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Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Editor: Franco Gori
  • Vol. 29, Iss. 11 — Nov. 1, 2012
  • pp: 2307–2313
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Stable algorithm for the computation of the electromagnetic field distribution of eigenmodes of periodic diffraction structures

Evgeni A. Bezus and Leonid L. Doskolovich  »View Author Affiliations


JOSA A, Vol. 29, Issue 11, pp. 2307-2313 (2012)
http://dx.doi.org/10.1364/JOSAA.29.002307


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Abstract

In the present work, a stable algorithm for the calculation of the electromagnetic field distributions of the eigenmodes of one-dimensional diffraction gratings is presented. The proposed approach is based on the method for the computation of the propagation constants of Bloch waves of such structures previously presented by Cao et al. [J. Opt. Soc. Am. A 19, 335 (2002)] and uses a modified S-matrix algorithm to ensure numerical stability.

© 2012 Optical Society of America

1. INTRODUCTION

In [4

4. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000). [CrossRef]

7

7. 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, 1364–1371 (2011). [CrossRef]

], modifications of the method aimed at simulating the diffraction on nonperiodic structures have been proposed. These modifications are based on artificial periodization in conjunction with the addition of absorbing layers on the boundaries of the introduced period or with using a special coordinate transformation. It was shown that the proposed modifications can also be used to calculate propagation constants of quasiwaveguide modes of periodic diffraction structures [4

4. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000). [CrossRef]

,8

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]

,9

9. G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express 15, 11042–11060 (2007). [CrossRef]

]. It should be noted that the calculation of the eigenmode electromagnetic field distributions was not considered in these works. The ability to calculate the eigenmode field can yield additional useful information; in particular, it allows for investigating the localization of the mode energy and separating the absorption and scattering (leakage) losses. A stable algorithm for calculating the field has been proposed for the modes of laser resonators [10

10. T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A 23, 906–911 (2006). [CrossRef]

], but its construction was based on the finiteness of the resonator in the longitudinal direction, which hinders its application to the analysis of eigenmodes of periodic diffraction structures.

In the present work, a stable algorithm for the calculation of the electromagnetic field distributions of the eigenmodes of one-dimensional periodic diffraction structures is presented. The proposed algorithm is based on the computational method for finding the propagation constants of the Bloch modes of such structures [8

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]

] and uses a modified S-matrix algorithm to ensure numerical stability at large truncation orders. Let us note that algorithms based on other matrices describing the propagation of electromagnetic waves in layered media such as the so-called hybrid matrix [11

11. E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 53, 417–428 (2006). [CrossRef]

,12

12. J. Ning and E. L. Tan, “Generalized eigenproblem of hybrid matrix for Bloch–Floquet waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 26, 676–683 (2009). [CrossRef]

] can also be considered as possible stable candidates for the solution of the described problem; however, in the present work we restrict our consideration to the S-matrix algorithm.

The paper is organized as follows: Section 2 briefly outlines the RCWA technique and the scattering matrix formalism for multilayered structures. Section 3 describes the use of this technique for calculating the propagation constants and electromagnetic field distributions of modes in periodic diffraction structures. The cause of numerical instability arising when calculating the mode field distributions is outlined, and a stable modification of the algorithm is proposed. Section 4 presents a numerical example of the eigenmode field calculation.

2. RIGOROUS COUPLED-WAVE ANALYSIS AND MODE COMPUTATION

A. Geometry of the Structure and Field Representation

Geometry of the considered structure is shown in Fig. 1. The structure is periodic along the z axis with period d and consists of N domains (layers) within the period, in each of which the material parameters depend on x coordinate only. To calculate the eigenmodes of the structure using the RCWA, artificial periodization along the x axis with some period d is introduced [4

4. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000). [CrossRef]

,5

5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]

,8

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]

]. To eliminate the interaction between adjacent periods along the x axis, special absorbing layers are added. Among different types of absorbing layers, so-called perfectly matched layers (PMLs) are widely used [13

13. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995). [CrossRef]

]. PMLs are described by tensors of dielectric permittivity and magnetic permeability, which are calculated using the material parameters of the regions above and below the structure (at x>xmax and x<xmin, respectively).

