Broadband Gerchberg-Saxton algorithm for freeform ... - OSA Publishing

Broadband Gerchberg-Saxton algorithm for freeform diffractive spectral filter design Shelby Vorndran,1,* Juan M Russo,2 Yuechen Wu,2 Silvana Ayala Pelaez,2 and Raymond K. Kostuk1,2 2

1 College of Optical Sciences, The University of Arizona, 1630 E University Blvd, Tucson, Arizona 85721, USA Department of Electrical and Computer Engineering, The University of Arizona, 1230 E Speedway Blvd #ECE249, Tucson, Arizona 85721, USA * [email protected]

Abstract: A multi-wavelength expansion of the Gerchberg-Saxton (GS) algorithm is developed to design and optimize a surface relief Diffractive Optical Element (DOE). The DOE simultaneously diffracts distinct wavelength bands into separate target regions. A description of the algorithm is provided, and parameters that affect filter performance are examined. Performance is based on the spectral power collected within specified regions on a receiver plane. The modified GS algorithm is used to design spectrum splitting optics for CdSe and Si photovoltaic (PV) cells. The DOE has average optical efficiency of 87.5% over the spectral bands of interest (400-710 nm and 710-1100 nm). Simulated PV conversion efficiency is 37.7%, which is 29.3% higher than the efficiency of the better performing PV cell without spectrum splitting optics. ©2015 Optical Society of America OCIS codes: (350.6050) Solar energy; (040.5350) Photovoltaic; (050.1940) Diffraction; (050.1970) Diffractive optics; (090.1760) Computer holography; (150.1135) Algorithms.

References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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Received 10 Aug 2015; revised 18 Sep 2015; accepted 21 Sep 2015; published 8 Oct 2015 30 Nov 2015 | Vol. 23, No. 24 | DOI:10.1364/OE.23.0A1512 | OPTICS EXPRESS A1512

15. T. R. Sales and D. H. Raguin, “Multiwavelength operation with thin diffractive elements,” Appl. Opt. 38(14), 3012–3018 (1999). 16. J. Z. Wang, J. S. Ye, Q. L. Huang, X. Xu, D. M. Li, Q. B. Meng, and G. Z. Yang, “Design optimization of highly efficient spectrum-splitting and beam-concentrating diffractive optical element for lateral multijunction solar cells,” Chin. Phys. B 23(4), 044211 (2014). 17. Q. Huang, J. Wang, B. Quan, Q. Zhang, D. Zhang, D. Li, Q. Meng, L. Pan, Y. Wang, and G. Yang, “Design and fabrication of a diffractive optical element as a spectrum-splitting solar concentrator for lateral multijunction solar cells,” Appl. Opt. 52(11), 2312–2319 (2013). 18. G. Kim, J. A. Domínguez-Caballero, and R. Menon, “Design and analysis of multi-wavelength diffractive optics,” Opt. Express 20(3), 2814–2823 (2012). 19. G. Kim, J. A. Dominguez-Caballero, H. Lee, D. J. Friedman, and R. Menon, “Increased photovoltaic power output via diffractive spectrum separation,” Phys. Rev. Lett. 110(12), 123901 (2013). 20. N. Mohammad, P. Wang, D. J. Friedman, and R. Menon, “Enhancing photovoltaic output power by 3-band spectrum-splitting and concentration using a diffractive micro-optic,” Opt. Express 22(106), A1519–A1525 (2014). 21. J. A. Domínguez-Caballero, “Optimization of the holographic process for imaging and lithography,” Ph.D. Thesis, Massachusetts Institute of Technology (2010). 22. R. Menon, “Ultra-high efficiency multi-junction solar cells using polychromatic diffractive concentrators,” United States Patent 8669461 (March 11, 2014). 23. C. H. Henry, “Limiting efficiencies of ideal single and multiple energy gap terrestrial solar cells,” J. Appl. Phys. 51(8), 4494–4500 (1980).

