The coupled wave theory dealing with optics of discontinuous two-dimensional (2D) periodic structures is reformulated by using Fourier factorization with complex polarization bases, which is a generalized implementation of the fast Fourier factorization rules. The modified approach yields considerably improved convergence properties, as shown on an example of a 2D quartz grating. The method can also be applied to the calculation of 2D photonic band structures or nonperiodic cylindrical devices, and can be generalized to elements with arbitrary cross-sections.
© 2009 Optical Society of America
Fourier-expansion-based electromagnetic theories treating the optical response of periodic structures such as diffraction gratings (hereafter referred to as the coupled wave theory [1, 2, 3]) or photonic crystals [4, 5] have been the subject of considerable development for past several decades. In the case of discontinuous structures the method suffered from poor convergence until the correct Fourier factorization (FF) rules were introduced by Li . Those rules have then been successfully applied to anisotropic , slanted , arbitrary-relief , two-dimensional (2D) gratings , and their various combinations [11, 12, 13], and are also used for other applications such as nonperiodic cylindrical devices [14, 15] or nonlinear optical systems .
As regards 2D structures [17, 18, 19], considerable troubles appeared in the case of elements with circular (or elliptical) cross-sections for which Li’s method of successive one-dimensional (1D) expansions  was not convenient. For this purpose David et al.  applied a method treating independently the tangential and normal components of fields on the edges of circular (or elliptical) holes, forming 2D photonic crystals. Similarly, Schuster and colleagues [21, 22] employed the normal vector method to 2D gratings with various distributions of polarization bases. However, these approaches, always dealing with linear polarizations, ignored the fact that the transformation matrix between the Cartesian and the normal/tangential component bases of polarization became discontinuous at the center and on the boundaries of the periodic cell, which slowed down the convergence of the numerical implementation.
In this article the theory is reformulated via FF with complex polarization bases, which avoids the points of discontinuity of the polarization transformation by employing generally elliptical polarization bases, simply by using complex-valued Jones matrices . The FF approach is incorporated into the Airy-like series propagation algorithm (which can be regarded as a physically transparent variant of the scattering-matrix algorithm [24, 25, 26]), following a 1D analogy described in ; the methodology for obtaining matrix elements is in .
2. Theoretical models
The coupled wave theory of 2D gratings is based on treating the equations (note that the space coordinates and the magnetic field [Hx; Hy] are scaled)
in the Fourier space by expanding the bi-periodic relative permittivity function ε(x,y) and the transverse (to the z axis) components of quasi-periodic electric field E x(y)
(and analogously Hx, Hy), where ∂x = ∂/∂x etc., ∇t = [∂x,∂y], pm = sinϑicosφi + mλ/Λ, qn = sinϑisinφi + nλ/Λ, with λ denoting the wavelength, Λ the grating period, ϑi and φi the spherical angles of the incident wave, and s the eigennumber of the wave equation (1). The configuration of the optical problem is displayed in Fig. 1(a), and the principle of the Airy-like series propagation algorithm is depicted in Fig. 1(b), where the propagation matrix P is determined from the eigensolutions of Eq. (1) and where the reflection and transmission matrices on interfaces R jk, T jk follow from boundary-matching conditions using Eq. (2).
In the Fourier space the operators ∂x, ∂y become diagonal matrices -i p, -i q with particular arrangements of the pm, qn values on their diagonals , whereas the multiplication εEx (or εEy) can be treated by either the Laurent factorization rule [εEx] = [[ε]][Ex] (matrix version of the convolution rule between the Fourier coefficients of the multiplied functions) or by the inverse rule [εEx ] = [[1/ε]]-1 [Ex], where [f] denotes a column vector of the Fourier coefficients of a function f, while [[g]] denotes a matrix composed of the Fourier coefficients of a function g . In this article we will define three models according to applying different factorization rules; first (Model A) assumes the Laurent rules [εEx ] = [[ε]] [Ex] and [εEy] = [[ε]] [Ey].
According to  the Laurent rule is valid (for the reason of uniform convergence) for multiplied functions possessing no concurrent discontinuities, whereas the inverse rule is used for functions whose product is continuous. Obviously, neither the Laurent rule nor the inverse rule is correct for both products Dx = εEx and Dy = εEy, because both pairs of functions have concurrent discontinuities and both products D32 and Dy (the electrical displacement) are discontinuous as well. On the other hand, by a clever change of the polarization bases at all points (using a space-dependent Jones matrix transform F), we can treat independently the normal and tangential components of the fields by the correct rules, i.e., [Du] = [[1/ε]]-1[Eu] and [Dv] = [[ε]][Ev], where Eu, Du are field components normal to the discontinuities of the permittivity function, while Ev, Dv are tangential; the reason for this is because Ev and Du are continuous. This idea was applied in  for 1D gratings, and proposed in  and applied in  for 2D periodicity.
