(il) .

The critical optical density is the radiative analog to the Forster

(63)

-199critical concentration (see 3. II. ). It is useful because it prescribes
the efficiency of an LSC as a function of the size or geometric gain.
In Figure 66 we have plotted the collection efficiency of an arbitrary
LSC plate as a function of self-absorptio n probability, using Eq. (10).
The index of refraction is assumed to be 1. 49.

The solid lines refer

to the case where there is no self-absorptio n in the critical cones,
and the dotted lines are an opposite limit where the self-absorptio n
rates are equal inside and outside of the critical cones.

The vertical

line originating at r = 0. 5 intersects the curves at the CODE operating point (by· definition).

For example, a typical LSC plate will have

an index of refraction of 1. 5, or an escape probability of 0. 26, a
quantum efficiency of 0. 9, and negligible self-absorptio n in the critical cone. If the self-absorptio n probability is 0. 5, we find from
Figure 65 that the collection efficiency of the plate is also about 0. 5.
To obtain reasonably good absorption of sunlight, or of emission from
other dyes, the peak optlcal density of the final dye must be at least
one across the thickness of the LSC. In this case, the critical optical
density will be the number of plate thicknesses which the emission can
traverse before fifty percent is self-absorbed. Since the geometric
gain of a plate is roughly the width of a plate divided by its thickness,
the critical optical density is roughly the maximum geometric gain of
a plate, using the particular luminescing center as a single or final
dye in a cascade, which produces a collection efficiency of 50%.
Calculating the performance of an LSC using this procedure is
simple. We initially restrict the geometric gain of the plate to be
equal to the measured CODE of the dye used (or of the final dye used

-200-

Figure 66.

Collection efficiency vs self-absorption probability. We

have calculated the collection efficiency of an arbitrary LSC, assuming P = 0. 26.

The solid lines are for self-absorption in the critical

cones being set to zero, and the dotted lines being set equal to the
self-absorption probability outside of the critical cones.

The vertical

line at r = 0. 5 indicates the collection efficiency at the critical
optical density (CODE).

-201-

~ 1.00

-r = r

.......
_J
........

r =o

~ 0.80
CD

················•······ ·••·····•··•·····••••·•·•·•·•·• 77

= 9.5

··················· .. ..

CL 0.60

··.....

t-t

...

.__ O.L!O

77 = • 7

~ 0.20

O.DB.oo

0.20

O.L!O

0.60

0.80

1.00

R - SELF-ABS~RPTI~N PR~BABILITY

-202-

in a multiple dye plate). We also require that the concentration of
this dye is such that the peak optical density across the thickness of
the plate is one. Under these conditions we have just shown that the
collection efficiency will be

Q = 11 /2

(64)

where n is the quantum efficiency of luminescence. We then need to
know the fraction of the solar flux which is absorbed by the dye.

This

is shown in 3. I to be S/I. S/I is typically about 15% per dye for the
very narrow band dyes such as rhodamine-590, and 30% for broad
band dyes such as DCM.

The total collector efficiency is given in the

usual way to be

7JLSC = 11cell. S/I. 11/ 2

where 11cell is the AMl efficiency of the cells used.

(65)

The flux gain is

Gnux = Ggeom · S/I · 111 2

(66)

VIII. SUMMARY
1. We derive the performance of a plate containing an isotropically scattering plane as a limiting case of a self-absorbing system.
2. We model the intensity and spectral dependence of selfabsorbed emission in the quasi-one-dimensional case of a rod of LSC
material.

-203-

The polarization component intensities of emission result-

ing from polarized excitation are related to self-absorption probabilities.

Emission from higher order generations is shown to be negligi-

bly polarized.
4.

Changes in the measured total lifetime in transient emission

experiments is related to self-absorption probabilities.

The masking

of first generation emission is shown to produce a transient emission
spectrum which is the difference of two exponentials.
5. We show that in a regime typical of actual devices the
average trajectory length traveled by emission in the plate is on the
order of the dimensions of the plate.

This leads to a simple charac-

teristic length approximation for the self-absorption probability.
6. We assign a critical optical density (CODE) to each dye.
The CODE is the peak extinction coefficient times some pathlength and
concentration such that the probability of self-absorption is 50%. We
also show that the CODE. is the geometric gain of a plate containing
the dye at a peak concentration such that the peak optical density
through the thickness of the plate is one, so that the collection efficiency of the plate is 50% times the quantum efficiency of luminescence.

-204-

CHAPTER 5

-205I.

ANALYSIS OF EXPERIMENTAL RESULTS
In this chapter we will analyze the data described in Chapter 2

using the techniques developed in Chapters 3 and 4. Degradation rates
for a variety of dyes and vehicles are tabulated.

Focusing on the

particular dye rhodamine-575, we find that the spectra are predominantly inhomogeneously broadened only in the cast PMMA samples.
We compared measured self-absorbed emission spectra with theoretical models.

The self-absorption probability of rhodamine-575 in

solution is measured using three independent techniques.

The full

analytical model from Chapter 2 is applied to the single dye rhodamine575 system.

Finally, the critical optical densities of all of the dyes

surveyed are tabulated.
Dye Degradation Rates
Table 4 is a compilation of observed degradation rates for dyes
measured by ourselves and others.

The error bars on measurements

made here are large due to uncertainties in the amount of light absorbed by the sample.

The average number of molecules degraded

per absorbed solar photon was computed by computing S for the
samples under an AMO spectrum at 1, 000 watts per square meter for
eight hours per 24 hours of total exposure.

The absorption spectrum

used was the initial dye spectrum before exposure. This overestimates the flux absorbed, since the absorption coefficient decreases
with exposure. We estimate that the resulting calculated number of
photons absorbed is accurate to within an order of magnitude.

The

change in the number of dye molecules in the sample was assumed to

-206-

Table 4. Measured quantwn efficiencies for photodegradation due
to solar exposure.

The number of photons absorbed by the sample

was estimated to be the fraction of a 5, 800° K black body spectrwn
absorbed by the sample's initial absorption spectrum times
0. 017 photon moles per square centimeter per hour times the
number of hours of effective full sunlight absorbed by the sample
at 5~ degradation.

The number of molecules reacted was assumed

to be given by the change in peak optical density. We estimate that
this technique of calculation is only accurate to within an order of
magnitude.

-207Table 4.

Quantum Efficiencies for Photodegradation due to
Solar Exposure.
Molecules bleached t
Absorbed photon

Dye

Solvent

Coumarin-500

methanol

3 x 10- 6

Coumarin-535

methanol

4 x 10- 6

Coumarin-540

methanol

6 x 10- 7

methanol in quartz

2 x 10- 5

PMMA

2 x 10- 6

DCM

methanol

2 x 10- 6

Rhodamine- 590

methanol

2 x 10- 7

methanol in quartz

3 x 10- 6

PMMA

1 x 10- 6

PMMA with acetic acid

1 x 10- 5

methanol

5 x 10- 6

Rhodamine- 640

Sulforhodamine-640 methanol

1 x 10- 6

methanol in quartz

6 x 10-

Cresyl Violet

methanol

2 x 10- 7

Oxazine- 720

methanol

1x10- 7

Oxazine- 750

methanol in quartz

1 x 10- 5

t The accuracy is estimated to be within an order of magnitude.

-208be proportional to the change in the peak optical density of the sample.
In general we found no dyes which are stable against photodegradation in PMMA plates for periods longer than a year. Dyes in
methanol solutions showed somewhat better stability when exposed in
sode-lime glass bottles. Dyes in degassed methanol in quartz cuvettes
degraded substantially in several hours. Our results for quantum
efficiencies for photodegradation are consistent with the results of
others.

Under optimal circumstances the quantum efficiences for

photodegradation for typical efficient organic laser dyes are on the
-6

order of 10 .
Dye stability will have to be improved about two orders of magnitude to obtain acceptable stability levels.
following way.