Fig. 1. Geometry of the structure.

Let us consider the case when the TM-polarized eigenmodes of the structure are computed (the TE-polarization case can be treated similarly by the following formal substitutions: EH, εμ). If the PML absorbers are used, the nonzero magnetic field component Hy satisfies the modified Helmholtz equation in each of the N domains (at zn1<z<zn) [5

5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]

]:
2Hy(n)z2+εxx(n)(x)x[1εzz(n)(x)Hy(n)x]+k02μ(n)(x)εxx(n)(x)Hy(n)=0,
(1)
where n is the domain number, k0=2π/λ, λ is the free-space wavelength, εxx(n) and εzz(n) are the diagonal components of the dielectric permittivity tensor, and μ(n) is the magnetic permeability (scalar in this case). Outside the PML, μ(n)(x)1, εxx(n)(x)=εzz(n)(x)=ε(n)(x). Inside the absorber matched to a material with dielectric permittivity εd, μPML=a(1+i), εzz,PML=εdμPML, εxx,PML=εd/μPML, where a is a positive real number [5

5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]

,13

13. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995). [CrossRef]

]. In the numerical simulations in the present work, a=5 was used.

Since the structure is periodic along the x axis, Eq. (1) can be solved using rigorous coupled-wave analysis [1

1. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). [CrossRef]

,5

5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]

]. Under this approach, the field in the nth domain can be represented in the following way:
Hy(n)(x,z)=l{cl,n,L+exp[k0ql,n(zzn1)]mwm,l,nexp(ikx,mx)+cl,n,Lexp[k0ql,n(zzn1)]mwm,l,nexp(ikx,mx)},
(2)
where cl,n,L+ and cl,n,L are the unknown coefficients (wave amplitudes), wm,l,n and ql,n are the elements of the eigenvector matrix Wn and positive square roots of the eigenvalues of the matrix An=1/εxx(n)(x)1(Kxεzz(n)(x)1Kxμ(n)(x)), respectively, and kx,m=2πm/d. The subscript L in cl,n,L± denotes that the coefficients correspond to the wave amplitudes at the left boundary of the domain (at z=zn1). In the expression for matrix An, Kx is a diagonal matrix composed of kx,m/k0 values, f is a Toeplitz matrix associated with the Fourier coefficients of the function f(x). The size of these matrices is M×M, where M is the number of Fourier harmonics retained in the expansion (2) (truncation order).

Expression (2) shows that inside each domain the field is represented as a superposition of waves with amplitudes cl,n,L±, the transverse field profile of which is described by a Fourier expansion.

B. Scattering Matrix Formalism

Let us briefly outline the derivation of the S-matrix, as it is important for further discussion. Suppose that at some step the scattering matrix SL(n1) relating the field amplitudes in the domains 1 and n1 is known. Next, we have to find the matrix SL(n) connecting the field amplitudes in the domains 1 and n:
(cn,L+c1,L)=SL(n)(c1,L+cn,L),
(3)
where ci,L± are column vectors consisting of the cl,i,L± values.

The unknown scattering matrix SL(n) has to be expressed in terms of the known matrices sL(n) and SL(n1). It should be noted that the expression (6) contains potentially ill-conditioned diagonal matrix Xn1 (the values exp[k0ql,n(zizi1)] are close to zero if the corresponding ql,n values have large real part), so the inversion of sL(n) matrix or its blocks while obtaining expression (3) can lead to numerical instabilities. In [3

3. 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]

], numerically stable recursion formulas for S-matrix blocks that do not involve inversions of ill-conditioned matrices have been derived:
T++,L(n)=t++,L(n)(IR+,L(n1)r+,L(n))1T++,L(n1),R+,L(n)=r+,L(n)+t++,L(n)R+,L(n1)(Ir+,L(n)R+,L(n1))1t,L(n),R+,L(n)=R+,L(n1)+T,L(n1)r+,L(n)(IR+,L(n1)r+,L(n))1T++,L(n1),T,L(n)=T,L(n1)(Ir+,L(n)R+,L(n1))1t,L(n),
(7)
where the following block representations are used:
SL(i)=(T++,L(i)R+,L(i)R+,L(i)T,L(i)),sL(i)=(t++,L(i)r+,L(i)r+,L(i)t,L(i)).
(8)
At the first step, according to (3), the recursion is initialized with SL(1) being the identity matrix.

