Design of wavelength selective concentrator for micro PV/TPV systems using evolutionary algorithm

Design of wavelength selective concentrator for micro PV/TPV systems using evolutionary

algorithm

Noboru Yamada1,*

and Toshikazu Ijiro1

1Department of Mechanical Engineering, Nagaoka University of Technology, Nagaoka, Japan *[email protected]

Abstract: This paper describes the results of exploring photonic structures that behave as wavelength selective concentrators (WSCs) of solar/thermal radiation. An evolutionary algorithm was combined with the finite-difference time-domain method (EA-FDTD) to determine the optimum photonic structure that can concentrate a designated wavelength range of beam solar radiation and diffusive thermal radiation in such a manner that the range matches the photosensitivity of micro photovoltaic and thermophotovoltaic cells. Our EA-FDTD method successfully generated a photonic structure capable of performing wavelength selective concentration close to the theoretical limit. Our WSC design concept can be successfully extended to three-dimensional structures to further enhance efficiency.

©2011 Optical Society of America

OCIS codes: (220.1770) Concentrators; (040.5350) Photovoltaic; (230.7408) Wavelength filtering devices; (350.6050) Solar energy; (350.5610) Radiation; (220.4298) Nonimaging optics.

References and links

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#146229 - $15.00 USD Received 19 Apr 2011; revised 13 Jun 2011; accepted 15 Jun 2011; published 22 Jun 2011(C) 2011 OSA 4 July 2011 / Vol. 19, No. 14 / OPTICS EXPRESS 13140

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1. Introduction

Photovoltaic (PV) and thermophotovoltaic (TPV) cells are promising devices that can generate electricity directly from solar radiation and thermal infrared radiation, respectively. Further, the rapid progress in nano/micro-fabrication technologies has made PV/TPV cells and/or their systems incredibly small [1]. For example, a workable PV cell with nanowires averaging 400–1000 nm in length and 30–50 nm in diameter has been reported [2]. Nanowires are also efficient for light trapping [3] and as thermoelectric materials [4] that boost electricity generation. A TPV system in which the gap between its emitter and the TPV cell is less than 1 μm has been proposed in order to exploit an evanescent effect that enhances the radiation flux into the TPV cell [5]. Furthermore, microcrystalline silicon solar cells with 14.9% efficiency have been presented [6]. Such research trends have motivated and inspired us to develop next-generation micro PV/TPV systems that include microconcentrators. Microconcentrators contribute to reductions in the PV/TPV cell areas in conventional scale concentrating photovoltaic (CPV) systems; moreover, similarly, they can also contribute to reductions in the PV/TPV cell areas, i.e., scarce material consumption, in micro PV/TPV systems. It is important to design microconcentrators to perform wavelength selective concentrations that can concentrate lightwaves within a designated wavelength range of solar/thermal radiation and ensure that the range matches the spectral response of PV/TPV cells. However, it is quite difficult to realize such wavelength selective concentrators (WSCs) in ordinary scale ray optics-based systems using mirrors and lenses. On the other hand, the potential to realize WSCs in nano/microscale wave optics-based systems using photonic structures is high because not only reflection and refraction but also diffraction can be used for the design. In fact, photonic structures exhibit some extraordinary optical properties; for example, two-dimensional photonic structures with air holes embedded in flat slabs have negative refractive indexes at specific wavelengths [7–9]. Burgos et al. have reported negative-refractive-index structures operating at visible wavelengths [10]. Photonic structures can also control thermal emission spectra [11–14]. Further, photonic structures act as optical filters and improve the efficiency and power density of TPV systems [15].

These state-of-the-art researches imply that there is great potential for realizing micro PV/TPV systems with WSCs. However, the flexibility of photonic structure designs also leads to difficulties in determining their optimal and efficient structures; time-consuming trial-and-error methods are typically used but they prevent quick exploitations of the WSC concept. Therefore, it is necessary to develop an efficient method to explore the optimum designs. Evolutionary algorithm (EA) is an optimization method that is suitable for the design of optical structures/geometries. Shirakawa, et al. employed a type of EA called genetic algorithm (GA) to design binary diffractive microlenses with subwavelength structures [16].


GAs have also been used to design a silicon integrated photonic lens operating at 1550 nm [17] and multilayered antireflection coatings [18]. One of the authors of the present study combined a GA with an immune algorithm (IA) to accelerate the optimization speed of nonimaging Fresnel lens geometry [19].

