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A variant on eigenmode method in periodic crossed gratings
ARTICLE IN PRESS
Optik
Optics
Optik 119 (2008) 511–514 www.elsevier.de/ijleo
A variant on eigenmode method in periodic crossed gratings Pierre Hillion Instiut Henri Poincare´, 86 Bis Route de Croissy, 78110 Le Ve´sinet, France Received 3 July 2006; accepted 17 January 2007
Abstract The eigenvalue problem discussed in Noponen and Turunen [Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles, J. Opt. Soc. Am. A 11 (1994) 2494–2502] for crossed gratings is analyzed in a different way using the three components of the electric field instead of the components Ex,y, Hx,y. As a result, half as many eigenvectors are needed. r 2007 Elsevier GmbH. All rights reserved. Keywords: Periodic media; Crossed grating; Wave propagation; Modal method
1. Introduction
r:H ¼ 0;
The methods developed to describe wave propagation in crossed gratings [1] are still confronted to the problem of computer resources [2]. We present a variant of the rigorous theory of binary-surface relief gratings [3] developed to accomodate three-dimensional (3D) modulated profiles and, this variant has the advantage to halve the dimensions of the matrix eigenvalue problem generated by Maxwell’s equations in such media. The numerical implantation of the corresponding formalism is not discussed in this theoretical work and we closely follow the notations used in [3], the modulated grazing region periodic in x and y with dx, dy periods, respectively, is located in the slab 0ozoh. We work with the three components of the electric field while the four components Ex,y, Hx,y are used in [2,3]. With the time dependence exp(iot) assumed, the Maxwell equations are inside the slab
and the permittivity e is [3] with p, q arbitrary integers ¼ 0
X
r:E ¼ 0
p;q exp½2iðpx=d x þ qy=d y Þ.
(1b)
(2)
p;q
We get from (1a) the Helmholtz equation satisfied by the electric field DE þ o2 m0 E rðr:EÞ ¼ 0
(3)
we look for the solutions of Maxwell’s equations in the form, m, n being arbitrary integers fE; HgðxÞ ¼
X
fE mn ; H m;n gfm;n ðxÞ;
m;n
fm;n ðxÞ ¼ exp½iðam x þ bn y þ gzÞ
ð4Þ
with am ¼ 2pm/dx, bn ¼ 2pn/dy. Then r ^ E iom0 H ¼ 0;
r ^ H ioE ¼ 0,
(1a) DE ¼
XX m;n
E-mail address: [email protected]. 0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.03.005
m;n
G2m;n E m;n jm;n ðxÞ,
G2m;n ¼ a2m þ b2n þ g2 ,
ð5aÞ
ARTICLE IN PRESS 512
P. Hillion / Optik 119 (2008) 511–514
r:E ¼ i
X ðam E m;n;x þ bn E m;n;y þ gE m;n;z Þjm;n ðxÞ,
Eq. (10) becomes
m;n
(5b) E ¼ 0
XX
¼ 0
(12)
mp;nq E p;q jm;n ðxÞ.
ð5cÞ
Substituting (5a–c) into (3) gives the set of equations in which k2 ¼ o2m0e0 X G2mn E mn;x k2 mp;nq E pq;x am ð. . .Þ ¼ 0, pq
2
X
2
X
k
Ar;s ¼ ðGys Þ2 dyrs k2 yrs
r ¼ 1; 2; . . . ; L;
p;q E m;n jm;n ðxÞ exp½2iðpx=d x þ py=d y Þ
m;n p;q
G2mn E mn;y
Ar;s Cys ¼ 0;
s¼1
m;n p;q
XX
L X
that we may write ACy ¼ 0 with nonnull solutions if detA ¼ 0, a condition supplying L eigen values (Gpq,l)2 from which we get according to the definition (5a) of G2p,q, L expressions for g noted from now on gl, l ¼ 1,2,y,L. As a simple illustration, suppose M ¼ N ¼ 1 so that L ¼ 9. Since m, n, p, q take the values 1, 0, 1, the nine components of Cy are Cy1 ¼ c1;1 ;
mp;nq E pq;y bn ð. . .Þ ¼ 0,
Cy2 ¼ c1;0 Cy8 ¼ c1;0 ;
Cy9 ¼ c1;1
pq
G2mn E mn;z
k
(13) mp;nq E pq;z gð. . .Þ ¼ 0,
ð6Þ
pq
ð. . .Þ ¼ am E pq;x þ bn E pq;y þ gE pq;z .
