Luminescent solar concentrators. 2: Experimental andtheoretical analysis of their possible efficiencies
J. S. Batchelder, A. H. Zewail, and T. Cole
Experimental techniques are developed to determine the applicability of a particular luminescing centerfor use in a luminescent solar concentrator (LSC). The relevant steady-state characteristics of eighteen
common organic laser dyes are given. The relative spectral homogeneity of such dyes are shown to depend
upon the surrounding material using narrowband laser excitation. We developed three independent tech-niques for measuring self-absorption rates; these are time-resolved emission, steady-state polarization an-isotropy, and spectral convolution. Preliminary dye degradation and prototype efficiency measurementsare included. Finally, we give simple relationships relating the efficiency and gain of an LSC to key spectro-scopic parameters of its constituents.
1. Introduction
The luminescent solar concentrator (LSC) offers thepromise of reducing the cost of photovoltaic energyconversion by the use of high gain concentrators whichdo not require tracking. The conceptual operation ofthe LSC is based on light pipe trapping of luminescenceinduced by the absorption of solar radiation. A trans-parent material, such as Plexiglas, is impregnated withguest luminescent absorbers such as organic dye mole-cules. Solar photons entering the upper face of theplate are absorbed, and photons are then emitted.Snell's law dictates that a large fraction of these lumi-nescent photons will be trapped by total internal re-flection; for example, -74% of an isotropic emission willbe trapped in a PMMA plate having an index of re-fraction of 1.49. Successive reflections transport theluminescent photons to the edge of the plate where theycan enter an edge-mounted array of solar cells.
We previously discussed in LSC-11 the primary issuesgoverning the applicability of LSC devices; these issuesare solar absorption bandwidth, self-absorption (or thereabsorption of emission from a particular dye or inor-ganic ion by another similar emitting center), andphotodegradation. Our primary concern in this paperis to demonstrate experimental techniques and appro-priate theoretical interpretations for determining theefficacy of a given dye in an LSC device. We will also
All authors are with California Institute of Technology, A. A. NoyesLaboratory of Chemical Physics, Pasadena, California 91125.
investigate the effects of the relative spectral homoge-neity, as determined by the type of substrate materialused.
This paper is organized as follows: Section II de-scribes spectroscopic techniques and results of mea-surements made on organic laser dyes. Such mea-surements include absorption, emission, and excitationspectra, relative spectral homogeneity in a variety ofhost materials, spatial filtering effects, and the polar-ization dependence of the output emission. We als3utilize transient spectroscopic methods to find the ob-served lifetime of dye excitations as a function of con-centration in the presence of self-absorption. Prelim-inary dye degradation data and prototype efficienciesare also described. Section III continues the treatmentof our previous paper (LSC-1), extending the analysisof the self-absorption phenomenon to model the ob-served spectral shifts, the rate of depolarization of theoutput emission, and the lifetime of the emission withconcentration. In Sec. IV we show how these resultsallow us to determine the efficiency of a single dye LSCplate as a function of the size or geometric gain of theplate. We conclude in Sec. V with our perceptions ofthe impact of the above developments on the potentialutility of the LSC.
II. Experimental Techniques and Results
A. Materials
We restricted our study of luminescent centers to theorganic laser dyes because of their high quantum effi-ciency of luminescence, their solubility in methylmethacrylate and organic solvents, and their readyavailability. The source of the dyes was ExcitonChemical Co.2 Dyes were used as received without
further purification. Liquid samples were made atknown concentrations by dissolving quantities of dyeweighed on a Cahn-25 electrobalance in reagent grademethanol. These samples were stored in soda-limeglass bottles and kept in darkness. The concentrationof the samples for absorption measurements was chosenso that the peak optical density of a 1-cm path lengthof the solution between 10,000 and 30,000 cm-' wasbetween 0.5 and 1.5.
Spectra were also taken of some of these dyes in avariety of other hosts. The principal solid matrix ma-terial was polymethyl methacrylate (PMMA), whichusually contained 5% hydroxy ethyl methacrylate byweight to increase the solubility of the more polar dyes.Large plates (1 m square X 0.3 cm thick) were com-mercially made to our specifications by Acrilex Inc.3
Smaller test samples were fabricated in our laboratoryin the following manner: Aldrich monomer, containinghydroquinone monomethyl ether as an inhibitor, waspurified by fractional distillation in a nitrogen atmo-sphere using a vacuum-jacketed vigreaux column. Thisdistilled monomer was combined with technical gradehydroxy ethyl methacrylate, and the desired dyes weredissolved therein. Small amounts of methanol and/oracetic acid were added to the solution to increase the dyesolubility if rhodamine dyes were to be used. Somesamples were prepared by adding the concentrateddye-monomer solution to prepolymerized PMMA andcontinuing the polymerization. However, best resultswere usually obtained by polymerizing just the mono-mer-dye solution. Two percent by weight of azobis-isobutyronitrile was added as an initiator, and themixture was poured into a mold formed by two glassplates. The plates were separated by a polyethylenetubing gasket and by aluminum spacers around theperiphery to maintain a constant plate thickness. Avery thin coat of silicon vacuum grease on the glassplates acted as mold release agent. These molds were
200.000
100,000
0X720CV670S640 c
KR620R6101R590R575R560C540C500C1460
10 . 000 WRVENUMBER(S (/CM) 30.000
Fig. 1. Composite of extinction coefficient spectra for a variety ofrepresentative organic laser dyes. Vertical axis is extinction coeffi-cient (liter/mole/cm), and horizontal axis is in wave numbers. All
spectra are from low concentration methanol solutions.
then immersed in a water bath with an oil surface filmand placed in a convection oven. Polymerization wasinitiated at -85 0C, when a noticeable increase in vis-cosity occurred, at which time the temperature waslowered to 550 for 48 h, followed by a final curing at 950for 4 h. Typically, significant fractions of the dye didnot go into solution, so that the dye concentrations inthe final plates were assayed by measuring the peakoptical density and assuming that the peak extinctioncoefficient was that of the methanol solutions. Wefound that dye concentrations in excess of 10 AM causedsignificant amounts of monomer to remain unpoly-merized in the cured plates in the case of rhodamine andoxazine dyes, and that this monomer could be slowlydriven out by vacuum degassing at 50°C. After curing,the plates were removed from the molds and werescribed and broken to size. The edges were polishedwith a sequence of grits; the final buffing compound wasa cerium oxide rouge.
We also developed an alternative technique of dopingthe plastic which has the great advantages of not re-quiring distillation, casting, or curing. If a commercialtransparent PMMA plate or rod is immersed in amethanol solvent containing the dye of interest, the dyewill diffuse into the plate along with the methanol. Asolution of 9% dichloromethane by volume in methanolwas found to be the best compromise between speed ofinfusion and maintaining good surface finish on theplate. The time required to achieve useful dye con-centrations in the plastic for a 20-cm (8-in.) rod was 15min, and for a 40-cm (16-in.) plate the time was .12 h,both at room temperature. Coumarin-540 infusedfaster than rhodamine-640, possibly due to the differ-ence in molecular weight. It appeared to the eye thatthe dyes typically resided in a film between 0.1 and 1mm from the surface of the plate, depending on tem-perature and soak time, so that the infusion techniquedoes not yield a uniform dye concentration across thethickness of the plate. It is unlikely that the dye con-centration in the film would be greater than that of thesoaking solution, so that the solution concentrationyields an upper bound on the local dye concentration.Measuring the peak optical density and assuming auniform dye dispersion in the plate gives a lower limiton the concentration in the region of the plate wheremost of the dye reside.
B. Steady-State SpectroscopyAbsorption spectra of all samples were made using
either a Cary-14 or a Cary-17 dual-beam spectropho-tometer. These spectra and their associated base lineswere digitized using a Houston Instruments Hipaddigitizer. The digitized spectra were corrected for baseline drift and stored on diskette using a Terak packagedPDP 11-03/2. Figure 1 shows a composite of a numberof these absorption spectra plotted by a High Plot dig-ital plotter on the same system. The digitization ac-curacy was five-hundredths of an inch, which is finerthan the pen line of the chart recorder output. Thebase line noise in the spectra is typically 1000 liters/mole/cm (SNR 100).
LAMP g OR W STEPPER MOTOR NTR 11-03/2SUPPLY LAMP CONTROLLER
CHOPPER
I LOCK-IN AMP D
MONOCHROMATOR R;
oSTEP PERl MOOR eRAHC
SAMPLE MONO-
CHROMATOR
H .V. |CHART l
Fig. 2. Schematic of the apparatus for emission and excitation
measurements. Regulated mercury or tungsten source was focused
down, chopped, and monochromated prior to exciting the sample.
Resulting emission was monochromated and detected by a PMT, and
the signal was amplified with phase-sensitive detection. Remote
computer controlled both monochromators and recorded the mea-sured spectra.
Emission and excitation spectra of the dyes weremade at micromolar concentrations using a computer-controlled apparatus, as shown in Fig. 2. The excitationsource was either a 200-W Oriel 6323 tungsten lamp ora 200-W Oriel 6137 high pressure mercury lamp. Thelight was collimated with quartz optics, chopped by aPAR 191 chopper, and monochromated by a Jarrell-Ash(model 82-410, f/3.5, 0.25-m) monochromator with ei-ther a 6000-A blazed grating with 1180 grooves/mm ora 3000-A blazed grating with 2365 grooves/mm. For thephase sensitive detection to produce accurate excitationand emission spectra when chopping the excitationsource, the lifetimes of the dyes must be short comparedwith the chopping period (2.5 msec in these experi-ments). This condition was fulfilled in all these mea-surements. The excitation linewidth was fixed at 90 Aacross the tuning range.
The emission 90° from the excitation was analyzedby a similar Jarrell-Ash monochromator, in this casewith an adjustable resolution of 2-90 A. The outputlight was detected by a 928 Hamamatsu PMT biased at900 V; the PMT output was terminated by 100k Qparallel with a PAR-HR8 lock-in amplifier. The analogoutput of the lock-in was digitized recorded by the PDP11-03/2. Care was taken to keep the PMT current atleast 20% below its rated output current of 100,glA. TheRC time constant of the lock-in was kept at least asshort as the time between digital sampling of theemission. These samples were taken typically every 10A. Both monochromators were driven by a Slo-SynSP151 driver so that either one or both monochromatorscould scan under the control of the PDP 11-03/2. Thisallowed acquiring spectra in cm-1 even though themonochromators had wavelength drives. Computercontrol also provided automatic backlash correction.Frequency calibration of the system was done with the
5451-A line from a mercury germicidal lamp, and thesystem response was measured using an Eppley Labo-ratory calibrated EPI-1669 quartz halogen lamp in acustom-made housing. The output of the HR-8 wasdigitized in a 12-bit analog-to-digital converter builtaround the Intersil ICL-7107. This generated a paralleldigital input to the remote PDP 11-03/2. The systemresponse function was stored on diskette; this was de-termined to be the response of the system to the cali-brated tungsten lamp divided by the known spectrumof the tungsten lamp. Emission spectra were correctedby dividing out this system response and were normal-ized so that the luminescence integrated over all wavenumbers equals 1. The peak position of the emissionspectra for the dyes tested is given in Table I. Cor-recting the excitation spectra for the response of thesystem requires a calibrated detector as well as a cali-brated source. Using both it is possible to determinethe response of the monochromator that is scanning theexcitation. We made the approximation that our PMTresponse.was flat over the region of interest. In thisapproximation, the variation of the intensity of thescanning excitation is the measured response to thecalibrated tungsten lamp. Excitation spectra werecorrected by dividing out this measured response.