According to , the simplest distribution of the matrix F, whose columns are the basis polarization vectors u = [ξ,; ζ], v = [-ζ *; ξ *] (mutually orthogonal ), can be defined as the rotation by a polar angle ϕ, i.e., ξ = cos ϕ, ζ = sinϕ, within a single cell where we define polar coordinates reiϕ = x + iy, and then by periodic repeating over the entire 2D space. The distribution of the basis vector u in the periodic cell is depicted in Fig. 2(a), from where it is obvious that the matrix function F(x,y) has no concurrent discontinuities with the electric field, so that we can use the Laurent rule for the transformation of polarization
Unfortunately, this approach (here referred to as Model B) only deals with linear polarizations and thus suffers from the fact that the matrix function F(x,y) has a singularity at the center of the periodic cell and other discontinuities on the cell boundaries, which slows down the convergence of the numerical implementation, as will be evidenced below.
To avoid the discontinuities, we develop the method of Fourier factorization with complex polarization bases (referred to as Model C) by using generally elliptic polarizations, so that the matrix function F(x,y) becomes completely continuous. We still use Eqs. (5) and (6) but with
is the space distribution (in the polar coordinates defined in one periodic cell) of the azimuth and the ellipticity of the polarization ellipse of the basis vector u, as depicted in Fig. 2(b), with R denoting the radius of the cylindrical element and D(ϕ)= Λ/2max(∣cos ϕ∣, ∣sinϕ∣) being the distance from the cell’s center to its edge. In Eq. (7) the Jones vector on the right represents a polarization ellipse (with ellipticity η) oriented along the x coordinate, the matrix in the middle rotates this polarization by the azimuth θ, and the factor eiθ preserves the continuity of the phase at the center and on the boundaries of the cell. Similarly to Model B, the azimuth of the polarization ellipse is constant on the lines coming from the cell’s center, but now the ellipticity is zero (corresponding to a linear polarization) only on the dot’s edges, has the maximum value (π/4 for the circular polarization) at the cell’s center and on its boundaries, and is continuously varying (with a smooth sine dependence) in the intermediate ranges [Fig. 2(c)]. The continuity at the center and boundaries of the periodic cell can be easily checked by evaluating limr→0 u = limr→D(ϕ) u = (1/2)1/2[1; i], which is independent of ϕ. Hence we obtain a smooth and completely continuous matrix function F(x,y) corresponding to Model C.
3. Numerical example and discussion
Let us examine the numerical performance of this approach for a 2D grating made as bi-periodically arranged holes patterned on the top of a quartz substrate. Suppose cylindrical holes of the diameter 2R = 200 nm, depth d = 100 nm, square periodicity Λ = 300 nm, an incident plane wave with the wavelength λ = 500 nm, the angle of incidence ϑi = 60°, and the angle of the plane of incidence φi = 0 [the configuration of Fig. 1(a)], with the values of permittivity ε a = 1 for vacuum inside holes and ε b = 2.138 for quartz .
In Fig. 3 we compare calculations using Models A (dashed curves with crosses), B (dotted curves with empty squares), and C (solid curves with filled circles). We plot the energy reflectance in the specular reflection [Figs. 3(a) and 3(b)] and in a first-diffracted order [Figs. 3(c) and 3(d)], more precisely the diffraction efficiencies in the [0,0] and [-1,0] orders of reflection. The efficiencies are determined for the incident s and p polarizations, denoted Rss (m,n) and Rpp (m,n) for the [m,n] order. Note that in the configuration of Fig. 1(a) the mixed-polarization (sp and ps) efficiencies of the two diffracted orders of interest are zero. We display the dependences of the quantities according to the maximum indices M = max ∣m∣ and N = max ∣n∣ of the Fourier harmonics retained inside the periodic medium. For simplicity we keep M = N, so that the order of the [[ε]]-like matrices is (2N + 1)2. Due to the limited memory of the computer used, calculations were possible to perform for N ≤ 16.
As obvious from the plotted curves, all the three models converge to the same limit, but with considerably different convergence performances. While the values yielded by Model B possess precision of about 10-4 for N ≥ 7, Model C yields precision of about 10-5 for the same N (except the sensitive case of Rpp (0,0) with particularly low reflection). The values yielded by Model A are even one order worse than those of Model B, so that they are not visible in Figs. 3(a)–3(c); therefore, insets are included with longer ranges for comparison.
The coupled wave theory of bi-periodic diffraction gratings was reformulated by employing the FF method with complex polarization bases, which enabled considerable enhancement of the numerical capabilities, as evidenced from the comparison with previous approaches. For most practical purposes (e.g., for studying the tendency of the optical quantities while some parameter is varied, or for comparison with an experiment affected by measurement errors) the new method provides sufficient precision for N = 4 (in the presented configuration), which considerably saves the computer memory and the time of calculations. It is worth pointing out that the essential difference between the presented Model C and the previous Model B is (from the mathematical viewpoint) the fact that the matrix transformation of polarization in the former method contains complex-valued elements, so that this generalization is surprisingly simple.