(7)
Now, let Cm,n denote the two-dimensional (2D) vector with the components cmn,x, cmn,y cmn;x ¼ E mn;x am g1 E mn;z , cmn;y ¼ E mn;y bn g1 E mn;z .
ð8Þ
Then substituting into the first two Eqs. (6) the term (y) obtained from the third one and taking into account (8), we get the eigenvalue problem we have now to solve G2mn Cmn k2
X
while the 9 9 matrix A is 2
C ¼ 0. pq mp;nq pq
(9)
G21;1 k2 0;0 6 6 k2 0;1 6 6 6 ... 6 4 2 k 2;2
3
k2 0;1
k2 0;2
...
k2 2;1
k2 2;2
G21;0 k2 0;0 ...
k2 2;0 .. .
... ...
k2 2;0 .. .
k2 2;1 .. .
k2 2;1
k2 2;0
...
k2 0;1
G21;1 k2 0;0
7 7 7 7 7 7 5
(14) Let Cl denote the L eigenvectors of (12). Then, with u written for x,y,z we may expand the components Emn,u of the vector field Emn, on the Cl basis so that E u ðxÞ ¼
X0
E mn;u ðxÞ ¼
mn
X0 XL mn
e f ðxÞCl , l¼1 mnl;u mnl
fmnl ðxÞ ¼ exp½iðam x þ bn y þ g1 zÞ,
(15) (15a)
with according to (8) emnl;x ¼ omnl þ am g1 1 emnl;z ,
2. Eigenvalues and eigenvectors
emnl;y ¼ omnl þ bn g1 1 emnl;z
In a numerical implementation of (9), the integers m, p and n, q would be compelled to take finite values from M to M for m, p; from N to N for n, q and, changing S into S0 to mark this limitation, we may write (9) (dmpdnq are Kronecker symbols)
so that for m,n fixed, the fields (15) depend on 2L arbitrary amplitudes omnl and emnl,z. But these fields are solutions of the Helmholtz Eq. (3) and we have still to impose that they satisfy the divergence Eq. (1b) r.eE ¼ 0. According to (5c) and (15) we have
X0
ðG2pq dmp dnp k2 mp;nq ÞCpq ¼ 0.