1. Introduction Diffractive Optical Elements (DOEs) are essentially transmittance filters with lateral and depth spatial features on the order of a wavelength. These features modify phase or amplitude of the incident field. The DOE is designed to produce a desired spatial and spectral light distribution at a specific distance. Computational algorithms can be used to design DOE surface relief patterns. Algorithms investigated for this purpose include geometrical transformation [1], direct binary search [2], simulated annealing [3], iterative Fourier transform [4], Gerchberg-Saxton [5], Yang-Gu [6], genetic [7], and various combinations of these methods [8–11]. Several groups have expanded these algorithms to include multiple design wavelengths [12–15]. The resulting DOEs can focus and spatially separate discrete wavelengths simultaneously. However, average optical efficiency within desired regions decreases significantly as the number of design wavelengths increases [14]. Recently, there has been interest in the design of DOEs for photovoltaic (PV) spectrum splitting [16–22]. The optical requirements in this case emphasize optical efficiency and control of the broad spectral bandwidth of the incident solar spectrum. Wang et. al. [16] designed a DOE which combines a diffractive lens and blazed grating for photovoltaic spectrum splitting. Kim et. al. [18] expanded the direct binary search algorithm to design a DOE which diffracts multiple wavelengths into specific regions. The algorithm was used in later work to design spectrum splitting DOEs for 2-bandgap [19] and 3-bandgap [20] PV systems. These spectrum splitting designs are promising, but also show room for improvement. Key challenges are obtaining high optical efficiency over the broad PV spectral range, minimizing system depth, and simplifying the design process. In this work, a modified version of the Gerchberg-Saxton algorithm is proposed for broadband DOE design. This algorithm was chosen for its efficiency: the entire DOE is modified after each diffraction calculation in contrast to pixel-by-pixel adjustment and evaluation. For DOEs with thousands of pixels and height levels, this aspect is critical to reduce computation time. Additionally, this algorithm does not require a pre-optimized phase profile. It produces consistent and acceptable convergence values, and is described at the end of Section 2. Lastly, the spectrum-splitting optical performance of the Broadband GerchbergSaxton algorithm is shown to equal or exceed that of other methods discussed recently [17, 19, 20]. More detail on this comparison is shown in Section 5. The original Gerchberg-Saxton algorithm, which assumes a single wavelength, has been modified to optimize over a broad wavelength range. Specific alterations in this work include:

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• Expansion to multiple design wavelengths and target irradiance regions; • DOE height selection process based on multiple wavelengths; • Generalization to the Fresnel diffraction domain; • Techniques to avoid stagnation specific to multi-wavelengths design: o Non-uniformity irradiance allowed within target regions; o Selection of best rather than final iteration for the selected phase pattern; o Weighting factor to adjust emphasis on design wavelengths. 2. Description of the broadband Gerchberg-Saxton algorithm The single-wavelength Gerchberg-Saxton algorithm performs forward and backward propagation of the light field between two planes: the DOE plane,

u1 ( x1 ) = a1 ( x1 ) ⋅ eiφ1 ( x1 )

(1)

and the diffracted plane is located a distance zd from the DOE plane, u2 ( x2 ) = a2 ( x2 ) ⋅ eiφ2 ( x2 )

(2)

The goal of the algorithm is to approximate a desired irradiance pattern I 2 ( x2 ) in the diffracted plane by transforming the phase of the incident solar illumination to eiφ1 ( x1 ) after passing through the DOE. In the broadband expansion of the Gerchberg-Saxton algorithm, each wavelength has a iφ ( x , λ ) different optical path length through the DOE, with corresponding phase function e 1 j 1 j . DOE phase is optimized to simultaneously diffract distinct irradiance patterns I 2 j ( x2 , λ j ) onto the diffracted plane. Figure 1 illustrates the system layout.

Fig. 1. System layout. Multiple wavelengths (λi) are incident on the DOE and diffract from the DOE plane to the diffracted plane located at distance zd.