Both the emission and excitation spectra of these dyeswere taken only at micromolar dye concentration tominimize the red-shifting effect of self-absorption. Theoptical density of the sample in the region where theabsorption and emission overlap must be sufficientlylow that the blue tail of the emission spectrum is notartificially filtered out. The excitation samples mustbe dilute for a second reason. Away from the peak ofthe absorption, where the absorption per unit length issmall, the exciting beam intensity does not vary greatlyalong the beam length in the sample. However, as theexcitation is scanned across the peak of the absorption,most of the exciting light is absorbed in a thin surfacelayer in the sample if the concentration is too high.This effect changes the spatial distribution of theemission and alters the observed emission spectrum.
Measurements were also made on the spectral char-acteristics of multiple dye solutions. Dyes could not becombined indiscriminately, possibly due to agglomer-ation phenomenon between some dyes which quenchedthe emission when they were in solution together. Forexample, our oxazine-720 or oxazine-750 solutionsshowed quenching of their emission when they were ina solution with rhodamine-610 or rhodamine-590. Anexample of a successful multiple dye combination issulforhodamine-640, rhodamine-590, and coumarin-540. In Figures 3 and 4 we show emission and excita-tion spectra, respectively, for an oxazine-720 methanolsolution and for an oxazine-720-rhodamine-640-cou-marin-540 methanol mixture at micromolar and hun-dred micromolar concentrations. The emission spectraof the multiple-dye solution at low concentration aredominated by rhodamine and coumarine emission,while at high concentration the oxazine dominates theemission. If we detect the emission intensity of oxazine,as shown in Fig. 4, as a function of excitation energy, the
RG. A. Reynolds, K. H. Drexhage, Optics Comm., 13, No. 3, 222 (1975).bP. R. Hammond, Optics Comm. , 29, No. 3, 331 (1979).
CK. H. Drexhage, "Structure and Properties of Laser Dyes, "ed. F. P. Schafer, Topics in Dye Lasers, Applied Physics I (SpringerNew York, 1977) p. 144.
dA. Baczynski, T. Marszalek, H. Walerys, B. Zietek, Acta Phys. Polo., A44 , No. 6, 805 (1973).
eT. Karstens, K. Kobs, J. Phys. Chem., 84, No. 14, 1871 (1980).1D. Magde, J. H. Brannon, T. L. Cremers, J. Olmsted I, J. Phys. Chem., 83, No. 6, 696 (1979).
J. M. Drake, R. T. Morse, R. N. Steppl, D. Young, Chem. Phys. Lett., 35, No. 2, 181 (1975).
hC. F. Rapp et al., Final Report of Owens Illinois, Sand77-7005, p. 40.
Fig. 3. Three emission spectra of methanol dye solutions resultingfrom 22,220-wave number (4500-A) excitation. Top spectrum is froma micromolar oxazine-720 solution, and the bottom two are micro-molar and hundred micromolar solutions, respectively, of couma-
rin-540, rhodamine-640, and oxazine-720.
ONE DYE TI I I LOW CONC.
3 DYES T
,{A ,HIGH, COIGH CONC.
3 DYES ATI , ,J , , ~ LOW CONC.
13,000 18,000 23,000
WRVENUMBERS
Fig. 4. Three excitation spectra of methanol dye solutions due to15,630-wave number (6400-A) detection. Three spectra correspond
to the same three solutions described in Fig. 3.
Dewar
Fig. 5. Diagram of the apparatus used to measure the luminescencespectra of liquid, cast plastic, and diffused plastic samples as a func-tion of excitation energy and temperature. Either the argon-ion laseror the tunable dye laser was used as the excitation source. Apparentsample temperature was maintained at either 77 or 300K. Resulting
emission was analyzed by a Spex 14018 double monochtomator.
HN C2 H5 ,#
I
C3
\ /O K, H C N 2H5
1 I I I
K~ ~~ "_. .; ~C/
l I I
Fig. 6. Structure of rhodamine-575.
low concentration multiple dye solution is dominatedby the pure oxazine absorption peak, while the highconcentration multiple dye solution gives a nearlyconstant response across the visible spectrum.
C. Spectral InhomogeneityA primary objective of our work was determination
of the homogeneity of the absorption and emissionspectra in solution, in cast plastic, and in diffused plasticand to determine the temperature dependence of theanti-Stokes shifted emission. We shall call this thespectral inhomogeneity of the dye in these various cases.To achieve these objectives the luminescence spectrawere taken using the apparatus shown in Fig. 5. ASpectra-Physics 160 argon-ion laser was used directlyas the exciting source or served as a pump for a Spec-tra-Physics 375A dye laser. The dye laser operatedwith a glycol solution of rhodamine-590. When weexcited the sample at an energy considerably below itspeak absorption, it was important that very littlespontaneous emission of a higher energy than theprincipal lasing wavelength be present in the sampleexcitation beam. The dye laser was therefore followedby several sharp cut filters, a dispersive prism, and a slitaperture. Neutral density filters were used to adjustthe input power to the sample. The samples were heldin a glass Dewar with windows positioned for both theexcitation beam and for the resulting right-angleemission. The luminescence spectra were taken bothat room temperature and with the samples immersedin liquid nitrogen. Since the excitation was notchopped, it was important to eliminate stray room lightas a source of excitation. The luminescence resultingfrom the laser excitation was collected by a Spex 1419Asample illuminator and then analyzed by a Spex 14018double monochromator with 2400-groove/mm holo-graphic gratings. The sample illuminator was a col-lection optics stage, originally designed for Ramanspectroscopy, which contained the dispersive prism, slit,sample translation stage, and matched f/No. optics forthe spectrometer. A polarizing filter in the collectionoptics was used to suppress scattered laser light, fol-lowed by a depolarizing prism to compensate for thepolarization response of the gratings. The spectrallyresolved output was measured by a Hamamatsu 955photomultiplier and a Spex DPC-2 photon counter.The final chart recorder output was digitized and storedin the PDP 11-03/2. System response correction waseffected by digitizing the measured spectrum of a cali-brated tungsten lamp, as described in the previoussection.
Rhodamine-575 was chosen for study because it canbe excited on the extreme red edge of its absorptionband by the rhodamine-590 dye laser. The structureof rhodamine-575 shown in Fig. 6 is also very similar toother xanthene dyes, so that the results of these mea-surements should be representative of most of that classof laser dyes. The liquid sample studied in this mannerwas 18-gM rhodamine-575 in methanol. This wasmounted in a 1.3-mm o.d. capillary tube, oriented so asto minimize the path length of the output emission.
Fig. 7. Emission spectra of rhodamine-575. Lower three plots showthe room temperature emission of rhodamine-575 in methanol dueto excitation at 16,509, 17,164, and 20,492 wave numbers (6075,5826,and 4880 A, respectively). Upper three plots show similar emissionspectra taken at 77 K using excitations at 16,877, 17,163, and 20,492wave numbers (5925, 5826, and 4880 A, respectively). Room tem-perature spectra show that the dye emits with essentially the same
spectrum for all the excitation energies.
When the methanol is cooled to liquid nitrogen tem-peratures it forms a fractured glass. The cast plasticsample was a 20- X 25-mm PMMA rod containing aneffective concentration of 14-buM rhodamine-575. Thediffused sample was a 1.6-mm diam rod of Rohm &Haas Plexiglas that had been soaked for 4 h in a 100-,gMsolution of rhodamine-575 in a mixture of 0.91 methanoland 0.09 dichloromethane by volume.
The results from the liquid sample are shown in Fig.7. The 200-cm-1 wide notches in the spectra are dueto a shutter which blocks the photomultiplier tube asthe spectrometer is scanning across the laser frequency.The lower three spectra correspond to the room tem-perature emission for laser excitations at 16,509, 17,164,and 20,492 cm-' (6057, 5826, and 4880 A, respectively).Note that the peak of the emission for the lowest energyexcitation is at the same frequency as that of the highestenergy excitation, despite the fact that some of theemitted photons were blue-shifted as much as 1700cm-' up from the excitation frequency. This anti-Stokes shift in energy is >5 kT for room temperaturespectra. Introducing sharp-cut filters into the excita-tion beam centered at the lasing wavelength caused areduction in the overall emission intensity in proportionto the reduction in the excitation intensity but causedno shift in the observed spectral shape. We concludedthat these emission spectra did not include stray exci-tation light of higher energy. The upper plot showsemission spectra at 77 K due to excitations at 16,877,17,163, and 20,492 cm-' (5925, 5826, and 4880 A, re-spectively). The luminescence spectrum due to thehighest energy excitation is a slightly narrower versionof the room temperature spectrum. The low energyexcitation at low temperature produces negligibleemission at higher energies, which agrees with the pre-vious result that the emission at room temperature from
Fig. 8. Emission spectra of rhodamine-575. Lower three plots showthe luminescence of rhodamine-575 cast in PMMA at room temper-ature for excitations at 16,273, 17,060, 19,436, and 20,492 wavenumbers (6145, 5826, 6145, and 4880 A, respectively). Notice thepronounced skewing of the emission toward lower energies by the lowenergy excitations. Upper plot shows luminescence at 77 K for ex-citations at 17,423, 19,436, and 20,492 wave numbers (5740,5145, and
4880 A, respectively).
low energy excitation was due to anti-Stokes shifting ofthe excitation.
Figure 8 gives corresponding results for rhodamine-575 cast in PMMA. The lower plot shows the roomtemperature luminescence spectra normalized to unitarea. These spectra differ from the methanol solutionsin that the emissions produced by low energy excitationsat 16,273 and 17,060 cm-' (6145 and 5862 A) are greatlyskewed toward the red, whereas the higher energy ex-citations as 19,436 and 20,492 cm-' (5145 and 4880 A)produce emission with the same general shape and po-sition as the methanol room temperature emission.The upper plot shows the low temperature emission dueto excitations at 17,423, 19,436, and 20,948 cm-' (5740,5145, and 4880 A, respectively). As in the liquid sam-ples, the anti-Stokes shifted emission is suppressed at77 K. However, the emission shape is still dependenton the position of the excitation.
The diffused plastic sample shows anti-Stokes be-havior intermediate between the cast plastic and theliquid solution. Figure 9 shows the room temperature
SURFRCE IFFUSED DYE
300 KELVIN
EXCIT.' 16116. 171449. -191136.
204192.
1.39 1.69 1Q
WVENUMBERS
Fig. 9. Emission of rhodamine-575 diffused into PMMA. Emissionspectra were taken at room temperature. Excitation energies were16,116, 17,449, 19,436, and 20,492 wave numbers (6205, 5731, 5145,
Fig. 10. Apparatus for measuring the intensity and spectral shiftsof the sample emission as a function of the path length through thesample traveled by the emission. The 22,940-wave number (4359-A)line from a mercury lamp was monochromated, chopped, and focusedonto an optical fiber. The opposite end of the fiber was scannedhorizontally with a micrometer stage along the rod-shaped sampleof LSC material. Output was detected by a scanning mono-
chromator.
LU
1.32 1.72 2.12E 4
WIVENUMBERS
Fig. 11. Emission spectra sequence from a 92-,uM methanol solutionof rhodamine-575. Sample path lengths for each spectrum in orderof the most to least intense were 0.3, 1.0, 3.0, 10.0, and 30.0 cm.Spectra have been corrected for the system response of the detectionmonochromator and the PMT, so that the amplitude of each spectrum
corresponds to the intensities emerging from the end of the rod.
