Near- to far-field transformation in the aperiodic Fourier modal method

Transcription

1 Near- to far-field transformation in the aperiodic Fourier modal method Ronald Rook, 1 Maxim Pisarenco, 1,2, * and Irwan D. Setija 2 1 Department of Mathematics and Computer Science, Eindhoven University of Technology, Den Dolech 2, 5600MB Eindhoven, The Netherlands 2 ASML Research, De Run 6501, 5504DR eldhoven, The Netherlands *Corresponding author: m.pisarenco@tue.nl Received 17 July 2013; revised 22 August 2013; accepted 29 August 2013; posted 5 September 2013 (Doc. ID ); published 1 October 2013 This paper addresses the task of obtaining the far-field spectrum for a finite structure given the near-field calculated by the aperiodic Fourier modal method in contrast-field formulation (AFMM-CFF). The AFMM-CFF efficiently calculates the solution to Maxwell s equations for a finite structure by truncating the computational domain with perfectly matched layers (PMLs). However, this limits the far-field solution to a narrow strip between the PMLs. The Green s function for layered media is used to extend the solution over the whole super- and substrate. The approach is validated by applying it to the problem of scattering from a cylinder for which the analytical solution is available. Moreover, a numerical study is conducted on the accuracy of the approximate far-field computed with the super-cell Fourier modal method by using the AFMM-CFF with near- to far-field transformation as a reference Optical Society of America OCIS codes: ( ) Numerical approximation and analysis; ( ) Computational electromagnetic methods; ( ) Forward scattering Introduction The classical Fourier modal method (FMM) is an efficient simulation tool for scattering problems when periodic structures, such as gratings, are concerned. The reader is referred to [1,2] for a description of the method and to [3, Section 4.1] for a history of improvements introduced in the FMM. One of the advantages of the FMM is that besides the near-field data, also far-field data (the relevant quantity in optics applications) are directly available. This is due to the fact that the field is decomposed into a discrete spectrum of plane waves, a mathematical description which naturally fits with the physical phenomenon of scattering from periodic scatterers. In the past few years, the aperiodic FMM in contrast-field formulation (AFMM-CFF) has been X/13/ $15.00/ Optical Society of America introduced [4 7]. It is based on the classical FMM and allows simulation of scattering from finite structures. This is achieved by placing perfectly matched layers (PMLs) [8,9] at the lateral sides of the structure and reformulating the governing equations. As a consequence of these modifications the far-field is not directly available from AFMM-CFF. By using a Green s function approach, we provide the steps for computing the field at an arbitrary point in space as well as obtaining the far-field spectrum. Other works on the near- to far-field transformation include [10 14]. In [10 12], the authors are concerned with the computation of the far-field by measuring the near-field of the structure. In order to minimize the number of measurements, optimal sampling constitutes an important issue. In [13], a parallel implementation of the near- to far-field transformation is presented, while [14] focuses on the comparison of the surface and volume integral approaches for the calculation of far fields. In our 6962 APPLIED OPTICS / ol. 52, No. 28 / 1 October 2013