The algorithm is written to vary the wavelength distribution in one dimension along the x direction. (It is possible to extend control of the wavelength distribution in 2D and is the subject of future work). The surface relief height varies along the x-direction and is constant along the y-direction. Height levels are not quantized and can take any value. Periodic boundary conditions are assumed: the DOE and target plane repeat along the x-direction. The optimization assumes normally-incident light, but the effect of off-axis incidence on spectral filtering properties will be discussed in Section 6. A step-by-step description of the broadband Gerchberg-Saxton algorithm follows: 1. Target diffraction regions are defined and assigned to each design wavelength λj. #247608 © 2015 OSA


2. The diffracted field, u2 j ( x2 , λ j ) = a2 j ( x2 , λ j ) ⋅ e

iφ2 j ( x2 , λ j )

, is initialized with the following

values: 1 a. a2 j ( x2 , λ j ) =  0

within target region outside of target region

b. φ2 j ( x2 , λ j ) = random value between 0 and 2π 3. u2 j ( x2 , λ j ) is back-propagated to the DOE plane using an inverse Fresnel transform to obtain the initial field u1 j ,initial ( x1 , λ j ) :  −iπ x12 u1 j ,initial ( x1 , λ j ) = exp   λz  j d

 iλ j z d   −i 2π zd exp   IFT   λ  Δx j  

  −iπ x2 2  exp    λ j zd

   u2 j ( x2 , λ j )  (3)  

IFT indicates inverse Fourier transform, which is calculated numerically as a discrete operation. Δx is sampling width in the DOE plane. Decreasing Δx increases accuracy of the calculation at the expense of computing time. 4. The height of each DOE pixel is determined: a. Optimum phase delay at the pixel for each design wavelength is obtained from the inverse Fresnel transform in step 3: φ1 j ,initial ( x1 , λ j ) b. This phase delay is translated into an optimum height level for each wavelength:

h j ( x1 , λ j ) =

λ j ⋅ φ1 j ,initial ( x1 , λ j )

(4)

2π ( n(λ j ) − 1)

c. Equivalent height levels are generated (up to specified maximum height hmax) by adding steps of 2π phase delay. h jk ( x1 , λ j , k ) =

λ j ⋅ (φ1 j ,initial ( x1 , λ j ) + 2π k ) 2π ( n(λ j ) − 1)

k = 1, 2,...kmax

(5)

d. Considering all possible combinations of height levels for each design wavelength hj(λj), the set of heights with the smallest range is selected. Figure 2 shows an example of this selection process. Final pixel height is calculated as a weighted average of this set of heights: h( x1 ) =  j w j ⋅ h j

(6)

Weight values wj are used to bias the algorithm towards design wavelengths of importance. The sum of weight values  j w j = 1 . The larger the value of hmax, the greater the chance of converging on a small range of heights and achieving a near-optimum value for multiple design wavelengths. If hmax is too small, a large range of h values will be averaged into the single height, leading to phase error and reduced spectrum splitting capability. 5. The field at the DOE plane u1 j ( x1 , λ j ) is determined.

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a. Phase is calculated based on the height profile in step 4d.

φ1 j ( x1 , λ j ) =

2π h( x1 )

λj

( n(λ ) − 1)

(7)

j

b. Amplitude = 1 to maximize optical efficiency: u1 j ( x1 , λ j ) = exp ( iφ1 j ( x1 , λ j ) )

(8)

6. u1 j ( x1 , λ j ) is forward-propagated to the diffracted plane using a Fresnel transform to obtain the diffracted field u2 j ,initial ( x2 , λ j ) : u2 j ,initial ( x2 , λ j ) =

 i 2π zd −i ⋅ Δx exp   λ λ j zd j 

  −iπ x2 2  exp    λ j zd

   iπ x12  FT  exp     λ j zd

 u2 j ( x2 , λ j )   (9)  Δx  

7. The amplitude for u2 j ( x2 , λ j ) is set to 0 outside of the target region. Amplitude and phase within the target region are not modified: 0  u2 j ( x2 , λ j ) =  a2 j ( x2 , λ j ) exp ( iφ2 j ( x2 , λ j ) )

x 2 outside of target region x 2 within target region

(10)

8. Convergence to a solution is monitored using a signal to noise ratio (SNR): SNR =  λj

 