Az
Sample 0g11 verO
I exc / A
x
Fig. 12. Principal direction for polarization anisotropy measure-ments. Monochromated light excites rhodamine-575 in an ethyleneglycol solution, and the resulting emission is detected 900 to the axisof excitation. Excitation is polarized with its electric field eitherhorizontal or vertical, and the emission is similarly polarized hori-
zontally or vertically (perpendicular or parallel, respectively).
emission of a PMMA sample soaked in a methanol so-lution of rhodamine-575. The excitation positions are16,116, 17,449, 19,436, and 20,492 cm-1 (6205, 5731,5145, and 4880 A, respectively). The emission spectrahave been normalized to unit area. While there is somevariation of the peak position with excitation energy,we find that the spectra are dominated by large anti-Stokes shifts for low excitation energies, as was the casefor the methanol solutions.
D. Spatial Filtering Effects
We measured the changes in the intensity and spatialcharacteristics of dye emission by scanning the positionof a small excitation spot with respect to the sample andthen observing the resulting emission from a fixed pointon the sample. Preliminary results from such an ex-periment were given in LSC-1. Figure 10 depicts theapparatus used. The 22,940-cm-1 (4359-A) radiationfrom a 200-W Oriel 6137 high pressure mercury lampwas filtered by a Bausch & Lomb 33-86-25 monochro-mator, modulated with a PAR-BZ-1 chopper, and fo-cused by a X40 microscope objective onto one end of a1-m section of a Math Associates OC-1200 glass opticalfiber bundle. Since we were interested in the changeof intensity as well as spectral shape of the emission withincreasing path length through the sample, it was im-portant that the position of the excitation source at thesample be adjustable without any variation in intensity.Small changes in the curvature of an optical fiber causelittle change in its transmission, so that the free end ofthe fiber was used as the mobile excitation. The freeend was mounted on a micrometer stage that wouldtrack parallel to the rod-shaped sample of LSC material,as shown in Fig. 10. Three different types of sampleswere tested: methanol solutions; cast PMMA; anddiffused PMMA. Samples were mounted inside 7-mmo.d. (5-mm i.d.) borosilicate glass tubes to maintain auniform contact between the excitation fiber and thesample. The end of the tube adjacent to the detectingmonochromator was sealed with a 1-cm glass plug. Thedetection system was a 0.25-m Jarrell-Ash spectrometer,a Hamamatsu 928 PMT, and a PAR HR-8 lock-in am-plifier. The detection monochromator was scanned,and the output of the lock-in amplifier was sampled bythe PDP 11-03/2, as described in the previous sec-tion.
The results from one such measurement are shownin Fig. 11. In this case the borosillicate glass tube wasfilled with a 92-gM solution of rhodamine-575 inmethanol. Spectra were taken for the following dis-tances between the fiber excitation and the glass plugseal at the end of the tube: 0.3; 1.0; 3.0; 10.0; and 30.0cm. System response corrections were made to com-pensate for the detection monochromator and PMT, asdescribed in the previous section. Apart from thiscorrection, the amplitudes of each spectrum correspondto the actual intensities emerging from the end of thesample rod. The predominant observed effect was thatthe higher energy portion of the band gradually disap-pears.
E. Steady-State Polarization MeasurementsWe measured the relative polarization of emission
from a polarized excitation. Light from a tungstenlamp was monochromated at 20,200 cm-' (4950 A),chopped, and passed through a polarizer (PolaroidHN38). This light was focused onto an ethylene glycolsolution of rhodamine-575. The concentration wasvaried during the course of the experiment by changingthe dye solution. The resulting emission 900 from theaxis of the excitation was passed through a second po-larizer, and the intensity at 17,860 cm-' (5600 A) wasmeasured with a second monochromator, a PMT, anda lock-in amplifier. The lamp, the monochromators,the chopper, the PMT, and the lock-in have been de-scribed previously. Figure 12 shows a schematic of thesample cell with the various combinations of light po-larizations of the excitation and the emission.
Ethylene glycol was chosen because it has a viscosityof -20 cP at room temperature and should produce arotational diffusion time of the dye long enough thatemission from a polarized excitation would also bepartially polarized. If this is the case, emission polar-ized in the Z direction should be more intense thanemission polarized in the X direction for a Z-polarizedexcitation (Fig. 12). Y-polarized excitation shouldproduce equal intensities for both polarizations ofemission.4 The expected polarization intensities arediscussed in the theory section and in Appendix B. Dyesolutions were used in place of cast plates to facilitatechanging the dye concentrations.
We varied the dye concentration of the sample andmeasured the intensity of the Z- and X-polarizedemissions at each concentration. The sample dyeconcentrations were varied from 0.2 to 100 gM by fac-tors of 2. The path length in the sample traversed bythe emission en route to the detection monochromatorwas 0.7 cm. The observed intensities were corrected forbackground and the polarization response of thesystem.
Figure 13 shows the variation between the Z- andX-polarized emissions for Z-polarized excitation as afunction of concentration. The plotted value is thepolarization anisotropy, which is the emission intensityof the Z minus the X-polarized emission intensities,both divided by the sum of the Z plus two times theX-polarized emission intensities. For very low con-centrations, the anisotropy plateaus at -0.18 orequivalently the ratio of the parallel to the perpendic-ular emissions are -1.65. As the concentration in-creases, the apparent polarization of the emission de-creases. The error bars reflect variations of repetitivemeasurements at a single concentration.
F. Transient Emission Measurements
We measured the time evolution of the emission in-tensity resulting from very short pulse excitation of dyesin a variety of hosts. Time-resolved fluorescencemeasurements were performed with a mode-lockedargon-ion laser (Spectra-Physics model 171/342) anda single photon counting apparatus described recentlyby Robbilns et al.6 of this laboratory. Samples were
2.00E -1
a_cc 1.500U,
Z 1.00DuJL)J
::)0.50uJ
0.00I10-7 10-6 10-5 10-4 10-3
CONC. X PTHLENGTH
Fig. 13. Reduced polarization anisotropy vs sample concentration.Polarization anisotropy of the emission from rhodamine-575 in eth-ylene glycol as a function of concentration for parallel polarized ex-citation is plotted. Reduced polarization anisotropy is defined to bethe difference between the parallel and perpendicular emission in-tensities divided by the sum of the parallel and twice the perpendic-
ular intensities. Solid line connects data points.
SYSTEM RESPONSE
U,z ,\ . , , I
UN-MASKED EMISSION
-10. -5. 0. 5. 10. 15. 20. 25.
NANOSECONDS
Fig. 14. System response and transient lifetime measurement.Upper plot shows a typical histogram of the system response to a5145-A pulse from a mode-locked argon-ion laser with a duration of<200 psec scattered off of a dilute coffee creamer solution. Widthand symmetry of the response were predominantly due to the PMTresponse. Lower plot shows the observed emission of a 4.6-,uM rho-damine-575 methanol solution superimposed on a best fit single ex-ponential convoluted with the system response. Exponential fit gives
a lifetime of 4.1 nsec.
illuminated with vertically polarized 19,436-cm-1(5145-A) light pulses at a repetition rate of -100 kHzwith a typical pulse halfwidth of <200 psec. Emissionfrom the sample was monitored by a Philips XP2020Qphotomultiplier. Cutoff filters at 5000 and 5700 A toeliminate scattered laser light, plus an analyzing po-larizer to sample different emission polarization in-tensities, were placed between the sample and the de-tector when luminescent samples were measured. Thesystem response, measured by laser light scattered froma dilute coffee creamer solution sample, is shown in theupper plot in Fig. 14 and usually had a response time of<250 psec.
I . .I J - , . 1- . . . ., 1 I . 1 . . II -. 10 o8
6.00E 0
5.00
LoD 4.00z
I 3.00
0 2.00z1.00
0.00
1
.3 CM PTHLENGTH IN
19,440 WVENUMBER EXC
. . .. ,,1 , ... I. -J , . . .10-7 10-6
CONC. X PTHLEN
Fig. 15. Measured lifetime of rhodanfunction of dye concentration is plotted.
data points.
Filterand
Polarizer
Mask
C uvette
The first type of measurement we made was to findthe total lifetime of the dye as a function of concentra-tion. In this case the dye emission was filtered by apolarizer at 54.70 from the laser's original vertical po-larization to average between the emission of the par-allel and perpendicular polarization components. Thelower plot in Fig. 14 shows typical transient emission
SPMPLE data for a 4.6-IuM solution of rhodamine-575 in meth-anol. The dye was contained in a 1.0- X 0.5-cm cuvette,
:ITPT ION ~ so that on average this emission passed through 0.5 cml1. I l l i,. . of the sample before arriving at the collection optics.
-5 -4 -3 The extinction coefficient for rhodamine-575 at 514510H 10 10 A was -90,000 moles/cm liter, so that, even at this lowIGTH concentration, -40% of the laser excitation was ab-
sorbed by the sample. Superimposed on the raw datanine-575 in methanol as a in Fig. 14 is a best fit convolution of the system response
in the upper plot and a single exponential. The bestexponential fit gave an observed lifetime of 4.13 nsec atthis concentration.
We measured apparent lifetimes for a sequence of dyeconcentrations from 0.2 to 100,uM for rhodamine-575in both methanol and ethylene glycol. Figure 15 showsthe fitted lifetimes as a function of concentration inmethanol. The highest concentration used was a factorof 10 below the critical concentration for nonradiativetransfer among similar dye molecules as defined byFbrster. At low concentrations, the lifetime of rhoda-mine-575 approached an asymptotic value of 3.7 nsec,while at high concentrations it increased to three timesits low concentration value. The error bars indicate theuncertainty of the lifetime in the numerical fitting of thedata. As in the previous case, the emission was filteredwith a polarizer 54.7° from the vertical to sampleequally the parallel and perpendicular polarizations.The emission traveled through -0.5 cm of sample beforebeing detected, and care was taken not to mask anyportion of the emission from the cell from illuminatingthe PMT.
' 200PicosecondLaser Pulse5145 A
Fig. 16. Position of the first generation emission mask. The 5145-Alaser excitation is 99% absorbed in the first 0.05 cm of the 460-,4Mrhodamine-575 methanol solution. Blocking just the leftmost portionof the cell, as shown, prevents the emission due to the initial excitation
from being directly detected.
To help determine the mechanism for the lifetimelengthening of the dye with increasing concentration,we performed a measurement with part of the samplecuvette blocked so that it could not illuminate the de-tector. Figure 16 shows a schematic of the sample ge-
MRSKED / ometry with respect to the excitating laser and theoutput emission. The sample used in this case wasrhodamine-575 in methanol at 460 ,uM. At this con-centration 99% of the excitation was absorbed in thefirst 0.05 cm of the sample. We were not interested in
UN-MRSKED measuring the emission resulting directly from thisexcitation but in the emission that. followed the self-absorption of this first emission. The mask shown in
-10. -5. 0. 5. 10. 15. 20. 25. Fig. 16 achieved this condition. The transient emission
NANOSECONDS spectrum resulting from this configuration is the upperplot in Fig. 17. When the mask was removed, we ob-
Unmasked and masked transient emission measurements. tained the lower plot shown in Fig. 17. The solid curvesplot shows the transient emission spectrum from 460-liM superimposed on these two spectra will be discussed in
mine-575 methanol solution for the case where a mask is posi- the theoretical section. It was evident that there werebetween the region of the sample excited by the laser and de- significant distortions of the single exponential decay
When the mask is removed, we obtain the spectrum shown when the initially excited portion of the sample wasin the lower half of the figure. masked.
