# Analysis of Faceted Gratings Using C-Method and Polynomial Expansion

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## Abstract

**:**

## 1. Introduction

## 2. Statement of the Problem

#### 2.1. Geometry of the Profile

#### 2.2. Overview of C-Method

## 3. Spectral Formulation of the Problem with C-Method

#### 3.1. Maxwell’s Equations under the Covariant Form

#### 3.2. 2D Operator

## 4. Numerical Solution

#### 4.1. Method of Moments

#### 4.2. Polynomial Basis

#### 4.2.1. Legendre Polynomials

#### 4.2.2. Expansion Basis and Test Basis

#### Properties of the New Basis

- It has to be emphasized that the test functions ${B}_{n}\left(x\right)$ are orthogonal to themselves but also to the ${\tilde{B}}_{m}\left(x\right)$. Indeed we have:$$\begin{array}{c}<{B}_{n},{B}_{m}>=<{B}_{n},{\tilde{B}}_{m}>={\delta}_{mn}{\displaystyle \frac{2}{2q+1}}{\displaystyle \frac{2}{{x}_{q}-{x}_{q-1}}}\phantom{\rule{4.pt}{0ex}}\mathrm{with}\hfill \\ {M}_{q}\le q<{M}_{q+1}\hfill \\ m=q+\sum _{k=1}^{q}{M}_{k-1},\phantom{\rule{1.em}{0ex}}{M}_{0}=0,\hfill \end{array}$$
- The relation (16) between the Legendre polynomials and their derivative may be generalized to the new basis. The inner product $<{B}_{m}\left(x\right),d{\tilde{B}}_{n}/dx>$ generates the following matrix:$$\tilde{\mathbf{D}}={\mathbf{K}}^{t}diag\left(\mathbf{d}\right)\tilde{\mathbf{K}}$$

#### 4.3. Algebraic Eigenequation

#### 4.4. Application to Diffraction Gratings

## 5. Results

#### 5.1. Validation by Comparison with Published Data

#### 5.2. Comparison of Convergence between FMM and CPE

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

- Song, Q.; Pigeon, Y.E.; Heggarty, K. Faceted gratings for an optical security feature. Appl. Opt.
**2020**, 59, 910–917. [Google Scholar] [CrossRef] - Song, J.; Ding, J.F. Echelle diffraction grating demultiplexers with a single diffraction passband. Opt. Commun.
**2010**, 283, 537–541. [Google Scholar] [CrossRef] - Miller, J.M.; de Beaucoudrey, N.; Chavel, P.; Turunen, J.; Cambril, E. Design and fabrication of binary slanted surface-relief gratings for a planar optical interconnection. Appl. Opt.
**1997**, 36, 5717–5727. [Google Scholar] [CrossRef] [PubMed] - Mattelin, M.A.; Radosavljevic, A.; Missinne, J.; Cuypers, D.; Steenberge, G.V. Design and fabrication of blazed gratings for a waveguide-type head mounted display. Opt. Express
**2020**, 28, 11175–11190. [Google Scholar] [CrossRef] [PubMed] - Popov, E. (Ed.) Gratings: Theory and Numeric Applications, 2nd ed.; AMU, CNRS, Institut Fresnel: Marseille, France, 2014. [Google Scholar]
- Bernd, H.; Kleemann, A.M.; Wyrowski, F. Integral equation method with parametrization of grating profile theory and experiments. J. Mod. Opt.
**1996**, 43, 1323–1349. [Google Scholar] [CrossRef] - Chandezon, J.; Raoult, G.; Maystre, D. A new theoretical method for diffraction gratings and its numerical application. J. Opt.
**1980**, 11, 235. [Google Scholar] [CrossRef] - Chandezon, J.; Dupuis, M.T.; Cornet, G.; Maystre, D. Multicoated gratings: A differential formalism applicable in the entire optical region. J. Opt. Soc. Am.
**1982**, 72, 839–846. [Google Scholar] [CrossRef] - Moharam, M.G.; Gaylord, T.K. Rigorous coupled-wave analysis of metallic surface-relief gratings. J. Opt. Soc. Am. A
**1986**, 3, 1780–1787. [Google Scholar] [CrossRef] - Moharam, M.G.; Grann, E.B.; Pommet, D.A.; Gaylord, T.K. Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings. J. Opt. Soc. Am. A
**1995**, 12, 1068–1076. [Google Scholar] [CrossRef] - Lalanne, P.; Morris, G.M. Highly improved convergence of the coupled-wave method for TM polarization. J. Opt. Soc. Am. A
**1996**, 13, 779–784. [Google Scholar] [CrossRef] - Granet, G.; Guizal, B. Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization. J. Opt. Soc. Am. A
**1996**, 13, 1019–1023. [Google Scholar] [CrossRef] - Li, L. Use of Fourier series in the analysis of discontinuous periodic structures. J. Opt. Soc. Am. A
**1996**, 13, 1870–1876. [Google Scholar] [CrossRef] - Li, L.; Chandezon, J. Improvement of the coordinate transformation method for surface-relief gratings with sharp edges. J. Opt. Soc. Am. A
**1996**, 13, 2247–2255. [Google Scholar] [CrossRef] - Morf, R.H. Exponentially convergent and numerically efficient solution of Maxwell’s equations for lamellar gratings. J. Opt. Soc. Am. A
**1995**, 12, 1043–1056. [Google Scholar] [CrossRef] - Edee, K.; Fenniche, I.; Granet, G.; Guizal, B. Modal method based on subsectional Gegenbauer polynomial expansion for lamellar gratings: Weighting function, convergence and stability. Prog. Electromagn. Res.
**2013**, 133, 13–35. [Google Scholar] [CrossRef] - Granet, G.; Randriamihaja, M.H.; Raniriharinosy, K. Polynomial modal analysis of slanted lamellar gratings. J. Opt. Soc. Am. A
**2017**, 34, 975–982. [Google Scholar] [CrossRef] - Edee, K.; Plumey, J.P. Numerical scheme for the modal method based on subsectional Gegenbauer polynomial expansion: Application to biperiodic binary grating. J. Opt. Soc. Am. A
**2015**, 32, 402–410. [Google Scholar] [CrossRef] - Plumey, J.P.; Guizal, B.; Chandezon, J. Coordinate transformation method as applied to asymmetric gratings with vertical facets. J. Opt. Soc. Am. A
**1997**, 14, 610–617. [Google Scholar] [CrossRef] - Preist, T.W.; Harris, J.B.; Wanstall, N.P.; Sambles, J.R. Optical response of blazed and overhanging gratings using oblique chandezon transformations. J. Mod. Opt.
**1997**, 44, 1073–1080. [Google Scholar] [CrossRef] - Li, L. Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings. J. Opt. Soc. Am. A
**1999**, 16, 2521–2531. [Google Scholar] [CrossRef] - Ming, X.; Sun, L. Simple reformulation of the coordinate transformation method for gratings with a vertical facet or overhanging profile. Appl. Opt.
**2021**, 60, 4305–4314. [Google Scholar] [CrossRef] [PubMed] - Li, L.; Chandezon, J.; Granet, G.; Plumey, J.P. Rigorous and efficient grating-analysis method made easy for optical engineers. Appl. Opt.
**1999**, 38, 304–313. [Google Scholar] [CrossRef] [PubMed] - Plumey, J.P.; Granet, G. Generalization of the coordinate transformation method with application to surface-relief gratings. J. Opt. Soc. Am. A
**1999**, 16, 508–516. [Google Scholar] [CrossRef] - Harrington, R.; Antennas, I.; Society, P. Field Computation by Moment Methods; IEEE Press Series on Electromagnetic Waves; IEEE: New York, NY, USA, 1996. [Google Scholar]

