Open Access
Add to BibSonomy
Abstract
Due to the wide range of applications of plasmonic diffraction gratings, it has become essential to provide an analytical method for modeling performance of the devices designed based on these structures. An analytical technique, in addition to greatly reducing the simulation time, can become a useful tool for designing these devices and predicting their performance. However, one of the major challenges of the analytical techniques is to improve the accuracy of their results compared to those of the numerical methods. So, here, a modified transmission line model (TLM) has been presented for the one-dimensional grating solar cell considering diffracted reflections in order to improve the accuracy of TLM results. Formulation of this model has been developed for the normal incidence of both TE and TM polarizations taking into account diffraction efficiencies. The modified TLM results for a silicon solar cell consisting of silver gratings considering different grating widths and heights have shown that lower order diffractions have dominant effects on the accuracy improvement in the modified TLM, while the results have been converged considering higher order diffractions. In addition, our proposed model has been verified by comparing its results to those of the finite element method-based full-wave numerical simulations.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Photovoltaics is an ideal candidate for meeting the ever-increasing global energy demand and reducing carbon emissions [1]. To date, silicon solar cells are dominant devices in the photovoltaic technology, sharing 90% of photovoltaic market investment due to their near-optimal bandgap, high efficiency, proven stability, mature fabrication technology, and wide availability of silicon [2–6]. In recent years, some plasmonic light trapping techniques have been introduced to increase the optical absorption by reducing the high reflectance and increasing the optical path length, especially in the thin layers of the solar cells across a wide range of wavelengths [4,7,8]. With these techniques, the physical thickness of the photovoltaic absorbing layer is reduced while its optical thickness remains the same [9]. As an advanced light trapping technique, periodic 1-D plasmonic diffraction gratings are one of the most efficient and standard methods that can be easily fabricated by lithography [3,6,9–14]. These periodic gratings on the back surface of the absorber layer couple sunlight into surface plasmon polariton modes at the metal/semiconductor interface as well as cavity modes in the semiconductor slab [9,15–19].
The finite element method (FEM), the finite difference time domain (FDTD) method [8,12,13,20,21], the finite difference frequency domain (FDFD) method [22], and the rigorous coupled wave analysis (RCWA) [14,23,24] have been previously used in order to simulate the performance of the solar cells and obtaining their absorption spectrum, field distribution, and power conversion efficiency. However, these are time-consuming numerical techniques that cannot give a complete physical insight into the performance of the structure. One of the fast and efficient analytical methods for analyzing the microwave and optical components is the transmission line model (TLM) [25–28]. However, the TLM as an electromagnetic model can not consider charge transport and recombination or collection losses in the solar cell [29]. In 2010, Stathopoulos et al. applied the TLM to calculate the optical electric field, the light absorption distribution, and the maximum short-circuit photocurrent in a multilayer hybrid polymer solar cell under normal incidence [29]. In 2011, they modified the aforementioned model to calculate the optical field distribution, short-circuit photocurrent, and reflectivity of an isotropic active material under oblique illumination [30]. Subsequently, in 2012, Polemi and Shuford developed this model to calculate the light absorption in a solar cell containing 1-D plasmonic grating nano-surface for both TE and TM polarizations under the circumstances of normal incidence [17]. However, to the best of our knowledge, effects of the light diffraction on the absorption of solar cells have not been studied by the TLM.
Thus, in this paper, a modified TLM has been proposed for accurate analyzing the light absorption in the grating solar cells; taking into account diffraction efficiencies for both TE and TM polarizations. According to the numerical results obtained by this model for different grating widths and heights, it has been shown that lower order diffractions have dominant effects on the accuracy improvement; while, convergence of the results realizes considering higher order diffractions. Good agreement between these results and those of the FEM-based full-wave numerical simulations verifies the accuracy of the presented modified TLM.
2. Plasmonic grating silicon solar cell structure
Figure 1 shows the 3-D schematic structure of the considered solar cell consisting of an Ag reflective mirror with the thickness of hAg = 100 nm (larger than its skin depth) and a Si absorber layer with the thickness of hSi = 1 µm. The interface between Si and Ag has been patterned with periodic Ag gratings with the period of Λ, the height of d, and the width of w. Throughout this study, Λ = 2 µm, d = 50, 200, and 300 nm, and w = 400, 700, and 1000 nm (equivalent to duty cycles of DC = w/Λ = 20%, 35%, and 50%).