G. Dye Degradation Measurements: PreliminaryResults
We have previously reported results for apparentdegradation rates of rhodamine-575 and coumarin-540in PMMA under accelerated and simulated solar ex-posure in LSC-1. We were interested in observing ratesof degradation under actual solar exposure. Subse-quently we have performed several experiments topursue the phenomenon further.
The first sample tested was a 76-rim rhodamine-590PMMA plate 3 mm thick. Acetic acid was added to themonomeric solution prior to polymerization to increasethe solubility of the dye. A 2-week continuous exposure(336 total h) caused a decrease in the peak opticaldensity from 3.98 to 2.79, as measured on a Cary-14spectrophotometer. From the change in peak o.d. weestimate that the bleaching rate is -10-6 molecules/photon. This is typical of photodegradation ratesmeasured under different circumstances for similardyes. 6
Multiple dye plates were manufactured to our spec-ifications. A two-dye plate composed of 140-juM cou-marin-540 and 97-AIM rhodamine-590 in PMMA wasexposed for 1 yr (we estimate -1000 h of direct sunlight)to the Michigan sun filtered by a plate glass window.We measured the emission of this exposed plate by il-luminating the sample with a dispersed tungsten lampsource. The spectral distribution of the tungsten lampclosely approximated that of a 3000 K blackbody. Asilicon cell was brought into optical contact with theedge using ethylene glycol as an index matching fluid.The output short circuit current on the cell was mea-sured directly with an HP-3466A digital multimeter.The emission of the plate exposed behind plate glasshad dropped -30% compared with a similar sample thathad been stored in the dark. The peak optical densityof the rhodamine peak had also been reduced from 2.7to 2.2 during exposure or a drop of 20%, while the cou-marin peak had been reduced from 2.3 to 1.7 or a dropof -30%. A section of the same unexposed plate wasexposed to direct sunlight for 243 total h, with a re-sulting drop in peak emission of -30% and of the peako.d. of the coumarin peak of -20%. It should be em-phasized that these are results from a particular LSCsample.
A PMMA plate containing 210-gm coumarin-540,81-,uM rhodamine-590, and 22-/iM sulforhodamine-640was also exposed to direct insolation for 243 continuoush. The total emission from the plate decreased 50%, asmeasured using the above technique. The peak o.d. ofthe three dyes decreased 52, 50, and 51%, respectively.Additional samples of this plate material were exposedunder the same conditions. However, they were pro-tected by either a Schott GG400 UV cutoff filter or byan 20-cm (8-in.) square soda-lime glass cover. Both ofthese protected samples were also sealed against theoutside atmosphere with polyvinylchloride electricaltape. Figure 18 shows a plot of the absorption spectraof the three samples described plus an unexposed con-trol. While the soda-lime glass and the 4000-A cutofffilter absorb more than 90% of the UV radiation content
L4 .00E
)
-jH-
I~
2.00
2.10 3.00E 4
WRVENUMBERS
Fig. 18. Solar bleaching of a multiple-dye LSC plate. These are theabsorption spectra for a PMMA plate containing 210-, 81-, and 22-,uMconcentrations of coumarin-540, rhodamine-590, and sulforhodam-ine-640, respectively. Four spectra correspond to 243 continuous hof exposure through a 4000-A cutoff filter, a soda-lime glass plate with
no filter, and a control which was unexposed.
in sunlight, they had only a moderate effect at slowingthe rate of photodegradation.
We made single-dye plates containing coumarin-460,500, and 540, rhodamine-590 and 610, and oxazine-725in PMMA. After -250 h of continuous exposure, theemission had dropped 13% for coumarin-540, which wasthe best case, and the emission had dropped 66% foroxazine-725, which was the worst case, with the otherdyes falling between these two limits. If the emissionis assumed to decrease in a roughly exponential manner,these correspond to half-lives ranging from 300 to 3000h (12-120 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 pre-pared in soda-lime glass vials 2.5 cm in diameter.Oxygen was removed from the samples by bubblingnitrogen gas through the solution for 1 h in a nitrogenatmosphere. The screw-on caps were sealed with RTVsilicon rubber sealant. The optical density of thesamples was measured every few days without dis-turbing the seals. Figure 19 shows the measured peakoptical densities of these solutions as a function of totalexposure time, which included hours of darkness. Themeasured optical densities typically did not decay in anexponential manner, but instead they decayed at ratesthat usually accelerated with time. The maximumexposure time was 432 h (18 days). This was insuffi-cient time to obtain accurate decay rates for the longerlifetime dyes. Using an exponential as a rough ap-proximation to the time-dependent optical density, thelifetime for the solutions ranged from 50 h for rhoda-mine-640 to -10,000 h (1 yr) for rhodamine-590.Among the relatively stable dyes were coumarin-540(3000 h), oxazine-720 (2000 h), and cresyl violet (7000h).
Finally we measured photodegradation rates for fiveof the most stable dyes in a degassed environment.Methanol solutions of coumarin-540, rhodamine-590,sulforhodamine-640, cresyl violet-670, and oxazine-750
Fig. 19. Peak optical densities of methanol dye solutions undercontinuous solar exposure: 2.5-cm i.d, soda-lime glass bottles werefilled with solutions of oxazine-720 perchlorate, LD-700 perchlorate,cresyl violet-670 perchlorate, rhodamine-590 perchlorate, couma-rin-540, coumarin-535, or coumarin-500. Screw-on caps were sealedwith RTV. Absorption spectra were taken without disturbing the
seals.
Table II. Performance Parameters of LSCs
AssumedGeometric FluX Cell Collector
Device No. Dyes Matrix Gain Gain Eff. Eff.
A 2 PMMA 23. 2.1 18% 1. 9%
Bt thin film 11. 1.3 21% 2. 5%
C 3 Glycol 36. 3.8 18% 1. 9%
Dt 3 thin film 11. 1. 7 21% 3. 2%
E 1 PMMA 68. 5.1 18% 1. 3%
Present work.
'P. S. Friedman, LSC Contract Report, Owen-Illinois, SERI Contract XS-9-82161, 1980.
were placed in quartz cuvettes with outer dimensionsof 1.3 X 1.3 X 6 cm. The concentrations of the solutionswere adjusted so that the peak optical density across thecuvette was between 0.5 and 1. These samples weredegassed by six freeze-pump-thaw cycles on a vacuummanifold and then flame sealed. The resulting lifetimeswere 3.3, 10., 36., 0.3, and 1.0 direct sunlight h, respec-tively. The lifetimes could have been affected by thehigh UV transmission of the quartz cuvettes.
H. Prototype Efficiency Measurements
A variety of LSC devices have been tested by our-selves and others. Table II is intended to be a repre-sentative list of typical performance parameters. Thegeometric gain is again the ratio of the area exposed tothe sun to the active area of the edge. The flux gain isthe factor by which the short circuit current increaseswhen attached to the plate, as opposed to facing .the sundirectly. The cell efficiency is the measured or assumedAMI efficiency of the solar cells used (which in all cases
were silicon). The collector efficiency is the totalelectrical power out divided by the total sunlight powerincident on the plate.
In Table II devices B and D were built and tested byOwens-Illinois. The dyes were contained in thin plasticfilms attached to the surface of a clear substrate.Measurements were made under actual insolation, withthe plate edges roughened and blacked where cells werenot mounted. These plates have achieved the highestefficiencies, but their small geometric gains make themsomewhat ineffective as concentrators. For example,cells mounted on LSC device D will have only a 70%increase in output over an equal area of cells facing thesun directly.
Device C was constructed by glueing two 110 X 110X 0.3-cm Plexiglas plates on opposing sides of 0.08-cmPlexiglas spacers. The resulting gap between the plateswas filled with a glycol solution containing 5.7 X 10-5,9.4 X 10-5, and 5.1 X 10-5 M of coumarin-540, rhoda-mine-590, and rhodamine-640, respectively. The edgesof the sandwich assembly were blacked with electri-cian's tape except for the portion coupled to the cell.The flux gain was measured under actual insolation byfirst measuring the short circuit current on a standardcell under direct insolation and then coupling the samecell to the plate, using a glycol film for optical matchingand measuring the new short circuit current. We cal-culate that this dye combination will absorb -30% of anAMO spectrum in a two-pass geometry. The collectorefficiency given in Table II is the cell AMI efficiencytimes the flux gain divided by the geometric gain. Thisassumes a similar efficiency of the solar cell with andwithout the LSC (see LSC-1).
Device E was a 120- X 100- X 0.4-cm PMMA platecontaining 2.2- X 10- 4 -M DCM (see later sections).The edges were taped as before except where the cellswere mounted. The measurements were made usingtwo cells with AMI efficiencies matched to within 5%and both with an active area of 0.4 X 2 cm. One cell wasmounted coplanar to the LSC surface as a reference, theother was contacted with the LSC edge using glycol asan index-matching fluid. The tests were made usinga xenon flashlamp AMO simulator at JPL. The fluxgain was determined to be the ratio of the peak powerof the LSC-mounted cell and the reference cell, cor-rected for flash-to-flash variations. We calculate thatthis plate should absorb 30% of an incident AMOspectrum. The collector efficiency was determined inthe same manner as for device C.
DCM and its possible variations appear to be apromising LSC dye due to its broad bands and largeStokes shift. The absorption, emission, and excitationspectrum for DCM cast in PMMA is shown in Fig. 20.Arrows A and E indicate the absorption and emissionpeaks in methanol, respectively. We observed that theStokes shift is considerably diminished in PMMA.[The critical optical density (CODE) for DCM inPMMA is about 20.] We are presently investigating thesuggestion of Sah et al.
7 that DMSO added to themonomer prior to polymerization restores much of thesolution Stokes shift of similar dyes.
Fig. 20. Absorption, emission, and excitation spectra for DCM inPMMA. Arrows A and E indicate the absorption and emissionmaxima, respectively, for DCM in methanol. Adding DMSO to themonomer prior to polarimerization should substantially increase the
Stokes shift in PMMA. 7
Ill. Theory
We will first provide the principal relations derivedin LSC-1 for predicting the performance of an LSC. Todetermine a figure or merit for LSC performance, wewill develop similar relations for a purely scatteringplate. Finally we will restate our self-absorption modelfor the large anti-Stokes shifted limit, and we will deriverelations for calculating self-absorption rates frommeasured steady-state spectroscopic and polarizationdata and from transient emission measurements.
A. Preliminaries
By way of review, we note that if Af is the total areaof the face exposed to direct sunlight and Ae is the areaof the edge contacting the solar cells, the geometric gainof the concentrator is Ggeom = Aface/Aedge. If n is theindex of refraction of the LSC plate material, by Snell'slaw we find that the probability that isotropic lumi-nescence (for the more general case see Ref. 12) willescape out of the surfaces of the plate rather than beingtrapped by total internal reflection is P = 1 - (1 - 1/n2)1/2. Light which escapes from the LSC plate becauseit was incident to the surface at less than the criticalangle of incidence is said to escape through the criticalescape cones. The fraction of the solar flux initiallyabsorbed by the plate which is transmitted to the solarcells is Q [LSC-1, Eq. (23)]:
Q= 1- [FP+ (1 -P) -rj
predominantly homogeneously broadened luminescentspecies are used in the plate. The result applies to acascade of luminescing centers in the limit that the ef-ficiency of energy transfer to the final or lowest energycenter is 100%.