3. COMPUTATION OF THE PROPAGATION CONSTANTS AND THE FIELD DISTRIBUTIONS OF THE EIGENMODES

A. Computation of the Propagation Constants

Let us consider the application of RCWA to the calculation of propagation constants of eigenmodes of the considered periodic structure [8

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]

]. As described above, the amplitudes of the waves in the domains 1 and N+1 can be linked by a scattering matrix:
(cN+1,L+c1,L)=SL(c1,L+cN+1,L).
(9)

Since the structure is periodic in z direction and consists of N domains (layers) along this axis, the N+1-th layer coincides with the first layer. Consequently, for the eigenmodes of the structure, the following equality holds:
(cN+1,L+cN+1,L)=β(c1,L+c1,L),
(10)
where β=exp(ik0neffd), neff is the mode effective index (normalized propagation constant). Let us transform the expression (9) using the block representation (8):
(T++,L0R+,LI)(c1,L+c1,L)=(IR+,L0T,L)(cN+1,L+cN+1,L).
(11)
Substituting (10) into (11), we arrive at the following generalized eigenproblem:
(T++,L0R+,LI)(c1,L+c1,L)=β(IR+,L0T,L)(c1,L+c1,L).
(12)
By solving the eigenproblem (12), a set of β values is found, from which neff values can be determined. Representing neff as a sum neff+neffi, we have
β=exp[ik0(neff+neffi)d]=exp(ik0neffd)exp(k0neffd).
(13)

From the expression (13) we immediately obtain
neff=ln|β|k0d.
(14)
Expression (13) also implies that
neff=2πl+argβk0d,
(15)
where l is an integer. At l=0 the mode propagation constant k0neff lies in the range [π/d,π/d], i.e., in the first Brillouin zone.

B. Computation of the Field Distribution and Numerical Instability

By solving the generalized eigenproblem (12), a set of eigenvectors describing the mode amplitudes at the left boundary of the first domain (at z=0) is also found. The found values of c1,L± can then be used in the expressions (2), (4) and a similar expression for Ez component to calculate the eigenmode field distribution in domain 1. Furthermore, the amplitudes cN+1,L± can also be found using expression (10). Using the amplitudes c1,L± and cN+1,L±, the amplitudes in the other domains can be sequentially calculated in a stable way beginning from the domain N. Indeed, let us suppose that at some step the amplitudes cn+1,L± are known. It follows from expression (5) rewritten for the layers n and n+1 that
cn,L=12Xn{(Wn1Wn+1Vn1Vn+1)cn+1,L++(Wn1Wn+1+Vn1Vn+1)cn+1,L}.
(16)

From Eq. (3),
cn,L+=T++,L(n)c1,L++R+,L(n)cn,L.
(17)
Equations (16) and (17) are stable because they do not contain inversions of potentially ill-conditioned matrices. Let us note that the obtained expressions together with the field representation (2), (4) are very similar to Eqs. (2)–(6) and (14) of [5

5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]

], the only difference being the term T++,L(n)c1,L+ absent in [5

5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]

] because of the zero values of the corresponding wave amplitudes. Knowing the ci,L± values, we can calculate the field distribution in the whole structure using the expressions (2) and (4). However, despite the fact that the ci,L± amplitudes are calculated in a stable way, the computation of the field distribution using Eqs. (2), (4) can still lead to numerical instability. Indeed, the expressions (2), (4) contain exponential functions exp[k0ql,n(zzn1)] with positive real parts of the argument. As shown below in Section 4, the calculation of field distributions of modes using expressions (2), (4), (16) and (17) actually leads to numerical overflow and appearance of numerical artifacts in the calculated field at large truncation order. Thus, to calculate the field distributions of the eigenmodes without numerical overflow, it is necessary to construct a stable algorithm.

