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The role of light scattering in the performance
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of fluorescent solar collectors
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Nazila Soleimani,a Sebastian Knabe,b Gottfried H. Bauer,b Tom
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Markvart,a Otto L. Muskensc
a
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Materials Research Group, School of Engineering Sciences, University of Southampton,
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Highfield, Southampton. SO17 1BJ
b
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Institute of Physics, Carl von Ossietzky University of Oldenburg, D-26111 Oldenburg,
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Germany
c
SEPnet and Department of Physics and Astronomy, University of Southampton, Highfield,
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Southampton. SO17 1BJ
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Nazila.Soleimani@soton.ac.uk; T.Markvart@soton.ac.uk; O.Muskens@soton.ac.uk
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Abstract. A fluorescent solar collector (FSC) is an optoelectronic waveguide
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device that can concentrate both diffuse and direct sunlight onto a solar cell.
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The electrical output of the device depends strongly on the photon fluxes that
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are absorbed, emitted and trapped inside the FSC plate; for this reason, it is
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important to study the photon transport losses inside the collector. One of the
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losses in FSC is investigated to be scattering which increases the probability
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of the escape cone losses. The purpose of this work is to determine the
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scattering contributions in FSC by using angle dependence of light internally
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reflected in the FSC. The cause of the scattering in spin-coated PMMA on
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top of the glass collector is identified to be roughness from the top surface
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rather than bulk losses. This loss can be suppressed using an index-matching
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planarization layer.
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Keywords: Fluorescent solar collector; Solar energy; Scattering; Scattering
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measurements.
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1 Introduction
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In a typical fluorescent solar collector (FSC), the luminescent light is trapped by total internal
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reflection (TIR) and guided to the edge of the plate where it can be harvested by a solar cell.
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One objective of the collector design is to reduce the size of the solar cell with respect to the
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area which collects light. Unlike their geometric counterparts, fluorescent collectors can
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accept light from a wide angle of incidence and are thus able to make use of the diffuse light.1
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The photon flux at the edge of an idealized FSC is the product of the absorbed solar flux, the
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fraction of the trapped luminescence, and the geometric ratio of the area of the face directly
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exposed to sunlight Af divided by the area Ae of the edge that is covered by the solar cell. The
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geometric gain Ggeom of the FSC is then given by:
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G geom 
Af
(1)
Ae
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Weber and Lambe (1976) initially suggested the concept of luminescence trapping for solar
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concentrators, which they named the luminescent greenhouse collector2. In experimental FSC
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devices, the emitted photon flux is less than ideal due to several loss processes, including
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reabsorption and scattering.1,2,3,4,5
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In this work, we investigate the effect of surface scattering in realistic FCS devices based on a
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dye-doped PMMA layer on top of a glass slide. To characterize the photon transport inside the
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collector we monitor the angular distribution of a collimated light beam which enters the
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collector from the edge, after propagation and total internal reflection. We find that the
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surface scattering process is described well by Fraunhofer diffraction at surface
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inhomogeneities of a size of several m.
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2
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2 Theory
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We calculate the effect of diffraction due to surface roughness on the FSC efficiency using the
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model developed by Weber and Lamb.2 In their model, they included the guiding of light in
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the FSC at angles larger than the critical angle for total-internal-reflection, c. Attenuation of
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reflected light by reabsorption is taken into account through an effective absorption
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coefficient e, while Fresnel reflection losses at the interface between the low-index FSC and
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the high-index solar cell is included through Fresnel’s reflection coefficients for s- and p-
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polarized light.
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In addition, we include here losses caused by angular spread of the reflected light due to
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surface roughness. The effect is included as a divergence of angles with a Gaussian profile of
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1/e half-width . Around the critical angle, this additional spread in angles results in part of
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the light to be lost within the escape cone. The angular distribution of light remaining after
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one reflection from the rough surface can be approximated from the convolution of the unit
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step function with the Gaussian diffraction function, resulting in the error function
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P()=erfc[(c-)/]/2. Under the condition that the scattering is only a small correction on
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the distribution function, the sequence of total internal reflections can be written as a
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multiplication of a number of error functions where the number is given by the amount of
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internal reflections from the rough surface. Inclusion of this effect in the Weber and Lambe
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model results in a total collection efficiency Q C given by
QC  (2L)
1
L
 dy
0
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 /2
 d
0
 /2


1
2
erfc ( c   ) /  
N R ( , )
sin d
c
 exp   e ( L  y ) / sin  sin    exp   e ( L  y ) / sin  sin  
2
 [2  rs ( ,  )  r p ( ,  ) ].
