This paper considers the electromagnetic scattering problem of periodically corrugated surface with local imperfection of structural periodicity, and presents a formulation based on the coordinate transformation method (C-method). The C-method is originally developed to analyze the plane-wave scattering from perfectly periodic structures, and uses the pseudo-periodic property of the fields. The fields in imperfectly periodic structures are not pseudo-periodic and the C-method cannot be directly applied. This paper introduces the pseudo-periodic Fourier transform to convert the fields in imperfectly periodic structures to pseudo-periodic ones, and the C-method becomes then applicable.
© 2012 OSA
The periodic structures are widely used for optical, photonics, and electronics applications. The most representative example of the periodic structures may be the diffraction grating ruled on a plane substrate, and it plays important roles in spectroscopic measurements or optical communications. Recent improvement of the technology requires highly accurate numerical modeling in some cases. However, the real structures of gratings are not perfectly periodic due to the finite extent and fabrication error, and specifying the influence of the imperfection is sometimes important . Besides, small collapses of the periodicity are sometimes useful to control the electromagnetic wave propagation in the periodic structures. Defects in the photonic crystals work as resonators or waveguides , and modulations of the local period in the gratings yield chirped gratings or the Fresnel lenses .
The aim of this paper is to present a rigorous spectral-domain approach to the electromagnetic scattering from periodically corrugated surface in which the structural periodicity is locally collapsed. The present formulation is based on the differential method of Chandezon et al. (C-method) , which is a well-known approach to the plane-wave scattering from surface-relief grating. When the plane-wave illuminates a perfectly periodic structure, the Floquet theorem asserts that the scattered fields are pseudo-periodic (namely, each field component is given by a product of a periodic function and an exponential phase factor). This implies that the fields have equidistant discrete spectra in the wavenumber space and can be expressed in the generalized Fourier series expansions, and many numerical approaches to periodic structures, including the C-method, utilize this property . However, the structure considered in this paper is not perfectly periodic and the fields generally have continuous spectra. The spectral-domain approach introduces a discretization in the wavenumber space, and the present approach uses the pseudo-periodic Fourier transform (PPFT)  to consider the discretization scheme. The PPFT makes it possible to consider the scheme only inside the Brillouin zone because the transformed fields have a periodic property in terms of the transform parameter related to the wavenumber. Also, the transformed fields are pseudo-periodic in terms of the spatial parameters, and the conventional approaches to the periodic structures become possible to be applied for the scattering problem of imperfectly periodic structures. In Ref. , the scattering problem of the lamellar grating with defects is formulated by the rigorous coupled-wave analysis (RCWA) [8, 9] with the help of PPFT. RCWA is certainly suitable to analyze the lamellar gratings with deep depth. However, it is not always easy to apply for arbitrary shaped gratings [10, 11]. This paper combines the PPFT with the C-method and represents a part of wide applicability.
2. Settings of the problem
The present paper considers the scattering problem of electromagnetic fields with a time-dependence exp(−iωt) from imperfectly periodic surfaces, in which structural periodicity is locally collapsed. Figure 1 shows an example of the structures under consideration. The structure is uniform in the z-direction and the y-axis is perpendicular to the periodicity direction though the periodicity is not perfect. The equation of the corrugated surface is given by y = g(x), where g(x) is supposed to be a continuous function. The minimum and the maximum values of g(x) are denoted by h1 and h2, respectively. The surface profile function g(x) is supposed to be decomposed into periodic and aperiodic parts:
First, we consider the TE-polarized fields. From Maxwell’s curl equations and the constitutive relations, the Cartesian components of electromagnetic fields are described by the following relations:Eqs. (2)–(4) are transformed as follows:
Here, we introduce the PPFT. Let f(u) be a function of u and d be a positive real constant. Then the PPFT and its inverse transform are formally defined byEqs. (5)–(7), we obtain the following relations: Eqs. (10)–(12) yield the following relations: Eqs. (10) and (12) are products of pseudo-periodic functions with no concurrent jump discontinuities, and the Laurent rule  has been used to derive Eqs. (15) and (17).