This can be seen in the

In order to minimize self-absorption, we can require

that a typical plate will have a peak optical density of one. If we
assume that a typical peak extinction coefficient for a dye is 50, 000
20

liters per mole centimet~r, then there will be about 10
per square meter.

molecules

(This argument will also pertain to the final dye

in a multiple dye plate.) If the dye absorbs 30% of the useful visible
solar spectrum, in 20 years the plate will have absorbed 10
per square meter.

photons

Acceptable performance, therefore, required that

the quantum efficiency of photodegradation be at most 10- •
Spectral Homogeneity
In some host materials, the absorption and emission spectra of
all of the dye molecules in a sample appear to be approximately the
same. In such a case the dye ensemble is said to be spectrally homogeneous.

For example, in Figure 44 the maximum of the emission

-209-·

spectrum for rhodamine-575 in methanol at room temperature is independent of the excitation energy, even for excitations which are
2, 000 wavenumbers lower in energy than the emission peak.

This

'blue-shifted' emission disappears when the sample is cooled to liquid
nitrogen temperatures, so that this extra energy in the emission must
come from thermal energy in the molecule or from the surrounding
material. If the same dye is infused into a PMMA plate, essentially
the same results are produced, as shown in Figure 46. We, therefore,
conclude that, for these two host materials, the absorption and emission spectra are predominantly homogeneously broadened.

By contrast,

dyes which were dissolved in methyl methacrylate monomer and then
polymerized displayed characteristics of inhomogeneous broadening.
As shown in Figure 4 5, the emission spectrum depends on the precise
energy of the excitation. It was thus possible to select portions of the
dye ensemble which displayed significantly different spectral characteristics than the ensemble average.
We have seen that dye molecules in a host material at room temperature are able to emit light several thousand wavenumbers greater
in energy then that of the initial excitation.

This is not surprising,

since these molecules typically have a large number of internal degrees
of freedom.

At liquid nitrogen temperatures, KT is reduced by a fac-

tor of four, which apparently is sufficient to decrease the available
energy in the molecule or in the surrounding bath sufficiently so that
these large blue-shifted emissions do not appear.

This temperature

sensitivity would imply that even larger blue-shifts of emission would
be possible in plates at the elevated temperatures of roof-mounted

-210collectors.
Self-Absorbed Emission Spectrum
Figure 67 shows the analytic results for the collection efficiency
Q(i/) as a function of wavenumber for a semi-infinite rod.

The absorp-

tion and emission spectra used were numerically generated spectra
similar in form to those of rhodamine-575 .

The pathlengths, rod

diameter, and dye concentrations are the same as for the experimental data shown in Figure 41.

The scattering coefficient used in this

calculation is 0. 0035 cm~ The absorption spectrum is the sum of two

gaussians with peak positions of 19, 500 and 20, 300 cm- 1 , peak widths
of 840 and 680cm- 1 , and peak heights of 94, 000 and 27,000 liters per
mole centimeter, respectively.

The emission spectrum was the

same sum of two gaussians reflected about 19, 000 cm- 1 and normalized
to unity.

The shift in spectral position with increasing self-absorption

is accompanied by a decrease in collected intensity, in good agreement
with the experimental data in Figure 41.

Figure 68 shows a similar

calculation, except that the Stokes shift between the absorption and the
emission spectrum has been increased from 1, 000 to 2, 500 cm -l (to
lower their overlap). While the nonzero scattering losses cause an
overall decrease in the intensity with increasing pathlength, the emission spectrum is seen not to shift appreciably.
Self-Absorptio n Probability Measurements
We have shown results on three different measurements of the
effects of self-absorptio n in rhodamine-575 : spectral overlap, emission depolarization , and lifetime broadening. We are interested in
determining the consistency of these different measurements in

-211-

Figures 67 and 68. Analytic result for emission spectra as a function
of sample pathlength. Experimental absorption and emission spectra
were each fitted as the sum of two gaussians. These are used with
the model from 4. III to imitate the experiment shown in Figure 41.
The Stokes shift used in Figure 67 is 1, 000 cm -i.
The calculation was repeated in Figure 69, except that the
Stokes shift was increased to 2, 500 cm -i.

15,000

r-z

z I

(f)

>r-I

17,000

er:

FLUL'JRESCENCE

snC"'J
......

19,000

WRVENUMBERS

::=tr

21,000

23,000

FLUL'JRESCENCE

CrJEFFICIENT ~

EXTINCTIL'JN

EMISSIL'JN SPECTRA WITH VARIABLE PATHLENGTH IN SAMPLE.

ANALYTIC ABSrJRPTirJN AND FLUrJRESCENCE SPECTRA.

.....

15,000

r-z

>r-en
_.........

17,000

19,000

WRVENUMBERS

I ........ ,..,., I

I ............,

21,000

1·-=

Q)
+-

23,000

FLUeJRESCENCE

EXTINCTieJN

C~EFFICIENT ';v

ANALYTIC ABS~RPTI~N AND FLU~RESCENCE SPECTRA.
EMISSieJN SPECTRA WITH VARIABLE PATHLENGTH IN SAMPLE

c:..:i

......

-214-

determining the actual self-absorption rates.
For a square cuvette sample geometry, the probability of selfabsorption in the critical cones is about the same as the probability
of self-absorption outside of the critical cones.

This self-absorption

probability, r, can be calculated in three different ways. We will
assume that the characteristic length approximation of Eq. (61) applies
for the spectral overlap calculation, with the length determined by the
pathlength through the sample from the excitation source to the detector.

This is actually the weakest approximation in all of the three

techniques, and will lead to discrepancies at very low and high selfabsorption rates. We will use Eq. (54) to find r-17 from the reduced
anisotropy measurements. If RAmax is the maximum reduced anisotropy (2e/5) measured at low concentrations and short pathlengths,
then the product of the self-absorption probability and the quantum
efficiency, r · T/ , from Eq. (54) is
r17

1 - RAexp

(67)

RAmax

Similarly, from Eq. (59), if Tmin was the shortest total lifetime measured at low concentrations and short pathlengths, the same result
should be given by the relation

r71

mm
exp

(68)

-215where 7

is the measured lifetime. If we apply the preceding two
exp
relations to the measured transient emission and steady state depolar-

ization measurements on rhodamine-575, we obtain the results shown
in Figure 69. We have plotted the calculated product of the quantum
efficiency time the concentration of the sample through which the
emission had to travel

The solid line is the result from the charac-

teristic length approximation.

(This is actually just the self-absorption

probability, and not the product of the self-absorption with the quantum
efficiency.) There is very good agreement between the lifetime lengthening and the emission depolarization measurements. The measured
value of RAmax = 0. 18 for rhodamine-575 in glycol is consistent with
a rotational diffusion time of several nanoseconds, which is the rate
observed for similar dyes in glycol (Von Jena and Lessing, 1979). We
conclude that there is good accord between the three different measurement techniques.
Optimal Efficiency
We believe we know all of the characteristics of rhodamine- 575
which determine the efficiency of a single dye LSC.

As a test, we will

calculate the optimal efficiency of an LSC containing rhodamine-575 as
a function of the size and concentration. We use the PSC geometry
formalism from Eq. (12) to compute the first generation collection
efficiency.

The self-absorption rate in the critical cones is computed

using the characteristic length approximation.

These are combined

using Eq. (13) to give the total collection efficiency, with the approximations of no reflection losses and only homogeneous spectral broadening. We will assume edge mounted silicon cells having 18% AMl

-216-

Figure 69. Self-absorption probabilities for rhodamine-575.

This

shows a juxtaposition of predicted self-absorption probabilities for
three measurement methods: spectral overlap convolution (solid),
emission depolarization (boxes), and transient lifetime (bars).

The

second two techniques are plotted assuming the quantum efficiency
of luminescence is one.

E -1

()_

Cf)

1 0 -B

o.oo l

Li • 2.00

Cf)

gs l!.00

()_

I-

z 6.00

........