C. Modification of the Algorithm for the Stable Field Computation

To construct a numerically stable algorithm for the computation of the field distribution, let us use the following field representation instead of the representation (2):
Hy(n)(x,z)=l{cl,n+exp[k0ql,n(zzn1)]mwm,l,nexp(ikx,mx)+cl,nexp[k0ql,n(zzn)]mwm,l,nexp(ikx,mx)}.
(18)
In expression (18), the coefficients cl,n± correspond to the wave amplitudes at the domain boundaries where they reach the maximal values: amplitudes cl,n+—at the left boundary, cl,n—at the right boundary:
cn+=cn,L+,cn=cn,R,
(19)
cn,R± being the wave amplitudes at the right boundary of the domain n. Hence, the real parts of the arguments of the exponential functions in (18) are always negative. Let us note that the same field representation was used in [14

14. 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, 1077–1086 (1995). [CrossRef]

] to obtain stable solution of diffraction problem in multilayer structures.

However, in order to use stable expression (18) (and similar expressions for the electric field components) for the calculation of the field distributions, it is necessary to find the amplitudes ci± defined by Eq. (19) in each domain of the structure. For this purpose, let us reformulate the generalized eigenproblem (12) with respect to the amplitudes of the waves in the first layer c1± (c1,L+ and c1,R) instead of the amplitudes c1,L±, and obtain stable expressions for the amplitudes ci±, i=2N in the other domains in terms of the c1± values.

Let us introduce another scattering matrix SR(n), which relates the wave amplitudes ci,R± at right boundaries of the domains n and N+1:
(cN+1,R+cn,R)=SR(n)(cn,R+cN+1,R).
(20)

The calculation of the matrix SR(n) is similar to the calculation of SL(n) described above. For the blocks of SR(n), the following stable recursion formulas can be derived:
T++,R(n)=T++,R(n+1)(Ir+,R(n)R+,R(n+1))1t++,R(n+1),R+,R(n)=R+,R(n+1)+T++,R(n+1)r+,R(n)(IR+,R(n+1)r+,R(n))1T,R(n+1),R+,R(n)=r+,R(n)+t,R(n)R+,R(n+1)(Ir+,R(n)R+,R(n+1))1t++,R(n),T,R(n)=t,R(n)(IR+,R(n+1)r+,R(n))1T,R(n+1),
(21)
where t++,R(i), r+,R(i), r+,R(i) and t,R(i) are the blocks of the matrix sR(i), which can be obtained similarly to the matrix sL(i) defined by expression (6).

Let us derive stable expressions relating the amplitudes of waves cn± in different domains for some chosen eigenmode with the amplitudes c1± in the first layer (at this stage assuming that the latter are known). Taking into account the block representations and the definitions (19), the expressions (3) and (20) can be written as
(I00X1)(cn+c1)=(T++,L(n)R+,L(n)R+,L(n)T,L(n))(I00Xn)(c1+cn),
(22)
(XN+100I)(cN+1+cn)=(T++,R(n)R+,R(n)R+,R(n)T,R(n))(Xn00I)(cn+cN+1).
(23)

Substituting the second equation of (23) into the first equation of (22) and transforming the result, we obtain
cn+=(IR+,L(n)XnR+,R(n)Xn)1(T++,L(n)c1++R+,L(n)XnT,R(n)cN+1).
(24)

Let us now substitute the first equation of (22) into the second equation of (23). After expressing the value cn, we get
cn=(IR+,R(n)XnR+,L(n)Xn)1(R+,R(n)XnT++,L(n)c1++T,R(n)cN+1).
(25)

Let us rewrite the expressions (24) and (25), taking into account the equality cN+1=c1β:
cn+=(IR+,L(n)XnR+,R(n)Xn)1(T++,L(n)c1++R+,L(n)XnT,R(n)c1β),cn=(IR+,R(n)XnR+,L(n)Xn)1(R+,R(n)XnT++,L(n)c1++T,R(n)c1β).
(26)