The method can also be advantageously applied to photonic band calculations [by reformulating the permittivity matrix in the formula ε -1∇ × (∇ × E) = (ω/c)2 E], nonperiodic cylindrical devices (by using a large periodic cell), etc., and can be straightforwardly generalized to elements with arbitrary cross-sections [by replacing the constant value of the radius R in Eq. (9) with a function R(ε) and by generalizing the azimuth θ(r,ε) so that the linear polarization on the edge of the element becomes again normal to it].
This work is part of the research plan MSM 0021620834 financed by the Ministry of Education of the Czech Republic and was supported by the Marie Curie International Reintegration Grant (no. 224944) within the 7th European Community Framework Programme. The author thanks Stefan Visnovsky, Martin Veis, and Kamil Postava for fruitful discussions.
References and links
1. R. Petit (ed.), Electromagnetic Theory of Gratings (Springer, 1980).
2. M. Neviere and E. Popov, Light Propagation in Periodic Media: Diffraction Theory and Design (Marcel Dekker, New York, 2003).
3. D. Maystre, “Rigorous vector theories of diffraction gratings,” Prog. Opt. 21, 1–67 (1984). [CrossRef]
4. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light ( Princeton Univ., 1995)
6. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]
7. L. Li, “Reformulation of the Fourier modal method for surface-relief gratings made with anisotropic materials,” J. Mod. Opt. 45, 1313–1334 (1998). [CrossRef]
8. B. Chernov, M. Neviere, and E. Popov, “Fast Fourier factorization method applied to modal analysis of slanted lamellar diffraction gratings in conical mountings,” Opt. Commun. 194, 289–297 (2001). [CrossRef]
9. E. Popov and M. Neviere, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000). [CrossRef]
10. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997). [CrossRef]
11. K. Watanabe, R. Petit, and M. Neviere, “Differential theory of gratings made of anisotropic materials,” J. Opt. Soc. Am. A 19, 325–334 (2002). [CrossRef]
12. K. Watanabe, “Numerical integration schemes used on the differential theory for anisotropic gratings,” J. Opt. Soc. Am. A 19, 2245–2252 (2002). [CrossRef]
13. L. Li, “Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,” J. Opt. A 5, 345–355 (2003). [CrossRef]
14. P. Boyer, E. Popov, M. Neviere, and G. Tayeb, “Diffraction theory in TM polarization: application of the fast Fourier factorization method to cylindrical devices with arbitrary cross section,” J. Opt. Soc. Am. A 21, 2146–2153 (2004). [CrossRef]
15. N. Bonod, E. Popov, and M. Neviere, “Light transmission through a subwavelength microstructured aperture: electromagnetic theory and applications,” Opt. Commun. 245, 355–361 (2005). [CrossRef]
16. N. Bonod, E. Popov, and M. Neviere, “Fourier factorization of nonlinear Maxwell equations in periodic media: application to the optical Kerr effect,” Opt. Commun. 244, 389–398 (2005). [CrossRef]
17. E. Noponen and J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994). [CrossRef]
18. S. Visnovsky and K. Yasumoto, “Multilayer anisotropic bi-periodic diffraction gratings,” Czech. J. Phys. 51, 229–247 (2001). [CrossRef]
19. R. Antos, S. Visnovsky, J. Mistrik, and T. Yamaguchi, “Magneto-optical polar-Kerr-effect spectroscopy on 2D-periodic subwavelength arrays of magnetic dots,” International Journal of Microwave and Optical Technology 1, 905–909 (2006).
20. A. David, H. Benisty, and C. Weisbuch, “Fast factorization rule and plane-wave expansion method for two-dimensional photonic crystals with arbitrary hole-shape,” Phys. Rev. B 73, 075107 (2006). [CrossRef]
21. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007). [CrossRef]
22. P. Gotz, T. Schuster, K. Frenner, S. Rafler, and W. Osten, “Normal vector method for the RCWA with automated vector field generation,” Opt. Express 16, 17295–17301 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17295. [CrossRef] [PubMed]
23. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North Holland, 1997).
24. N. P. K. Cotter, T. W. Preist, and J. R. Sambles, “Scattering matrix approach to multilayer diffraction,” J. Opt. Soc. Am. A 12, 1097–1103 (1995). [CrossRef]
25. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024–1035 (1996). [CrossRef]
26. S. Kaushik, “Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface-relief gratings,” J. Opt. Soc. Am. A 14, 596–609 (1997). [CrossRef]
27. R. Antos, J. Pistora, J. Mistrik, T. Yamaguchi, S. Yamaguchi, M. Horie, S. Visnovsky, and Y. Otani, “Convergence properties of critical dimension measurements by spectroscopic ellipsometry on gratings made of various materials,” J. Appl. Phys. 100, 054906 (2006). [CrossRef]
28. E. D. Palik (ed.), Handbook of Optical Constants of Solids (Academic, 1998).