In this study, we present the first results of our investigations into photonic structures that operate as WSCs for solar/thermal radiations in micro PV/TPV systems. The finite-difference time-domain (FDTD) method [20] is used with an EA to determine the optimum photonic structure. Our EA successfully generates photonic structures that perform wavelength selective concentrations.

2. Computational model and method

2.1 Computational model

Figure 1 shows the computational model of (a) micro PV system and (b) micro TPV system (hereafter called the PV model and TPV model, respectively). In the PV model, the radiation source is a parallel plane wave with a broadband wavelength of the AM1.5D standard spectrum [21]. Further, we have considered a micro Si cell whose photovoltaic conversion has a spectral response ranging from 0.4 to 1.2 μm. The photonic structure material, i.e., WSC, is poly methyl methacrylate (PMMA) with a constant refractive index nPMMA = 1.49 over the simulated wavelength range. On the other hand, in the TPV model, the radiation source is an isotropic pointwise source that emits blackbody (BB) radiation at 1500 K spectrum. The photonic structure material is Si with a constant refractive index of nSi = 3.4. We have also considered a micro GaSb cell with a spectral response of photovoltaic conversion ranging from 0.8 to 1.8 μm. The geometric concentration ratio Cg is 12 and 9.6 for the PV and TPV models, respectively. A perfectly matched layer (PML) absorbing boundary condition [22] has been applied to the boundary surrounding the computational domain in both models.

In the present computations, two types of photonic structures have been examined as the WSC candidate; they are shown in Fig. 2. The circular-gap structure (Fig. 2(a)) comprises circular air gaps with a refractive index of nAir = 1.0 embedded in a staggered array in the slab material (PMMA). The distances between the centers of each circular gap were fixed as a in the x and y directions. The diameter of each circular gap, D, was varied by using the EA. At the edge of the photonic structure surface (aperture side), the gaps were truncated at the center of the gap in order to achieve antireflection and high transmission. On the other hand, the layered binary structure (Fig. 2(b)) comprises square segments arranged in an array. The attribute of each segment is either a square air gap (assigned 0) or square material (assigned 1). During the computations, i.e., the evolution processes, EA changes the diameter of the individual circular gap, D, and the segment attribute 1 or 0 in order to determine the optimum structure. The other parameters, i.e., the total number of air gaps and segments, were fixed during the computation. Table 1 summarizes the fixed and varied parameters. It is noteworthy that the fixed parameters listed in Table 1 are the best values in our investigations over a certain possible range. The FDTD method was used to rigorously solve Maxwell’s equations for the lightwave propagation. All the computations were performed in a two-dimensional spatial domain. Lightwave behaviors in the TE and TM modes were examined by the FDTD.


Si cell

SR: 0.4–1.2 μm

Perfectly matched layer

x

y

Photonic structure

nPMMA = 1.49

Pla

ne w

ave s

ourc

e

GaSb cell

SR: 0.8–1.8 μm

Perfectly matched layer

x

y

Photonic structure

nSi = 3.4

Pointwise source

Fig. 1. Two-dimensional computational model. (a) A parallel plane wave source emulating the AM1.5D solar spectrum was applied; the spectral response (SR) of the Si cell was set as 0.4–1.2 μm. (b) A pointwise source emulating perfectly diffuse blackbody radiation at 1500 K was applied; the SR

of the GaSb cell was set as 0.81.8 μm.

Air gap

nAir = 1.00

0

0

0 1

1

a

Air gap

nAir = 1.0

Photonic structure

a

w

Fig. 2. Two types of photonic structures. Both (a) circular-gap and (b) layered binary structures were individually examined in the PV/TPV model. The blue region represents poly methyl methacrylate (PMMA) or silicon (for the PV or TPV model, respectively), while the white region represents the air gap.

Table 1. Parameters for Two Types of Photonic Structures

Fixed parameters Varied by EA

Circular gap

Aperture width: 9.6 μm Si cell width: 1.0 μm Thickness of photonic structure: 1.8 μm Total number of air gaps: 34 × 7 in staggered arrangement Distances between centers of individual air gaps: a = 0.3 μm Position of centers of individual air gaps

Diameter of each air gap, D

0.1a ≦ D ≦ 0.9a

Layered binary

Aperture width: 12 μm GaSb cell width: 1.0 μm Thickness of photonic structure: 2.0 μm Total number of square segments: 30 × 5 in grid arrangement Side length of square segments: w = 0.4 μm Position of centers of individual square segments

Attribute of individual segment 1 (material) or 0 (air gap)


2.2 Evolutionary algorithm (EA)

In order to explore the optimum photonic structure under the given conditions, an EA was implemented with the FDTD method (EA-FDTD) in the form of a hybrid algorithm that comprised GA and IA. After conducting numerous preliminary trials, this hybrid algorithm performed the fastest evolution of the structure. Figure 3 shows the flowchart of the present EA-FDTD. The major steps of the EA-FDTD are as follows:

(1) Generate initial structures.