Target x2

I 2 j ( x2 , λ j )dx2

Non-Target x2

I 2 j ( x2 , λ j )dx2

where I 2 j ( x2 , λ j ) is diffracted irradiance, target region,



Non-Target x2



Target x2

(11)

dx2 is spatial integration within the

dx2 is spatial integration outside of the target region,

λ

j

is

a summation of design wavelengths. Higher SNR indicates closer approximation of I 2 j ( x2 , λ j ) to its target value for all design wavelengths. 9. Steps 3-8 are repeated for a fixed number of iterations, or until the optical performance of the DOE reaches a desired SNR value. An illustrated block diagram of the algorithm is shown in Fig. 3.

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Fig. 2. Set of potential height levels for a DOE pixel with 4 design wavelengths and maximum height of 6µm. The set of heights with minimum range is circled. Final pixel height is a weighted average of these values.

Fig. 3. Block diagram of broadband Gerchberg-Saxton algorithm.

Convergence of the Broadband Gerchberg-Saxton algorithm is demonstrated by monitoring SNR over multiple iterations. Each simulation has unique starting conditions due to the random initial phase term. Convergence occurs when SNR reaches a constant value for each simulation, as shown in Fig. 4(a). If the system is overly-constrained (by attempting to find a single DOE height for too many design wavelengths, for example), SNR is not as stable. This case is shown in Fig. 4(b). For consistent design results, conflicting constraints must be reduced.

Fig. 4. Progression of SNR metric for two optimization runs. a. Stable SNR for a system with a 1 design wavelength assigned to each target region. b. Unstable SNR for a system which is

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overly-constrained with 2 design wavelengths assigned to each target region (0.4 and 0.6 μm; 0.8 and 1.0 μm).

3. DOE evaluation metrics

Several metrics are defined to quantify performance of the DOE. These metrics are utilized in Section 4 to make comparisons between different DOE designs. 3.1 Spectral optical efficiency (SOE)

SOE is defined as the percentage of incident spectral irradiance collected within each target region as a function of wavelength:

 SOE (λ ) =

Target

x2



x1

I 2 ( x2 , λ )dx2

I1 ( x1 , λ )dx1

⋅100

(12)

where I1 ( x1 , λ ) is the incident spectral irradiance, I 2 ( x2 , λ ) is the diffracted spectral irradiance,



x1

dx1 is spatial integration over the DOE plane, and



Target x2

dx2

is spatial

integration over the diffracted plane within the target region. 3.2 Average optical efficiency (OAvg)

OAvg quantifies filter performance with respect to an ideal rectangular spectral filter. This metric calculates average SOE(λ) over the spectral band of interest, and its value ranges from 0 to 100%: λ2

OAvg

λ =

SOE (λ )d λ

1

100 ⋅ (λ2 − λ1 )

⋅100

(13)

3.3 Conversion efficiency (η)

In a spectrum splitting PV application, conversion efficiency is a key evaluation metric. OAvg alone does not consider the wavelength-dependent effect of solar spectral irradiance, SOE, and spectral responsivity on PV output. Two DOEs with identical OAvg values but different spectral filtering shapes can lead to different conversion efficiencies. The DOE with higher long-wavelength SOE leads to higher conversion efficiency because it is better-matched to the spectral responsivity of the PV cell. 3.4 Improvement over best bandgap (IoBB)

The final evaluation is a comparison of spectrum splitting conversion efficiency with that of the highest-efficiency PV cell in the system under the full solar spectrum. This metric, called Improvement over Best Bandgap (IoBB), quantifies whether or not the spectrum splitting system is worth pursuing.  η SS  IoBB =  − 1 ⋅100  η Best − PV 

(14)

4. Effect of design parameters on DOE performance

A DOE for two-bandgap spectrum splitting is designed. The selected PV materials are silicon (λbandgap = 1.11 µm) and CdSe (λbandgap = 0.71 µm). PV electrical properties are determined using detailed balance analysis [23]. The cells have equal target areas in the diffraction plane. The configuration is shown in Fig. 5.

