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Pseudo-disordered structures enable additional design freedom for photon management. However, the optimization and interpretation is challenging when the large number of degrees of freedom encounters computationally intensive electromagnetic simulation method. Here we propose a novel one-dimensional multi-periodic pattern generation method to help us squeeze the disorder design space before performing rigorous calculation, by making use of the periodic attribute of the pattern. Consequently, thanks to the pre-filtered design space, it typically relieves us from computational burden and enables us to ‘globally’ optimize and study pseudo-disordered patterns. As an example, we show how this approach can be used to comprehensively optimize and systematically analyze generated disorder for broadband light trapping in thin film.
© 2016 Optical Society of America
1. Introduction
Photonic Crystal (PC) structures are well known to be able to control light propagation and possibly light-matter interaction for applications such as guided optics [1], microcavities [2], Light-Emitting Devices [3], Surface-Enhanced Raman Scattering [4]. As PC structures are highly efficient for selected wavelengths or angles, it has been proposed to resort to random [5–7], quasi-crystal [8–12], or multi-periodic [13–20] structures to control light properties over a broadband spectrum or angle. Heuristic optimization methods, such as Genetic Algorithm (GA) [21–23] and Particle Swarm Optimization (PSO) algorithm [24] that can be used to design and optimize periodic structures become of less interest for pseudo-disordered structures since the degrees of freedom or the number of parameters greatly increase. In order to analyze such structures one can resort to statistical study as it can be done on nanopatterned [25] or rough absorber [5].
In this paper, we propose to apply to multi-periodic patterns a mathematical scheme based on a combinatorial problem named free necklace problem [26] to partially conquer the conflict between the design freedom and vast search space. The basic idea is to eliminate patterns which seem to be unique in short range, yet they are replications of other patterns in long range with periodic boundary conditions. The detailed explanation is presented in section 2. Then these intentionally created unrepeated disordered patterns can be used to systematically and comprehensively study the effect of short range disorder for photon management. We illustrate its usefulness with broadband light trapping using a crystalline Silicon (c-Si) substrate patterned with stripe arrays, which is shown in section 3. Through numerical simulations, we demonstrate in section 4 that, by adding ‘appropriate’ engineered fine stripes to each long period, these so-called multi-periodic patterns achieved high broad-band light trapping efficiency.
2. Free necklace (or bracelet) problem
The pattern generation method is based on a combinatorial problem named necklace problem. It is a general method to construct unique patterns which are infinitely repeated. Specifically, we focus on the free necklace problem or bracelet problem, which means opposite orientations (mirror images) are regarded as equivalent.
Let us first consider a string of n characters where each character can take a value among k possibilities, the total number of all possible strings is:
From combinatorics, if we replace the string by a necklace, thanks to the rotational symmetry, the number of possible necklaces reduces to:where di are the divisors of n, v(n) is the number of divisors of n, and φ(di) is Euler’s totient function. In the case of free necklace (or bracelet), the mirror symmetry decreases this value to:Let us focus on arrangements with a length of 4 digits that can take 2 different values (‘0’ and ‘1’). Table 1 shows all the possible strings as well as the possible bracelets. For instance, the binary quadruple {1100} is the representative of set {1100, 0110, 0011, 1001} and {1010} is the representative of set {1010, 0101}. In this case it appears that only 6 bracelets are needed, instead of 16 strings, to represent all the possibilities so 2.7 times less.
Table 1. Construction of strings and bracelets for a binary quadruple.
The algorithm explained in [26] was used to efficiently generate all the possible bracelets with different lengths of code sequences (up to 20). Table 2 illustrates the comparison of the number of strings and bracelets for different lengths of binary code sequences [27]. As we can see, the reduction factor drastically increases with the increase of the length of the binary code sequences (more than 30 for length longer than 17 characters).
Table 2. Comparison of the number of strings (S2(N)) and bracelets (B2(N)) for different lengths of binary code sequences (N).
In the following of the paper we will combine the bracelet generation algorithm with binary PC. As we can simulate all the possible arrangements without losing any time due to ‘multiple simulations’, we argue that in contrast with heuristic algorithms (e.g. PSO and GA), our optimization process always leads to the overall optimum among all the PCs with a given width for all the binary slices. Beyond that, our approach remains very flexible, as various unrepeated patterns can be easily constructed by using the general formation (Eq. (3)).
3. Multi-periodic structures and numerical method
Let us define the device that we will study. A 1 μm thick c-Si layer was chosen to study light trapping under un-polarized normal incident light. All the patterns were generated based on this single layer. No antireflective coating and back reflector were used for the sake of simplicity and clarity. In order to give a general demonstration of our approach, the simple integrated absorption (Iabs) in a wide wavelength range (from 300 to 1100 nm) was calculated as figure of merit. Our calculations were done using rigorous coupled wave analysis (RCWA) [28] with improvement for TM polarization [29]. Sufficient spatial harmonics were used to guarantee the accuracy and a reasonable calculation time (few minutes for a typical desktop computer).