Before treating the real case, it is instructive to cal-culate the collection efficiency of a purely scatteringplate. Consider the planar solar concentrator (PSC)geometry of LSC made from a ribbon of clear materialhaving a thickness T and width W. We hypothesizethat there is a plane of isotropic scattering centers lo-cated a distance T/2 from either exposed surface, so thatlight impinging upon this scattering plane is isotropi-cally scattered, with a scattering efficiency of ?lscat.Cells are mounted along both edges. The probabilitythat scattered light from a point on the scattering planea distance L from one of the edge mounted array of cellswill be incident on that array of cells is (1/7r) tan'l(T/L)(see Appendix A). Ignoring multiply scattered light,the fraction of uniform scattering by the entire scat-tering plane that is collected by both edge-mounted cellarrays is
F = W *' tan-1(T/W) = 1/2 ln[1 + (WIT)2}-Wr T
(2)
So if light is incident on this scattering plate, we findthat the initial fraction of that flux which is collectedand transferred directly to the cells without furtherscattering, called the first generation collection proba-bility, is
Q(1) = 7lscat * F. (3a)
Light will return to the scattering plane if it is not col-lected by the cells and if it does not escape out to theface. If (1 - F) (1 - P) is the probability that scatteredlight is trapped by total internal reflection and is notcollected by the edge-mounted cells, we find that thesecond generation collection probability is
Q(2) = ~1cat F(1 -F)(1 - P). (3b)
In a similar manner, we find that the ith generationcollection probability is
Q(i) = 71scat F[?scat(1 - F)(1 - p)]i-1 (3c)
The total collection probability is the sum of the indi-vidual probabilities for each generation:
-(1 77scat F -(d1 - F)(1 - P)ilscat (3d)
The average number of times that collected light hasbeen scattered from the scattering plane is the sum ofthe collection probabilities times their generationnumber:
where r is the average probability that a luminescentphoton will be self-absorbed while traversing the LSCplate, is the probability that a similar self-absorptionwill occur within the critical escape cone, and 7 is thequantum efficiency of luminescence for the given lu-minescing center. The approximations made in Eq. (1)are that trapped luminescence is not scattered by bulkor surface imperfections, that the level of excitation ofthe plate is well below the coherence limit, and that only
Ne = Fi iQ(i/Q.i=1
(4)
If we make the approximations that except for thescattering plane the matrix is perfectly transparent, thatthe AM1 efficiency of these photovoltaic cells ?7pvc isindependent of the light intensity, and that the scat-tering plane scatters all the incident sunlight, the totalcollector efficiency ij, (electrical power out per solarpower in) is
The potential broadband spectral characteristics andinsensitivity of scattering centers to degradation suggestthat a scattering plate might have interesting applica-tions for systems requiring only modest flux gains.
B. Steady-State Self-Absorption
In LSC-1 we derived a recursion relation for com-puting the intensity and spectrum of self-absorbedemission in the limit that the anti-Stokes shift is small.We will begin by presenting a similar formalism for thelimiting case of unrestricted anti-Stokes shifting of theemission with respect to the energy of the absorbedexcitation, i.e., the emission spectrum of the luminescingspecies is independent of the excitation energy. Forclarity, we will restrict ourselves to the case of a semi-infinite LSC rod. Next we will provide a simplifiedform for calculating the self-absorption probability fromknown absorption and emission spectra. This proba-bility can also be deduced experimentally from the an-isotropy of the polarization of the emission from apolarized excitation or from the change in lifetime as afunction of the concentration and path length throughwhich the emission travels.
Consider a semi-infinite rod of LSC material havinga diameter d and containing luminescing centers at aconcentration of C moles per liter, with an extinctioncoefficient E(17)and a normalized luminescence spectrumf(v). SC is the scattering coefficient corresponding tolosses in the rod other than self-absorption (e.g., matrixabsorption and scattering). P is the probability thatluminescence will escape out of the side of the rod,which has the form P = /n in the case of a cylindricalrod, where n is the index of refraction of the rod mate-rial. We define A(x,y) to be a function specifying theprobability that an emission originating at a position xwill be absorbed at a position y (see Appendix C):
2n10 -ifi) exp(-z)A(x,y) = 2 C Jsw SAdz -^ (6)
wherez = x-yj .[lnlO* C*E(v)+SC],
A = [n(10) C e(v) + SC] x - l,
[ln(10)*C*e(v)+SC]*Ix-yI/sinOclx-yI > tanO,2
B=|[Cn(10) C (v) + SC] -+ (x -y)
2X -yj < 2 tanoc.1 ~~ ~ ~4 2
Equation (6) assumes that absorbed excitation any-where in an infinitesimal disk at x is emitted from apoint at the center of the disk. Next we define S tobe the spatial distribution of the ith generation of ex-citations along the rod. For example, if the rod is ini-tially excited by a focused light source at a position x,
is a delta function at position x:
(1)(X ) = Io0(x). (7a)
Excitations directly due to absorption of the externallyincident light are first generation excitations. Second
IH-
znuJHz
E
0
EN._)
15,000 17,000 19,000 21,000 23,000
W VENUMBERS
Fig. 21. Calculated spectral and intensity changes due to self-ab-sorption. Experimental absorption and emission spectra of rhoda-mine-575 were approximated by the sum of two Gaussians. Stokesshift of 1000 wave numbers reproduces the typical overlap betweenthese two spectra. Using Eqs. (6)-(8), we have analytically imitatedthe experiment shown in Fig. 11 using the same geometry, concen-
tration, and path lengths.
generation excitations arise from the self-absorption offirst generation emissions. For example,
,, (2)(x= n f. dy Z(')(y)A(x,y),
and in general
i)(X) = n dy --(i-Z)(y)A(x,y),
(7b)
(7c)
where is the quantum efficiency of luminescence. Theluminescence spectrum observed at the end of the rod(x = 0) is the sum of the intensities due to all the dif-ferent generations of excitations in each element:
Q0i = n dx flr/2
-Oc
X sinOdO expj-x[ln(10) C * c() + SCJ/cosOj
(8)X f P )* E A (ii=l
In the limiting case that the Stokes shift is large, theprobability for reabsorbing the emission is approxi-mately zero everywhere or A(x,y) = 0. Therefore,
(i) (x) = 0 for all i > 1, by Eq. (7c), and only the firstgeneration emission is collected.
Figure 21 shows the results for Q () vs v for two nu-merically generated absorption and emission [(iv) andf(v) of Eq. (6)] spectra similar in form to the absorptionand fluorescence spectra of rhodamine-575. The pathlengths, rod diameters, and dye concentrations are thesame as for the experimental data shown in Fig. 11.The scattering coefficient used in this calculation is SC= 0.008 cm-1 . The absorption spectrum is the sum oftwo Gaussians with peak positions of 19,500 and 20,300cm-l, peak widths of 840 and 680 cm 1, and peakheights of 94,00 and 27,000 liters/mole/cm, respec-tively. The emission spectrum was the same sum of twoGaussians reflected in a mirror line at 19,000 cm-' andnormalized to unity. Notice the shift in spectral posi-tion with increasing self-absorption is accompanied by
Fig. 22. Calculated spectral and intensity changes due to self-ab-sorption. Same calculation was made as in Fig. 21, except that the
Stokes shift was increased to 2500 wave numbers.
a decrease in collected intensity. Figure 22 shows theresults of a similar calculation in which the Stokes shiftbetween the absorption and emission spectrum has beenincreased from 1000 to 2500 cm- to lower their overlap.While the nonzero scattering losses cause an overalldecrease in the intensity with increasing path length,the emission spectrum is seen not to shift appreciably.[Note: In LSC-I an additional (/cosO) factor is requiredin the finite element analysis of Appendix II in orderthat each element have the same initial intensity withrespect to a point on the edge.]
C. Self-Absorption ProbabilityWhile the preceding technique is useful in obtaining
spectral information about the emission from an LSCrod, a simpler calculation appears to be a reasonablygood approximation for calculating self-absorptionprobabilities. In the simplest 1-D case, the probabilitythat self-absorption will occur is found by the Beers-Lambert law:
r = dif(5)[1 - 1 0 -CO)], (9)
r is the probability that an emission, having the nor-malized distribution f(V), will be self-absorbed by theluminescing centers having an extinction coefficient E(i)and a concentration C over a path length x. The ap-proximation that we will make in this section is that theprobability of self-absorption for emission in an LSCplate is given by Eq. (9), where C is the concentrationof dye in the plate, and x is the characteristic length8 ofthe LSC plate. The characteristic length of a squareLSC is the length of the side of the square. The char-acteristic length for self-absorption in the critical coneis the thickness of the LSC plate containing the lumi-nescing centers.
Clearly this is a gross simplification of the properaveraging of the self-absorption probability over all pathlengths and energies of possible emission. We justifythis approximation by showing that the self-absorptionprobability has roughly a logarithmic dependence onpath length (see Fig. 25, for example). However, it isuseful to explicitly calculate the weighted average pathlength traveled by light collected in an LSC and com-
I0 z
: CWbJ _J11 JI
- aLL c_
0 Ld
<-J
wo
1,000
100
10
10 100
PSC WIDTH
- [C] = 105M
104M
103M
lo- M
1,000
Fig. 23. Average path length of collected emission vs width of a PSCplate. Using the full PSC collection efficiency calculation fromLSC-1, we have calculated the average distance through the platetraversed by emission which is collected at the edge of the plate forfour different concentrations of rhodamine-575. Cells were assumedto be mounted on both edges of the ribbon. Plate thickness was as-sumed to be 1 cm so that the 10-5 -M concentration corresponds toa peak o.d. of 1. Dashed line indicates the characteristic length ap-proximation that the average path length is equal to the plate
width.
pare this result with that predicted by the characteristiclength approximation.
We used the collection efficiency formalism of LSC-1[Eq. (20)] to compute the first generation collectionefficiency Q() for a PSC geometry LSC. We assumedthat the width was variable, the thickness was 1 cm, theindex of refraction was 1.5, and cells were mounted onboth edges of the ribbon. Since Q() = (1 - r)(1 -P),we found the self-absorption probability r. Using Eq.(9), we found the average path length of the 'collectedlight in the plate that would result in the computed r.Figure 23 shows the results of this calculation, where wehave calculated the average path length as a functionof the plate width for four different concentrations ofrhodamine-575. The dashed line indicates the char-acteristic length approximation, where the average pathlengths are assumed to be the width of the plate. At lowconcentrations and PSC widths, the average path lengthis greater than the width, while at high concentrationsand PSC widths, the average is less than the width.
D. Polarization Anisotropy: a Second Self-Absorption Measurement
Equation (1) tells us that the collection efficiency ofthe plate can be easily found once four parameters areknown: P the probability of escape out of the criticalcones; r, the probability of self-absorption for trappedemission; , the probability of self-absorption within thecritical cones; and , the quantum efficiency of lumi-nescence. P is known from the index of refraction ofthe host material, while the quantum efficiency can bemeasured using standard techniques.9 Equation (9)gives a reasonably simple approximate result for nu-merically calculating the self-absorption probabilitiesif the average path length of travel of the emitted lightin the plate is known. As we just showed, an exact
calculation of the self-absorption probabilities is trac-table for a few simple LSC geometries. In general,however, a simpler or more direct measurement of r andT is desirable. Two techniques providing this mea-surement are emission polarization anisotropy and totallifetime lengthening.