Expressions (26) for the wave amplitudes cn± are stable because they do not contain inversions of potentially ill-conditioned matrices containing the Xi matrices as multiplicative terms. The found amplitudes can be used in the field representation (18) for the stable computation of the field inside the structure. However, we still have to reformulate the generalized eigenproblem (12) to find the unknown c1± and β values. For this, we write the expression (24) in matrix form at n=N+1:
cN+1+=(IR+,L(N+1)XN+1R+,R(N+1)XN+1)1(T++,L(N+1)R+,L(N+1)XN+1T,R(N+1))(c1+cN+1).
(27)
Expression (25) at n=1 takes the form
c1=(IR+,R(1)X1R+,L(1)X1)1(R+,R(1)X1T++,L(1)T,R(1))(c1+cN+1).
(28)

Combining (27) and (28) and taking into account that R+,R(N+1)=0, R+,L(1)=0, T++,L(1)=I, T,R(N+1)=I, we obtain:
(cN+1+c1)=(T++,L(N+1)R+,L(N+1)XN+1R+,R(1)X1T,R(1))(c1+cN+1).
(29)
Because of the periodicity of the structure under consideration, X1=XN+1. In addition, an equality similar to (10) holds for the eigenmodes of the structure:
(cN+1+cN+1)=β(c1+c1).
(30)

Using (30) we obtain from (29) the following generalized eigenproblem from which the values of c1± and β can be found:
(T++,L(N+1)0R+,R(1)X1I)(c1+c1)=β(IR+,L(N+1)X10T,R(1))(c1+c1).
(31)

Expressions (26) and (31) represent the main result of this work. It should be noted that the derived expressions (26) and (31) involve only the following S-matrix submatrices: T++,L(n), R+,L(n), R+,R(n) and T,R(n). It follows from the recursion formulas (7) and (21) that other blocks of the scattering matrices SL(n) and SR(n) are not needed for their computation; thus, the constructed algorithm requires computation of only a half of each of the scattering matrices. Let us also note that the proposed algorithm not only removes the instability associated with the field representation described by Eqs. (2) and (4) but also allows parallel calculation of the amplitudes of the waves cn± as opposed to sequential calculation described by Eqs. (16) and (17). Parallel calculation can be useful when analyzing eigenmodes of continuous-relief gratings, which are represented by a large number of domains (staircase approximation) in the RCWA technique.

It is worth mentioning that the use of two types of scattering matrices (SL(n) and SR(n)) in the proposed modified algorithm is somewhat similar to two types of recursion (downward and upward) that are presented in [15

15. E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157–1161 (2002). [CrossRef]

] and described by Eqs. (16) and (20) therein. However, for the initialization of the recursion schemes presented in [15

15. E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157–1161 (2002). [CrossRef]

], a presence of a semi-infinite layer in the direction z is needed, so these schemes cannot be applied to the problem considered in the present manuscript without modification.

4. NUMERICAL EXAMPLES

To verify the correctness of the developed modification of the algorithm, the propagation constants of TE- and TM-polarized modes of a periodic waveguide studied in [4

4. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000). [CrossRef]

,8

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]

,16

16. K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980). [CrossRef]

] were calculated. The simulation parameters (period along x axis d and the parameters of PML absorbers) were chosen to coincide with the parameters in [8

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]

]. The calculated values were found to be in perfect agreement with the converged values given in Table 2 of [8

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]

].

As an example for the calculation of the electromagnetic field distributions of eigenmodes in periodic structures, a diffraction grating studied in [17

17. E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008). [CrossRef]

] was considered (Fig. 2). The parameters of the grating are listed in the figure caption.

Fig. 2. Studied diffraction grating. Parameters of the structure: εsuper=1, εgr=εsub=2.56, εm=12.9222+0.4473i (Ag), d=1539nm, w=d/2, hgr=435nm, hm=65nm.