**Figure 1.**Illustration of faceted gratings. (

**a**): blazed grating, (

**b**) trapezoidal grating, (

**c**) overhanging grating, (

**d**) Omega-like grating.

**Figure 2.**Illustration of an inclined translation coordinate system. Two coordinate surfaces ${x}^{3}=cte$ are translated from each other parallel to ${x}^{3}$-axis.

**Figure 3.**Illustration of the incident and diffracted waves on a surface relief faceted grating. The incident medium has a real optical index. $\left(1\right)$ and $\left(2\right)$ refer to medium 1 and medium 2 respectively.

**Figure 4.**Overhanging gratings. The endpoint coordinates of the facets of the grating (

**a**) are $(0,0)$, $(1.1,1)$, and $(1,1)$. Gratings (

**b**) and (

**c**) have similar profile shapes with different parameters. The endpoint coordinates of the facets of the grating (

**b**) are $(0,0)$, $0.75,1.25$, $(1,1.25)$, $(0.72,0)$, $(1,0)$ whereas the endpoint coordinates of the facets of the grating (

**c**) are $(0,0)$, $(3/8,1)$, $(5/8,1)$, $(4/8,0)$, $(1,0)$. All units are arbitrary.

**Figure 5.**Error in the calculation of the $T{e}_{+1}$ efficiency of grating (b) in Figure 4. N is the number of Floquet harmonics, M is the number of slices. The grating is enlightened from the vacuum under normal incidence. The parameters are $\lambda =1.031$, ${\nu}_{1}=1$, ${\nu}_{2}=1.5$ and its end facets coordinates in arbitrary units are: $(x,z)$ are $(0,0)$, $0.75$, $1.25$, $(1,1.25)$, $(0.72,0)$, $(1,0)$.

**Figure 6.**Error in the calculation of the $T{e}_{+1}$ efficiency of grating (b) in Figure 4. The grating is enlightened from vacuum under normal incidence. The parameters are $\lambda =1.031$, ${\nu}_{1}=1$, ${\nu}_{2}=1.5$ and its end facets coordinates in arbitrary units are: $(x,z)$ are $(0,0)$, $0.75$, $1.25$, $(1,1.25)$, $(0.72,0)$, $(1,0)$.