Suppose we excite an LSC with vertically polarizedlight and detect either the vertically or horizontallypolarized emission intensities emitted in a horizontaldirection, as shown in Fig. 12. If the orientation of theabsorption dipole moments of the luminescing centersare initially isotropic, the probability that any centerwill absorb an excitation as a function of its absorptiondipole orientation is proportional to [cos(o)]2. Usinga more concise notation, 4 1 0 if ej is a unit vector in thedirection of the electric field of the polarized excitation,and v-i-a is the absorption dipole moment, the angulardependence of the excitation probability is given by
jia]2 If we assume that the emission dipole momentjie is parallel with the absorption dipole and that theorientation of these moments is fixed on the time scaleof the total lifetime, the intensity of emitted lightpolarized in the direction ef is given by (apart from aconstant)
I=([e' - 1a2[ef. - 2) (
The brackets () denote an average over the unit sphere.The emission intensity polarized parallel to the exci-tation is therefore given by
2 2 2 rJ doJ dO sinO COS4047r 0 0
= e2a2/5. (lOb)
Similarly the emission intensity polarized in a directionperpendicular to the excitation is
III = 1/2 ( [i * - 1]2[e X U 2)
pa e 2do dO sin2O COS20
87r o
_ I/aiU e
15
The reduced anisotropy is defined to be
(lo )
RA = I - I . (1Od)III + 2I~
The initial excitation has no perpendicular component,so its reduced anisotropy is 1. From the intensitiescalculated in Eqs. (10c) and (Od), we see that the par-allel polarized emission has three times the intensity ofthe perpendicular component giving RA = 0.4 for thefirst generation emission. If the orientation of the lu-minescing dipole is different from that of the absorptiondipole, the reduced anisotropy decreases by a factor e(Ref. 4):
RA = 2e/5 0 < e 1. (1Oe)
Emission from an LSC due to polarized excitationdisplaying a reduced anisotropy of 2e/5 can thereforebe identified as emission from the first generation ofexcitations. The reduced anisotropy due to the self-
absorption of this emission and its subsequent reemis-sion are given by (see Appendix B)
RA(2) = (2/5)3 (lOf)
in the limit of parallel moments fixed in space. Ingeneral, the reduced anisotropy of the ith generation'semission is
RAW = (2 /5)(2i-1) (log)
Suppose we measure the reduced anisotropy of theemission from an LSC due to a polarized excitation.From Eq. (Og), we know that the measured RA is thesum of RA( ) for each generation times the collectionprobability for that generation; assuming Q2 is polar-ization independent,
RAexp = E RA(')Q(')/Q.i=l
(11)
In the limit of very low concentration luminescingcenters, there is no self-absorption, so Q(l) = Q Q(i)(i> 1) = 0, and RAexp = 2e/5. For higher concentrationswe use the technique of averaging over each generation,as in LSC-1, Eq. (23). If we assume by analogy that theith generation has RA W = (2e/5)2i-1,
RAexp = [e (1 -r)(1 -P) + (-)3
X 2(- r)(- P)[-P + r(l-P)]
+ (-)e 5n3( - r)( - P)[!rP + r(l p)]2 + * /
2e 1 - if7P +r(l -P))
1-7T5) [P + r(l -P)]
If the excitation is near enough to the edge of the plateso that the probability of self-absorption inside andoutside of the critical cones is about the same, r = ,then
RAexp = -nr 2e(1-r).
5
(12a)
Similar simplifications arise if the measurement is ar-ranged so that P is 0.
E. Lifetime Lengthening: a Third Self-AbsorptionMeasurement
The rate of self-absorption both inside and outsideof the critical cones can be calculated directly frommeasured transient emission spectra. At t = 0, supposethat there are no excitations in a dye ensemble. Therate of change of this population is proportional to thepopulation:
a n(')(t) = - n)(t) n(l)(o) = no,at
(13a)
where r is the total lifetime. We assume that there isno transit time dispersion due to emission from differentparts of the sample or that the lifetime is long comparedwith the transit time for the light in the sample. Sincetypical fluorescent dyes have lifetimes of the order ofnanoseconds, the applicability of this formalism is re-
stricted to devices a foot across or smaller. If r is theprobability that the emission is self-absorbed if it isemitted outside of the critical cones, and T is the prob-ability of self-absorption within the critical cones, P isthe probability of trapping in the critical cones, and 7)is the quantum efficiency of luminescence, then thekinetic equation for the number of second generationexcitations as a function of time n(2)(t), resulting fromfirst generation emissions, is
Note that T = Trad 7), where Trad is the radiative life-time.
We will assume that the probability of self-absorptionis independent of the number of emissions previouslyexperienced by the excitation. This is equivalent toassuming the limit of a predominantly homogeneouslybroadened ensemble at temperatures sufficiently highto allow anti-Stokes shift of the emission which is largecompared with the emission linewidth. The thirdgeneration excitations therefore are
Summing the various generations gives the number ofexcitations in the ensemble as a function of time:
n(t) = EJ n(i)(t) = no exp (- 41 - 77[FP + r(l -P). (14)
Thus self-absorbed emission will have an apparentlifetime which is lengthened by a factor 1/11 - 7 [FP +r( - P)]}. This factor is equal to one if r = = 0.Measurements of the experimental transient emissiondecay from an excitation which is short compared withthe total radiative lifetime at low concentrations cantherefore give r, r/(1 - r7), and r/1 - 7[FP + ri - P)]}for the three cases of an extremely dilute sample, edgeemission due to an excitation near to the edge, and fromgeneral illumination of the plate, respectively.
If each generation of excitations is distributedthroughout the sample in such a way that the averageprobability of emission from each generation reachingthe detector is the same, the transient emission spec-trum should have the simple exponential behavior ofEq. (14). It is possible, however, to configure the ex-citing source and sample in such a way that the firstgeneration emission has a very different probability ofreaching the detector. For example, if the exciting laserpulse is at an energy corresponding to the absorptionmaximum for the emitting center, nearly all the laserlight will be absorbed in typically less than the first fewmillimeters of the sample. A mask can then be placedbetween this excited region and the detector, so that theexcited region of the sample can illuminate the rest of
the sample without illuminating the detector (Fig. 16).Assuming that none of the first generation emission il-luminates the detector due to scatter in the rest of thesample and r = T, we find that the measured transientspectrum should be proportional to the usual totaldecay of the population minus the missing first gener-ation emission
If the probability of self-absorption is high enough, asignificant fraction of the first generation emission canbe reabsorbed in the masked region. In this case, somefraction of the higher-order generations will have to besubtracted. Because of the sensitivity to scattering andinadvertent masking of higher-order generations, thistechnique is inferior to the simple lifetime measure-ments for determining self-absorption probabilities.
F. Effects of Homogeneous BroadeningThe experimental results of the previous section en-
courage us to assume that the absorption and emissionspectra are predominantly homogeneously broadened,at least for the case of organic dyes dissolved in solventsor diffused into plastics. More specifically, this ap-proximation allows a complete calculation of the effi-ciency of a PSC or ribbon geometry device using the firstgeneration collection efficiency [LSC-1, Eq. (20)].
Since the first generation collection efficiency is de-fined to be (1 - r)(1 - P), where P is the critical coneescape probability and r is the self-absorption proba-bility outside of the critical cones, we can solve for thetotal collection efficiency from Eq. (1). Solving for r interms of Q(1 )(PSC) and substituting in Eq. (1), wefind
QPSc = QI()C/t1 + Q )c - 7[1 - P(l - T], (16)
where r is the self-absorption probability inside thecritical cones. Equation (16) gives a tractable calcu-lation of the efficiency of the PSC device with only twofurther approximations. The first approximation isthat the self-absorption probability in the critical conescan be computed using Eq. (9). The second is that wemust assume that the self-absorbed emission is dis-tributed isotropically across the plane of the PSC plate.The latter approximation should be expected to breakdown only for high concentrations of luminescingspecies and then only in the region near the solar cells.The reason one would expect the approximation tobreak down in this case is that if the concentration ishigh, self-absorption occurs in a path length which isshort compared with the width of the plate, so that it ispossible for one region of the plate to contain a differentexcitation density than another region. Since the solarcells do not emit light, the region near the cells is notilluminated from all sides, as are other portions of theplate. If the luminescing species are predominantlyinhomogeneously broadened, appropriate averagingmust be done over all homogeneous subensembles andover all cm-1 .
In summary, we have suggested three ways to deter-
mine self-absorption rates in an LSC: spectral convo-lution; steady-state emission polarization anisotropy;and transient lifetime lengthening. Additional tech-niques exist. For example, the peak of the emissionspectrum in Fig. 11 moves toward lower energy as self-absorption increases. This allows for an indirectmeasure of the self-absorption probability if these peakpositions are calibrated using one of the above tech-niques.
IV. Discussion of Results
A. Spectral Homogeneity
As mentioned before, the relative homogeneity ofabsorption and emission spectra of organic dyes is de-pendent on the host material. For example, in Fig. 4we see that the maximum of the emission spectrum forrhodamine-575 in methanol at room temperature isindependent of the excitation energy, even for excita-tions which are 2000 cm-1 lower in energy than theemission peak. This blue-shifted emission disappearswhen the sample is cooled to liquid nitrogen tempera-tures, so that this extra energy in the emission mustcome from thermal energy in the molecule or from thethermal bath. If the same dye is diffused into a PMMAplate, essentially the same results are produced, asshown in Fig. 6. We therefore concluded that, for thesetwo host materials, the absorption and emission spectraare predominantly homogeneously broadened. Dyeswhich were dissolved in methyl methacrylate monomerand then polymerized displayed characteristics of in-homogeneous broadening. As shown in Fig. 5, theemission spectrum is dependent on the precise energyof the excitation; it was possible to select portions of theensemble of dye molecules which displayed significantlydifferent spectral characteristics than the ensembleaverage.
We have seen that dye molecules in a host materialat room temperature are able to emit light of severalthousand wave numbers greater energy than the initialexcitation. This is not surprising, since these moleculestypically have a large number of internal degrees offreedom. At liquid nitrogen temperatures, kT is re-duced by a factor of 4, which is sufficient to decrease theavailable energy in the molecule so that these largeblue-shifted emissions do not appear. This tempera-ture sensitivity would imply that even larger blue-shiftsof emission would be possible in plates at the elevatedtemperatures of roof-mounted collectors.
B. Self-Absorption
We can also infer that the cast PMMA samplesshowed stronger absorption in the extreme red tails thanthe solution or diffused samples. This can be seen bycomparing the shape and the signal to noise for emissionspectra in Figs. 4 and 5. This is very important foroptimizing LSC efficiency, because it is the red ab-sorption tail which leads to self-absorption losses.
A comparison of Figs. 11 and 21 shows that the nu-merical technique described in the theory section iscapable of accurately modeling the attenuation andspectral shift of the emission due to self-absorption.
We observe that most of the emission spectrum overlapssufficiently with the red absorption tail, so that most ofthe emission can be self-absorbed for concentrationsand path lengths typical of LSC devices. The modelalso verifies that a large Stokes shift will eliminateself-absorption losses. This experiment also suggeststhat the probability of self-absorption might be mea-sured simply by observing the peak position of theemission as a function of concentration or pathlength.
