On error estimation in the fourier modal method for diffractive
gratings
Citation for published version (APA):
Hlod, A., & Maubach, J. M. L. (2010). On error estimation in the fourier modal method for diffractive gratings. (CASA-report; Vol. 1067). Technische Universiteit Eindhoven.
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EINDHOVEN UNIVERSITY OF TECHNOLOGY
Department of Mathematics and Computer Science
CASA-Report 10-67
October 2010
On error estimation in the fourier modal
method for diffractive gratings
by
A. Hlod, J.M.L. Maubach
Centre for Analysis, Scientific computing and Applications
Department of Mathematics and Computer Science
Eindhoven University of Technology
P.O. Box 513
5600 MB Eindhoven, The Netherlands
ISSN: 0926-4507
On error estimation in the Fourier Modal
Method for diffractive gratings
A. Hlod, J.M.L. Maubach
Abstract The Fourier Modal Method (FMM, also called the Rigorous Coupled
Wave Analysis, RCWA) is a numerical discretization method which is often used to calculate a scattered field from a periodic diffraction grating. For 1D periodic gratings in FMM the electromagnetic field is presented by a truncated Fourier series expansion in the direction of the grating periodicity. The grating’s material proper-ties are assumed to be piece-wise constant (called slicing), and next per slice the scattered field is approximated by a truncated Fourier series expansion. The trunca-tion representatrunca-tion of the scattered field and the piece-wise constant approximatrunca-tion of the grating’s material properties cause the error in FMM.
This paper presents an analytical estimate/bound for the FMM error caused by slic-ing.
1 Introduction
Fourier Modal Method or Rigorous Coupled Wave Analysis is a well known numer-ical method to model diffraction from an infinitely periodic grating. This method was introduced at the beginning of the 1980 by Moharam and Gaylord [13] and since then had a wide area of applications and became a classical method to com-pute diffracted fields [1, 3, 14]. These fields are approximated by a truncated Fourier series and are calculated based on the assumption that all material properties are piecewise constant (called slicing). In this paper we address a problem of the error
A. Hlod
Dept. of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513 5600 MB Eindhoven The Netherlands, e-mail: avhlod@gmail.com
J.M.L. Maubach
Dept. of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513 5600 MB Eindhoven The Netherlands, e-mail: name@email.address
2 A. Hlod, J.M.L. Maubach bound in FMM due to the finite number of harmonics (truncation) and slicing of the grating profile.
The question about an error bound is closely related to the convergence of FMM. The convergence of the FMM for laminar gratings and TE polarization was proven by Lifeng Li in [2, Chapter 4] by using the theory of determinants of infinite order. To the authors knowledge there is no proof of FMM convergence in more general sit-uation. Moreover, providing a priori estimate for the number of slices and/or Fourier harmonics in FMM remains an open problem; cf. [11, pages 356-357].
In this paper we present an analytical approach of the error estimation for FMM, more precisely we look at the error due to slicing. We do this for TE polarization in case of 1D periodic grating. The cases of TM or conical polarizations for 1D periodic gratings, and the case of 2D periodic gratings are done similarly.
An error estimation due to slicing can be put in more general context of continu-ous dependence of the reflected filed on the grating parameters e.g. shape of grooves and /or electric permittivity of the grating material. Some advance in this question was done by A Kirsch; see [12] and the references therein.
In our approach we use a regularity estimate obtained in the context of existence, uniqueness, regularity, and variational formulation of a solution to the diffraction problem; see [4–10, 15]. To obtain the bound for the case of TE polarization we use the estimate from [2, Chapter 2].
2 Errors in FMM
The diffraction from the periodic grating with the periodΛ in the case of TE
polar-ization is described by the Helmholtz equation for the transverse component of the electric field u(x, z)
∆u(x, z) +ω2µε(x, z)u(x, z) = 0, 0 < x <Λ, and−∞<z <∞. (1)
Hereω is the angular frequency of the field,εis the electric permittivity, andµis
magnetic permeability. The domain of interest is one period of the grating depicted
in Figure 1. In the domain we distinguish three regions one boundedΩ0(b > z >
−b) whereε(x, z) is allowed to vary, and two unboundedΩ1(b < z)andΩ2(z <−b)
whereε(x, z) is constantε1andε2, respectively; see Figure 1. The incident wave is
uinc= e−iω
√ε
1µ(x sin(θ)+z cos(θ)),(x, z) ∈Ω
1
The boundary conditions are quasiperiodic in the x direction
u(x +Λ,z) = u(x, z)e−iαΛ,α=ω√ε1µsin(θ),
and the outgoing wave conditions at z= ±∞. The field inΩ1andΩ2, which satisfy
the outgoing wave condition, is presented as a Rayleigh expansion. The expansion
On error estimation in the Fourier Modal Method for diffractive gratings 3 Λ
x
z
b
-b
0
ε
1ε
2Ω
0Ω
2Ω
1ε(x,z)
Fig. 1 One period of the grating
u(x, z) = uinc+ ∞
∑
n=−∞
rnXn(x)eikI,znz,Xn(x) = e−ikxnx,(x, z) ∈Ω1 (2)
where rnare the reflected field amplitudes to be found later, kxn=ω√ε1µsin(θ) − 2πn
Λ , kI,zn=
p
ω2ε
1µ− k2xn. InΩ0we approximateε(z, x) by a piecewise function
of z
ε(z, x) ≈εM(z, x) =εm(x) zm<z < zm+1, m= 1..M, (3) where the rectangular regions {(x,z) : 0 < x <Λ and zm<z < zm+1} are called
slices often having the same size. In each slice um(x, z) =∑nn=N=−NZm,n(z)Xn(x) and εm(x) =∑nn=N=−Nεm,nXn(x) are represented as a finite series expansion in the Fourier
basis Xn(x), n = −N..N. In such way one obtains an ODE system with constant
coefficients for Zm,n(z).