2 work, the near- to far-field transformation is specifically tailored for the near-field representation in terms of Fourier modes used in the AFMM-CFF. Note that the approach presented here applies both for propagating the near-fields to far-fields as well as for propagation to any intermediate location. On the other hand, because of the approximations applied to the surface integrals, the standard transformation typically used within the finite-difference timedomain (FDTD) method [15] is strictly a near- to far-field transform. The proposed approach is validated by using a case for which the analytical solution is known. Subsequently, a numerical study on the accuracy of the approximate far-field computed with the super-cell FMM is performed. To our knowledge, this is the first study of this type. This paper is structured as follows. In Section 2,we discuss the limitations of the AFMM-CFF concerning far-field data. Section 3 gives a technical description of far-field recovery from the near-field data using Green s functions. Numerical results and conclusions are presented in the last two sections. 2. Far-Field Limitations of the AFMM-CFF We consider a periodic structure with period Λ in a two-layer medium with electric permittivities ϵ 1 and ϵ 2 (see the top part of Fig. 1). Nonmagnetic materials are considered. Propagation of electromagnetic fields within and around the structure is governed by Maxwell s equations e t x k 0 h t x ; h t x k 0 ϵ x; z e t x ; (1a) (1b) where x x; y; z is the position vector, e t e x ;e y ;e z is the total electric field, and h t h x ;h y ;h z is the total magnetic field scaled by p i ϵ 0 μ 0. The temporal frequency ω is incorporated p into the constant k 0 ω ϵ 0 μ 0. The electric permittivity ϵ is assumed y-invariant. The structure is illuminated by a plane wave e inc x ae ikinc x ; (2) where k inc k inc x ;k inc y ;k inc z is the wavevector and a a x ;a y ;a z is the amplitude vector. Fig. 1. Problems P 1 and P 2 have equal solutions on Ω 0 for reflectionless PMLs (indicated by hatched stripes). In the classical FMM, the geometry is divided into M slices along the z direction and the field above the structure (slice 1) is written as cf. [1] where k xn k inc x e r x; y; z XN n N r n e i k xnx k y y k 1;zn z ; n 2π ; n N; ; N; (3a) Λ k y k inc y ; (3b) q k 1;zn k 2 0 ϵ 1 k 2 y k 2 xn; n N; ; N: (3c) The square root in the last expression is defined such that Ik 1;zn 0 (here I denotes the imaginary part of a complex number). This requirement is necessary in order to guarantee that evanescent waves decay in the direction of propagation. The electric permittivity ϵ 1 1 corresponds to free space (air). The terms in the summation are plane waves which coincide with the orders of diffraction from the given periodic structure. Thus, this representation directly gives the discrete far-field spectrum r n for a periodic structure. Also, the diffraction efficiencies, which are often the quantities of interest, may be easily computed. A similar representation of the field is used in AFMM-CFF for a finite structure of width Λ. However, in this case, the coefficients r n do not directly correspond with the far-field spectrum of that finite structure. As a first indication serves the fact that a discrete spectrum is obtained, while a continuous one is expected for a structure which is not periodic. A more detailed explanation follows. The extension of the FMM to AFMM-CFF is based on the observation that the solution of problem P 1 on the domain Ω 0 (see Fig. 1) is equivalent to the solution of P 2 on Ω 0 [4]. Note that the hatched layers represent the PMLs which act as absorbing layers and decouple the fields in the two neighboring cells. Although the fields in Ω 0 are equal for the two problems, the far-fields are different. Thus, instead of the desired continuous spectrum of P 1 we obtain the discrete spectrum of the artificially periodized problem P 2. In the next section we discuss the computation of the far-field spectrum of P 1 based on the solution of P Far-Field Recovery We will use the Green s function formalism in order to compute the field outside the scatterer when the field inside the scatterer is given. In order to apply this procedure, the magnetic field in Eq. (1) is eliminated and the so-called double-curl equation is obtained 1 October 2013 / ol. 52, No. 28 / APPLIED OPTICS 6963