First we employed PSO algorithm [30] to optimize the absorption of the c-Si layer by varying 3 parameters (period, groove depth and filling fraction). After the whole optimization process, the best performed pattern (so-called mono-periodic pattern) with etching depth of 106 nm, period of 311 nm, and filling factor of 0.502 yielded an integrated absorption of 37.8%. For the sake of clarity we fixed the etching depth to 100 nm, period to 300 nm and filling factor to 0.5. It yielded an optimum integrated absorption of 37.7% (Fig. 1(a)), which is 75.3% higher than that of the bare silicon slab (Iabs = 21.5%).
Fig. 1 Absorption spectra of a 1 μm thick c-Si layer under un-polarized normal incident light, (a) for the optimized mono-periodic PC with lattice length of 300 nm (inset shows a typical cell) and (b) for the optimized multi-periodic PC with lattice length of 2550 nm (inset shows a typical cell).
Then the multi-periodic patterns considered in this paper were generated by setting an identical length of ridge and groove according to the generated code sequences (‘1’ represents a ridge and ‘0’ represents a groove). The etching depth was the same as the mono-periodic pattern and the length of ridges and grooves was set to be 150 nm. As the size of ‘1’ and ‘0’ was set, patterns represented by long code sequences ended up with long lattice length.
4. Results and discussion
In this section we study the PC obtained using the bracelet generation algorithm to optimize and analyze light trapping in the c-Si layer. All the multi-periodic patterns listed in Table 2 have been simulated. The code sequences of different lengths that maximize Iabs are listed in Table 3. Large Iabs (>40%) is achieved for patterns containing at least 12 characters. When the period of the multi-periodic pattern equals 2550 nm (containing 17 characters), Iabs is as large as 42.2% (Fig. 1(b)), which is 11.9% higher than that of the optimized mono-periodic pattern (Fig. 1(a)).
Table 3. The optimized code sequences with different lengths of binary code sequences (N). The all-zero code sequences were excluded. Patterns with lattice length of 2700 nm and 3000 nm were optimized with constraint to 50% filling fraction because of the heavy computational burden.
To simplify the analysis and discussion, we define the number of ‘1’s in the code sequence as number of ridges, which is directly linked to filling fraction ff (ff = number of ridges/N). It is noticeable that for all the optimal code sequences listed in Table 3, ff is close or equal to 50%. More precisely, ff = 50% for even values of N except for N = 12 where ff = 5/12, and ff = 6/15 for N = 15 and 8/17 for N = 17.
Then, the pattern generation method is used to comprehensively study multi-periodic patterns since each generated pattern is unique. To understand the absorption improvement, we analyze the patterns with lattice length of 2550 nm. Figure 2 shows the Iabs map of all unrepeated multi-periodic patterns. Each point in the map represents a unique design of the pattern as a function of the number of ridges. Moreover, block number (BN) is defined as the number of ridge groups in a multi-periodic pattern, which is represented by different colors in the image. Obviously it can be noted that the performance of the optimized pattern increases and then decreases with the number of ridges. As previously mentioned, the best performed pattern appears in the group with approximately ff = 50%, and patterns with similar ff perform also quite well (Fig. 2). Another overall property is the outperformed patterns appear in the patterns with relatively large BN. Associated to the moderate ff, it avoids the clustering of ridges, and thus enriches the diversity of the generated patterns. Similar conclusions derived from a statistical study [31].
Fig. 2 Integrated absorption achieved for all the unrepeated representations of multi-period patterns with lattice length of 2550 nm. Block numbers are represented by different colors. The horizontal dashed line shows the Iabs of the optimized mono-periodic pattern under un-polarized illumination.
To analyze further in details, we choose patterns with the number of ridges of 8 which exhibit the highest Iabs for detailed discussion. Figure 3 shows the relation between Iabs and BN under TE (electric field parallel to the groove lines), TM (electric field perpendicular to the groove lines), and un-polarized illuminations. For the viewpoint of morphology, this map exhibits two features that can be correlated with large absorption enhancement.
Fig. 3 Integrated absorption for all different multi-periodic patterns with lattice length of 2550 nm and number of ridges of 8. The horizontal dashed line shows the Iabs of the optimized mono-periodic pattern under un-polarized illumination.