Fig. 1. (a) 3-D schematic of the proposed 1-D plasmonic grating Si solar cell. (b) 2-D side view of a single unit cell used for the electromagnetic simulations of the structure.
The optical constants for Ag and Si have been taken from [31] considering their material dispersions. As depicted in Fig. 1(b), in the numerical simulations, periodic boundary conditions have been applied along the y and z directions of the unit cell for modeling the periodicity of the solar cell; while, perfectly matched layers have been considered along the x-axis to prevent boundary scattering. The unit cell has been illuminated under normal incidence of TE and TM polarized light.
3. Diffraction efficiency
The metallic diffraction gratings shown in Fig. 1 are reflection-type gratings realizing diffracted reflections [32] illustrated in Fig. 2(a). Diffraction angles can be calculated as follows by the grating’s law [33],
Fig. 2. (a) Diffracted reflections pattern. (b) Diffraction orders spectrum. (c) Diffraction angles spectrum. αc = sin−1(1/nSi) and α = tan−1(0.5Λ/(hSi – d)). Λ = 2 µm, d = 50 and 300 nm.
This equation is the foundation for the description of redirecting light, in which λ is the optical wavelength, θin is the angle of incidence, m is an integer called the diffraction order with values of m = 0, ± 1, ± 2, …, Λ is the grating period, and nSi is the refractive index of Si. Throughout this study, the normal incidence has been considered.
It should be noted that the diffraction angles are independent of the grating’s depth and width. For m = 0, the diffraction grating is purely reflective and exhibits no spectral properties, and diffraction properties occur for m ≠ 0.
From Eq. (1), it can be concluded that values of the diffraction orders are limited by the following relation [23],
As shown in Fig. 2(b), the maximum value for m in the wavelength spectrum from 300 to 1000 nm is equal to 36. Diffraction angles spectrum for these diffraction orders (θm), the critical angle for the Si-Air interface (αc), and angles that for values larger than them the diffracted beams exit from the unit cell and enter the side cells (α1 for d = 50 nm and α2 for d = 300 nm) have been demonstrated in Fig. 2(c). It has been shown that the θm increases non-linearly with increasing m. Also, from Figs. 2(b) and 2(c) it is evident that higher diffraction orders appear for the short wavelengths. Therefore, it is expected that they improve the accuracy of the modified TLM for these wavelengths.
Modification of the TLM formulation requires applying the diffraction efficiency (DE) that has been defined as the percentage of incident light that is redirected by the diffraction angle [34]. Considering the scalar diffraction approximation, its formulas for m = 0 and m = ±1, ± 2, … are as follows [35],
These diffraction efficiencies are functions of the diffraction order m, duty cycle DC, and phase difference for back gratings Δφ. The phase difference can be calculated as [36],
4. Transmission line model
The normal incidence of the TE and TM polarizations to the 1-D plasmonic grating Si solar cell have been shown respectively in Figs. 3(a) and 3(b). According to these figures, the unit cell has been divided into Region I and Region II, respectively, with the heights of hSi – d and hSi. Also, input impedances for both of the mentioned modes at the Air/Si interface, i.e., $Z_{in,\textrm{TE}}^m$ and $Z_{in,\textrm{TM}}^m$, have been considered for the diffraction order m.
Fig. 3. The normal incidence to the 1-D plasmonic grating Si solar cell and input impedances at the Air/Si interface for the diffraction order m. (a) TE polarization and (b) TM polarization, considering sub-regions I and II, respectively, with the heights of hSi – d and hSi for the unit cell. TLM for (c) Region I and (d) Region II, containing two transmission lines with the characteristic impedances of Z0,Air and Z0,Si terminated to the load impedance of ZAg. TLM for the unit cell (e) TE polarization and (f) TM polarization.