(2) Evaluate the performance of each photonic structure using the FDTD as the WSC by calculating an evaluation index that should be appropriately defined before the computation.

(3) Select photonic structures that will exist in the next generation by comparing the evaluation indexes.

(4) Generate new structures via GA manipulations—crossover and mutation—using the information of two photonic structures that were randomly chosen from the surviving structures.

Figure 4 shows an example of crossover and mutation for both types of structures. In crossover, some diameters of the gaps are randomly exchanged between two structures according to a predefined probability termed as the ―crossover rate‖ (= 0.5 in the present study). In mutation, some diameters of the gaps in a single structure are randomly changed according to a predefined probability termed as the ―mutation rate‖ (0.05 in the present study). GA manipulations also change the segment of the attributes in the layered binary structure.

(2) EvaluationEvaluate optical performance by FDTD

The most optimal structure

(1) Start calculationGenerate initial structures

(6) Elimination and supplementationEliminate similar structures

and supply eliminated

(3) SelectionSelect good structure

based on evaluation index

(4) ManipulationGenerate new structures

by crossover and mutation

(5) Find similar structuresCalculate similarityamong structures

Go to IA?

GA IA

YesNo

Exceeded maximumgeneration limit?

No

Yes

Fig. 3. Flowchart of EA with finite-difference time-domain (EA-FDTD) for design of wavelength selective concentrator (WSC).


A typical GA process that involves repetitions of steps (1)–(4) automatically improves the optical performance of structures. However, the abovementioned GA process tends to converge to a local solution that is often not the best possible solution. Therefore, we added IA process (steps (5)–(6)) to the GA process in order to mitigate the convergence to the local solution and to accelerate the speed of evolution. Some previous reports have described the advantages of combining IA with GA [18]. The IA process steps are as follows:

(5) Find similar structures by evaluating the structural similarities among the survived structures, and subsequently, create groups of the similar structures.

(6) Eliminate all the structures except one from each group of similar structures and randomly supply new structures to maintain the total number of structures.

In this manner, the IA process contributes to maintaining a variety of structures. This results in an evolution speed that is faster than that achieved by only the GA process.

0

0

1

1

0

1

0

0

0

0

0

0

1

1

1

1 1

1

0

1

1

1

0

0

0

0

0

0

1

1

1

1 0

1

0

0

1

1

1

0

and

and

Fig. 4. Genetic manipulations that generate new structures. (a) crossover and (b) mutation. The manipulated parts are shown in red. Upper: circular-gap structure. Lower: layered binary structure.

2.3 Evaluation index

Defining an appropriate evaluation index is an important procedure in EA computations. The desirable characteristic of the WSCs considered for the PV/TPV model is that they should concentrate lightwaves only within a limited wavelength range that matches the range of the spectral responses of the applied PV/TPV cell. The spectral optical efficiency ηλ_is defined as the ratio of the lightwave energy that is successfully concentrated onto the cell to the total lightwave incident onto the aperture of the concentrator, i.e., the photonic structure. In the PV model, ηλ_ideal should be unity in the Si cell’s spectral response range of 0.4–1.2 μm, whereas ηλ_ideal should be zero outside this range. However, obviously, even if lightwaves outside this range are concentrated onto the Si cell (ηλ > 0), the lightwave energy is not converted into electricity but into heat, and this decreases the photovoltaic conversion efficiency. An identical scenario is valid for the TPV model. Another factor that should also be considered in the case of the evaluation index is the spectrum of the radiation source, i.e., solar radiation and BB radiation. The concentrator should concentrate the maximum possible energy from the lightwave source onto the cell. Consequently, the evaluation index F can be defined as the integral of the deviation from the ideal ηλ_ideal weighted by the normalized wave source spectrum as follows:

_

0idealF E d

(1)


Here Eλ is the normalized wave source spectrum. AM1.5D spectrum was used for the PV model, and BB radiation spectrum at 1500 K was used for the TPV model. From the definition, the lower the F value, the better the WSC, and for an ideal WSC, F = 0.