RRTES F~R R-575 IN S~LUTI~N.

1 0 -S

X PATHLENGTH

1 0 -S

:ff:::

C~NC.

l [I~
10 - 7

i' I

CJF TRANSIENT EMISSICJN ~

I LIFETIME BRCJADENING

STATE
PCJLARIZATICJN

~ STEADY

1 0-Y

STEADY STATE
SPECTRAL OVERLAP

10-3

s.oo--~~____,,.--~~--.~~~_..,.~~~--~~~--.

SELF-RBS~RPTI~N

.......

-218efficiency are coupled to a plate with an index of refraction of 1. 49.
The quantum efficiency of the dye is assumed to be 0. 9.
of the calculation is given in Figure 70.

The result

For the sake of comparison,

the system efficiency of a scattering plate (4. II) with an identical
geometry is included.

The single dye plate outperforms the scatter-

ing plate for geometric gains greater than 10.

From Table 1, we

find that the CODE of rhodamine-575 is about 20. Thus, for a plate
with a geometric gain of 20, we expect a collection efficiency of about
0. 9 ... 0. 5, and that about 30% of the solar flux to which the cell would
be sensitive would be absorbed by the plate.

The estimate for the

efficiency for such a plate from a CODE calculation is, therefore,
2. 4%.

The plate concentration for the code calculation is 10- M.

The optimized plate actually has 100 times this dye concentration for
a gain of 20, giving an efficiency of 4%.
CODES of the Dyes
Table 1 gives the critical optical densities for the organic laser
dyes tested.

The error bars are due principally to uncertainty in the

measurement of the absorption tails.

Typical values for the critical

optical densities are about 20. Such dyes, therefore, produce collection efficiencies of less than 50% if incorporated in plates with geometric gains greater than 20 due to self-absorption effects. Several
dyes have critical concentrations around 80.

These tend to be the

U. V. and blue absorbing coumarin dyes. While they have much improved CODEs, their high energy absorption bands are not effective
in absorbing sunlight. DCM (4-dicyanomethylene-2-methyl-6-pdimethylaminostyryl-4H-pyran) is by far the most promising dye

-219-

Figure 70.

Efficiency of a single dye LSC. We assume that

rhodamine-575 is the only dye in a 1 cm thick infinite ribbon (PSC)
geometry LSC. We have calculated the optimal dye concentration
using a binary search routine.

The coarseness of the concentration

plot reflects time restraints on the search.

The efficiency was cal-

culated using the full PSC calculation described in Chapter 3. An
average matrix optical density of 4 x 10- per centimeter was assumed.

18% efficiency AMl Si cells were assumed to be mounted on both
edges of the ribbon.

-220-

SINGLE DYE PSC - RH~DRMINE-575
-1

>- 10
t:: 10-2
0:::
<[
_J

~PTIMRL

DYE

-3

M~LRRITY

0 10
-s

w 10

b 10-6
>-

§E 8~ ..

~PTIMRL
C~LLECT~R

~ 6~

EFFICIENCY

w Li~
0:::

guw 2~ SCATTER!~
..
PLATE

-'
_J

GE~METRIC

...

GRIN

-221found to date. It has a moderately blue absorption, which allows for
30% absorption of an AMO spectrum, as well as a CODE of 250.

The

performance of DCM is considerably reduced when it is case directly
in PMMA, which is probably due to the decreased Stokes shift.

The

methanol performance can be partly restored by adding solvent to the
monomer prior to polymerization.
II.

SUMMARY
1.

The best case quantum efficiency for photodegradation was

found to be on the order of 10- 6 molecules per absorbed photon, in
agreement with literature values.

Rates on the order of 10- are

required for a twenty year useful life.
2.

Rhodamine-575 in cast PMMA was found to be predominantly

inhomogeneously broadened, while the same dye in methanol and in
diffused PMMA showed spectral homogeneity.

At room temperature

the dye could emit at enE:'.rgies over 2, 000 cm -i (0. 25 ev) greater than
the exciting energy.
3.

Good agreement was obtained between experimentally mea-

sured self-absorbed emission spectra and numerically modeled selfabsorption for the LSC rod geometry.
4.

The self-absorption probability for rhodamine- 575 as a func-

tion of concentration and pathlength was computed from the characteristic pathlength approximation (spectral information) from emission
polarization measurements, and from transient emission measurements.

Good agreement was obtained between the three techniques,

with very good agreement between the second two techniques.

-222-

5. We calculated the efficiency of a rhodamine- 575 plate as a
function of size and optimized concentration. The results are in
general agreement with prototype measurements.
6.

Critical optical densities (CODEs) were calculated for the

dyes surveyed.
Table 1.

The CODEs for methanol solutions are tabulated in

Casting the dyes in PMMA can reduce the Stokes shift,

which reduces the CODE.

The Stokes shift in PMMA can be increased

by adding solvent such as DMSO to the monomer prior to polymerization.

-223-

CHAPTER 6

-224I.

INTRODUCTION
Winston derived (Winston, 1974) the maximum gain for a mirror

concentrator using geometric optics arguments.

Rahl later found

(Rabl, 1976) the same results from the second law of thermodynamics.
The basic result is that an optimal concentrator on the earth can reproduce the original brightness at the surface of the sun. Since the
geometric dispersion experienced by sunlight in traveling to the earth

is a factor of sin (88 ), where 08 is the half-angle of the solar disk
(0. 267 degrees) the maximum gain is, therefore, the inverse of that

dispersion, or 1/sin (O. 267) = 46, 050.

This is also an application of

the brightness theorem from geometric optics (Born and Wolf, 1975).
It is tempting to try to develop similar relationships for the

performance of an LSC. Since the energy of the incident light is not
preserved in an LSC, the above result must be generalized appropriately. We will first give the results of a detailed balance calculation
which gives the intensity ·of the plate output as a function of size, ignoring self-absorption effects. We then repeat the result due to
Yablonovitch which generalizes the brightness theorem from geometric
optics to include nonconservative systems. This approach incorporates
self-absorption effects.

The Yablonovitch result is applied to the

spectra of all of the dyes surveyed, and the resulting prediction of
the maximum concentration attainable is compared with the results of
the CODE calculation.

Finally, we incorporate these results in a

simple model of the optimal efficiency of a self-absorbing LSC.

-225II. DETAILED BALANCE CALCULATION
Consider an LSC as shown in Figure 1. A ribbon of thickness
D and width L is mirrored on one edge, and has a black oody absorber
cavity mounted on the other edge. The geometric gain of the ribbon is
L/D. We define a thermodynamic gain of the plate to be the fourth
power of the ratio of the black body absorber energy to the sun's temp2

erature, divided by sin (ec):

(69)

This is equivalent to the concentration factor for the device.

The

energy radiated and absorbed by the black body cavity in equilibrium
is proportional to the fourth power of its temperature. If the temperature of the absorber is equal to the sun's temperature, then Eq. (69)
reduces to Winston's result. We will assume spectral properties for
the LSC ribbon to calculate the equilibrium temperature of the absorber.
This temperature then determines the maximum concentration }X>Ssible
by such a ribbon.
The energy input into the system is from the absorption of sunlight.

The total solar energy per unit length of the ribbon is assumed

to be

(70)

-226where a is the Stephan Boltzman constant (5. 67 · 10..a joule meter- 2
second -1 K-4 ) . The spectral intensity of the sunlight per unit length
is
. 15 (~)4

= I

1TKTs

v dv

(71)

e hcv/KTs _ l

(Landau and Lifshitz, 1977). We approximate the absorption spectrum
as being zero everywhere except between v1 and v2 , where the absorption across the thickness of the plate is assumed to be 100%.
emission is assumed to be monochromatic at v0 •

The

The energy which

is absorbed, luminesced, and trapped by our idealized LSC is, therefore

(72)

where P is the critical cone loss and n is the quantum efficiency of
luminescence. If v0 < v1 <

v;, there is no self-absorption.