3 e t x k 2 0 ϵ x; z et x 0: (4) The AFMM-CFF reformulates this equation in terms of a contrast field e c and the incident field e inc is incorporated into the background field e b [4] e c x k 2 0 ϵ x; z ec x k 2 0 ϵ x; z ϵb z e b x ; (5) where ϵ x; z is the permittivity profile of the problem and ϵ b z is the permittivity of the background multilayer. Since the Green s function for an arbitrary permittivity profile ϵ x; y is not available, we add k 2 0 ϵ x; z ϵb z e c x to both sides of Eq. (5) resulting in e c x k 2 0 ϵb z e c x k 2 0 ϵ x; z ϵb z e t x ; (6) where e t e b e c. In operator form Eq. (6) is written as with Le c x f x (7) L k 2 0 ϵb z ; (8) f x k 2 0 ϵ x; z ϵb z e t x : (9) Note that we only need the value of e t x within the scatterer, that is in the region where ϵ x; z ϵ b z 0. The Green s function is defined as the tensor function satisfying with and LG x; x 0 Iδ x x 0 (10) 0 G xx x; x 0 G xy x; x 0 1 G xz x; x 0 G x; x G yx x; x 0 G yy x; x 0 G yz x; x 0 A G zx x; x 0 G zy x; x 0 G zz x; x 0 0 Iδ x x δ x 1 x δ x x 0 0 A: 0 0 δ x x 0 Multiplying Eq. (10) byf x 0, integrating in x 0 and using Eq. (7) we observe that LG x; x 0 f x 0 dx 0 Le c x : (11) Since the L operator is independent of x 0 it can be taken outside the integral and we obtain the expression for the contrast field in any point x R 3, e c x G x; x 0 f x 0 dx 0 : (12) In terms of electromagnetic quantities this expression states that the field due to a radiating source can be computed as a projection of the source on the fundamental solution (Green s function) of Maxwell s equations. Our goal is to present the principle of far-field calculation with AFMM-CFF while keeping the exposition simple. For this reason we will consider a two-dimensional geometry, isotropic materials, and planar TE-polarized incident light. The latter implies that only the yy-component of the Green s tensor [as used in Eqs. (13) and (14)] is needed, while the former simplifies the equations by considering scalar permittivity instead of a tensor one. Generalization to TM-polarization in planar incidence as well as to conical incidence is straightforward and requires the full 3-by-3 Green s tensor (see [16,17]). Extension to three-dimensional configurations requires minimal modifications: the integration domain is extended by one dimension while the layered media Green s function remains unchanged. Now we can use Eq. (12) to compute the field in an arbitrary point in space. The Green s function G is the fundamental solution in layered media (see [16]) and the source term is defined by Eq. (9). Since ϵ x; z ϵ b z 0 outside the scatterer, we only need the field within a bounded domain that is accessible to the AFMM-CFF. In the case of a planar incident wave and TE-polarization, the electric field and the Green s function are fully determined by their y components (e y and G yy, respectively). The contrast field is given by e c y x G yy x; x 0 ϵ x 0 ϵ b x 0 e y x 0 dx 0 : (13) The numerical evaluation of the spatial Green s function can be avoided if Fourier modes are desired only. The Fourier modes of the contrast field on a plane z const can be obtained via a similar projection on the spectral Green s function ê c y k x ; z Ĝ yy k x ; x 0 ;z;z 0 ϵ x 0 ;z 0 ϵ b z 0 e y x 0 ;z 0 dx 0 dz 0 ; (14) where Ĝyy k x ; x 0 ;z;z 0 is the Fourier transform of the spatial Green s function. We separate the x 0 -dependent part from the z 0 -dependent part, Ĝ yy k x ; x 0 ;z;z 0 G ~ yy k x ; z; z 0 e ik xx 0. If the scatterer is situated in the top layer of a multilayer medium, we have [16, Appendix 1] ~G yy k x ; z; z 0 1 2ik 1;z e ik 1;zjz z 0j ~ R 12 e ik 1;z0 z z 0 2d ; (15a) 6964 APPLIED OPTICS / ol. 52, No. 28 / 1 October 2013

4 where the generalized Fresnel reflection coefficient ~R 12 is given by the recursive relation ~R l;l 1 R l;l 1 ~ R l 1;l 2 e 2ik l 1;z0 d l 1 d l 1 R l;l 1 ~ Rl 1;l 2 e 2ik l 1;z0 d l 1 d l 15b for l 1; ;M 1. In the above expression the local reflection coefficients are defined as with R l;l 1 k l;z k l 1;z k l;z k l 1;z 15c q k l;z k 2 0 ϵ l k inc y 2 k 2 x; (15d) q k l;z0 k 2 0 ϵ l k inc y 2 k inc x 2 : (15e) The recursion is initialized with ~R M;M 1 0: (15f) 4. Results Three configurations are addressed in this section: a dielectric cylinder in free space, a single rectangular line on substrate, and an ultra small grating (with four lines) on substrate. The purpose of the first configuration is to validate the results of our far-field