(10)
pq
mn
Then, we introduce the direct sums r ¼ m n;
s¼pq
Cys ¼ Cpq
mp;nq ðam epql;x þ bn epql;y þ g1 epql;z Þ
l¼1
ð11aÞ
ð17Þ
implying X0 mp;nq ðam epql;x þ bn epql;y þ g1 epql;z Þ ¼ 0, pq
l ¼ 1; 2; 3; . . . L,
dyrs ¼ dmp dnq ,
yrs ¼ mp;np ;
pq
fmnl ðxÞC1 ¼ 0 (11)
the integers r, s take L ¼ (2M+1)(2N+1) values and with the notations Gys ¼ Gpq ;
L X0 X0 X
ð16Þ
ð17aÞ
which are L constraints on the field amplitudes. Then, for m,n fixed, the electric field has L and on the whole L2
ARTICLE IN PRESS P. Hillion / Optik 119 (2008) 511–514
independent amplitudes since the couple m,n takes L values. Remark. The expressions (17) is obtained with a permutation of the derivatives qu and of the truncated sums S0 which is sound but would need justification for the infinite series S since nothing is known on the convergence of this series with as consequence that S0 is not necessarily a consistent approximation of S. We shall discuss this point elsewhere together with the convergence of the Fourier series in (15), a question also investigated in [2]. Similar to (15), the components of the magnetic field have the expansions H u ðxÞ ¼
X0
H mn;u ðxÞ ¼
mn
L X0 X mn
hmnl;u fmnl ðxÞCl
(18)
l¼1
and the amplitudes hmnl,u using the Maxwell Eqs. (1a) and (15) are obtained in terms of emnl,u by the relations hmnl;x ¼ ðbn emnl;z g1 emnl;y Þ=m0 o,
X0
E r ðxÞ ¼
Rmn exp½iðam x þ bn y rm;n zÞ,
mn
X0
E t ðxÞ ¼
T mn exp½iðam x þ bn y þ tm;n fz hgÞ,
ð21Þ
m;n
where am ¼ a0 þ 2pm=d x ; 2
rmn ¼ ½ðn1 kÞ
a2m
bn ¼ b0 þ 2pn=d y ,
b2n 1=2 ;
tmn ¼ ½ðn2 kÞ2 a2m b2n 1=2 ð22Þ
which is real or purely imaginary according to the value of am2+bn2. Finally H i;r;t ðxÞ ¼ ir ^ E i;r;t ðxÞ=m0 o.
(23)
Then, according to (19), (21), the continuity of the omponents Ex,y Hx,y at z ¼ 0 gives the relations L X
½emnl;x þ expðig1 hÞZmnl;x Cl ,
l¼1
ð18aÞ
Finally, according to (15) and (18), taking into account the fields reflected on the boundaries of the crossed gratings, the general solution of Maxwell’s Eqs. (1) is L X0 X ½emnl;u fmnl ðxÞ þ Zmnl;u fymnl ðxÞCl , E u ðxÞ ¼
L X
ðb0 uz r00 uy Þdm0 dn0 þ bn Rmn;z þ rmn Rmn;y L X
½hmnl;x þ expðig1 hÞymnl;x Cl ,
ðr00 ux a0 uz Þdm0 dn0 þ am Rmn;z þ rmn Rmn;x ð19Þ ¼
2
in which Zmnl,u is a further set of L independent amplitudes while ymnl,u is deduced from Zmnl,u by relations similar to (18a). fmnl(x) is the exponential (15a) and
L X
½hmnl;y þ expðig1 hÞymnl;y Cl
while at z ¼ h we have T mn;x ¼
L X
½emnl;x expðig1 hÞ þ Zmnl;x C1 ,
l¼1
T mn;y ¼
L X ½emnl;y expðig1 hÞ þ Zmnl;y C1 , l¼1
bn T mn;z tmn T mn;y ¼
3. Boundary conditions
L X ½hmnl;x expðig1 hÞ þ ymnl;x C1 , l¼1
We use the notations [3] for the incident (from zo0), reflected and transmitted fields (characterized by the superscripts i, r, t) in the semi-infinite media with the refractive indices n1 when zo0 and n2 for z4h.
a0 ¼ n1 k sin y cos j; b0 ¼ n1 k sin y sin j, r0;0 ¼ n1 k cos y; k ¼ 2p=l
ð24Þ
l¼1
(19a)
To sum up, the fieds (19) depend on 2L2 independent amplitudes to be determined by the boundary conditions on the faces z ¼ 0 and h of the modulated grazing region.