2
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Here the number of total internal reflections from the rough surface is given by
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 2L  y

N R ( ,  )  
cot  cot   ,
 2T

(2)
(3)
3
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where we have introduced the FSC thickness T. The brackets indicate the floor function which
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rounds NR to the largest previous integer.
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Results were calculated for the collection efficiency of the FSC for a representative collector
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with index nc=1.5 coupled into a semiconductor of index ns=3.35. We assumed a thickness
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given by T=L/50, which is typical for a medium-scale FSC device with a geometric gain of
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50. Different values of T will only change the total amount of scattering loss, not the
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qualitative response. Figure 1a shows computed values obtained by numerical integration of
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Eq. (2). The curve in absence of surface scattering (closed circles) corresponds to the Weber
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and Lambe result.2 Introduction of a surface roughness characterized by a =1.5° divergence
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results in a reduction of the efficiency by several percent (open circles). Also shown are the
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total (triangles) and relative (diamonds) scattering losses, where the latter is normalized to the
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collection efficiency for a smooth film (=0). The right-hand panel in Fig. 1 shows the
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absolute and relative scattering losses as a function of for a value of eL=10-2, i.e. low
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reabsorption. Above =1°, the scattering loss increases roughly linear with the diffraction
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width . The loss percentage of 5% for 2.5° indicates the importance of surface
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roughness optimization in experimental FSC. For comparison we have also shown the results
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for T=L/5.2, which corresponds to the small-scale experimental configuration in this work. As
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can be seen the scattering loss does not scale linearly with the system size, i.e. the number of
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internal reflections, indicating that the majority of the loss is accumulated during the first few
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reflections. This can be understood from the fact that the angular range around the critical
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angle gets depleted of photons after the first few internal reflections and therefore the effect
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saturates.
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Figure 1. (Left) collection efficiency for the FSC as a function of eL calculated using the
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modified Weber and Lambe model Eq. (2), for a perfectly smooth surface (, closed dots)
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and a surface roughness characterized by ° and T=L/50 (open dots). (Triangles, blue)
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Absolute scattering loss and (diamonds, red) relative scattering loss normalized to the result for
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. (Right) (Triangles, blue) Absolute and (diamonds, red) relative scattering loss at eL =
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10-2 as a function of the 1/e half width of the diffraction cone, for T=L/50 (symbols) and for
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T=L/5.2 (dashed lines).
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2 Method
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2.1 Sample
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The substrate used in this study was a BK7 glass slide with dimensions of 26×26×5 mm3.
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Before depositing the layers the slides were thoroughly cleaned. A solution of 9% - 10%
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PMMA dissolved in Chlorobenzene (Micro Chem) was spin-coated (Laurell EDC-650-
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15TFM) at a speed of 4000 rpm for 1 minute. The thickness of the spin-coated layer was
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measured by using a profilometer (Talysurf-120 L, Taylor Hobson Ltd.) and was found to be
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24.85 μm. The surface morphology of the PMMA layer was characterized using Atomic
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Force Microscopy. Figure 4 shows a representative AFM image revealing that the layer shows
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a roughness of around 48 nm over a typical correlation length of several m. This roughness
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may be associated with the relatively large thickness of PMMA required for fluorescent
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conversion in a realistic FSC, resulting in height variations during the spin-coating process.
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However, fluctuations may also be intrinsic to the polymer itself. Large scale heterogeneities
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have been observed even in very pure PMMA, and its nature may be related to local
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alignment of polymer chains and strain produced during the drying process.6
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Figure. 2. Atomic Force Microscopy measurement of the PMMA layer,
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showing height fluctuations of ~48 nm over a distance of several m.
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2.2 Scattering setup
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For the characterization of light scattering, the samples were illuminated with a laser beam
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impinging from one side-edge of the glass. The sample was placed at the center of a circular
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goniometer table illuminated with a laser. A silicon photodiode (SM05PD1A, Thorlabs) was
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mounted at the edge of the table as shown in Fig. 3. The distance between the sample and the
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detector was 27 cm. A pinhole with a diameter of 1 mm was placed in front of the silicon
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photodiode which gives a solid angle for collection of 0.2°. A frequency of 185 Hz was
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applied via a chopper to the beam and the current was detected by a photodiode connected via
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a lock-in amplifier (SR810). The lock-in amplifier was used to decrease the noise of the
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measurement by locking in to the chopper frequency and the average of two current
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measurements, captured 1 second apart, was taken. The interface of this setup was
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programmed in Labview (8.6) software. A sample holder was fixed to the center of the
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goniometer table. The laser beam was aligned perpendicular to the surface of the sample, and
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the photodiode could be rotated around the table for measuring the scattered intensities at
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different angles. The scattered light intensity was measured between 0° and 180° with a
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scattering angle increment of 0.5°.