Transmission line models for the introduced regions have been shown in Figs. 3(c) and 3(d) containing two transmission lines with the characteristic impedances of Z0,Air (= 120π (Ω)) and $Z_{0,Si}^m$ terminated to the load impedance of ZAg = Z0,Air/nAg [17], considering nAg as the refractive index of the Ag. $Z_{0,Si}^m$ can be calculated for the diffraction order m as below
Note that the length of the equivalent lines for Region I and Region II is equal to their heights. $Z_{in,\textrm{I}}^m$ and $Z_{in,\textrm{II}}^m$ have been defined as the input impedances of Region I and Region II for the diffraction order m that can be calculated for both TE and TM polarizations as [17]
For TE polarization, the electric potential difference between the two regions is zero; so, $Z_{in,\textrm{I}}^m$ and $Z_{in,\textrm{II}}^m$ can be treated as two parallel impedances as shown in Fig. 3(e). On the other hand, TM polarization has a continuous tangential component of the magnetic field throughout the interface. This means that their current is equal; therefore, $Z_{in,\textrm{I}}^m$ and $Z_{in,\textrm{II}}^m$ can be connected in series as illustrated in Fig. 3(f). It should be noted that the effect of the duty cycle on TLM formulation can be taken into account considering the weighting factors of ${f_\textrm{I}} = {w / \Lambda }$ and ${f_{\textrm{II}}} = {{({\Lambda - w} )} / \Lambda }$.
As a result, for normal incidence of EM waves, the reflection coefficient at the Air/Si interface for the diffraction order m is calculated by [37,38],
Therefore, considering the reflective mirror with a thickness larger than its skin depth, the light absorption for the diffraction order m can be calculated by
Finally, total light absorption for the structure can be computed by
Obviously, the modified TLM has the same formulation to that of the TLM in the case without diffraction.
It will be shown in Section 5 that the formulation developed for the modified TLM leads to accurate results for the absorption spectrum of the grating solar cell.
5. Results and discussion
The simulated results obtained by the FEM and the modified TLM for the absorption spectra of the reference cell with hSi = 1 µm, hAg = 100 nm, and d = 0 (without grating) are shown in Fig. 4 considering normally incident TE/TM polarized plane wave. The complete agreement between the results of these numerical and analytical methods indicates that the modified TLM has an excellent accuracy for the full-wave analysis of the absorbers without gratings.
Fig. 4. Absorption spectra obtained by the FEM (solid line) and the modified TLM (dashed line) for the polarization-independent reference cell considering normally incident TE/TM polarized plane wave. The geometrical parameters are hSi = 1 µm, hAg = 100 nm, and d = 0 (without grating). Λ has an arbitrary value (for example, 2 µm) without any influence on the results.
In the following, the FEM and the modified TLM results for the absorption spectra of the plasmonic grating Si solar cell illustrated in Fig. 1 with Λ = 2 µm, hSi = 1 µm, and hAg = 100 nm are shown in Figs. 5 and 6 respectively for the normally incident TE and TM polarized plane waves considering d = 50 nm and w = 700 nm; and in Figs. 7 and 8 respectively for the normally incident TE and TM polarized plane waves considering d = 300 nm and w = 400 nm. The modified TLM results have been obtained for different diffraction orders.
Fig. 5. Absorption spectra obtained by the FEM (solid line) and the modified TLM (dashed line) considering normally incident TE polarized plane wave for (a) m = 0, (b) m = 0, ± 1, (c) m = 0, ± 1, ± 2, (d) m = 0, ± 1, ± 2, ± 3, (e) m = 0, ± 1, …, ± 4, (f) m = 0, ± 1, …, ± 5, (g) m = 0, ± 1, …, ± 6, and (h) m = 0, ± 1, …, ± 36. The geometrical parameters are Λ = 2 µm, d = 50 nm, w = 700 nm, hSi = 1 µm, and hAg = 100 nm.
Fig. 6. Absorption spectra obtained by the FEM (solid line) and the modified TLM (dashed line) considering normally incident TM polarized plane wave for (a) m = 0, (b) m = 0, ± 1, (c) m = 0, ± 1, ± 2, (d) m = 0, ± 1, ± 2, ± 3, (e) m = 0, ± 1, …, ± 4, (f) m = 0, ± 1, …, ± 5, (g) m = 0, ± 1, …, ± 6, and (h) m = 0, ± 1, …, ± 36. The geometrical parameters are Λ = 2 µm, d = 50 nm, w = 700 nm, hSi = 1 µm, and hAg = 100 nm.