The resulting approximated solution will depend on M and N is written as follows
uM,N(x, z) = n=N
∑
n=−N
Zn(z; M, N)Xn(x), 0 < x <Λ,and −∞<z <∞, (4)
where Zn(z; M, N) is a piecewise function equal to the reflected and transmitted
am-plitudes inΩ1andΩ2, and Zm,n(z) in each slice. The error due to finite N and M is defined as follow
E(M, N) = ku∞,∞− uM,Nk, (5)
wherek · k is some norm to be chosen later. The norm follows from a regularity
estimate and in this paper we use L2(Ω0). By using the triangular inequality we
separate the total error as follow
4 A. Hlod, J.M.L. Maubach
The first term in the right-hand-side of the separation (6) is equal to E(M,∞), and
represents the error due to slicing, whereas the second term represents the error due to finite amount of harmonics N as well as the interplay between the errors
due to finite N and M. Next, we focus on E(M,∞) and interpret it as the difference
between the exact fields for the grating with the original profile, and the grating with the approximated profile due to slicing.
3 Error estimation due to slicing
To estimate the error due to slicing E(M,∞) we consider the diffraction problem for
the gratings with the permittivities in the grating regionε(x, z) andεM(x, z) ∆uexact+ω2µεuexact= 0 and∆usliced+ω2µεMusliced= 0, (x, z) ∈Ω0.
Note that here we consider a bounded domainΩ0, with the quasiperiodic boundary
conditions at x= 0,Λ, and at z= ±b the outgoing wave conditions plus incoming
wave become ”transparent” boundary conditions described in [2, Section 2.2.2]. The equation for the difference udiff= uexact− uslicedbecomes
∆udiff+ω2µεMudiff=ω2µ(ε−εM)uexact, (7)
together with the quasiperiodic boundary conditions at x= 0,Λ, and the
”transpar-ent” boundary conditions without the incoming wave at z= ±b. Here the
”transpar-ent” boundary conditions are derived e.g. in [8], and allow to consider the diffraction problem on a bounded domain. A source term at the right-hand-side of (7) represent the difference between the electric permittivities due to slicing of the grating profile,
and tends to zero if M→∞.
Using the regularity estimate derived in the proof of [2, Theorem 2.1, pages 45-47] we obtain kudiffeiαxkL2(Ω 0)≤ C(ω)k(ε−εM)uexacte iαxk L2(Ω 0), (8) where C(ω) = 1
(c(ω)−cω2). The estimate (8) is only possible if c(ω) − cω2>0. The
constant c(ω) becomes zero if kI,znor kII,znare zero for some n. Because of that if
kI,znor kII,znapproach to zero for some n then the right-hand-side of the estimate (8) blows up to infinity.
Finally, we estimate the reflected amplitudes. By adding an extra sliceΩdof the
height d on the top of Ω0, and using the estimate (8) for the regionΩ0∪Ωd we
On error estimation in the Fourier Modal Method for diffractive gratings 5 C(ω)k(ε−εM)uexacteiαxkL2(Ω 0∪Ωd)≥ kudiffe iαxk L2(Ω 0∪Ωd) ≥ kudiffeiαxkL2(Ω d) = s ∞
∑
n=−∞ cn(rn− rn,M)2,where rn and rn,M are the nth order reflected amplitudes for the grating with the
exact and the sliced grating profiles, respectively. Here, for the last step we have used the Rayleigh expansion (2). In such a way one gets an estimate for the error in reflected amplitudes due to slicing, which goes to zero as the number of slices approach to infinity.
4 Results and conclusions
Two relevant question for the error estimation due to slicing arise from the estimate
(8). How sharp the estimate is and for which parameters the constant C(ω) is
posi-tive and finite? To illustrate that we present an example forΛ= 1m, b = 1m,µ=µ0,
andε0≤ε(x, z) ≤ 2ε0. In this case C(ω) is strictly positive forω<4.218106Hz that
corresponds to the wavelengthes larger than 446m. For that wavelength only 0 re-flected order propagates towards infinity. For future development of this approach it is necessary to find more optimal constants and/or estimates using different norm.
In this paper we present an approach how to estimate the error due to slicing
in FMM. By using a regularity estimate we obtain an estimate of the error in L2
which converges to zero as the difference in permittivities due to slicing approaches to zero. This estimate is not possible for all parameters which correlates with the uniqueness results; see e.g. [8].
Acknowledgements The authors would like to acknowledge Marjan Aben, Maxim Pisarenco,
Georg Prokert, and Mark van Kraaij for valuable contributions and suggestions.
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