E i ðxÞ ¼ u exp½iða0 x þ b0 y þ r0;0 zÞ,
½emnl;y þ expðig1 hÞZmnl;y Cl ,
l¼1
l¼1
l¼1
fymnl ðxÞ ¼ exp½iðam x þ bn y gl fz hgÞ.
uy dm0 dn0 þ Rmn;y ¼
¼
l¼1
L X0 X ½hmnl;u fmn;l ðxÞ þ ymnl;u fymnl ðxÞCl H u ðxÞ ¼ mn
and
ux dm0 dn0 þ Rmn;x ¼
hmnl;y ¼ ðg1 emnl;x am emnl;z Þ=m0 o, hmnl;z ¼ ðam emnl;y bn emnl;x Þ=m0 o.
mn
513
(20)
ð20aÞ
tmn T mn;x am T mn;z ¼
L X ½hmnl;y expðig1 hÞ þ ymnl;y C1 . l¼1
ð25Þ
For m,n fixed, eliminating Rmn,u from (24) and Tmn,u from (25) and taking into account (16a) we get in each case L relations between the amplitudes emnl,x and Zmnl,x that is on the whole 2L2 relations needed to fully determine the expressions (19) of the electromagnetic field in the modulated grazing region and once this field known it is easy to get Rmn,u and Tmn,u.
ARTICLE IN PRESS 514
P. Hillion / Optik 119 (2008) 511–514
4. Conclusion
References
This variant of the method developed in [3] needs only L eigenvalues identified with the propagation constants gl instead of 2L for the same Rayleigh orders in the x and y directions. So, this variant could be computationally more efficient requiring, according to [3], approximatively an eight-fold reduction of computations to get the numerical solution. But, the L2 constraints imposed by the relations (17a) on the amplitudes emnl,u and the further L2 constraints on Zmnl,u have to be taken into account. I regret that the absence of computer resources available prevents me to present numerical results.
[1] M. Nevie`re, E. Popov, Light Propagation in Periodic Media, Marcel Dekker, Basel, 2003. [2] L. Li, New formulation of the Fourier modal method for crossed surface relief gratings, J. Opt. Soc. Am. A 14 (1997) 2758–2767. [3] E. Noponen, J. Turunen, Eigenmode method for electromagnetic synthesis of diffractive elements with threedimensional profiles, J. Opt. Soc. Am. A 11 (1994) 2494–2502.
Optik
Optics
Optik 119 (2008) 511–514 www.elsevier.de/ijleo
A variant on eigenmode method in periodic crossed gratings Pierre Hillion Instiut Henri Poincare´, 86 Bis Route de Croissy, 78110 Le Ve´sinet, France Received 3 July 2006; accepted 17 January 2007
Abstract The eigenvalue problem discussed in Noponen and Turunen [Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles, J. Opt. Soc. Am. A 11 (1994) 2494–2502] for crossed gratings is analyzed in a different way using the three components of the electric field instead of the components Ex,y, Hx,y. As a result, half as many eigenvectors are needed. r 2007 Elsevier GmbH. All rights reserved. Keywords: Periodic media; Crossed grating; Wave propagation; Modal method
1. Introduction
r:H ¼ 0;
The methods developed to describe wave propagation in crossed gratings [1] are still confronted to the problem of computer resources [2]. We present a variant of the rigorous theory of binary-surface relief gratings [3] developed to accomodate three-dimensional (3D) modulated profiles and, this variant has the advantage to halve the dimensions of the matrix eigenvalue problem generated by Maxwell’s equations in such media. The numerical implantation of the corresponding formalism is not discussed in this theoretical work and we closely follow the notations used in [3], the modulated grazing region periodic in x and y with dx, dy periods, respectively, is located in the slab 0ozoh. We work with the three components of the electric field while the four components Ex,y, Hx,y are used in [2,3]. With the time dependence exp(iot) assumed, the Maxwell equations are inside the slab
and the permittivity e is [3] with p, q arbitrary integers ¼ 0
X
r:E ¼ 0
p;q exp½2iðpx=d x þ qy=d y Þ.