Firstly, TE and TM modes respond differently to patterns with the same BN. As we can see in Fig. 3, the points that represent Iabs for the TM mode with the same BN are more compact than those for the TE mode, reflecting that the first one is less sensitive to patterns with the same BN than the second one. In addition, the optical performances for TM and TE polarizations are not matched for a given pattern. As such, it is necessary to take into account the influence of the polarization. Secondly, TE and TM modes share the same tendency despite of a non-perfect matching. Iabs grows with increasing BN then slows down or drops for large BN. The best performed multi-periodic pattern appears in patterns with relatively large BN. We believe that this kind of spatial arrangement of stripes can efficiently couple both TE and TM modes into the pseudo-guided modes supported by the silicon slab when the reciprocal lattice vector of the grating matches the propagation constant of these modes [31].
In reciprocal space, the relationship between the stripe arrangement and its optical performance is more profound and clear. Patterns with appropriate large BN tend to provide affluent Fourier components and the absorption is linked with the richness of Fourier components [17,32,33]. Hence, Fourier transform was performed for all the patterns (represented by code sequences) illustrated in Fig. 3. The Fourier coefficient maps are shown in Fig. 4(a) and 4(c). We notice a significant correlation between the position of the spatial Fourier coefficients and the absorption. Generally, patterns with their Fourier components shifted to higher orders tend to outperform other patterns in both TE and TM modes. In order to quantitatively clarify our argument, a light trapping range in Fourier space is defined. The lower boundary of the spatial frequency that is required for light trapping is defined by light line. Here, 600 nm is chosen to be the critical wavelength because at this wavelength the absorption in c-Si becomes critically weak [33]. It corresponds to a spatial frequency of 10.5 μm−1. The upper limit of spatial frequencies is set to be 31.4 μm−1, as for larger frequencies the Fourier coefficients become negligible in the interested region [25]. Figure 4(b) and 4(d) show the integrated Fourier coefficients below the lower boundary (from 0 to 10.5 μm−1) and in the light trapping range (from 10.5 to 31.4 μm−1) for all the generated patterns. The outperformed patterns tend to shift the spatial frequencies into the light trapping range for both modes. As the lower spatial frequency limit is set to the critical wavelength of 600 nm, the coupling into the quasi-guided modes at longer wavelengths contributes more to the light trapping enhancement.
Fig. 4 Fourier analysis of all the patterns with lattice length of 2550 nm and number of ridges of 8. Fourier transform of the PCs code sequences sorted by increasing absorption for (a) TE mode, and (c) TM mode. The magnitude of Fourier components lying in the leakage range (from 0 to 10.5 μm−1 in blue) and in the light trapping range (from 10.5 to 31.4 μm−1 in red). To compare the Fourier coefficients of different patterns, the Fourier amplitudes are normalized with respect to the constant zeroth order coefficient. High orders are not taken into account.
Previous optimization study reveals how the multi-periodic pattern can beat others. Then we focus on the absorption mechanism. Compared with mono-periodic PC, multi-periodic PC excites a larger number of resonances, since more peaks appear in the absorption spectrum of the multi-periodic pattern (Fig. 1), which is the reasoning behind multi-periodic design concept. Another point is that the optimum multi-periodic pattern is non-symmetric, since the non-symmetric patterns exhibit two times more modes than the symmetric ones [16]. Figure 5(a) and 5(b) show the correlation between the Fourier components of the PC code sequences and their corresponding Fourier components of the fields from 600 to 1100 nm. The optimized mono- and multi- periodic patterns diffract incident light into different orders and each order has its corresponding Fourier components of the code sequences. In addition, the amplitudes of the Fourier components of the fields (integrated from 600 to 1100 nm) are closely related to the amplitudes of the Fourier components of code sequences [25,33]. It becomes clear that the optimized multi-periodic pattern diffracts and couples broadband light incident into more supported modes of the beneath silicon slab.
Fig. 5 Analysis of the optical mode decomposition in Fourier space. Fourier transforms of the PC code sequences and electromagnetic fields (TE-TM averaged) for (a) the optimized mono-periodic pattern, (b) the optimized multi-periodic pattern, and (c) the mono-periodic pattern with the same lattice length and ff as the best performed multi-periodic pattern (code sequence: 11111111000000000). The Fourier transforms of the electromagnetic field is calculated by integrating the Fourier components of the electromagnetic field from 600 to 1100 nm. For easier comparison, the Fourier amplitudes of the PCs code sequences and the fields are normalized according to the strength of zeroth order. As such, the zeroth order is not taken into account. Decomposition of the electromagnetic fields in Fourier space associated to the three selected structures (d, e and f) from 600 to 1100 nm.