We include three mechanisms by which energy can leave the
system.

The spectral distribution of the light emerging from the

cavity per unit length is

he )4
v dv
n7TKTa ehcil/nKTa _ 1

= 1 . 15 (

(73)

-227where the total emerging intensity per unit length is

(74)

n is the index of refraction of the ribbon. A fraction of this emission,
GF, is immediately lost out of the critical cones:

GF = 2
1(

,... Be

dB cos (8) sin (e) cos-

cos(e )
[ _ _ £._

7f /2-Bc

(75)

sin (8)

The fraction which is then lost to matrix absorption is o (1- GF).
Finally, there is energy lost through absorption by the dye.

This has

two components: the fraction of the absorbed emission which is radiated out of the critical cone, and the fraction of the photon's energy
which is lost in the Stokes shift.

The total energy lost out of the sys-

tem is given by
" V2

= Ia(GF+c5(1-GF)] + (1-c5)(1-GF)

J_ Ia(v) di.T
ll1

,--. V2

- (1- P)(l-GF)(l-c5) 11 Jz.>1

Ia(v) dv

(76)

We find the temperature of the absorber by numerically equating
the energy in and out of the system. We have shown in Figure 71 the
results from performing the above calculation for an LSC absorbing in

-228-

Figure 71.

Thermodynamic gain of an idealized LSC. We compute

the gain of an LSC (Planar Solar Collector geometry, PSC) by substituting a black body cavity for the edge mounted cells, and then
computing the temperature of the cavity by a detailed balance calculation.

The straight diagonal line indicates the operation of concentra-

tors using geometric optics such as mirrors and lenses.

-229-

THERMODYNAMIC GRIN OF
RN IDEALIZED LSCD

-t .-

::r
rn

,&.

GAIN CIF 2-DIM.
WINSTON COLLECTOR
PSC WITHOUT
INTERNAL
RBSCIRPTICIN

PSC WITH

INTERNAL
ABSORPTION

-< .....

::D (,.)
3:

GAIN CIF 1-DIM.
WINSTCIN CCILLECTCJR

OPTICAL DENSITY 0.0016
INDEX OF REF. 1.49

ABSORPTION FROM 3500.

~---~..........._._...._._._..~~~~__._.~.....___._._~

100

101

102

103

GEelMETRIC GAIN

10~

105

8000.

EMISSION AT 8300.
FINGSTRCJMS

-230-

the region from 3, 500.A to 8, ooo.A, emitting at 8, 300.A, anQ. for

geometric gains ranging from 1 to 10 . The two upper curved lines
show the thermodynamic gain (or energy density gain) for the cases of
no matrix absorption (upper curve) and for a matrix absorption corresponding to an optical density of 0. 0016 across the LSC thickness.

The

straight diagonal line represents all nonlossy geometric optics collectors, because in such collectors the thermodynamic gain can be
equal to the geometric gain.

The stars indicate the operating points

of the one-dimensional (trough) and two-dimensional Winston collectors.
Under the conditions of the model there is a considerable buildup
of energy in the plate-cavity system.

For example, an LSC with no

gain at all (Ggeom = 1) would cause the cavity to rise to a temperature of 1, 000°K.

A similar black body facing the sun directly only

would rise to about 400° K. While the technique can be elaborated to
include effects such as self-absorption in more realistic dye spectra,
the results in their present form are overly optimistic in the predicted light output.
Ill. GENERALIZED BRIGHTNESS THEOREM OF YABLONOVITCH
A more generalized approach to a thermodynamic limit to the
performance of an LSC would be to consider the incident sunlight and
the light trapped in the plate as two systems of photon gasses in
equilibrium.

This system was first studied by Kennard (Kennard,

1918), and by Ross (Ross, 1961).

Recently Yablonovitch (Yablonovich,

1980) has applied these results to the LSC.

The simplest result from

·his paper is the most useful, and that is a generalization of the

-231-

brightness theore.m of optics to inelastic processes (optical .elements
which change the energy of the light).
We will follow closely the derivation of 'a generalized brightness
theorem' for optical systems, including inelastic processes given by
Yablonovitch. According to statistical mechanics, the entropy change
associated with the loss of a photon from the incident solar Bose field
is

= - K £n (1 + B 1T v;_ )

(77)

where n is the index of refraction of the LSC plate, v, is the wavenumber of the light, and B 1 is the brightness of the incident field in
photons per unit area, per wavenumber, per unit time, and per 47T
solid angle.

The entropy increase in the concentrated field due to the

fluorescent emission of one photon is
= K .fn(l + Brr v;2
B2

) + he (V,, - ii;)

(78)

where the additional term is due to the thermal dissipation of the
Stokes shift at an ambient temperature T.

By the second law of ther-

modynamics, .1. 8 1 + .1.8 2 =::::: 0, therefore

K £n (1 + 87T Vi2 K2 ) I .tn (1 + 87T v;2 n ) !S he (v1 - ii;)
B1
B2
nT

(79)

-232or the achievable concentration ratio or flux gain is

(80)

If v1 = 112 , this reduces to the brightness theorem from optics.

This limit as stated sets a rough upper bound for the possible gain of
an LSC, in that the details of the spectral structure are not incorporated. As a first approximation, we will assume that most emission
occurs at the peak of the emission spectrum and similarly most of the
absorption occurs at the peak of the absorption spectrum.

Table 5 is

obtained by plugging known values of the spectra maxima for the dyes
surveyed into Eq. (81 ).

The CODE calculation results are included in

Table 5 for comparison.
The reasonable correspondence between the spectrally derived
CODE and the thermodynamically derived maximum gain is somewhat
fortuitous.

Eq. (80} can be employed in a less approximate manner

by utilizing it for relating the achievable gain for emission at ii;_ due
to absorption at 112 • We assume that the incident solar flux has a flat
spectral response, and that the concentration of a dye in an I.SC plate
will be adjusted such that the peak optical density through the thickness
of the plate is one. The maximum light concentration for the dye will,
therefore, be the weighted average of this wavenumber dependent gain
times the normalized emission spectrum:

-233-

Table 5.
dyes.

Maximum light concentration for selected organic laser

The second column gives the Stokes shift in electron volts

between the peaks of the absorption and emission spectra.

The

third column gives the critical distance for dipole-dipole transfer
from a dye molecule to another similar dye molecule, as defined
by Eq. 20.

The fourth column contains the critical optical densities

for methanol solutions;

The fifth column contains the maximum

light concentration as defined by the Yablonovitch relation in Eq. 80,
where v 1 and V-2 are the wavenumber positions of the peak extinction and emission spectra values, respectively.

Finally the last

column is obtained using Eq. 81, which performs an average of
Eq. 80 over both the absorption and emission spectra.

-234Table 5.

Dye

Stokes shift
e. v.

Coumarin-480
0. 53
0. 63
Coumarin-500
Coumarin-535
0.31
Coumarin-540
0.25
0.67
DCM
Rhodamine- 5 60
0. 12
0. 117
Rhodamine- 5 75
Rhodamine- 590
0. 104
Rhodamine-610
0. 101
Kiton red- 62 0
0.086
Rhodamine- 640
0. 096
Sulforhodamine-640 0. 08
Cresyl violet-670 0.094
Oxazine- 720
0.07
Oxazine-750
0.079
0.073
DOD CI
0.089
DO TC I
IR-144
0. 18

Ro, A

CODE

CYabl.

GYabl.

25.3
26. 4
33.0
33.7
24.2
43.8
46. 1
48.7
49.4
52.4
53.1
54. 7
50.2
54. 4
55. 5
65. 1
69.8
61. 9

113.
25.7
31. 3
82.7
244.
17.0
34.2
25.3
36.3
15.8
16.6
17.0
16.8
16.9
24. 5
11. 0
7. 5
15.7

2.lxl0 9
11
1.3xl0
3. 7xl0 5
2. 2xl04
11
5.4xl0
162.
113.
69.
58.
32.
48.
25.
44.
16.