computation against the available exact solution. The other two configurations represent real-life applications for which the far-field behavior is studied. Although no exact solution is available for these cases, we may use the super-cell approach as a reference. This comparison is performed for the ultra small grating. In all cases, the object is illuminated by a normal planar incident wave, explicitly given in Eq. (2). We will compute the far-field in real space using Eq. (13) and the far-field in the Fourier domain (spectrum) using Eq. (14). The volume integrals are approximated by accurate quadrature rules. Fig. 2. Slicing of the cylinder with color-coded refraction indices. and the near-field e y inside the cylinder is obtained with the AFMM-CFF. The far-field recovered from the near-field is compared to the far-field given by the analytical solution for this problem. The most sensitive parameters are the number of slices and the number of harmonics used in the AFMM-CFF. Both parameters control the accuracy of the near-field needed as input for the far-field calculation. We define the far-field error in real space as E 1 e y e ref y 2 e ref y 2 ; (16) where e y is computed via Eq. (13) and e ref y is the exact field given by the analytical solution of plane-wave scattering by a cylinder [19, Section 11.2]. The fields are evaluated at observer points that are distributed on a line z 20, y 0 at an angle from 85 to 85 deg with respect to the z axis on top of the cylinder. The error E 1 (spatial far-field error) as a function of number of slices and number of harmonics is depicted in Fig. 3. As expected, the error decreases as the number of slices and harmonics are simultaneously increased. If one of the discretization parameters is kept constant then the error is dominated by the error due to this parameter (slicing/harmonics) and does not decrease. A. Cylinder Problem The geometry of the cylinder is approximated in the AFMM-CFF by multiple slices as depicted in Fig. 2. The permittivity of the cylinder is ϵ 2.25 and the wavevector of the incident plane wave is normal to the x y plane. The (contrast) far-field in real space is given by Eq. (13) with the closed-form Green s function for free space G yy x; z; x 0 ;z 0 i 4 H 2 0 q k 0 x x 0 2 z z 0 2 : The integral over a disk is numerically approximated by a cubature rule of degree 21 with 88 points [18] Fig. 3. Logarithmic plot (log 10 E 1 ) of the error in the spatial far-field for the cylinder problem as defined in Eq. (16). 1 October 2013 / ol. 52, No. 28 / APPLIED OPTICS 6965

5 The far-field error in Fourier space is defined as E 2 ê y ê ref y 2 ê ref y 2 ; (17) where ê y is computed via Eq. (14) and ê ref y are the exact propagating Fourier modes obtained from the stationary-phase approximation [20]. The error E 2 (spectral far-field error) as a function of number of slices and number of harmonics is depicted in Fig. 4. The behavior of the spectral far-field error is very similar to the spatial far-field error. The two convergence plots validate the correct numerical implementation of the calculations in Eqs. (13) and (14). The error decreases as the number of slices and harmonics in the preceding calculation involving AFMM-CFF is increased, and the rate of decrease is similar for the two calculations. This implies that the accuracy of the near-field obtained with the AFMM-CFF limits the accuracy of the far-field. The errors introduced by the numerical evaluation of integrals Eqs. (13) and (14) are negligibly small. B. Single Rectangular Line After validating the far-field recovery routine, we apply it to two other configurations. We consider a single rectangular line supported by a substrate. The AFMM-CFF discretizes the medium in three layers: the top is air, the middle is air with one resist line, and at the bottom is the substrate. In this case, the substrate and the scatterer are of the same material, ϵ As usual, the near-field is obtained with the AFMM-CFF. The Green s function is given by Eq. (15a) with ~ R 12 R 12. In the AFMM-CFF the number of harmonics is N 200 and the artificial period is Λ 5. The field