Clearly it is important to account for the self-ab-sorption rates for luminescing species which are to beincorporated into an LSC. In general, the self-ab-sorption rate is related in a complicated way to the ge-ometry, the dye concentrations, and their absorptionand emission spectra. For the purpose of designingdevices, this interrelationship usually does not suggest,in a transparent way, what the proper species are andwhat their concentrations should be. We will thereforedefine a self-absorption parameter, the critical opticaldensity, to be the peak optical density of a samplecontaining a luminescing species, so that the emissionfrom the species has a 50% chance of being self-absorbedwhile traversing the sample. Mathematically, if a lu-minescing species has a normalized luminescencespectrum f (v), a molar concentration C, and an extinc-tion coefficient E(H) with a maximum value of E(VM), andthis species is contained in a sample with an optical pathlength L, the critical optical density is
CODE = E(Vm)CL, (16a)
where the concentration and path length have beenadjusted so that
Fig. 24. Collection efficiency vs self-absorption probability. As-suming that P = 0.26, we have used Eq. (1) to calculate the collectionefficiencies for the cases where there is no self-absorption in thecritical cones (solid lines) and where the self-absorption rates insideand outside of the critical cones are the same (dotted lines). Uppertwo curves assume a quantum efficiency of luminescence of 0.9, whilethe lower two assume 0.7. Vertical line at r = 0.5 indicates the self-
The CODE is the radiative analog to the Forster criticalconcentration. It is useful because it characterizes theefficiency of an LSC plate as a function of size or geo-metric gain. In Fig. 24, we have plotted the collectionefficiency of an arbitrary LSC plate as a function ofself-absorption probability using Eq. (1). The indexof refraction is assumed to be 1.49, the solid lines referto the case where there is no self-absorption in thecritical cones, and the dotted lines are for the case whereself-absorption rates are equal inside and outside of thecritical cones. The vertical line originating at r = 0.5intersects the curves at the CODE operating point. Forexample, a typical LSC plate will have an index of re-fraction of 1.5 or an escape probability of P = 0.26, aquantum efficiency of luminescence of 7 = 0.9, andnegligible self-absorption in the critical cone. If theself-absorption probability is 0.5, we find from Fig. 24that the collection efficiency of the plate is also -0.5.To obtain reasonably good absorption of sunlight oremission from other dyes, the peak optical density of thefinal dye must be at least 1 across the thickness of theLSC plate. In this case, the CODE will be the numberof plate thicknesses which the emission can traversebefore being 50% self-absorbed. Since the geometricgain of a plate is roughly the width of a plate divided byits thickness, the CODE is roughly the maximum geo-metric gain of a plate, using the particular luminescingcenter as a single or final dye in a cascade, which pro-duces a collection efficiency of 0.5.
Table I gives the critical optical densities for the or-ganic laser dyes tested. The 30% errors are due to un-certainty in the measurement of the absorption tails.Typical values for the CODE are 20. Such dyes wouldproduce collection efficiencies of <0.5 if incorporatedin plates with a geometric gain greater than 20. Severaldyes have critical optical density around 80, whileDCM (4-dicyanomethylene-2-methyl-6-p-dimethyla-minostyryl-4H-pyran) has a critical optical density of-250.
We have shown results on three different measure-ments of the effects of self-absorption in rhodamine-575: spectral overlap; emission depolarization; andlifetime broadening. The critical optical density forrhodamine-575, from a spectral overlap calculationusing Eq. (16b), implies that self-absorption will havesubstantial effects on the performance of an LSC. Weare therefore interested in the consistency of thesedifferent measurement techniques in determining ac-tual self-absorption rates.
For a square cuvette sample geometry, the probabilityof self-absorption in the critical cones is about the sameas the probability outside of the critical cones. Theself-absorption probability r is calculated in three dif-ferent ways. We will assume that the characteristiclength approximation of Eq. (9) applies for the spectraloverlap calculation, with the length determined by thepath length through the sample from the excitationsource to the detector. We will use Eq. (12a) to find rqfrom the reduced anisotropy measurements. If RAmnais the maximum reduced anisotropy (2e/5) measuredat low concentrations and short path lengths, the
8.00E -1z~0
I-
0U)co
ILIj
bLJU)
Cci00ca-
6.001
4.00
2.00
10-7 10-6 10-5
CONC. X PTHLENGTH
10-4 10-3
Fig. 25. Self-absorption probabilities for rhodamine-575. Thisshows a juxtaposition of the predicted self-absorption probabilitiesfor the three measurement methods: spectral overlap convolution(solid curve); emission depolarization (error boxes); and transientlifetime (error bars). Lifetime measurements have the highest pre-cision, especially at low concentrations, and the lifetime and polar-ization experiments are in good agreement. These two techniquesactually measure the self-absorption probability times the quantumefficiency of luminescence, which we have approximated as being
unity.
product of the self-absorption probability and thequantum efficiency r7 from Eq. (12a) is
r -- 1- (RAexp/RAmax) 1- (RAexp/RAmax)-1 - (RAexp RAmax)
Similarly, from Eq. (14), if Tmin was the shortest totallifetime measured at low concentrations and short pathlengths, the same result should be given by the relation(r = ), r = 1 - Tmin/Texp, where rexp is the mea-sured lifetime. If we apply the preceding two relationsto the measured transient emission and steady-statedepolarization measurements on rhodamine-575, weobtain the results shown in Fig. 25. We have plottedthe calculated product of the quantum efficiency andself-absorption probability as a function of path lengthtimes the concentration of the sample through whichthe emission had to travel. The solid line is the resultfrom the characteristic length approximation of Eq. (9).This is actually just the self-absorption probability andnot the product of the self-absorption with the quantumefficiency. There is very good agreement between thelifetime lenghtening and the emission depolarizationmeasurements. The characteristic length calculationoverestimates the self-absorption rate at high opticaldensities, as we showed earlier in Fig. 23. The mea-sured value of RAma, = 0.18 for rhodamine-575 in glycolis consistent with a rotational diffusion time of severalnanoseconds, which is the rate observed for similar dyesin glycol.11
C. Optimal Efficiency
We know all the characteristics of rhodamine-575which determine the efficiency of a single-dye LSC, sothat now it is instructive to determine the optimal ef-ficiency of an LSC containing rhodamine-575 as afunction of its size. Since the optimal concentration of
Fig. 26.. Optimal efficiency and dye concentration for a single-dyeLSC. Geometry assumed is an infinite ribbon with 18% AMi siliconcells mounted on both edges. Plate has an index of 1.49 and containsrhodamine-575 with an assumed quantum efficiency of luminescenceof 0.9. Dotted line represents the performance of a scattering plate
with an identical geometry.
the dye will vary with size, we also must compute theconcentration. We use the PSC geometry formalismfrom LSC-1 [Eqs. (11) and (20)], to compute the firstgeneration collection efficiency Q(1). We used thecharacteristic length approximation of Eq. (9) to com-pute the self-absorption rate in the critical cones.These were combined in Eq. (1) to give the total col-lection efficiency with the approximation of no reflec-tion losses and only homogeneous spectral broadening.We will assume edge-mounted cells which have 18%AM1 efficiency coupled to a plate with an index of 1.49.The quantum efficiency of the dye is assumed to be 0.9.The result of this calculation is given in Fig. 26. For thesake of comparison, the system efficiency of a scatteringplate with an identical geometry is included. The sin-gle-dye plate outperforms the scattering plate for geo-metric gains greater than 10. From Table I, we findthat the CODE of rhodamine-575 is about 20. Thus,for a plate with a gain of 20, we expect a collection effi-ciency of -0.5 and that -40% of the solar flux to whichthe cell would be sensitive would be absorbed by theplate. The zeroth-order estimate for the efficiency ofthe plate for a gain of 20 is therefore 0.18 X 0.5 X 0.4 =0.036 (3.6%) in agreement with Fig. 26.
Finally, the simplest model for the efficiency of anLSC which includes self-absorption effects can be madeas an extension of a simplified solar cell efficiencymodel. We assume that all light with energy higherthan some absorption cutoff Ei is absorbed and re-emitted with an energy Ec minus the Stokes shift. Thisis transmitted to the cells, which then subtract another0.4 eV before producing useful electricity. If we assumea 0.7 eV Stokes shift (typical for DCM), this modelpredicts 9% maximum overall efficiency using eithersilicon or gallium arsenide cells.' 2
V. Conclusions and RamificationsWe can draw five principal conclusions from these
studies:(1) It is important to know the relative homogeneity
of the spectral broadening of the absorption and emis-sion lines for the luminescing species in an LSC. Wehave found that our organic laser dye rhodamine-575is markedly inhomogeneously broadened when dis-solved in methyl methacrylate monomer and poly-merized, while dye solutions and dye diffused intoprecast material are predominantly homogeneouslybroadened. The preferable case is the one that pro-duces the least low energy absorption tail. Materialsfor altering both homogeneous and inhomogeneouslinewidths are presently under investigation.
(2) We have demonstrated three different techniquesfor measuring the effect of self-absorption in sampleplates. These techniques are spatial filtering, emissiondepolarization, and lifetime lengthening. There is goodagreement between the techniques when measuring theself-absorption rates of rhodamine-575. Each tech-nique is best suited for different applications. Thelifetime lengthening measurement is the most accurate,especially at the low concentrations and path lengths.The spectral overlap is best suited for calculating theperformance of a plate. The emission depolarizationmeasurement requires a minimal expenditure forequipment and is quite accurate for high rates of self-absorption.
(3) Self-absorption plays a fundamental role in de-termining LSC efficiencies in large plates or plates witha high geometric gain. We defined the CODE to be thepeak optical density of a sample so that emission passingthrough the sample has a 50% chance of being self-absorbed in the sample. Tabulated critical opticaldensities for eighteen principal laser dyes are shown inTable I. We have shown that the CODE is roughly themaximum geometric gain at which the collection effi-ciency of the plate is greater than or equal to 0.5 fortypical dyes and matrix materials. None of the dyestested could successfully perform as single-dye LSCsor as final dyes in a multiple-dye LSC with a systemefficiency of >8% assuming a geometric gain of greaterthan 100 and that the edge-mounted solar cells have an18% AM1 efficiency.
(4) We suggest that the CODE allow for a roughcalculation of the efficiency of an LSC plate, includingself-absorption effects. Thus the CODE of a dye is animportant parameter in selecting a dye for use in anLSC. As an example of such a calculation, let us con-sider a plate of thickness T having a geometric gain(equal to the area of the face divided by the active areaof the edge) and containing a dye with a quantum effi-ciency of luminescence 7, a concentration C, a normal-ized luminescence spectrum f), and an extinctioncoefficient e(i) with a peak at v We will assume thatthe concentration of the dye is adjusted so that the peakoptical density across the thickness of the plate is one.The CODE of this dye is defined to be CODE =E(vm)CX, where X is an arbitrary path length chosenso that
We have shown that for a plate whose geometric gain isequal to this CODE and which has an index of refractionof -1.5, -50% of the emission can be collected byedge-mounted absorbers. The collection efficiencytherefore is approximately Q = -q/2. The fraction of thesunlight absorbed is S/I[LSC-1, Eq. (11)1:
S/ = I (;)di [l 10 -C,(-)T] / I(^),
where I(v) is the solar flux and a is the wave numbercorresponding to the low energy absorption limit for thesolar cell used. S/I can vary from <10% in single-dyeplates to 75% in multiple-dye plates. The total collectorefficiency is
Eff. = /2 ?AM1S/I (S/I)Qcei,1
where GAM1 is the air mass 1 efficiency of the solar cellsused.
For example, from Table I we find that rhodamine-590 has a CODE of 25, a quantum efficiency of 0.95, anda peak extinction coefficient of 107,000 moles/cm/liter.If we make a 3-mm thick Plexiglas plate with a geo-metric gain of 25 containing 30-AtM rhodamine-590, ifthe solar cells attached to the edge of the plate have an18% AM1 efficiency, and if we assume that the dye ini-tially absorbs 20% of the usable sunlight, the aboverelation lets us approximate the total collector efficiencyas 0.98 X 0.18 X 0.2/2 = 1.8%, which is typical for a plateof this type.