3. 6xld
1. 2xl018
17
3. 5xl0
13
5. 6xl0
13
4. 6xl0
14
6. 7xl0
11
3. 3xl0
2. 1x1a1°
11
1. lxl0
10
9. 2xl0
11
1. lxl0
11
1. 3x10
1. 2x10
8. 7x10
4. 6xl0
3. lxlO
2. lxl0
1. 5x10

24.
19.
36.
1417.

-235-

· ~ exp (he (v1 - v2 )) / \

,oo

(1 -10-E (v1) • C · x) di7i

.-,00

f ( v 2 ) v2

exp ( - -he
KT

v 2 ) d v2

(81)

[1- 10 - E (Vi)· C · x ] dv

Results for this calculation are also tabulated in Table 5.
The success of the Yablonovitch result lies in the choice of the
emission and absorption maxima as the wavenumber values used in the
calculation. When the entire fluorescence spectrum is included instead
of just the peak position, unphysically large light gains are predicted.
An obvious result from Table 5 is that the dye with the largest
Stokes shift, DCM in methanol, also has the highest possible light concentration potential.

This is experimentally verified in that a DCM

plate presently holds the record for the brightest light output of any
LSC.

The Stokes shift for this dye is 5, 760 wavenumbers or 0. 67 ev

(1 ev = 8, 654. 7 cm - 1 ), as shown in Table 5.

-236IV. OPTIMAL EFFICIENCY MODEL
We have seen that an LSC can achieve higher light concentrations
as the Stokes shift between the absorption and emission spectrum is
increased. Self-absorption ceases to limit the flux gain for a dye like
DCM, whose Stokes shift corresponds to about 0. 7 ev (peak absorption
to peak emission). In general, as the Stokes shift becomes larger
than 0. 7 ev, the efficiency of the plate will decline for two reasons.
The first is that less of the solar flux is utilized, the second is that
progressively lower voltage output cells must be used to convert .the
output light. We can extend the usual treatment of the optimal efficiency of a solar cell (Wolf, 1960) in a simple manner to include the
effect of the Stokes shift in an LSC on the overall efficiency of the
collector.
Consider the 5, 800 ° K black body spectrum in Figure 72. We
assume that the dye in an LSC absorbs all of the solar flux from the
peak of its absorption out to higher energies. We assume that there
is some gap between this absorption cutoff and the absorption edge of
the cell.

The energy per unit area incident on the LSC is

·'> 00

Ein

A·

he v s di/ (e hcv/KTs - 1 f i

(KT 8 )

(hc) 3

A is a constant that scales the incident solar intensity.

(82)

Let the output

-237-

Figure 72. Simple model of LSC efficiency.

The envelope function

is the flux distribution from a 5800°K black body.

The shaded region

is the flux above Eabs which is absorbed by the LSC.
absorption cutoff of the cell.

E cell is the

-2380

:::r

>c..!:>
a:

I-

:::>

I-

::>

(/)

I-

--'
_J

--'

CI:

c.b

:L
::J

(/)

(f)

zE)

t--t

I-

Cf)

a: m
I- a:
u LL.
0....

(/')

...J
...J

uLU

t--t
Cf)

0....

......•

Xnl_j titJlQS

a:
Iu
_J

-239energy of the cell be Eout"

The energy output of the LSC mounted

cells is maximally

= A· (1 - P) · Eout ·

., co

hcv /KTS

di/ ii

-1

-1)

(83)

Eab/hc

where (1- P) is the fraction trapped in the critical cone. We assume
that there is sufficient Stokes shift such that the only dominant loss
mechanism is light escaping out of the critical cones. Combining
Eqs. (82) and (83) gives an optimal efficiency for an LSC:

11 LSC =

15 Eout
(1 - P) 1T4

KTs

:-· co

dx x (e - lf

(84)

·ab/KTs

Silicon cells are the obvious candidates for the edge-mounted cells.
Their typical output voltage is about 0. 5 ev, with an absorption edge at
about 1. 0 ev. If we assume a Stokes shift of 0. 7 ev is required for
eliminating self-absorption losses, then the absorption edge of the dye
should be at 1. 6 ev. We assume 70% critical cone trapping.

Under

these assumptions we find that the maximum efficiency of an LSC is
8. 3%.

Figure 73 shows the efficiency of LSC plates utilizing silicon

cells or gallium arsenide cells as a function of the difference between
the absorption cutoff of the dye and the bandgap of the semiconductor.
Clearly this calculation gives an upper limit on the efficiency, since
we have ignored numerous loss mechanisms such as matrix absorption,

-240-

Figure 73. Collector efficiency vs Stokes shift. We assume the sun
is a 5800° K black body, and that the LSC plate absorbs all light above
a cutoff energy E ABS . We assume all the light is emitted just above
the band gap energy, Ecell' so that the Stokes shift is E ABS - Eceu ·
We also assume the output from the cells is 0. 5 ev below the band gap
energy.

The two cases plotted are for Ga As (upper) and Si (lower).

The dotted line indicates the operating point using a dye with a Stokes
shift similar to that of DCM.

E -1

0 0
· 8.20

:::1 a.so

~ 1.00

tt 1..50

.........

~ 2.00

EFFICIENCY VS. ST~KES SHIFT

__ -

7- -

=0.5 EV •

=0.6 EV.

a.so

0.80

E ABS - E CELL I N E• v.

O.llO

OUT

---~

~~UT =0.95 EV.

E CELL - Eour

1.00

2.50--~---~~--~---~~-,--~~r-~-,-~~,--~-,

C~LLECT~R

'"I"'"

-242emission at lower energies than the absorption edge of the cell, and
incomplete absorption of sunlight in the absorption band of the plate.
The computed efficiency also benefits from the use of a 5, 800° K
black body spectrum. If we repeat the calculation using a measured
solar spectrum, we find the efficiency is 9. 3% for an absorption cutoff at 1. 6 ev and an output voltage of 0. 5 v.
V.

SUMMARY
1. We developed a detailed balance calculation for the light out-

put from an LSC.

The edge mounted cell is replaced by a black body

absorber, and the temperature of this absorber is computed at equilibrium from a balance of the energy absorbed and emitted by the black
body.
2. We compute Yablonovitch 's prediction for the light concentrating ability of dyes as a function of the absorption and emission energies, and the temperature of the surrounding material. This model
gives reasonable results for dyes with Stokes shifts of less than 0. 2 ev
in the simple approximation that all light is absorbed at the peak of the
absorption spectrum and all light is emitted at the peak of the emission spectrum.
3. A very simple model for the ultimate efficiency of an LSC
including self-absorption effects is presented. We use the empirical
measurement that dyes with CODEs over 100 have Stokes shifts of
0. 7 ev or more.

This model predicts that plates with geometric gains

greater than 100 will have a maximum efficiency of 9%.

-243-

CHAPTER 7

-244-

INTRODUCTION
In this chapter we will briefly summarize the status of the LSC

technology and offer our opinions as to what the subsequent steps
should be in its development.
Prototype Performance
The most efficient LSC reported to date was a thin film multiple
dye plate made by Owens-Illinois. Its measured efficiency was 3. 2%.
However, the plate area was so small compared to its thickness that
even a purely scattering plate (one which contained scattering centers
instead of luminescing centers) could produce at least this efficiency
if it were made with the same geometry.

This is, therefore, not

representative of an LSC which is performing appreciable light
concentration.
The most efficient device made to date which outperforms a
scattering collector was a liquid LSC made by ourselves. Its efficiency was 1. 9%, and its· flux gain (the increase in output power from a
cell when it is attached to the plate) was 3. 8. We made the highest
flux gain device, which was a PMMA containing DCM. It was 1. 3%
efficient with a flux gain of 5. 1.