Figure 3 shows the electric field intensity distribution within one period of the grating at the resonance conditions (normal incidence of TM-polarized plane wave with free-space wavelength λ=550nm). The distribution was calculated using the conventional RCWA for the solution of grating diffraction problems [18

18. P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998). [CrossRef]

]. It is evident from Fig. 3 that the field distribution in the grating is close to the field distribution corresponding to the interference between two counterpropagating plasmonic modes. In addition, underneath the structure, a high-frequency interference pattern of ±5-th evanescent diffraction orders is generated. In [17

17. E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008). [CrossRef]

], it was suggested to use this effect in near-field photolithography for generating periodic structures with nanoscale features.

Fig. 3. Electric field intensity distribution in the considered structure at normal incidence of TM-polarized plane wave with λ=550nm (the grating is shown with white dashed lines).

Figure 4 shows the intensity distribution of the electric field |E|2 for the interference of two counterpropagating eigenmodes of the grating with the propagation constants closest to zero (which correspond to the excitation at normal incidence), calculated using a “standard” algorithm [Eqs. (2), (4), (12), (16), (17)]. In the simulations, the artificially introduced period along the x axis was chosen to be 2.5λ, and the thickness of each of the PML absorbers amounted to λ/4. All intensity distributions are normalized to the maximum value. Field distribution in Fig. 4(a) corresponds to the truncation order M=101, and Fig. 4(b) corresponds to M=201. Figure 4(a) shows that at M=101 the calculated field distribution is qualitatively consistent with the field distribution in the structure illuminated at the resonance conditions (Fig. 3); however, the number of retained Fourier harmonics is not enough for an adequate description of the field profile of the plasmonic modes. Figure 4(b) shows that increasing the truncation order leads to numerical overflow in a part of the central domain of the structure corresponding to the grating ridge (marked with white dotted rectangle), which is consistent with the description given in Section 3. Additional simulation results (not presented here) show that the further increase of the truncation order leads to the appearance of numerical overflow also in the other domains of the structure corresponding to the grating groove. We have also found that similar instability arises when analyzing modes of other structures, e.g., guided-mode resonant gratings [19

19. E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011). [CrossRef]

].

Fig. 4. Distributions of the electric field intensity |E|2 corresponding to the interference of two counterpropagating eigenmodes of the considered structure calculated using the standard algorithm at (a) M=101 and (b) M=201. Dotted rectangle in (b) denotes the area where the numerical overflow occurs.

Figure 5 shows the electric field intensity distributions of the eigenmodes interference pattern calculated by the proposed stable algorithm based on the expressions (18), (26), and (31) at the truncation order M=401 (a) and M=801 (b). In this case (as well as for other studied structures), the numerical instability is not observed, and for a sufficiently large number of Fourier harmonics the calculated eigenmode field distribution converges and is close to the field distribution in the structure illuminated at resonance conditions (Fig. 3).

Fig. 5. Distributions of the electric field intensity |E|2 corresponding to the interference of two counter-propagating eigenmodes of the considered structure calculated using the proposed stable modification of the algorithm at (a) M=401 and (b) M=801.

5. CONCLUSION

In the present work, a stable algorithm for the calculation of the electromagnetic field distribution of eigenmodes of one-dimensional periodic diffraction structures was proposed. The algorithm is based on a previously presented computational technique for calculating the propagation constants of eigenmodes of such structures [8

8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]

]. It was demonstrated that the use of a modified S-matrix algorithm is necessary to remove numerical instabilities arising when increasing the truncation order in the used Fourier expansions. The performance of the algorithm was illustrated with a numerical example.

ACKNOWLEDGMENTS

This work was supported by a Dynasty Foundation grant; Russian Foundation for Basic Research grants 12-07-00495, 12-07-31116, 12-07-90804, 11-07-12036, and 11-07-00153; Russian Federation Presidential grants NSh-4128.2012.9 and MD-1041.2011.2; and a Russian Federation State contract (agreement 8027).

REFERENCES

1.

M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). [CrossRef]

2.

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]

3.

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]

4.

P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000). [CrossRef]

5.

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]

6.

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010). [CrossRef]

7.

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, 1364–1371 (2011). [CrossRef]

8.

Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]

9.

G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express 15, 11042–11060 (2007). [CrossRef]

10.

T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A 23, 906–911 (2006). [CrossRef]

11.