(1b)
(2)
p;q
We get from (1a) the Helmholtz equation satisfied by the electric field DE þ o2 m0 E rðr:EÞ ¼ 0
(3)
we look for the solutions of Maxwell’s equations in the form, m, n being arbitrary integers fE; HgðxÞ ¼
X
fE mn ; H m;n gfm;n ðxÞ;
m;n
fm;n ðxÞ ¼ exp½iðam x þ bn y þ gzÞ
ð4Þ
with am ¼ 2pm/dx, bn ¼ 2pn/dy. Then r ^ E iom0 H ¼ 0;
r ^ H ioE ¼ 0,
(1a) DE ¼
XX m;n
E-mail address: [email protected]. 0030-4026/$ - see front matter r 2007 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2007.03.005
m;n
G2m;n E m;n jm;n ðxÞ,
G2m;n ¼ a2m þ b2n þ g2 ,
ð5aÞ
ARTICLE IN PRESS 512
P. Hillion / Optik 119 (2008) 511–514
r:E ¼ i
X ðam E m;n;x þ bn E m;n;y þ gE m;n;z Þjm;n ðxÞ,
Eq. (10) becomes
m;n
(5b) E ¼ 0
XX
¼ 0
(12)
mp;nq E p;q jm;n ðxÞ.
ð5cÞ
Substituting (5a–c) into (3) gives the set of equations in which k2 ¼ o2m0e0 X G2mn E mn;x k2 mp;nq E pq;x am ð. . .Þ ¼ 0, pq
2
X
2
X
k
Ar;s ¼ ðGys Þ2 dyrs k2 yrs
r ¼ 1; 2; . . . ; L;
p;q E m;n jm;n ðxÞ exp½2iðpx=d x þ py=d y Þ
m;n p;q
G2mn E mn;y
Ar;s Cys ¼ 0;
s¼1
m;n p;q
XX
L X
that we may write ACy ¼ 0 with nonnull solutions if detA ¼ 0, a condition supplying L eigen values (Gpq,l)2 from which we get according to the definition (5a) of G2p,q, L expressions for g noted from now on gl, l ¼ 1,2,y,L. As a simple illustration, suppose M ¼ N ¼ 1 so that L ¼ 9. Since m, n, p, q take the values 1, 0, 1, the nine components of Cy are Cy1 ¼ c1;1 ;
mp;nq E pq;y bn ð. . .Þ ¼ 0,
Cy2 ¼ c1;0 Cy8 ¼ c1;0 ;
Cy9 ¼ c1;1
pq
G2mn E mn;z
k
(13) mp;nq E pq;z gð. . .Þ ¼ 0,
ð6Þ
pq
ð. . .Þ ¼ am E pq;x þ bn E pq;y þ gE pq;z .
(7)
Now, let Cm,n denote the two-dimensional (2D) vector with the components cmn,x, cmn,y cmn;x ¼ E mn;x am g1 E mn;z , cmn;y ¼ E mn;y bn g1 E mn;z .
ð8Þ
Then substituting into the first two Eqs. (6) the term (y) obtained from the third one and taking into account (8), we get the eigenvalue problem we have now to solve G2mn Cmn k2
X
while the 9 9 matrix A is 2
C ¼ 0. pq mp;nq pq
(9)
G21;1 k2 0;0 6 6 k2 0;1 6 6 6 ... 6 4 2 k 2;2
3
k2 0;1
k2 0;2
...
k2 2;1
k2 2;2
G21;0 k2 0;0 ...
k2 2;0 .. .
... ...
k2 2;0 .. .
k2 2;1 .. .
k2 2;1
k2 2;0
...