in the scatterer is sampled in points and the Clenshaw Curtis integration rule [21] is used for the integral Eq. (14). Figure 5 shows the continuous Fourier modes. The wave numbers k x are defined similarly to Eq. (3) k x κ k inc x κ; Fig. 4. Logarithmic plot (log 10 E 2 ) of the error in the spectral far-field for the cylinder problem as defined in Eq. (17). Fig. 5. Reflective modes for different angle of incidence: θ 0 (solid line) to θ 1 3 (dashed dotted line). where κ is the continuous angular frequency. On the horizontal axis is k x κ with κ 3; 3 and on the vertical axis are the absolute value and the phase of the modes, respectively. The resulting modes in this problem have close resemblance to the modes of a single point source in layered media. A 90 deg phase transition of the reflected field occurs between the poles of expression (15a). C. Ultra Small Grating In this subsection we will investigate the far-field spectrum of an ultra small grating. For comparison, an approximate discrete far-field spectrum will be computed with the super-cell approach [22,23]. It consists of using the classical periodic FMM for the whole structure (instead of applying it to a single period) while including empty space on the sides where the field can decay such that the effect of the periodicity is minimized. Since the computational domain includes all lines of the structure plus the empty space, the number of harmonics has to be accordingly increased in order to represent the solution accurately. The modes computed by the far-field routine can be related to the modes obtained from a super-cell 6966 APPLIED OPTICS / ol. 52, No. 28 / 1 October 2013

6 Fig. 6. Multiple resist lines with color-coded refractive indices. simulation. To demonstrate this, we consider an ultra small grating with four rectangular lines on a silicon substrate. The geometry, p together with the refractive indices n ϵ of all materials, is shown in Fig. 6. The discrete spectrum from the super-cell simulations (pitch Λ 20, number of harmonics N 100) and the continuous spectrum obtained using the AFMM-CFF and the far-field routine are depicted in Fig. 7. The discrete modes follow quite accurately the continuous spectrum. However, it is not always easy to predict the behavior between the modes as can be seen at k x 1. Between modes 3 and 4 the continuous spectrum has a nonsmooth behavior (fast decrease followed by sudden increase). We define the super-cell far-field error as E 3 Iêsuper cell y ê ref y 2 ; (18) ê ref y 2 where I is an interpolation operator that interpolates the discrete super-cell modes, such that we can compare them to the continuous modes. Higherorder interpolation is not advised for this situation since the continuous modes exhibit discontinuous derivatives with respect to k x. Therefore, linear interpolation is used. The comparison is restricted to propagating modes only, i.e., to the plane-waves with Rk z > 0 (R denotes the real part of a complex number). The reference ê ref y is obtained numerically with AFMM-CFF via Eq. (14) using a large number of harmonics, N 150. The quadrature employed for the integral Eq. (14) uses integration points in each line. Figure 8 shows the super-cell far-field error E 3. The number of harmonics used in the super-cell FMM solution is increased to resolve the increasing pitch of the (artificially periodic) grating. We observe that in a super-cell approach, a high number of harmonics and a large pitch (N >100, Λ 150) are required for accurate results (E < 10 3 ). Such a calculation takes about 3 s for 100 harmonics. On the other hand, a typical AFMM-CFF calculation for Fig. 7. Super cell modes (open circle) compared to the continuous modes (solid line). Fig. 8. Logarithmic plot (log 10 E 3 ) of the error in the spectral far-field for the super-cell problem as defined in Eq. (18). the same problem requires harmonics and together with the far-field routine takes less than 1 s. 5. Conclusion The Green s function theory provides a tool for calculating the far-field solution, which is a projection of the total field (provided by the AFMM-CFF) on the spatial