(5) Dye degradation is now the most critical technicalbarrier to the practical utilization of the LSC. Thedegradation rates appear more dependent on the totalflux than on UV intensity alone. Preliminary studiesperformed here show results that are consistent withpreviously published quantum efficiencies for photo-degradation, which usually are within an order ofmagnitude of 10-6. We are optimistic about the pos-sibility of increasing the lifetime of the dyes, since insome cases, lifetimes of at least a year are measured.
A. H. Zewail is an Alfred P. Sloan Fellow and Camilleand Henry Dreyfus Teacher-Scholar. Part of this workwas supported by a contract from the U.S. Departmentof Energy and part by ARCO Solar, Inc. We thank R.J. Robbins and D. P. Millar for performing the transientlifetime measurements and for their stimulating dis-cussions. We particularly thank R. J. Robbins forproviding us with the anistropies for higher-order gen-erations. We thank Stuart Vincent for help in castingplates and in the degradation experiments and PrakashKasiraj for writing data handling and plotting software.We also thank Jim Liu, Bob Mueller, and co-workers atJPL for performing prototype measurements. J. S.Batchelder and A. H. Zewail also hold appointments inthe Department of Applied Physics.
This is Contribution 6455 from the Arthur AmosNoyes Laboratory of Chemical Physics, California In-stitute of Technology, Pasadena 91125.
w
Fig. 27. Schematic diagram of a scattering plate.
Appendix A: Scattering PlateWe wish to compute the collection efficiency of a
scattering plate. The geometry of interest is an infiniteribbon of thickness T and width W, as shown in Fig. 27.We shall assume that there is a plane of scatteringcenters located midway between the two faces of theribbon.
The fraction of an isotropically scattered light fluxincident on the scattering plane a distance x away fromone edge that is intercepted by that edge is
F(x) =-X df sinOdOT7 -x/2 a_ s
2 rx 2d7r/ c os4 = tan , (Al)
where a+ = r/2 + tan-l (T coso/x). We have allowedonly light trajectories which do not intercept the scat-tering plane.
If the scattering plate has solar cells mounted on bothedges, the fraction of the scattered incident light whichis collected without intercepting the scattering planea second time is
Ql) = 2 7lscat dxF(x)
= 77scat 5 dx tan- 1
=2 7scat J T tan1 (T + 1/2 n [1 + (Wi], (A2)17rWV~~ IT \WJv [ T/J')
where Nscat is the scattering effiency of the scatteringplane.
We note that if the scattered light does not escape outof the critical cones, and if it is not collected by the solarcells, it must intercept the scattering plane again. As-suming that the spatial distribution of this scatteredlight is isotropic, we find that the higher generationcollection efficiencies are
QS2) = (1 -P)[77.cat Q Q ,
Qs = (1 - )2 [iScat -Q()]2Q(1)
Therefore the total probability of the solar cells col-lecting the scattering incident light is
The dotted line in Fig. 26 shows the performance of ascattering plate as a function of size. The scatteringplane is in a 1-cm thick ribbon of infinite length andvariable width; 18% AMI cells are attached to bothedges. The scattering efficiency is assumed to be90%.
Appendix B. Higher Generation Anisotropies
We wish to derive the reduced polarization anisotropyor the fraction of the detected emission intensity whichis purely polarized in the direction of the polarizedinitial excitation for all generations of emission. Thiscan be computed for low rates of self-absorption withthe approximation that the absorption and emissiondipoles are parallel.
Following the notation of Gordon,4 let e be a unitvector in the direction of the incident electric field po-larization, TA a unit vector parallel to the absorptiondipole of an ith-generation excitation, A,&i a unit vectorparallel to the corresponding emission dipole moment,and ei the emission polarization direction. The angulardependence of absorption is given by the square of theprojection of the incident polarization on the dipoleaveraged over all orientations of both [i_ - ai]2. Inthe radiation field limit (average distance between in-teracting dipoles >>5000 A), the angular distribution ofthe emission is also proportional to the square of theprojection of the emitting dipole on a random vector ei[Aei * ej]2. Thus the second-generation parallel andperpendicular emission intensities are
I?'= K * ([f * 4aI2[4e* ]2[, - ,a2]2[Ae2 -.12)
2) = 2 ( a&[Ael - e1]2[e, * Aa2][lie X ]2).(Bi)
We use the identity that the cosine of the angle ft be-tween two vectors with orientations (01,01) and (02,02)is cosIP = cosO1 cos02 + sin 1 sinO2 cos(01 - 2). In thefar-field limit, the parallel and perpendicular intensitiesbecome
IV)= K , do f dO sino f do,
x f dOl sino, Hf0 d 2 X d0 2 sinO2
X cos2 0 [cosO cosOl + sinO sinkl cos(o - 1)]2
X [cosO1 cosO2 + sinOk sinO2 cos(01 - 2)I2 cos202
I) (2) (2)3I)+ 2IT ks
(B4)
Similarly the ith generation reduced anisotropy is
= (2)(2i-l) (B5)
The radiation field limit is applicable for dye con-centrations used in LSCs. The 50% self-absorptionprobability occurs for concentration times path lengthof typically 10-4 mole cm/liter, so that the averagedistance for self-absorption is of the order of a centi-meter for typical concentrations.
The first-generation reduced anisotropy is rM') = 2/5- 0.4, the second is r(2) = (2/5)3 = 0.064, and the thirdis r(3) = (2/5)5 0.01. Thus it is reasonable to ignorethe polarization contribution from higher-order gen-erations under these conditions.
Appendix C: Self-Absorption and CollectionEfficiency of an LSC Rod: Derivation of Eqs. (6)and (8)
In LSC-1 we developed analytical expressions for theintensity and spectrum of light emerging from ablackened LSC rod including self-absorption effects[Eqs. (32)-(38)]. We wish to extend this treatment tothe case where light incident on the surface of the LSCmaterial at an angle greater than the critical angle ac istotally internally reflected.
Consider a polished semi-infinite rod of LSC materialwith a diameter d. The probability P that isotropicemission in the rod will escape is
where n is the index of refraction.Suppose the rod contains a luminescing dye at a
concentration of c moles/liter, with an extinction coef-ficient e(v), with a normalized luminescence spectrumf(i), and with a quantum efficiency of luminescence 7).Furthermore, suppose that the matrix material has asmall and wave number independent absorption coef-ficient SC (or scattering coefficient for light whichscatters out of the rod). In general the transmissionthrough a length (a) of this material is given by theBeers-Lambert law:
T(a,i) = expl-a[ln(10) C E(v) + SC].(B2)= K 47/153;
We have assumed that the absorption and emissiondipoles are both parallel and fixed in space on a timescale of the excitation lifetime. Thus the second gen-eration reduced isotropy is
(C2)
If the scattering coefficient is small, it can be ignoredover short path lengths. Then the transmissionthrough an incrementally longer distance is
T(a + A, ) = T(a,i) exp[-A * ln(10) * C e(v)]= T(a,i)[1 - A ln(1 ) * C e(V)].
The probability of absorption in the incremental dis-tance is the difference in the two transmission proba-bilities:
T(a,) - T(a + A,5) = A. ln(10) C e(v)expl-a[ln(10) * C * e(v) + SC.
Now we go from a 1- to a 3-D case, which requires thatwe average over all possible paths. Suppose a molecule
located on the symmetry axis of the rod emits a photonwhose path makes an angle 0 with the symmetry axis.We will assume that all emissions from a disk elementoccur at the center of the disk. This is a reasonableapproximation for emission path lengths which arelonger than the diameter of the rod. If the emissionoccurs at point y from the end of the rod, and we areinterested in the absorption probability per unit lengthat x from the end of the rod, the length of a trajectorywill be a/cos0, where a = x-y . Similarly the incre-mental distance is increased by a factor of 1/cosO. Theprobability of absorption per unit length becomes
(fa v)-la +A v)T --a T , cosA(x,y,i) = I A sin~dO
- Us dO-ln(10) C ()o coso
X expl-Ix - yI [n(10) C e(v) + SC},
r/2-Oc for Ix -yI > 2tan,2
B' = (C3)
tan- 1 ___ I for Ix - < tank.
The upper limit of integration depends on the distanceIx -Y l. If, for example, x - yI < (d/2) tanO, lightfrom y with a plar angle greater than r/2 - , canreach x, even though it would not be trapped by totalinternal reflection. Now let
z = Ix - [ln(10) C e(i) +'SC]/cos0,dz sinO
- -dO.
z coso
Substituting in Eq. (C2) gives
A(x,y,i) = X dz Z ln(10) C e(),
A = x - YJ [ln(10) C (v) + SC],
I x -yI [ln(10) C e(V) + SC]/sinOc when x -yI >tanO.,Iln~lo) * c * f(V) + SC] W h e n Ix-Al < 2 tan~c-
t[ln(1O) .C eQ) + SC] -d2+ ( -y) 2 when Ix -yl <-dtano,,.I V ~~4 2Finally, we perform a weighted average over the lumi-nescence spectrum to arrive at Eq. (6):
A(xy) = Jo dif(i)A(xy,i).
To derive Eq. (8), we first observe that the emissionat x and along the rod is the sum of all generations ofexcitations at x times the normalized luminescencespectrum times the quantum efficiency
nf(i) E(¢ .
Emission off-axis by an angle of is attenuated by
T (- a', I) = expi-x[ln(1O) C e( + SC]/cosO]-
The output spectrum is the average of this emission overall elements x and all trajectories of trapped emission0 < 0 < 7/2 - 0c:
Q(0) = J dO sinO J dx
* expi-x [ln(10) C * e(i) + SC]/cos01
n-QU) - V O()(x).
This is Eq. (8). Summing over wave numbers gives theoverall collection efficiency:
Q = f d5Q(i). (C5)
Note added in proof. Recent experiments on somedyes in PMMA by A. Gupta and S. Vincent indicatethat the value of -0.2 for the reduced anisotropy ofemission in our samples is due to the presence of un-polymerized monomers in the vicinity of the dye. Thisallows rotational diffusion (D. Millar, R. Shah and A.Zewail, Chem. Phys. Letts. 66, 435 (1979)) during thefluorescence lifetime. The implications of these im-portant findings to photostability is currently underinvestigation.
(C4)
References1. J. S. Batchelder, A. H. Zewail, and T. Cole, Appl. Opt. 18, 3090
(1979) and references therein.2. Exciton Chemical Co., Inc., P.O. Box 3204, Overlook Station,
Dayton, Ohio 45431.3. Acrilex, Inc., 8 Hope Street, Jersey City, N.J. 07307.4. R. G. Gordon, J. Chem. Phys. 45,1643 (1966).5. R. J. Robbins, D. M. Millar, and A. H. Zewail, J. Chem. Phys. 75,
3649 (1981). [A brief description of the apparatus appears in R.J. Robbins, D. P. Millar, and A. H. Zewail, Picosecond Phenom-ena II, Springer Topics in Chemical Physics 14, R. M. Hoch-strasser, W. Kaiser, and C. V. Shank, Eds. (Springer, Berlin,1980).]
6. D. Beer and J. Weber, Opt. Commun. 5, 307 (1972).7. R. E. Sah, G. Baur, and H. Keller, Appl. Phys. 23, 369 (1980).8. For a planar LSC a characteristic path length can be defined that
reduces the 3-D problem to a pseudo-1-D problem.9. J. N. Demas and G. A: Crosby, J. Phys. Chem. 75, 991 (1971).
10. T. Tao, Biopolymers 8, 609 (1969).11. A. Von Jena and H. E. Lessing, Chem. Phys. 40, 245 (1979).12. J. S. Batchelder, Ph.D. thesis, Calif. Inst. Tech. (1981).