Considering the periodicity in terms of ξ, Eqs. (15)–(17) have to be satisfied for arbitrary ξ ∈ (−kd/2, kd/2]. However, we take L sample points inside the Brillouin zone and assume that Eqs. (15)–(17) are satisfied at these sample points. Also the integrations in Eqs. (15) and (17) are approximated by an appropriate numerical integration scheme using the same sample points. Let and be the sample points and the weights chosen by a numerical integration scheme. Then we may obtain the following relations:Eqs. (21)–(23) yield the following coupled differential-equation set: (28) can be obtained by solving the eigenvalue-eigenvector problems because the matrix of coefficients Mr is constant. The 2L(2N + 1) eigenvalues can be divided into two sets, each containing L(2N + 1) eigenvalues. The first set contains the negative real eigenvalues and the complex eigenvalues with positive imaginary parts, and the second set contains those with the opposite signs. We denote the reciprocals of the eigenvalues of Mr by , in which correspond to the first set and correspond to the second set. Let pr,n denote the eigenvector of Mr associating with the eigenvalue 1/ηr,n. Then the matrix for diagonalization is constructed as Eqs. (22), (23), and (31), the generalized Fourier coefficients of Ēz(u;ξl, v) and H̄t(u;ξl, v) are expressed in the following form:
For the TM-polarization, following the same process, the generalized Fourier coefficients of H̄z(u;ξl, v) and Ēt (u;ξl, v) are expressed in the following form:
The general solutions separately obtained in the regions s and c can be matched at the grating surface v = 0 by using the boundary conditions, which are given by the continuities of the tangential components of the fields. Then the scattering matrix that relates the amplitudes of the incoming and outgoing fields is derived asEq. (31), the amplitudes of the eigenmodes are written as
4. Numerical experiments
To validate the present formulation, we consider sinusoidal profile gratings that have local collapses of the structural perfection, and show the results of numerical experiments. The periodic part of the surface profile function g(p)(x) is given by
4.1. Treatment of line-source excitation
Since the line-source is located in the region s, the incident field ψ(i)(x,y) is expressed asEqs. (44) and (45) are not easy, and we rewrite these equations as 13]: Eqs. (48) and (49) can be written as Eq. (46) and, then, the Fourier coefficients appeared in Eq. (51) are given by Eqs. (52) and (53) by using the same truncation order N with Eq. (13) and obtain approximate values of the first integrals on the right-hand sides of Eqs. (48) and (49). On the other hand, the second integrals are with a finite interval, and can be evaluated by standard numerical integration.
4.2. Grating with a defect
The first example of the imperfectly periodic structure is the grating with a defect. Here, we define the aperiodic part of the surface profile function g(a)(x) as
The obtained field intensities at the observation point are shown in Fig. 2. Figure 2(a) shows the convergence characteristics in terms of the number of sample points L. The truncation order for each generalized Fourier series expansion is set to N = 4 to calculate these results. The dotted curves are the results of the trapezoidal scheme that uses equidistant sample points and equal weights. Since the transformed field ψ̄(x;ξ,y) is not a smooth function of ξ, they converge very slowly. On the other hand, for the solid curves, we split the Brillouin zone at the Wood-Rayleigh anomalies, which is the non-smooth points, and the double exponential scheme [14, 15] is applied for the subintervals to decide the sample points and the weights. This scheme was used in Ref. , which presents RCWA with the help of PPFT for the lamellar gratings with defects, and showed significant improvement of convergence. The solid curves in Fig. 2(a) show that the same scheme is also effective to improve the convergence. The results of convergence test with respect to the truncation order N are shown in Fig. 2(b). The values are computed with L = 80, and the sample points and the weights are determined by applying the double exponential scheme for the subintervals. The convergence with respect to the truncation order N is very fast. Since the present formulation does not include the generalized Fourier series expansion of discontinuous function and the surface profile under consideration is smooth, the convergence speed is slightly faster than that shown in Fig. 2(b) of Ref. . The distribution of the total field intensities near the defect are calculated with L = 80 and N = 4 by changing the observation point, and shown in Fig. 3. The positions of surfaces are indicated by the white dashed lines and the obtained results seem to be proper.