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Fig. 5. Schematic of spectrum splitting module. The unit cell contains a DOE of width w, and an arrangement of PV cells at the diffracted distance zd. Two equal-area PV cells fill the plane, and the larger bandgap cell is placed in the center. The device repeats in the lateral direction and extends out-of-plane.

In the next section, system parameters are explored, and their corresponding effect on performance metrics is described. 4.1 Baseline spectrum splitting design and parameter specification

For the baseline DOE, design parameters are set to the center of their practical range (Table 1). Figure 6 illustrates spectral optical efficiency, and Fig. 7 shows the DOE height profile. Table 1. Starting Parameters for a CdSe/Si Spectrum Splitting DOE Design Parameter

Value

Design wavelengths

0.6 μm, 0.8 μm

Weight factor of wavelengths

[0.5 0.5]

DOE aperture width

1 cm

Diffracted distance

50 cm (0.98 cm using 1 cm focal length lens)

Number of DOE pixels, pixel width

3000, 3.33 μm

Maximum DOE height

6 μm

DOE material

Norland Optical Adhesive #61

Number of iterations

1000

Fig. 6. Spectral optical efficiency for baseline DOE

Fig. 7. Height profile of baseline DOE

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4.2 DOE spatial feature size

Smaller spatial features (pixels) within the DOE enable larger diffraction angles and greater system design flexibility. On the other hand, large spatial features are easier and less expensive to fabricate. Therefore, an optimum feature dimension can be obtained. Two pixel widths are evaluated to observe their effect on SOE: 2.5 μm wide pixels in Fig. 8(a) and 28.6 μm wide pixels in Fig. 8(b). The analysis shows that relatively large pixel width can be chosen for ease of fabrication without a significant decrease of spectrum splitting performance.

Fig. 8. Spectral optical efficiency for a. decreased pixel width, b. increased pixel width.

4.3 Diffraction plane distance

The field transmitted by the DOE must propagate over a distance (zd) to form the desired light field. This distance should be minimized to reduce the depth of the spectrum splitting module. On the other hand, there is a minimum distance required to sufficiently separate diffracted spectral components. This minimum distance is based on minimum diffracted angle, and can be determined using the grating equation:  λmin    2Δx 

θ min = sin −1 

(15)

where index of refraction outside of the grating = 1, incident angle = 0°, order m = 1 , and Δx is DOE pixel width. This diffracted angle should direct short-wavelength light at the outermost edge of the DOE to the edge of the CdSe PV cell. For the layout in Fig. 5, this requires a lateral 1 translation of w (w = unit cell width). These constraints lead to a minimum required 4 diffracted distance dmin: d min =

w  −1  λmin   4 tan sin    2Δx   

(16)

dmin = 2.77 cm for the baseline DOE. Figure 9(a) shows SOE for dmin. Figure 9(b) shows SOE for a much larger diffracted distance of 200 cm.

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Fig. 9. Spectral optical efficiency for a. decreased diffracted distance, b. increased diffracted distance.

While small distance zd is desirable to minimize module depth, it also requires smaller pixel widths and more precise fabrication methods. An alternative approach is to choose a separation distance zd > dmin and use a lens to reduce system depth. The diffracted plane distance (zl) when a lens is combined with the DOE is calculated: zl =

zd ⋅ f zd + f

(17)

where f is focal length of the lens, and zd is diffracted plane distance without using a lens. 4.4 Maximum DOE height

As maximum surface relief height increases, it is possible to achieve a greater range of phase delay values and provide a better phase match for all design wavelengths. The improvement in SOE from increasing hmax depends on design wavelength: wavelengths assigned to the same target plane that are significantly “out-of-phase” (λ1 = 0.75·λ2 for example) show the most improvement. On the other hand, a small hmax with 1-2 equivalent height levels for each design wavelength produces a roughly periodic DOE structure with linear dispersion. This effect expands the SOE curve beyond the design wavelengths, which is desirable for a broadband PV application. Figure 10(a) shows SOE for a DOE with maximum surface relief height of 2.5μm. Figure 10(b) shows SOE for a DOE with larger maximum height of 9μm. In the second case, SOE is at the design wavelengths is higher, but it rapidly decreases at other wavelengths.