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Figure 3. Schematic of the experimental setup for the angle-dependent scattering
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measurements.
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2.3 Scattering measurement of FSC by angular dependence method
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This section describes the method for measuring the scattering of PMMA spin coated on top
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of the BK7 glass by illumination from one edge in different incident angles and detection
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from another edge.
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Figure 4. PMMA spin coated glass illuminate with laser light from one edge and the
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scattered light was detected from another edge at different angles, θ1=15° and
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N=14.21mm, M=11.79 mm. The error in this experiment was 0.8 mm which is the
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width of the laser beam.
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The distance of M (distance from detector inside of the glass) can be calculated by
sin 1
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n
 2 for n2=1.5 and n1=1
sin  2
n1
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tan 2 
D
2N
, N=26-M .
(4)
(5)
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Values for N for different incident angle from one edge are shown in Table 1. If N <26 mm,
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the probability that the light is totally internally reflected increases if 90-θ2 is more than the
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critical angle.
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Table 1. Distance from the incident beam inside of the glass for different incident angle, the
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error in this calculation is 0.8 cm, given by the width of the laser beam.
Angle of
incidence
No. of TIRS
Distance M(cm)
0°
0
N.A.
15°
1
14.21
35°
2
6.04
45°
3
4.77
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8
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For incident angles less than 8.3° there is no reflection inside of the collector. For incident
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angles between 8.3° to 24.6° there is one reflection inside of the collector, for the incident
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angles between 24.6° to 40.5° there are two reflections inside the collector, and for the
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incident angle of 45° there are three reflections inside the collector.
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3 Results
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Values of the intensity of scattered light measured over a range of angles are plotted in Fig. 5
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for three different angles of incidence of 15º, 35º and 45º. To compare the results the
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maximum intensity for all the measurement was defined as 0º. Clearly, the light cone
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transmitted through the PMMA-coated FSC is considerably broadened compared to a bare
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glass slide. As seen in Table 2, the angular width of the scattered intensity cone increases for
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larger angles of incidence up to 45°. The width here is determined as the length over which
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the intensity drops to 1/e of the maximum. The measurements at 15° and 35° show the same
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width, which can be understood from the fact that the second internal reflection takes place at
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the opposite glass surface without PMMA. As we will show below, the observed behavior is
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consistent with diffraction of the laser beam caused by the roughness of the PMMA layer.
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In a simple model, we include the diffraction by describing the reflected light as a collection
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of diffraction-limited cones with an aperture given by the correlation length of the surface
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roughness.
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Intensity (norm.)
1.00
-5
Glass
15
35
45
Gaussian
0.50
-4
-3
-2
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0.00
-1
0
1
Scattering angle ( )
2
3
4
5
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Figure. 5. Angular dependence of normalized intensity for different incident
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angles for 670 nm incident light. Lines: Gaussian fits.
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Table 2. Beam width for different incident angles of incidence.
Angle of incidence
0° laser beam
1/e half-width of the scattered
beam (º)
0.6
15°
1.3
35°
1.4
45°
1.6
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186
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Thus, if we consider the PMMA surface as a collection of uncorrelated areas, the angular
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distribution of the reflected intensity can be fitted to the Fraunhofer diffraction formula for the
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circular aperture, given by7
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I  I0
G 2 2 J 1 ( ka sin  )
ka sin 
2 r 2
2
(6)
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where the G=πa2, a is the radius of the aperture, r is the distance from observation point to
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shadow area, λ is the wavelength of the incident, J1 is the Bessel function of the first order and
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k is the optical wavenumber.
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Equation (6) gives us the intensity of the scattered light in different detection angles known as
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the Airy pattern. We note that the use of a Gaussian angular spread in Eq. (1) is a good
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approximation as the central lobe of the Airy pattern shows close agreement with a Gaussian
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profile.7 Good agreement is obtained for a radius of the aperture of 11 µm for the data taken at
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670 nm wavelength and at 15º angle of incidence, as shown in Fig. 7.
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200
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Figure 6. Scheme showing PMMA spin coated on top of the glass illuminate
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with laser light from one edge in incident angle of 15° and the forward
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scattered light detected at different angles. The roughness from spin coated
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PMMA is assumed to produce an effective circular aperture when the laser is
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reflected by the rough layer.
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1.00
15°, 670 nm
Intensity (norm.)
Fraunhofer diffraction, 670 nm
-5
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0.50
-4
-3
-2
0.00
-1
0
1
2
Scattering angle (degree)
3
4
5
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Figure 7. Normalized angular dependence of I in 15° incident angles in
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PMMA side for 670 nm. The solid line was fitted by using equation 4, circles
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indicate the experimental results.