We examine also the reciprocal property to validate the present formulation. We use the reciprocity error introduced in Ref.  here. Let ψ(xp, yp; xq, yq) denote the field observed at (xp, yp) for a line-source located at (xq, yq). Then, the reciprocity error for two points (xp, yp) and (xq, yq) is defined byFig. 4. The largest values are about 2.9 × 10−3 for the TE-polarization and 2.9 × 10−2 for the TM-polarization. The largest value for the TM-polarization does not seem to be small enough but the related field intensity is very small (about 5.3×10−3) for this point. The second largest value for the TM-polarization is about 3.4 × 10−3, and it can be said that the reciprocity relation is well satisfied.
4.3. Grating with period modulation
Next, we consider the grating in which the period is locally modulated. We define the aperiodic part of the surface profile function as
Figure 5 shows the results of the same convergence tests as in Fig. 2 though the values in Fig. 5(a) are computed with N = 8. The convergence characteristics in terms of the sample point number L are similar to those for the grating with a defect. In contrast, the convergence characteristics in terms of the truncation order N are slower than that for the grating with a defect. Since the surface profile function g(x) under consideration has larger components in high spatial frequency range than that for the grating with a defect, the generalized Fourier series expression requires more terms to construct the original function accurately. The field intensity distributions computed with L = 80 and N = 8 and shown in Fig. 6 and the obtained results seem to be proper for both of the TE- and the TM-polarized cases. Figure 7 shows the results of the same reciprocity test with Fig. 2 though the values are computed with L = 80 and N = 8. The largest values are about 1.5 × 10−3 for the TE-polarization and 1.2 × 10−3 for the TM-polarization, and the reciprocity relation is well satisfied.
5. Concluding remarks
This paper has dealt with the two-dimensional electromagnetic scattering from periodically corrugated surface in which structural periodicity is locally collapsed, and presented a spectral-domain formulation based on the C-method with the help of PPFT. The main problem on the spectral-domain analysis is generally summarized to the discretization scheme on the wavenumber space. In the present formulation, the discretization scheme can be considered only inside the Brillouin zone owing to the PPFT. To validate the present formulation, we considered two specific examples of imperfectly periodic structures: a sinusoidal profile grating with a defect and the same grating with period modulation, and showed some numerical results for a line-source excitation. A simple discretization scheme presented in Ref.  showed good convergences, and the convergence speeds in terms of the sample point numbers were similar for both examples. By contrast, the convergence characteristics in terms of the truncation order depend on the spatial frequency of the structure. The present formulation was also validated by the reciprocity tests.
This work has been partially supported by the Grant Agency of the Czech Republic (# P205/11/2137), by Structural Funds of the European Union and state budget of the Czech Republic ( NANOBASE project # CZ.1.07/2.3.00/20.0074), and by the IT4Innovations Centre of Excellence project, reg. no. CZ.1.05/1.1.00/02.0070.
References and links
1. T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. 46, 5435–5440 (2007). [CrossRef]
2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton Univ. Press, Princeton, 1995).
4. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982). [CrossRef]
5. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980). [CrossRef]
6. K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagnetic Res. PIER 74, 241–271 (2007). [CrossRef]
7. K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express 19, 25799–25811 (2011). [CrossRef]
8. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978). [CrossRef]
9. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982). [CrossRef]
10. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002). [CrossRef]
11. K. Watanabe, “Numerical integration schemes used on the differential theory for anisotropic gratings,” J. Opt. Soc. Am. A 19, 2245–2252 (2002). [CrossRef]
12. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]
13. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
14. H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974). [CrossRef]
15. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic Press, New York, 1984).