or spectral Green s function for layered media. The method has been validated using the cylinder problem in free space. The spatial and spectral far-fields computed numerically have been compared to the corresponding exact data. In the example with an ultra small grating it has been demonstrated that the AFMM-CFF combined with a postprocessing step involving the far-field routine is several times faster than the traditional super-cell approach. More importantly, the spectrum computed with the far-field routine based on the Green s function is continuous (and can be evaluated for any wavenumber), as opposed to the discrete spectrum computed with the super-cell approximation. The discrete spectrum can only be made denser by increasing the number of harmonics resulting in a significant impact on the computational cost. References 1. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings, J. Opt. Soc. Am. A 12, (1995). 2. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach, J. Opt. Soc. Am. A 12, (1995). 3. G. Bao, L. Cowsar, and W. Masters, eds., Mathematical Modeling in Optical Science (Frontiers in Applied Mathematics) (Society for Industrial Mathematics, 2001). 4. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures, J. Opt. Soc. Am. A 27, (2010). 5. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, Modified S-matrix algorithm for the aperiodic Fourier modal method in contrast-field formulation, J. Opt. Soc. Am. A 28, (2011). 1 October 2013 / ol. 52, No. 28 / APPLIED OPTICS 6967

7 6. M. Pisarenco, J. M. L. Maubach, I. D. Setija, and R. M. M. Mattheij, Efficient solution of Maxwell s equations for geometries with repeating patterns by an exchange of discretization directions in the aperiodic Fourier modal method, J. Comput. Phys. 231, (2012). 7. M. Pisarenco, Scattering from finite structures: an extended Fourier modal method, Ph.D. thesis (Eindhoven University of Technology, 2011). 8. J.-P. Berenger, A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys. 114, (1994). 9. J. P. Hugonin and P. Lalanne, Perfectly matched layers as nonlinear coordinate transforms: a generalized formalization, J. Opt. Soc. Am. A 22, (2005). 10. O. M. Bucci, C. Gennarelli, and C. Savarese, Fast and accurate near-field-far-field transformation by sampling interpolation of plane-polar measurements, IEEE Trans. Antennas Propag. 39, (1991). 11. A. Taaghol and T. K. Sarkar, Near-field to near/far-field transformation for arbitrary near-field geometry, utilizing an equivalent magnetic current, IEEE Trans. Electromag. Compat. 38, (1996). 12. T. K. Sarkar and A. Taaghol, Near-field to near/far-field transformation for arbitrary near-field geometry utilizing an equivalent electric current and MoM, IEEE Trans. Antennas Propag. 47, (1999). 13. P.-W. hai, Y.-K. Lee, G. W. Kattawar, and P. Yang, Implementing the near- to far-field transformation in the finitedifference time-domain method, Appl. Opt. 43, (2004). 14. P. Török, P. R. Munro, and E. E. Kriezis, Rigorous near- to farfield transformation for vectorial diffraction calculations and its numerical implementation, J. Opt. Soc. Am. A 23, (2006). 15. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd ed. (Artech House, 2005). 16. K. A. Michalski and D. heng, Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I Theory, IEEE Trans. Antennas Propag. 38, (1990). 17. J. R. Wait, Electromagnetic Waves in Stratified Media (IEEE/ OUP Series on Electromagnetic Wave Theory) (Oxford University, 1996). 18. R. Cools and K. Kim, A survey of known and new cubature formulas for the unit disk, J. Appl. Math. Comput. 7, (2000). 19. A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering (Prentice-Hall, 1990). 20. R. Courant and D. Hilbert, Methods of Mathematical Physics (Interscience, 1953). 21. C. W. Clenshaw and A. R. Curtis, A method for numerical integration on an automatic computer, Numer. Math. 2, (1960). 22. 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, (2006). 23. J. Saarinen, E. Noponen, and J. P. Turunen, Guidedmode resonance filters of finite aperture, Opt. Eng. 34, (1995) APPLIED OPTICS / ol. 52, No. 28 / 1 October 2013