Fig. 7. Absorption spectra obtained by the FEM (solid line) and the modified TLM (dashed line) considering normally incident TE polarized plane wave for (a) m = 0, (b) m = 0, ± 1, (c) m = 0, ± 1, ± 2, (d) m = 0, ± 1, ± 2, ± 3, (e) m = 0, ± 1, …, ± 4, (f) m = 0, ± 1, …, ± 5, (g) m = 0, ± 1, …, ± 6, and (h) m = 0, ± 1, …, ± 36. The geometrical parameters are Λ = 2 µm, d = 300 nm, w = 400 nm, hSi = 1 µm, and hAg = 100 nm.
Fig. 8. Absorption spectra obtained by the FEM (solid line) and the modified TLM (dashed line) considering normally incident TM polarized plane wave for (a) m = 0, (b) m = 0, ± 1, (c) m = 0, ± 1, ± 2, (d) m = 0, ± 1, ± 2, ± 3, (e) m = 0, ± 1, …, ± 4, (f) m = 0, ± 1, …, ± 5, (g) m = 0, ± 1, …, ± 6, and (h) m = 0, ± 1, …, ± 36. The geometrical parameters are Λ = 2 µm, d = 300 nm, w = 400 nm, hSi = 1 µm, and hAg = 100 nm.
It can be seen from Figs. 5(a), 6(a), 7(a), and 8(a) that considering only m = 0 in the modified TLM calculations (ignoring the diffracted reflections) does not lead to acceptable results. Also, according to the results shown in Figs. 5 to 8, it can be concluded that the modified TLM results converge gradually to those of the FEM taking into account the higher-order diffractions. Obviously, the best agreement between the FEM and the modified TLM results has been realized in the case that all of the diffraction orders (here, m = 0, ± 1, ± 2, …, ± 36) have been considered.
Although Figs. 5 to 8 visually represent the diffraction effect on the calculated absorption based on the modified TLM formulation, quantitative evaluation of deviations of the modified TLM results from those of the FEM is essential for a deeper understanding of the role of diffraction orders in improving the accuracy of the modified TLM results. The deviation can be calculated by
The calculated deviations in the TE and TM polarizations have been tabulated in tables 1 and 2 for various diffraction orders considering d = 50 and 300 nm, and DC = 20%, 35%, and 50%.
Table 1. Deviations in the TE and TM polarizations for various diffraction orders considering Λ = 2 µm,
d = 50 nm, w = 400, 700, and 1000 nm, hSi = 1 µm, and hAg = 100 nm.a
Table 2. Deviations in the TE and TM polarizations for various diffraction orders considering Λ = 2 µm,
d = 300 nm, w = 400, 700, and 1000 nm, hSi = 1 µm, and hAg = 100 nm.a
It is evident from these tables that for m = 0, there are significant deviations between the results of the analytical and numerical methods. However, the deviations decrease significantly taking into account the first-order diffraction. In the same way, the deviations are also reduced by considering the second-order diffraction. Gradually, influence of the diffraction orders in increasing the accuracy of the modified TLM results decreases and finally, convergence of the results and consequently minimizing the deviations are guaranteed when all the diffraction orders are considered in the calculations.
Moreover, by studying the results more closely, we notice that there are no changes in some cases that have been marked with underline. This issue can be easily analyzed through the term sin2(mπDC) in Eq. (3). For example, considering DC = 20%, 35%, and 50%, sin2(mπDC) = 0 respectively for m = ±5 k; k = 1, 2, …, 7, m = ±20, and m = ±2 k; k = 1, 2, …, 18. Therefore, according to Eqs. (3) and (9), these m’s do not play any role in determination of the total absorption.
On the other hand, for the minimum d (= 50 nm) and maximum DC (= 50%), greatest deviations have been realized for both polarizations. This phenomenon is also justifiable through analysis of Eqs. (3) and (9).
Here, an important question arising from investigation of Figs. 5(h), 6(h), 7(h), and 8(h) and tables 1 and 2 is why the TLM results, especially at longer wavelengths, do not completely agree with those of the FEM, despite taking into account all of the diffraction orders? Analysis of this issue is possible through study of Fig. 2(c). From this figure, three distinct regions can be considered:
Region 1 (θm < αc), where diffracted beams do not experience the TIR phenomenon at the Si/Air interface and consequently, do not return to the solar cell for further light trapping as they reach the Si/Air boundary. These beams have been considered in the calculations associated with the presented formulation for the modified TLM.