E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 53, 417–428 (2006). [CrossRef]

12.

J. Ning and E. L. Tan, “Generalized eigenproblem of hybrid matrix for Bloch–Floquet waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 26, 676–683 (2009). [CrossRef]

13.

Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995). [CrossRef]

14.

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, 1077–1086 (1995). [CrossRef]

15.

E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157–1161 (2002). [CrossRef]

16.

K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980). [CrossRef]

17.

E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008). [CrossRef]

18.

P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998). [CrossRef]

19.

E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011). [CrossRef]

OCIS Codes
(050.1950) Diffraction and gratings : Diffraction gratings
(050.1970) Diffraction and gratings : Diffractive optics
(130.2790) Integrated optics : Guided waves
(230.7390) Optical devices : Waveguides, planar
(050.1755) Diffraction and gratings : Computational electromagnetic methods

ToC Category:
Diffraction and Gratings

History
Original Manuscript: July 3, 2012
Revised Manuscript: September 13, 2012
Manuscript Accepted: September 17, 2012
Published: October 11, 2012

Citation
Evgeni A. Bezus and Leonid L. Doskolovich, "Stable algorithm for the computation of the electromagnetic field distribution of eigenmodes of periodic diffraction structures," J. Opt. Soc. Am. A 29, 2307-2313 (2012)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-29-11-2307


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References

  1. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995). [CrossRef]
  2. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996). [CrossRef]
  3. 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]
  4. P. Lalanne and E. Silberstein, “Fourier-modal methods applied to waveguide computational problems,” Opt. Lett. 25, 1092–1094 (2000). [CrossRef]
  5. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, “Use of grating theories in integrated optics,” J. Opt. Soc. Am. A 18, 2865–2875 (2001). [CrossRef]
  6. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010). [CrossRef]
  7. 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, 1364–1371 (2011). [CrossRef]
  8. Q. Cao, P. Lalanne, and J.-P. Hugonin, “Stable and efficient Bloch-mode computational method for one-dimensional grating waveguides,” J. Opt. Soc. Am. A 19, 335–338 (2002). [CrossRef]
  9. G. Lecamp, J. P. Hugonin, and P. Lalanne, “Theoretical and computational concepts for periodic optical waveguides,” Opt. Express 15, 11042–11060 (2007). [CrossRef]
  10. T. Vallius, J. Tervo, P. Vahimaa, and J. Turunen, “Electromagnetic field computation in semiconductor laser resonators,” J. Opt. Soc. Am. A 23, 906–911 (2006). [CrossRef]
  11. E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 53, 417–428 (2006). [CrossRef]
  12. J. Ning and E. L. Tan, “Generalized eigenproblem of hybrid matrix for Bloch–Floquet waves in one-dimensional photonic crystals,” J. Opt. Soc. Am. B 26, 676–683 (2009). [CrossRef]
  13. Z. S. Sacks, D. M. Kingsland, R. Lee, and J.-Fa Lee, “A perfectly matched anisotropic absorber for use as an absorbing boundary condition,” IEEE Trans. Antennas Propag. 43, 1460–1463 (1995). [CrossRef]
  14. 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, 1077–1086 (1995). [CrossRef]
  15. E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157–1161 (2002). [CrossRef]
  16. K. C. Chang, V. Shah, and T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980). [CrossRef]
  17. E. A. Bezus, D. A. Bykov, L. L. Doskolovich, and I. I. Kadomin, “Diffraction gratings for generating varying-period interference patterns of surface plasmons,” J. Opt. A Pure Appl. Opt. 10, 095204 (2008). [CrossRef]
  18. P. Lalanne and M. P. Jurek, “Computation of the near-field pattern with the coupled-wave method for transverse magnetic polarization,” J. Mod. Opt. 45, 1357–1374 (1998). [CrossRef]
  19. E. A. Bezus, L. L. Doskolovich, and N. L. Kazanskiy, “Evanescent-wave interferometric nanoscale photolithography using guided-mode resonant gratings,” Microelectron. Eng. 88, 170–174 (2011). [CrossRef]

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