This is a high concentration factor

for a device with nearly the same absorption dependence on the angle
of incidence of the input light as a black body absorber.
We have experimentally verified that the most important factor
limiting performance, especially flux gain, is self-abosrption.

For

example, the LSC with the highest flux gain contains the dye with the
highest Stokes shift of all the dyes we have measured.

(We also found

-245that the Stokes shift is nearly halved when DCM is cast in PMMA, as
compared to DMSO solutions).
How well can LSCs be expected to perform? A numerical model
incorporating the methanol spectra of a standard xanthene dye predicts
that 4% efficiency with a flux gain of 20 should be achievable. A CODE
calculation based on the DMSO spectra of the dye DCM predicts that
3% efficiency and a flux gain of 40 should be possible. If we assume
that a 0. 7 ev Stokes shift is required for high flux gains, then an upper
limit of about 9% efficiency results from a simple thermodynamic model.
Stability

We found lifetimes on the order of a month for dye photodegradation in PMMA plates. Dyes in solution typically did better, with
methanol samples having lifetimes between several hours and several
years.
Our measurements are in reasonable agreement with published
quantum efficiencies for photodegradation.

Typical optimal values for

such quantum efficiencies are 10-fuolecules per absorbed photon, which
corresponds to device lifetimes of about two months. We presently do
not know to what extent the maximum stability can be improved.

Twenty

year device lifetimes probably require quantum efficiencies for photodegradation of less than 10:

Future Development
Two areas of research emerge as the next logical step in LSC
development.

These are to examine the liquid LSC in more detail, and

to undertake the synthetic chemistry of dye optimization.

-246-

We feel that the homogeneous spectra and large Stokes· shifts
characteristic of dyes in solution are highly desirable.

Either the

liquid cell design, or some matrix-solvent combination, should supercede the cast plastic design.

The efficiency and gain should improve

due to the higher Stokes shift.

(The highest efficiency high gain plate

was a liquid cell containing dyes with only moderate Stokes shifts. )
In addition, the photodegradation problem is diminished due to both
more chemically inactive surroundings and by the ability to replenish
the dye.
None of the dyes available to us were satisfactory for use in an
LSC.

The closest approximation to the perfect dye is DCM.

The

princi}:lll problem with DCM is that both its emission and absorption
are about 2, 000 wavenumbers too high in energy.

A difficult but im-

portant task would be to try and synthesize new dyes with the right
spectra, a high quantum efficiency of luminescence, and good stability.

-247-

APPENDIX I

ECONOMICS OF LSC A PPIJCA TIONS

-248-

Research into LSCs is motivated by the need to increase the
affordability of solar power. Given our present understanding of the
technology, it is reasonable to perform at least a preliminary assessment of the LSCs economic viability. We do not have the data or the
expertise to rigorously model the manufacturing costs on a large
scale. Our aim is to discover whether the LSC is conspicuously promising

in comparison to other photovoltaic designs, and so we require

a result which is accurate to within roughly a factor of two. We will
use results of an LSC manufacturing analysis by Owens-Illinois
(Rapp et al, 1980) to estimate the module cost. We will compare the
price and performance of an LSC system with two other potential conversion techniques: high efficiency silicon cells with mirror concentrators, and thin film silicon cells.
The cost of 3 mm thick plexiglass plate from Du Pont is about

$8. 70 per square meter assuming very high volume manufacturing.
Approximately one-half gram of high purity dye is needed per square
meter. We will assume that the price of dye will drop a factor of 10
from the present average cost of about $20. 00 per gram because of
high volume production, so that dye will cost $1. 00 per square meter.
A tempered glass cover is required to prevent surface abrasion and

for UV protection. Owens estimates this should cost $4. 30 per square
meter. A 2 mm nontempered glass backing is probably also required,
for $2. 70.

Either aluminum foil or a mirrored surface on the backing

plate is probably required, at a cost of $. 50 per square meter. We
will assume a geometric gain of 50. Nonencapsulated 18% AMl silicon

-249-

cells at $ 500 per square meter will cost $10. 00. Owens estimates
assembly and packaging costs at $7. 00 per square meter.

The cost

of a square meter LSC module. at the factory is summarized as
follows:
3 mm plexiglass plate
$ 8. 70
0. 5 grams dye
$ 1. 00
3 mm tempered cover glass $ 4. 30
2 mm backing glass
$ 2. 70
mirroring
$ . 50
18% AMI silicon cells
$10. 00
assembly and packing
$ 7. 00
$34.20
The upper limit on the efficiency of the module. is 9%.

Prac-

tically speaking, we feel efficiencies of 3% are readily attainable by
properly casting DCM or one of its derivatives in PMMA in the presence of DMSO. We assume that the peak solar power is 1, 000 watts
per square meter.

Therefore, the cost per peak watt of the LSC

modual is $. 43 and $1.14 for 9% and 3% efficient modules, respectively.
For comparison, we will use Hovel's estimates (Hovel, 1978)
for the price and efficiency of a system using silicon cells and mirror
concentrators. We will assume that encapsulated 20% AMl efficiency
silicon cells cost $1, 250 per square meter, and that trough collectors
will produce geometric gains of 30 with 80% efficiency.

The cell cost

is $41. 70 per square meter, and the concentrator cost is estimated
to be $ 60. 00 per square meter.
$101. 70.

The total price per square meter is

The efficiency of the combination is about 16%, giving

$. 64 per peak watt. Estimates on the performance of thin film

-250silicon cells suggest that AMl efficiencies of 10% and costs per peak
watt of $. 50 are reasonable (Hovel, 1975).
The key performance parameters of the three systems is summarized in Table At. It is interesting to compare the noncell costs of

Comparison of photovoltaic converters.

Table AI.

Efficiency

9%- 3%

Silicon
Thin Film
10%

Cost per
peak watt

$. 43 - $1. 14

$.50

LSC

the LSC module and the mirror concentrator.

Silicon with
Mirror Concentrator
16%

$. 64

The cost of the LSC

module without the cells is $24. 20, and the cost of the mirrors alone
is $ 60.

The fractional loss in output introduced by the LSC is between

0. 17 and 0. 44 (for 3% and 9% system efficiencies, respectively),
against 0. 80 for the mirror collector.

The LSC is 2. 4 times cheaper

than the mirrors, and the mirror is between 1. 8 and 4. 7 times more
efficient.
The baseline for performance for utility power generation is
apparently set by coal technology.

For photovoltaic power to be com-

petitive, the converters themselves should cost between $. 10 and

$. 40 per peak watt, and should have an overall efficiency of at least
10% to offset noncell costs (Robinson, 1979). We can readily conclude that LSCs will not be competitive with coal because they cannot
achieve the required efficiency.

-251We can conclude that LSCs do not compare favorably with
mirror or lens concentrators due to the LSC 's power efficiency.

The

inefficiency of the LSC overcomes its cost advantage for practical
devices.

There is an exception to this rule, however.

Mirrors and

lenses are in general unable to concentrate diffuse light, whereas an
LSC can. Applications that require concentration of diffuse sources
will benefit from the use of LSCs.

A good example of such an applica-

tion is scintillation counting, where LSC-like devices are presently
being utilized very effectively.

However, the ability to concentrate

diffuse light is probably not a significant advantage for photovoltaic
power generation, since regions that receive predominantly indirect
sunlight usually also receive less total sunlight than would be desirable for power generation.
Summary
1. We estimate the cost of a square meter LSC module to be
$34. 20 based on a manufacturing analysis by Owens-Illinois. We
estimate the efficiency of a practical device to be 3%, and the maximum efficiency to be {%. Assuming the peak solar power is 1, 000
watts per square meter, the cost per peak watt is between $. 43 and
$1. 14 for 9% and 3% efficient devices, respectively.
2.

The low efficiency of LSCs prevent them from being compe-

titive with coal for utility power generation.
3.