Fig. 10. Spectral optical efficiency for a. decreased maximum DOE height, b. increased maximum DOE height.

Figures 11(a) and 11(b) show spatial distribution of spectral irradiance for the two DOE designs, illustrating how light is distributed across the PV plane in each case.

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Fig. 11. Spectral distribution of irradiance along diffracted plane for DOEs with maximum height of a. 2.5 μm and b. 9μm.

4.5. Specification of design wavelengths

An important step in the design process is the specification of the design wavelengths. Figure 12(a) corresponds to a DOE with a small difference in design wavelengths (0.69μm and 0.71μm), while Fig. 12(b) corresponds to a large difference (0.5μm and 0.9μm).

Fig. 12. Spectral optical efficiency for a. decreased difference in design wavelengths, b. increased difference in design wavelengths.

There is not a significant change in peak SOE, but there is a difference in uniformity and cut-off wavelength characteristics. The first case, shown in Fig. 13, exhibits a tightly focused dispersion line at the PV cell boundaries. This occurs because two closely-spaced design wavelengths are constrained to fall within separate target regions. In order to achieve this condition in a single DOE structure, light is diffracted toward the boundary between target regions and each wavelength is placed on the appropriate side of that border.

Fig. 13. Spectral distribution of irradiance along diffracted plane for DOE with a small difference in design wavelengths (0.69 μm and 0.71 μm).

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4.6 Weighting of design wavelengths

Adjusting the weight factor (part 4d of the algorithm) puts emphasis on increasing SOE of a particular design wavelength. Figures 14(a) and 14(b) show SOE for weight distributions favoring blue and red design wavelengths, respectively.

Fig. 14. Spectral optical efficiency for a. W = [0.95 0.05], b. W = [0.05 0.95].

4.7 Number of design wavelengths

Increasing the number of design wavelengths broadens the wavelength range collected in each target region, but also increases computational cost of the algorithm, particularly in step 4 where all height combinations are evaluated. Computation time is proportional to HN where H = number of equivalent height levels for each wavelength and N = number of design wavelengths. For a DOE with H = 3, N = 2 requires 9 evaluations/pixel while N = 4 requires 81 evaluations/pixel. Scaling this by the number of DOE pixels leads to a significant difference in computing time. Figure 15 shows SOE with the number of design wavelengths increased to 4 (0.4 μm, 0.6μm, 0.8 μm, and 1μm). Shorter wavelengths are assigned to the central region on the PV plane, and longer wavelengths are assigned to the outer regions. The optimization tends improve the SOE for long wavelengths at the expense of shorter wavelengths. To counteract this, a higher weight factor is assigned to the shorter wavelengths.

Fig. 15. Spectral optical efficiency for an increased number of design wavelengths [0.4 0.6 0.8 1.0] μm and W = [0.325 0.325 0.175 0.175].

4.8 Material dispersion

Secondary diffraction orders occurring at approximately half of the design wavelength can be detrimental in a spectrum splitting filter. The SOE curves contain a secondary peak at ≈400 nm which is collected by the Si cell rather than the CdSe cell. To explain secondary diffraction, consider a periodic grating. Bright diffraction orders occur when optical path difference is equal to an integer multiple of λ:

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OPD = h ( n(λ ) − 1)

(18)

For a given height level h and material index of refraction n(λ), a primary wavelength λP (OPD = 1λP) and secondary wavelength λS (OPD = 2λS) can be calculated: 1λP = h ( n(λP ) − 1)

(19)

2λS = h ( n(λS ) − 1)

(20)

 λP   n(λS ) − 1   ⋅  2   n ( λP ) − 1 

λS = 

(21)

If the DOE material has no dispersion: n(λP ) = n(λS )

→

λS =

λP 2

(22)

If the DOE material has normal dispersion, the secondary order is closer to λP: n(λP ) < n(λS )

→

λS >

λP 2

(23)

If the DOE material has anomalous dispersion, the secondary order is further from λP: n(λP ) > n(λS )

→

λS
0.4μm for normal material dispersion and λ