3. Results and discussion

Our inhouse source code was implemented and executed by employing the parallel computing technique. Figure 5 shows the manner in which the photonic structures evolved toward lower F values as the number of generation (iteration) steps for the PV/TPV model increased. The computation was terminated after the 10000th generation. From Fig. 5, in both PV/TPV models and with both types of photonic structures, the F value successfully decreased with the generation step. For the PV model, the layered binary structure exhibited the best performance, that is, the lowest resultant F value (= 36.0) at the 10000th generation. On the other hand, in the TPV model, the circular-gap structure exhibited the best performance at the 10000th generation. This implies that the layered binary structure is suitable for the wavelength selective concentrations of parallel plane lightwaves while the circular-gap structure is suitable for those of diffuse lightwaves. This result is true in the case of the TE mode. Since the polarization characteristics of solar radiation and thermal radiation have been typically dealt as the average of TE and TM modes, EA computations for not only TE-mode lightwaves but also for both TE and TM modes were conducted. The results for the average of the TE and TM modes exhibited a similar trend (not shown); however, the resultant F values were approximately 1.3 and 1.2 times higher than that for only the TE mode in the case of PV and TPV models, respectively. Furthermore, the resultant F values for only the TM mode were 1.05 and 1.06 times higher than the value for only the TE mode in the case of the PV and TPV models, respectively.

55

60

65

70

75

1 10 100 1000 10000

Circular gap

Layered binary

34

38

42

46

50

1 10 100 1000 10000

Generation

Evalu

atio

n ind

ex,

F

F = 45.8

F = 36.0

F = 71.8

F = 57.1

Fig. 5. Evaluation index F vs. generation (iteration) step for selective wavelength concentration (TE mode).

Figure 6 shows the obtained spectral optical efficiency ηλ of the 1st and 10000th generations for layered binary structures in the PV model (Fig. 6(a)) and circular-gap structures in the TPV model (Fig. 6(b)); here, the TE mode corresponding to Fig. 5 is shown in the upper part, and the average of the TE and TM modes is shown in the lower part. Figure 6 also shows the theoretical efficiency limit. In the present analysis, the theoretical limit of the optical efficiency was derived from the Fraunhofer diffraction pattern and Fresnel reflection. The Fraunhofer diffraction pattern gives the maximum spot size of the concentrated light at the focal plane for a certain wavelength [23]. Hence, we can calculate the theoretical maximum fraction of the energy contained within a cell area. An increase in the wavelength causes the spot size to increase, resulting in smaller optical efficiencies. In addition, the concept of Fresnel reflection leads to potential reflection losses occurring at the uniform planer interface. The generated structure clearly demonstrates wavelength selective concentrations. We first note the results for the TE mode. In the PV model (Fig. 6(a); upper), the peak value reaches ηλ = 0.55 at 1.0 μm within the range of the spectral responses of the Si cell, whereas, in contrast, ηλ is less than 0.1 outside the range. The results in the case of the TPV model (Fig. 6(b); upper) also exhibit a similar tendency. However, the peak value is ηλ =


0.36 at 1.58 μm, which is almost half of that in the PV model, while ηλ outside the range fluctuates and exceeds 0.15 at several wavelengths. This implies that it may be more difficult to achieve wavelength selective concentrations of diffuse radiations from a pointwise source than to achieve beam radiation. On the other hand, the lower figures in Fig. 6 show the results for the average of the TE and TM modes. The EA-generated structures also show wavelength-selective concentrations, but the performance of the wavelength selective concentration is less than that in the TE mode. This implies that a two-dimensional structure is not sufficient to achieve wavelength selective concentrations for the TE and TM modes. A three-dimensional structure design will improve the WSC performance for the average TE and TM modes, i.e., for actual solar/thermal radiations.

0 2 4 6

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4

0

0.2

0.4

0.6

0.8

1Ideal

1st generation

10000th generation

Theoretical limit

Wavelength, λ [μm]

Optical effic

iency,

ηλ

TE mode

TE and TM modes

TE mode

TE and TM modes

Fig. 6. Optical efficiency of EA-generated photonic structures and theoretical limits. (a) EA-generated layered binary structure (PV model); (b) EA-generated circular-gap structure (TPV model). Upper: TE mode. Lower: Average of TE and TM modes.

12 μm

2 μm

9.6 μm

1.8

μm

x

y

TE mode

TE and TM modes

TE mode

TE and TM modes

Fig. 7. EA-generated photonic structures at 10000th generation for selective wavelength concentrations (TE mode).(a) EA-generated layered binary structure in PV model (b) EA-generated circular-gap structure in TPV model.Upper: TE mode. Lower: Average of TE and TM modes.

The efficiency of the WSC in the TPV model was closer to the theoretical limit than that in the PV model but it was still not beyond the limit. A possible reason for this is that the circular-gap structure behaved like a negative-refractive-index layer that has been reported in some types of photonic crystals and metamaterials although this property is still debatable, particularly for photonic crystals.