To gain insight into how the spatial arrangement affects absorption in further detail, it is instructive to observe how the electromagnetic field distributes in the patterns for different wavelengths. Figure 5(d) and 5(e) show the field distribution in Fourier space. For the optimized mono-periodic pattern (Fig. 5(d)) the energy mainly remains in low order modes (0 and ± 1 orders) to enhance the absorption from 600 to 1100 nm. Compared with the optimized mono-periodic pattern, the optimized multi-periodic pattern (Fig. 5(e)) can couple light incident into a large amount of modes. The benefit of distributing energy into high order modes becomes obvious especially at large wavelengths. Long lattice length provides multi-periodic patterns access to a large amount of orders in reciprocal space, the structure on the surface leads the distribution of energy into different orders. Patterns that cannot only couple light incident into numerous modes but also distribute the energy into various high orders outperform others. In order to illustrate that, we calculated field distribution for a large mono-periodic pattern with the same lattice length and ff as the optimized multi-periodic pattern (code sequence: 11111111000000000), as shown in Fig. 5(c) and 5(f). Even through this pattern can couple incident into multiple orders, the energy is obviously mainly distributed in low orders. As such, the overall performance is not as good as the optimized multi-periodic pattern.
In light trapping, the angular performance is also a crucial criterion. Multi-periodic PCs are tolerant to different input angles because of the multi-mode absorption mechanism [34]. At a given illumination angle, the incident light can be decomposed into different orders with different prorogation angles. For each order, the propagation in the silicon slab might be constructive or destructive depending on the diffraction angle because of the Fabry-Perot phenomenon. The light trapping performance of a pattern at a given incident angle is the overall effect of all the diffraction orders. As the absorption of the optimized multi-periodic pattern at this angle depends on many diffraction orders, and it is less likely that all the diffraction orders propagate in either constructive or destructive way. Statistically speaking, distributing energy into more orders which might increase or decrease absorption stabilizes the overall absorption efficiency. Because of this, the optimized multi-periodic pattern is less sensitive to angular change. Figure 6(a) and 6(b) show the TE-TM averaged angular absorption spectra for the optimized multi- and mono-periodic patterns. Additional resonances introduced by the optimized multi-periodic pattern present at all angles, contributing to the increased absorption and angular tolerance. Note that the optimized multi-periodic pattern performs differently for positive and negative angles because of the asymmetric structural attribute. Figure 6(c) clearly shows the overall angular performance of the optimized multi- and mono- periodic patterns under un-polarized state. The Iabs oscillates and then drops for the optimized mono-periodic pattern when the incident angle increases. However, the Iabs of the optimized multi-periodic pattern remains relatively isotropic and asymmetric for the same amount of incident angle variation. For the incident angles ranging from −75° to 75°, the average Iabs for the optimized multi-periodic pattern is found to be 36.9%, whereas for the optimized mono-periodic pattern it is found to be only 35.0%, which quantitatively implies the benefit of multi-periodic pattern.
Fig. 6 TE-TM averaged angular absorption spectra for (a) the optimized mono-periodic pattern, and (b) the optimized multi-periodic pattern. (c) Comparison of integrated absorption between the optimized mono-periodic pattern (red line) and the optimized multi-periodic pattern (blue line) for different angles of un-polarized incident light.
From the above discussion, it is clear that patterns with rich Fourier spectra have more access to different modes and the distribution of energy to each mode is also of importance to enhance absorption. This phenomenon becomes even important for broadband spectrum at different illumination angles. Hence, the multi-periodic pattern benefiting from its flexible and rich Fourier spectra is desirable when broadband absorption for un-polarized incident light in different angles needs to be optimized. If this is the target, optimized multi-periodic patterns always outperform the periodic ones as they offer more flexibility to shift and balance spatial frequencies that facilitate efficient coupling for both TE and TM modes.
5. Conclusions
We have proposed to use free necklace or bracelet problem concept to squeeze the design space for one-dimensional multi-periodic pattern. This applicable approach partially relieved us from vast search space. As such, rigorous electromagnetic calculations can be used to comprehensive study the performance of multi-periodic structures. As an example, a thin film absorber was designed using this method to enhance the absorbance through a broadband spectrum (300 nm – 1100 nm). The results have shown that the optimized multi-periodic pattern, benefitting from engineered rich Fourier spectra, yielded the highest integrated absorption of 42.2%, which is 11.9% higher than that of the optimized mono-periodic pattern. Further, given the efficiency and simplicity of the ‘unique’ pattern design principle, this method inspires the design and optimization of periodic patterns in different disciplines [18,35]. Beyond that, such design concept could be even more efficient for two-dimensional photonic structures. Considering the complexity of two-dimensional problem and computational burden, a combination of stochastic optimization method and necklace problem concept might be feasible.
Acknowledgments
This work is supported by the European Union Seventh Framework Programme ‘PhotoNvoltaics’ (Grant Agreement No. 309127). Authors are grateful to Christian Seassal for the helpful suggestions. Jia Liu acknowledges the China Scholarship Council (CSC).
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