**Figure 7.**Computation time as a function of error with FMM and CPE. The computation concerns the $T{e}_{+1}$ transmitted order. The graphs plot 100 times the true computation time. For Fmm, the number of Floquet harmonics being fixed at 21, the error diminishes with the number of slices. For CPE the error diminishes with the number of basis elements.

**Table 1.**Comparison between FMM and C-method by Plumey et al. [19] on one hand and our implementation of C-method in an oblique coordinate system on the other hand. The exponent $\left(a\right)$, is for Plumey’s results, taken from Table 3 in [19]. The profile shape is that of grating $\left(a\right)$ in Figure 4. The optical indices are such that ${\nu}^{2}=2.25$ (D), and ${\nu}^{2}=-21-i60.4$ (LD). The incident wave parameters are $\theta ={25}^{\circ}$ and $\lambda =0.7$.

Orders | C-Method ^{(a)} | FMM ^{(a)} | CPE | |||
---|---|---|---|---|---|---|

D | LD | D | LD | D | LD | |

TE polarization | ||||||

${R}_{-2}$ | 0.04506 | 0.23791 | 0.04502 | 0.23602 | 0.45071 | 0.23869 |

${R}_{-1}$ | 0.00316 | 0.31502 | 0.00315 | 0.30576 | 0.00316 | 0.31410 |

${R}_{0}$ | 0.00019 | 0.10992 | 0.00019 | 0.10593 | 0.00020 | 0.10922 |

${T}_{-2}$ | 0.35193 | 0.35197 | 0.35199 | |||

${T}_{-1}$ | 0.02459 | 0.02459 | 0.02459 | |||

${T}_{0}$ | 0.5659 | 0.56594 | 0.56591 | |||

${T}_{1}$ | 0.00913 | 0.00912 | 0.00913 | |||

TM polarization | ||||||

${R}_{-2}$ | 0.03438 | 0.45797 | 0.03418 | 0.42275 | 0.03445 | 0.45382 |

${R}_{-1}$ | 0.00114 | 0.14879 | 0.00111 | 0.14854 | 0.00115 | 0.15215 |

${R}_{0}$ | 0.00004 | 0.00841 | 0.00004 | 0.00322 | 0.00005 | 0.00616 |

${T}_{-2}$ | 0.11085 | 0.11061 | 0.11087 | |||

${T}_{-1}$ | 0.16566 | 0.16515 | 0.16569 | |||

${T}_{0}$ | 0.68405 | 0.68501 | 0.68396 | |||

${T}_{1}$ | 0.00384 | 0.00389 | 0.00386 |

**Table 2.**Comparison between C-method by Ming et al [22] on the one hand and our implementation of C-method (CPE, for C-method by polynomial expansion) in an oblique coordinate system on the other hand. The exponent $\left(a\right)$, is for Mings’s results, taken from Table 3 in [22] The profile shape is that of grating $\left(c\right)$ in Figure 4. The optical index is ${\nu}_{2}=1.5$. The incident wave parameters are $\theta ={0}^{\circ}$ and $\lambda =1/1.5$.

Orders | C-Method ^{(a)} | CPE |
---|---|---|

TE polarization | ||

${R}_{1}$ | 0.1156 (−1) | 0.01162 |

${R}_{0}$ | 0.3981 (−2) | 0.00397 |

${R}_{1}$ | 0.2673 (−3) | 0.00270 |

${T}_{-2}$ | 0.9877 (−1) | 0.98399 |

${T}_{-1}$ | 0.4771 | 0.47659 |

${T}_{0}$ | 0.9975 (−1) | 0.09944 |

${T}_{1}$ | 0.2685 | 0.26885 |

${T}_{2}$ | 0.3831 (−1) | 0.03428 |

TM polarization | ||

${R}_{-1}$ | 0.5258 (−2) | 0.00523 |

${R}_{0}$ | 0.3197 (−2) | 0.00320 |

${R}_{1}$ | 0.5832 (−3) | 0.00058 |

${T}_{-2}$ | 0.3419 (−1) | 0.03405 |

${T}_{-1}$ | 0.7470 | 0.74813 |

${T}_{0}$ | 0.4760 (−1) | 0.04749 |

${T}_{1}$ | 0.1502 | 0.15022 |

${T}_{2}$ | 0.1240 (−1) | 0.01239 |

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**MDPI and ACS Style**

Granet, G.; Edee, K.
Analysis of Faceted Gratings Using C-Method and Polynomial Expansion. *Photonics* **2024**, *11*, 215.
https://doi.org/10.3390/photonics11030215

**AMA Style**

Granet G, Edee K.
Analysis of Faceted Gratings Using C-Method and Polynomial Expansion. *Photonics*. 2024; 11(3):215.
https://doi.org/10.3390/photonics11030215

**Chicago/Turabian Style**

Granet, Gérard, and Kofi Edee.
2024. "Analysis of Faceted Gratings Using C-Method and Polynomial Expansion" *Photonics* 11, no. 3: 215.
https://doi.org/10.3390/photonics11030215