Figure 7 shows the geometry of the 10000th

-generated layered binary structure for the PV model and the circular-gap structure for the TPV model, which were optimized for the TE


mode and the average of the TE and TM modes. The obtained EA-generated structures were non-periodic structures. In the PV model, the layered binary structure was superior to the circular-gap one. This is consistent with the trends of previous researches [for example 16,24, ] that have reported a binary type structure for a lens concentrating parallel plane lightwaves. On the other hand, in the TPV model, the circular-gap structure was superior to the layered binary one. This is also consistent with the trends in previous studies [for example 7–9, ] that reported circular and other periodic air gaps embedded in a flat slab, which led to a negative-refractive-index layer that can concentrate diffuse lightwaves from a pointwise source beyond the diffraction limit at a specific wavelength. However, the primary difference between the present and previous studies is that the present study has attempted to concentrate the broadband wavelength while previous studies have focused on monochromatic concentrations. We believe that this difference causes our structures to be more non-periodic than those of the previous studies, especially in the TPV model. However, despite our best effort, we were not able to determine the physical reasons for the distributions found for holes or squares. In fact, when we attempted to replace all the air holes in the final geometry (Fig. 7(b)) into squares with the equivalent gap area, the results did not significantly change. This could imply that the degrees of freedom, i.e., the allowed number and parameter ranges play an important role in the present design method. Nevertheless, further researches are required in order to clarify this further.

In terms of the applicability of the present design method, basically, the diameter in the algorithm does not have a lower limit. However, computational resources such as memory spaces may limit the diameter because the FDTD method requires far finer numerical grids than the diameter. Further, in terms of fabrication, small diameter is not practical even in the case of the current advanced fabrication technologies. Hence, in the present analysis, the diameter was varied from 0.1a to 0.9a where a was set as a = 0.1, 0.3, 0.5, and 0.7 μm. We only demonstrate the best performance case of a = 0.3 μm.

λ = 1 [μm]

ηλ = 54.6 [%]

Si cellSR: 0.4–1.2 [μm]

(a)

λ = 2 [μm]

ηλ = 2.9 [%]

(b)

λ = 1.58 [μm]

ηλ = 36.1 [%]

GaSb cell

SR: 0.8–1.8 [μm]

(c)

λ = 1.9 [μm]

ηλ = 0.4 [%]

(d)

Fig. 8. Two-dimensional simulation of the steady |Ez|2 field intensity using the 10000th-

generation photonic structure (TE mode). (a) λ within SR range of Si cell (Media 1); (b) λ outside SR range of Si cell (Media 2);(c) λ within SR range of GaSb cell (Media 3); (d) λ outside SR range of GaSb cell (Media 4).

Figure 8 shows the steady |Ez|2 electric field intensity of the 10000

th-generated layered

binary and circular-gap structures for the PV and TPV models, respectively. Movies showing time-varying electric field intensity have also been presented (supplement). Figures 8(a) and (c) show the electric fields when the wavelength of the lightwave is within the range for the PV and TPV models, respectively. The lightwave is concentrated on the Si and GaSb cells. In contrast, Figs. 8(b) and (d) show the electric fields when the wavelength is outside the range. In the PV model, most of the lightwave outside the range tends to miss the Si cell and pass through beside it. In the TPV model, most of the lightwave outside the range tends to be reflected from the GaSb cell. This is a preferable characteristic for TPV systems because the lightwave energy emitted from the lightwave (heat) source is not wasted, and the WSC acts as a thermal insulator to reduce the heating energy of the heat source. Furthermore, it is obvious that the circular-gap structure in the TPV model behaves like a negative-refractive-index layer for wavelength selective concentrations.


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4. Conclusion

In this paper, we have presented a conceptual design for developing WSCs for micro PV/TPV systems using EAs. Our design method employed an EA to generated non-periodic photonic structures that acted as broadband WSCs. For diffuse thermal radiation, the generated photonic structure exhibited remarkable optical performances close to the theoretical limit of the designated wavelength range. However, for actual solar and thermal radiation sources emitting lightwaves with both TE and TM modes, it is still possible to further explore the optimal structure to achieve better wavelength selective concentration performances. We think that three-dimensional designs will help to enhance the performances. Our design method can also be applied to three-dimensional designs and to different types of non-periodic photonic structures. Such an application will be the subject of our future study.


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Design of wavelength selective concentrator for micro PV/TPV systems using evolutionary algorithm