Region 2 (αc < θm < α), where diffracted beams experience the TIR phenomenon at the Si/Air interface and consequently, return to the solar cell. Some of these beams reach the Si/Ag interface and the rest enter the side cells. This phenomenon that leads to an increase in light absorption has not been considered in the formulation developed for the modified TLM.
Region 3 (θm > α), where diffracted beams, before reaching the Si/Air interface, exit the cell and enter the adjacent cells. These beams experience the TIR phenomenon at the Si/Air interface of the side cells and consequently, return to the solar cell. This phenomenon, which leads to an increased absorption, has not been included in the modified TLM formulation.
In fact, those diffracted beams that are reflected back from the Si/Air interface and also the diffracted beams that enter the adjacent cells, have not been considered in the calculations associated with the modified TLM formulation. According to Fig. 2(c), the number of the diffraction orders in Region 1 decrease with increasing wavelength. Therefore, as shown in Figs. 5(h), 6(h), 7(h), and 8(h), the accuracy of the modified TLM results is lower at long wavelengths compared to the short wavelengths.
On the other hand, as d increases, α increases as well, and the number of diffracted waves entering the side cells decreases; as a result, it is expected that the accuracy of the modified TLM will be improved. According to the results represented in Figs. 5 to 8 and tables 1 and 2, it is clear that for larger d, agreement between the modified TLM and the FEM results in both TE and TM modes has been improved and consequently, corresponding deviations have been reduced. In addition, as DC increases (with increasing w or decreasing Λ), α decreases as well, and the number of diffracted waves entering the side cells increases; as a result, it is expected that the accuracy of the modified TLM will be reduced.
Finally, it should be emphasized that fortunately ignoring the mentioned phenomena, i.e., returning the diffracted beams to the solar cell due to the TIR and entering the diffracted beams to the adjacent cells, in the formulation of the modified TLM did not lead to a significant error in the calculations. Because first, the main part of the power of diffracted beams is absorbed by the silicon layer, before reflection by the TIR or entering the adjacent cells. It is obvious that this absorption is calculated by the modified TLM formulation. Secondly, the higher-order diffractions, which are more affected by these phenomena than the lower ones, have much lower diffraction efficiencies and, as a result, have negligible contributions in increasing absorption.
6. Conclusion
An accurate modified TLM formulation was presented for the one-dimensional grating solar cell taking into account the diffracted reflections. This formulation was developed for the normal incidence of both TE and TM polarizations taking into consideration diffraction efficiencies based on the scalar diffraction approximation. According to the numerical results obtained for a silicon solar cell with a thickness of 1 µm consisting of silver gratings with a period of 2 µm considering different grating widths and heights, it was shown that lower order diffractions have dominant effects on the accuracy improvement in the modified TLM, while the results were converged considering higher order diffractions. Moreover, the proposed model was verified by comparing its results to those of the FEM-based full-wave numerical simulations. It was demonstrated that the modified TLM as a fast analytical method provides accurate results and a good physical insight about the performance of the plasmonic grating solar cells.
Disclosures
The authors declare no conflicts of interest.
Data availability
Datasets generated and analyzed during the current study are available from the corresponding author upon reasonable request.