Conventional concentrators such as mirrors appear to be

preferable to LSCs due to their higher efficiency, despite their higher
cost.

The exception is for applications requiring concentration of

-252diffuse or indirect light. In such applications an LSC can outperform
conventional concentrators.

-253-

APPENDIX II
MOLECULAR ABSORPTION AND LUMINESCENCE

-254-

We have considered organic laser dyes throughout the text as
though they were very small black boxes that absorb and emit light in
well characterized ways. In this appendix we will elaborate on the
operation of these dye molecules in a schematic fashion. Our purpose
is not to derive spectral characteristics from molecular parameters.
Instead we are interested in defining and motivating the existence of
terms like the Stokes shift and the extinction coefficient.
Absorption and emission by an organic dye molecule is a quantum
mechanical many body problem. If we consider the atoms of the molecule as masses interconnected by springs, and ignore the effect of the
surrounding material, there are in general 3N-6 independent internal
modes of oscillation (3N-5 for a linear molecule) where N is the number of atoms (Tinkham, 1964). A typical molecule will have on the
order of 30 atoms, or 100 normal vibrational modes. We will try to
get some understanding for this system by examining a particular
optically active mode (one with a high electrical polarizability in an
optical frequency).

There is a generalized interatomic spacing coor-

dinate associated with this mode (corresponding to the interatomic
spacing in a diatomic molecule, for example).

There is a coulombic

repulsion between the nuclei which dominates at very small distances.
Since the molecule is stable, there must also be an attraction which
dominates at longer distances.

Combining these two effects gives a

potential surface for the energy in the system as a function of this
generalized coordinate. If the electron cloud is in some excited state,
the form of the attractive interaction can change, so that in general

-255there is a different potential surface for each electronic state. Two
such potential surfaces are shown in Figure Al.
tions are not necessarily the same.

The equilibrium posi-

The lowest energy electronic

state or ground state usually has all the electron spins combined pairwise, forming a singlet state.

The lowest energy excited state can be

a triplet state, or one with unpaired electron spins, due to a more
strongly bonding character of this configuration.
The restoring force near the equilibrium position can be expanded
as a simple harmonic quantum mechanical oscillator. Near the minimum of the electron surface are quantized vibrational levels corresponding to simple harmonic oscillator-like wave functions.

As in the

case of the simple harmonic oscillator, the system is most probably
at the equilibrium position for low vibrational states, and is most
probably as far away as possible from the equilibrium position for the
higher vibrational states. The system is further complicated by a
sequence of rotational modes (in the gas or liquid phase) or librational
modes (in the solid phase) which are superimposed on each vibrational
state.
So far we have been cavalier about separating the electronic
motion from the nuclear motion.

The outer shell electrons that parti-

cipate in optical processes and chemical bonding have oscillation fre1s

quencies on the order of 10
13

occur on the order of 10

per second, while the nuclear vibrations

per second.

Thus it is reasonable to assume

that the nuclei experience some average force due to the electrons, but
that they are not sensitive to the particular position of the electrons.

-256-

Figure AI. Energy level diagram for absorption and emission.

The

ordinate is in increasing internal energy, and the abscissa is the
coordinate corresponding to the optically active mode.

(In a diatomic

molecule, this would correspond to the interatomic distance.) The
two curves are the ground and first excited electronic surfaces.
(a), (b), (c), and (d) correspond to absorption, relaxation, emission,
and final relaxation (Kar.plus and Porter, 1970).

-257-

Molecular

Coordinate

(Inter-atomic spacing)

-258This assumption is formalized in the Born-Oppenheimer approximation, which is that the electron wave functions can be calculated
assuming that the nuclear positions are fixed.
Suppose we subject this mode to an oscillating electric field.
The transition probability per unit time might be calculated using the
approach of Fermi's Golden Rule No. 2 (Schiff, 1968).

This requires

knowledge of the dipole coupling between initial and final states separated by an energy equal to hcv, as well as the density of states near
the final state. There are two reasons why the mode is most easily
excited by energies corresponding to higher level vibrational states in
the excited electronic state.

The first is that the density of vibrational

and librational states increases with higher energy.

The second is

that the nuclear positions are effectively stationary over the time required to undergo the transition.

This means that transitions are ver-

tical in Figure Al.

This second approximation is known as the Frank-

Condon principle.

Since the equilibrium positions are usually not the

same, a vertical transition from the equilibrium position of the ground
state will terminate at a vibrationally excited state on the excited
electronic surface.
The mode we have just excited is coupled to many others around
it, both in the molecule and in the surrounding material. Excess vibrational energy can be coupled out of this mode, bringing it into thermal
equilibrium in the excited electronic state. This relaxation process
takes place very quickly (on the order of 10-

seconds). At this point

there will be some probability per unit time that the mode will spontaneously

emit

(Yariv, 1975).

The rate and energy of the

-259-

emission is determined by the same sorts of considerations· as for
absorption.

The final state after emission will most likely be an

excited vibrational state in the electronic ground state. Again vibrational relaxation will occur.

The whole process of excitation, relaxa-

tion, emission, and final relaxation is depicted by processes (a), (b),
(c), and (d) in Figure AI.

The differences between the energy of (a)

and (c) is the Stokes shift of the emission with respect to the excitation
energy.

The lifetime for singlet fluorescence is typically about a nano-

second in these molecules, and the lifetime for triplet phosphorescence
is about a millisecond.

The combination of both singlet and triplet

emission is termed luminescence.
Throughout the text we used the extinction coefficient to represent the absorption characteristics of a molecule.

The extinction coef-

ficient can be related in a simple way to the absorption cross section

a(v) = fn(10) • 1000 • E (iJ")

(A.l)

C. Nav

Similarly, the extinction coefficient is related to the imaginary part of
the atomic electronic susceptibility (Yariv, 1975).

The electronic

displacement vector is defined to be D = (e + e 0 x(il)) E, where E 0
is the electric permeability of vacuum, E is the dielectric constant
far from the absorption (so that the index of refraction is n = ...fElE;),
and x (iT) is the susceptibility. If we rewrite x (iJ) = x' (V) - i x"(v),

-260-

then the extinction coefficient can be written in terms of x"(V):

E (v)

= 11CvX"(V)
n 2 in (10) · C

(A. 2)

Other possible modes of de-excitation from the excited electronic
state are possible apart from radiation. Let T be the lifetime of the
excited state, Tr be the radiative lifetime, and 'iir be the nonradiative
lifetime:

l/T = 1/7r + 1/Tnr

(A. 3)

We can define the quantum efficiency of luminescence, TJ, to be

(A. 4)

In other words, 11 is the fraction of the excitations that will produce
emissions.

-261-

APPENDIX ill

TOTAL INTERNAL REFLECTION AND WAVEGUIDES

-262-

We normally assume that light arriving at the LSC-air interface is very far away from its point of emission compared to the
wavelength of the light.

The problem can, therefore, be simplified to

the case of a plane wave incident on a plane boundary.

Let the incid-

ent, reflected, and transmitted electric fields be E 0 exp (i K· x - iwt),
E~ exp(iK'::X-iwt), and E~ exp(iK"· x-iwt),

(Jackson, 1975).

respectively

Let the boundary between a material with index n

and a material with index n' occur at the z = 0 plane. If the phase
of the wave is constant everywhere on the boundary, then (K · x)

(K,. x) = (K "· x) for z = 0. This relation tells us that the three wave
vectors are co-planar, and that the index times the sine of the angle
of incidence (or exit) is a constant (Snell's law}. If the index of one of
the materials is 1, then we have that the criteria for total internal
flection is that sin (8) ~ 1/n, because B would be imaginary for the
transmitted wave.

There is a nonpropagating evanescent wave outside

of the trapped material in this case which decays exponentially.
There is an interesting case which perhaps should receive more
attention. What happens if the incident light cannot be considered a
plane wave? Specifically, suppose the LSC is a thin dielectric slab,
whose thickness is on the order of the wavelength of light. In this case
the photons no longer propagate as a series of straight lines between
reflections.