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The validity of the diffraction model can be verified by changing the optical wavelength.
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Figure 8 shows the scattered light intensity measured over a range of angles when we
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illuminate the sample from one edge at 15 degree through the PMMA part at different
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wavelengths of respectively 401 nm, 532 nm, 633 nm , and 670 nm. Clearly, the scattered
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light cones show a decrease in width for decreasing wavelength.
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The Fraunhofer diffraction for the circular aperture was calculated when the size of the
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particle is fixed at 11 µm while the wavelength was decreased according to the used lasers.
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We find good quantitative agreement with our experimental results in Fig. 8 (lines), showing
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that the diffraction pattern gets more and more compressed into a narrow cone around the
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forward direction for shorter wavelengths.
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1.00
15°, 670 nm
Fraunhofer diffraction, 670 nm
15°, 633 nm
Intensity (norm.)
Fraunhofer diffraction, 633 nm
15°, 532 nm
Fraunhofer diffraction, 532 nm
0.50
15°, 401 nm
Fraunhofer diffraction, 401 nm
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0.00
-1
0
1
Scattering angle (°)
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Figure 8. Comparison of normalized scattering intensity at wavelengths of
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670 nm, 633 nm, 532 nm, and 401 nm detected over a range of angles with
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illumination from 15° incident angles through the PMMA part of the sample.
-5
-4
-3
-2
2
1.00
3
4
5
15 °, without optical gel
15 °, with optical gel
Intensity (norm.)
Gaussian
-5
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0.50
-4
-3
-2
0.00
-1
0
1
2
Scattering angle (degree)
3
4
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Figure 9. Comparison of intensity of scattered light at a wavelength of 670
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nm detected over a range of angles with illumination from 15° incident
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angles once when the PMMA top surface covered with the optical gel
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(circles, blue) and without optical gel (diamonds, green).
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To further investigate the cause of the scattering in spin coated PMMA on top of the BK7
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glass, the top surface in PMMA (n=1.49) part was covered by optical gel with refractive index
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of 1.4646(THORLABS). The surface tension of the gel produced a smooth film which
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compensated the surface roughness of the PMMA layer. Subsequently, the irradiance of
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scattered light as a function of angle was measured when illuminating the sample at a 15º
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angle of incidence. The results are compared with a measurement taken on the same sample
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before the gel was added. Results are shown in Fig. 9. As we can see, application of the
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optical gel results a decrease of the angular width of the scattering cone. This effect
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unambiguously shows that the cause of the scattering in spin coated PMMA on top of the
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glass is the surface roughness. Compared to the surface scattering contribution, the bulk
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scattering in this type of sample is negligibly small, i.e. of order 10-5 cm-1.6
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It is relevant to consider whether diffraction effects play a role in the situation that the
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illumination is provided by mutually incoherent fluorescent molecules. To assess this we have
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to compare the divergence of a point source with the diffraction provided by the surface
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roughness. For example, when a point source illuminates an area of 10 m at a distance of 1
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mm from the source, this corresponds to a divergence angle of 10-2 radians or 0.5º, i.e. smaller
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than the typical diffraction broadening in our samples. Therefore surface scattering losses are
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relevant for incoherent fluorescence emission at propagation distances larger than several
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hundred m, which is the case in FSC.
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4 Conclusion
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In summary, the contribution of light scattering in fluorescent solar collectors (FSC) based on
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a spin-coated dye/polymer layer was investigated. Measurements of the transmission cones as
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a function of total internal reflections (TIR) revealed that the predominant loss factor is
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surface scattering at the top layer, caused by roughness of the spin-coated PMMA.
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Good agreement is obtained when we consider the PMMA roughness as an coherent aperture
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with a circular shadow therefore the angular distribution of the scattering intensity can be
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fitted using far-field Fraunhofer diffraction. The width of the scattered beam increases when
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the wavelength is increased from 401 nm to 670 nm, in good agreement with Fraunhofer
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diffraction for a circular aperture at different wavelengths where the size of the aperture is
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fixed to the value of 11 µm. This value is of the same order as the correlation length of the
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height variations observed in AFM measurements. The value of the scattering was found to
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decrease significantly when we cover the top surface with optical gel, which unambiguously
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shows that surface scattering is the predominant loss factor in our FCS compared to bulk
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scattering effects. Surface roughness is caused by to the large required thickness of the
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PMMA-dye layer for the operation of the FSC. Our results indicate that the performance of
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the FSC can be improved by using a subsequent planarization with a thin index-matching
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layer of low roughness.
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Acknowledgements The authors would like to thank L. Danos for help with the
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experimental setup. This project was funded partly through UK SUPERGEN (Sustainable
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Power Generation and Supply) PV21 Programme.
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