References
1. Z. Yuan, P. C. Wu, and Y. C. Chen, “Optical Resonator Enhanced Photovoltaics and Photocatalysis: Fundamental and Recent Progress,” Laser Photonics Rev. 16(2), 2100202 (2022). [CrossRef]
2. X. Qu, Y. He, M. Qu, T. Ruan, F. Chu, Z. Zheng, Y. Ma, Y. Chen, X. Ru, and X. Xu, “Identification of embedded nanotwins at c-Si/a-Si: H interface limiting the performance of high-efficiency silicon heterojunction solar cells,” Nat. Energy 6(2), 194–202 (2021). [CrossRef]
3. M.-L. Hsieh, A. Kaiser, S. Bhattacharya, S. John, and S.-Y. Lin, “Experimental demonstration of broadband solar absorption beyond the lambertian limit in certain thin silicon photonic crystals,” Sci. Rep. 10(1), 11857 (2020). [CrossRef]
4. M. A. Elrabiaey, M. Hussein, M. F. O. Hameed, and S. S. Obayya, “Light absorption enhancement in ultrathin film solar cell with embedded dielectric nanowires,” Sci. Rep. 10(1), 17534 (2020). [CrossRef]
5. S. Bhattacharya and S. John, “Beyond 30% conversion efficiency in silicon solar cells: a numerical demonstration,” Sci. Rep. 9(1), 12482 (2019). [CrossRef]
6. C. E. Disney, S. Pillai, and M. A. Green, “The Impact of parasitic loss on solar cells with plasmonic nano-textured rear reflectors,” Sci. Rep. 7(1), 12826 (2017). [CrossRef]
7. H.-L. Chen, A. Cattoni, R. De Lépinau, A. W. Walker, O. Höhn, D. Lackner, G. Siefer, M. Faustini, N. Vandamme, and J. Goffard, “A 19.9%-efficient ultrathin solar cell based on a 205-nm-thick GaAs absorber and a silver nanostructured back mirror,” Nat. Energy 4(9), 761–767 (2019). [CrossRef]
8. M. H. Muhammad, M. F. O. Hameed, and S. S. Obayya, “Broadband absorption enhancement in modified grating thin-film solar cell,” IEEE Photonics J. 9(3), 1–14 (2017). [CrossRef]
9. H. A. Atwater and A. Polman, “Plasmonics for improved photovoltaic devices,” Nat. Mater. 9(3), 205–213 (2010). [CrossRef]
10. M. H. Elshorbagy, P. A. Sánchez, A. Cuadrado, J. Alda, and Ó. Esteban, “Resonant nano-dimer metasurface for ultra-thin a-Si: H solar cells,” Sci. Rep. 11(1), 7179 (2021). [CrossRef]
11. P. Yu, Y. Yao, J. Wu, X. Niu, A. L. Rogach, and Z. Wang, “Effects of plasmonic metal core-dielectric shell nanoparticles on the broadband light absorption enhancement in thin film solar cells,” Sci. Rep. 7(1), 7696 (2017). [CrossRef]
12. J. Poursafar, M. Bashirpour, M. Kolahdouz, A. V. Takaloo, M. Masnadi-Shirazi, and E. Asl-Soleimani, “Ultrathin solar cells with Ag meta-material nanostructure for light absorption enhancement,” Solar Energy 166, 98–102 (2018). [CrossRef]
13. X. Meng, E. Drouard, G. Gomard, R. Peretti, A. Fave, and C. Seassal, “Combined front and back diffraction gratings for broad band light trapping in thin film solar cell,” Opt. Express 20(S5), A560–A571 (2012). [CrossRef]
14. J. Wu, “Absorption enhancement in thin-film solar cells based on periodically chirped structure,” Solar Energy 165, 85–89 (2018). [CrossRef]
15. M. Mirsafaei, A. H. Fallahpour, P. Lugli, H.-G. Rubahn, J. Adam, and M. Madsen, “The influence of electrical effects on device performance of organic solar cells with nano-structured electrodes,” Sci. Rep. 7(1), 5300 (2017). [CrossRef]
16. F. Enrichi, A. Quandt, and G. C. Righini, “Plasmonic enhanced solar cells: Summary of possible strategies and recent results,” Renewable Sustainable Energy Rev. 82, 2433–2439 (2018). [CrossRef]
17. A. Polemi and K. L. Shuford, “Transmission line equivalent circuit model applied to a plasmonic grating nanosurface for light trapping,” Opt. Express 20(S1), A141–A156 (2012). [CrossRef]
18. P. Mandal and S. Sharma, “Progress in plasmonic solar cell efficiency improvement: A status review,” Renewable Sustainable Energy Rev. 65, 537–552 (2016). [CrossRef]
19. P. Prabhathan and V. Murukeshan, “Surface plasmon polariton-coupled waveguide back reflector in thin-film silicon solar cell,” Plasmonics 11(1), 253–260 (2016). [CrossRef]
20. Y. Da and Y. Xuan, “Role of surface recombination in affecting the efficiency of nanostructured thin-film solar cells,” Opt. Express 21(S6), A1065–A1077 (2013). [CrossRef]
21. A. Pahuja, M. S. Parihar, and D. Kumar, “Performance enhancement of thin film solar cell using two-dimensional plasmonic grating in rear electrode,” IEEE Trans. Nanotechnol. 18, 626–634 (2019). [CrossRef]
22. F. Taghian, V. Ahmadi, and L. Yousefi, “Enhanced thin solar cells using optical nano-antenna induced hybrid plasmonic travelling-wave,” J. Lightwave Technol. 34(4), 1267–1273 (2016). [CrossRef]
23. M. Wellenzohn and R. Hainberger, “Insights in the light trapping effect in silicon solar cells with backside diffraction gratings,” J. Photonics Energy 3(1), 034595 (2013). [CrossRef]
24. D. N. Weiss, B. G. Lee, D. A. Richmond, W. Nemeth, Q. Wang, D. A. Keszler, and H. M. Branz, “Diffractive light trapping in crystal-silicon films: experiment and electromagnetic modeling,” Appl. Opt. 50(29), 5728–5734 (2011). [CrossRef]