Instead the slab acts like a waveguide, and will support

TE and TM modes in certain wavelength regimes (Yariv, 1975). Only
TE 0 and TM 0 modes can propagate if the thickness is less than about
a third of a wavelength.

(The slab begins to act like the usual LSC

-263-

plate for thicknesses greater than about three wavelengths. ) There
are two prominent disadvantages with this geometry.

The layer would

have to have a dye concentration on the order of 1 molar, or in other
words be solid dye.
quenching.

This would probably lead to severe emission

The second problem is that devices with geometric gains

of 100 would only have a characteristic largest dimension of about 10
microns.

-264-

APPEND IX IV
BLACK BODY RADIATION

-265-

In Chapter 6 we use several results on the properties· of a photon gas in thermal equilibrium.
manner (Yariv, 1975).

These can be derived in the following

The allowed electromagnetic radiation modes

depend in general on the geometry of the cavity which contains them.
However, if the cavity size is large compared to the wavelength of the

radiation, then we can assume a cubic cavity with a volume L •

This

imposes the boundary condition that the fields be periodic in L.

For

example, the allowed wavenumbers in the x

direction will be k.x =

21Tm/L, for some interger m. Each mode, therefore, has a volume

(211 /L) .

The number of modes between 0 and k is the volume of a

sphere of radius k, times the number of modes per unit volume, times

2 (for the two possible polarization directions): Nm (ii) = 4 1Tk /3 ·
(L/211)

2 = k L /37T • Since k = 21fv, Nm (v) = 8JTil L /3.

The

density of states is the number of modes per wavenumber per unit
volume:

6(v)

-4
_2

= 87T v

(A. 5)

The average number of photons in any mode is given by the Bose distribution n(v) = [exp (hcv/nkT) - 1]. n is the index of refraction of
the medium.

The total number of photons, N (ii), and the energy in

the cavity, E (ii) , is given by

-266-

N (ii) d ii = L p ( v ) n (v ) d ii

= B71

_2
11

L d ZJ
ehvc/nkT _1

E(v)dil = N(il)dv · hcv

(A. 6)

(A. 7)

The energy emitted into a 21f solid angle by a surface of area A at

a temperature T is given by integrating E (ii) over all wavenumbers

and multiplying by c/4nL (Landau and Lifshitz, 1977):

E =

-. 00

A.~
4nL

E (ii) dv

= An T

2 7T k

15c h

= An Ta

(A. 8)

a is the Stefan-Boltzmann constant, which is most easily remembered
as 5. 67 x 10

watts per square meter per K .

Eqs. (70) and (74).

This is used in

The brightness of the emission, in photons per

wavenumber per second per unit area per solid angle, is given by

N (il)/L (Planck, 1959):

-267_2

B (v) ==

811 v

(A. 9)

e hci7/nKT - l

This can be rewritten as

hell== K .£n [l+ 87Tv

(A. 10)

In general, the energy in a thermodynamic system is related to the
other variables by (Goodstein, 1975)

(A. 11)

o E == To S - Po V + µo N

µ == 0 for an ideal gas.

The entropy change resulting from removing

a photon of wavenumber v from a photon gas with brightness B is

.6.S
.6.N

1 .6.E

811 v
= - K in [ 1 +

T .6.N

This result is used in Eq. (77).

(A. 12)

-268-

APPENDIX V

SOLAR CELLS FOR USE WITH LSCs

-269-

In the text we have treated the edge-mounted solar cells as devices which convert light above some absorption cutoff energy with a
constant energy efficiency.

This efficiency is taken to be character -

ized by the measured AMl efficiency of the cell.

We ignore the effect

of the spectral position of the LSC light output (as long as it is above
this cutoff energy), the intensity and the angular distribution of the
light, as well as parameters such as the cell temperature. In other
words we have treated the LSC as a lens-like system with low enough
light concentration so as not to heat the cell {and lower its efficiency)
or substantially increase the incident flux (and increase its efficiency).
These approximations require some additional justification.
Measured spectral responses for both p on n silicon cells and
Ga 1 -x Al x As - GaAs cells are typically flat within ± 10% from 6, OOOA
to 9, OOOA

(1. 5 to 2. 25 ev) (Hovel, 1975).

Since this is the most

probable spectral band for the light output from an LSC, it is a good
approximation to ignore the effect of the output wavelength on the cell
response.

The LSC plates are, in general, more sensitive to the blue

end of the spectrum than either type of cell. We have assumed that the
resulting increase in the conversion efficiency was ignorable due to
both the relatively low flux at these high energies, and to the low conversion efficiency produced by a low voltage output from a high energy
photon.
Increasing the concentration of the light incident on the cell increases the open circuit voltage, and, therefore, can increase the
efficiency.

The open circuit voltage is given by

-270-

= A 0 KT

fn [ sc + 1 ]

(A. 13)

T is the junction temperature, K is Boltzmann's constant, q is the
electronic charge, Isc is the short circuit current, and Ao and I 0
are device

parameters. I 0 is the dark or reverse saturation cur-

rent. A typical value of I 0 for a room temperature silicon cell is a
microampere per square centimeter. If the short circuit current is
increased by increasing the light on the cell the open circuit voltage
increases logarithmically. A typical value for the short circuit current
from a silicon cell under one sun is about 30 milliamps.

Therefore, a

cell mounted on an LSC with a flux gain of 10 might be expected to have
an increased open circuit voltage by about 20%.
The increased concentration usually also brings about an increase
in the cell temperature.

The cell temperature on the highest efficiency

LSC claimed to date was reported to be 60°C (Friedman, 1980).

Ther-

mally generated carriers cause a strong increase in the dark current
with temperature, which in turn lowers the output voltage.

Thus the

higher temperature of an LSC-mounted cell can cause a decrease in the
cell efficiency which usually more than compensates for the increase in
efficiency due to light concentration.

The IR absorption of PMMA is

problematic in this regard because the plate itself can get hotter when
exposed to the sun than a cell separate from an LSC.

Thermal contact

between the plate and the cell can, in this case, lower the cell's

-271-

performance.
We originally expected GaAs cells to perform better than silicon
cells when mounted on LSCs.

This is because there are no dyes to our

knowledge that will efficiently absorb and emit in the far red, so as to
take advantage of silicon's smaller band gap.

However, we measured

60% higher flux gains for a silicon cell than for a gallium arsenide cell
when interfaced with the fluid LSC. It should be emphasized that this
was one measurement made with one particular cell. Owens-Illinois
also reported better performance with silicon cells than with gallium
arsenide. It is possible that silicon cells might perform better with

LSCs than GaAs cells will. One reason might be that the silicon cells
can be etched to have a rougher surface.
transmission into the cell.

This is done to improve light

The gallium arsenide cells may not have

been treated in a similar way.

-272-

APPEND IX VI
THE BRIGHTNESS THEORE M

-273-

Born and Wolf derive the following inequality for the brightness
B1 in energy per area per 4 7T steradians produced by light from a
source with brightness B 0 passing through an arbitrary optical
apparatus:

(A. 14)

n 0 and n1 are the indices of refraction at the source and detector,
respectively.

This result is obtained by ray tracing techniques.

We note that the same result can be obtained for nonimaging
optics for the special case of black body radiators using the second
law of thermodynamics.

Suppose a spherical black body radiator at a

temperature T 0 is radiating in a medium with index n 0 •

Using

Eq. (A. 8), we find that the brightness of the sphere is B 0 = n;

Let this sphere communicate with a second sphere with a temperature
T1 in a medium with an index n1 •

The brightness of this second sphere

will be B1 = n1 T1 a • By the second law of thermodynamics, at
equilibrium we have that T 1 :::::; T 2 , or that B 1 :s (~ fB 0 •
no

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