25. D. M. Pozar, Microwave Engineering (John Wiley & Sons, 2011).
26. M. Shahabadi, S. Atakaramians, and N. Hojjat, “Transmission line formulation for the full-wave analysis of two-dimensional dielectric photonic crystals,” IEE Proceedings-Science, Measurement and Technology 151(5), 327–334 (2004). [CrossRef]
27. A. Habibzadeh-Sharif and M. Soleimani, “Efficient Lanczos–Fourier expansion-based transmission line formulation for full-wave modal analysis of optical waveguides,” J. Opt. Soc. Am. B 29(6), 1296–1304 (2012). [CrossRef]
28. A. Habibzadeh-Sharif and M. Soleimani, “Analysis and design of a rib-like-based slot waveguide for nonlinear silicon nanophotonics,” J. Mod. Opt. 60(11), 891–899 (2013). [CrossRef]
29. N. A. Stathopoulos, L. C. Palilis, S. P. Savaidis, S. R. Yesayan, M. Vasilopoulou, G. Papadimitropoulos, D. Davazoglou, and P. Argitis, “Optical modeling of hybrid polymer solar cells using a transmission-line model and comparison with experimental results,” IEEE J. Sel. Top. Quantum Electron. 16(6), 1784–1791 (2010). [CrossRef]
30. N. A. Stathopoulos, L. C. Palilis, S. R. Yesayan, S. P. Savaidis, M. Vasilopoulou, and P. Argitis, “A transmission line model for the optical simulation of multilayer structures and its application for oblique illumination of an organic solar cell with anisotropic extinction coefficient,” J. Appl. Phys. 110(11), 114506 (2011). [CrossRef]
31. E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1998), Vol. 3.
32. N. Bonod and J. Neauport, “Diffraction gratings: from principles to applications in high-intensity lasers,” Adv. Opt. Photonics 8(1), 156–199 (2016). [CrossRef]
33. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (CRC Press, 2018).
34. G. J. Swanson, “Binary optics technology: the theory and design of multi-level diffractive optical elements,” (MIT Lexington Lincoln Lab, 1989). [CrossRef]
35. Y.-C. Chang, P. Zhou, and J. H. Burge, “Analysis of phase sensitivity for binary computer-generated holograms,” Appl. Opt. 45(18), 4223–4234 (2006). [CrossRef]
36. D. Madzharov, R. Dewan, and D. Knipp, “Influence of front and back grating on light trapping in microcrystalline thin-film silicon solar cells,” Opt. Express 19(S2), A95–A107 (2011). [CrossRef]
37. S. Hannan, M. T. Islam, A. F. Almutairi, and M. R. I. Faruque, “Wide bandwidth angle-and polarization-insensitive symmetric metamaterial absorber for X and Ku band applications,” Sci. Rep. 10(1), 10338 (2020). [CrossRef]
38. A. Hoque, M. T. Islam, A. F. Almutairi, M. E. Chowdhury, and M. Samsuzzaman, “SNG and DNG meta-absorber with fractional absorption band for sensing application,” Sci. Rep. 10(1), 13086 (2020). [CrossRef]