k2 0;1
G21;1 k2 0;0
7 7 7 7 7 7 5
(14) Let Cl denote the L eigenvectors of (12). Then, with u written for x,y,z we may expand the components Emn,u of the vector field Emn, on the Cl basis so that E u ðxÞ ¼
X0
E mn;u ðxÞ ¼
mn
X0 XL mn
e f ðxÞCl , l¼1 mnl;u mnl
fmnl ðxÞ ¼ exp½iðam x þ bn y þ g1 zÞ,
(15) (15a)
with according to (8) emnl;x ¼ omnl þ am g1 1 emnl;z ,
2. Eigenvalues and eigenvectors
emnl;y ¼ omnl þ bn g1 1 emnl;z
In a numerical implementation of (9), the integers m, p and n, q would be compelled to take finite values from M to M for m, p; from N to N for n, q and, changing S into S0 to mark this limitation, we may write (9) (dmpdnq are Kronecker symbols)
so that for m,n fixed, the fields (15) depend on 2L arbitrary amplitudes omnl and emnl,z. But these fields are solutions of the Helmholtz Eq. (3) and we have still to impose that they satisfy the divergence Eq. (1b) r.eE ¼ 0. According to (5c) and (15) we have
X0
ðG2pq dmp dnp k2 mp;nq ÞCpq ¼ 0.
(10)
pq
mn
Then, we introduce the direct sums r ¼ m n;
s¼pq
Cys ¼ Cpq
mp;nq ðam epql;x þ bn epql;y þ g1 epql;z Þ
l¼1
ð11aÞ
ð17Þ
implying X0 mp;nq ðam epql;x þ bn epql;y þ g1 epql;z Þ ¼ 0, pq
l ¼ 1; 2; 3; . . . L,
dyrs ¼ dmp dnq ,
yrs ¼ mp;np ;
pq
fmnl ðxÞC1 ¼ 0 (11)
the integers r, s take L ¼ (2M+1)(2N+1) values and with the notations Gys ¼ Gpq ;
L X0 X0 X
ð16Þ
ð17aÞ
which are L constraints on the field amplitudes. Then, for m,n fixed, the electric field has L and on the whole L2
ARTICLE IN PRESS P. Hillion / Optik 119 (2008) 511–514
independent amplitudes since the couple m,n takes L values. Remark. The expressions (17) is obtained with a permutation of the derivatives qu and of the truncated sums S0 which is sound but would need justification for the infinite series S since nothing is known on the convergence of this series with as consequence that S0 is not necessarily a consistent approximation of S. We shall discuss this point elsewhere together with the convergence of the Fourier series in (15), a question also investigated in [2]. Similar to (15), the components of the magnetic field have the expansions H u ðxÞ ¼
X0
H mn;u ðxÞ ¼
mn
L X0 X mn
hmnl;u fmnl ðxÞCl
(18)
l¼1
and the amplitudes hmnl,u using the Maxwell Eqs. (1a) and (15) are obtained in terms of emnl,u by the relations hmnl;x ¼ ðbn emnl;z g1 emnl;y Þ=m0 o,
X0
E r ðxÞ ¼
Rmn exp½iðam x þ bn y rm;n zÞ,
mn
X0
E t ðxÞ ¼
T mn exp½iðam x þ bn y þ tm;n fz hgÞ,
ð21Þ
m;n
where am ¼ a0 þ 2pm=d x ; 2
rmn ¼ ½ðn1 kÞ
a2m
bn ¼ b0 þ 2pn=d y ,
b2n 1=2 ;
tmn ¼ ½ðn2 kÞ2 a2m b2n 1=2 ð22Þ
which is real or purely imaginary according to the value of am2+bn2. Finally H i;r;t ðxÞ ¼ ir ^ E i;r;t ðxÞ=m0 o.
(23)
Then, according to (19), (21), the continuity of the omponents Ex,y Hx,y at z ¼ 0 gives the relations L X
½emnl;x þ expðig1 hÞZmnl;x Cl ,
l¼1
ð18aÞ
Finally, according to (15) and (18), taking into account the fields reflected on the boundaries of the crossed gratings, the general solution of Maxwell’s Eqs. (1) is L X0 X ½emnl;u fmnl ðxÞ þ Zmnl;u fymnl ðxÞCl , E u ðxÞ ¼
L X
ðb0 uz r00 uy Þdm0 dn0 þ bn Rmn;z þ rmn Rmn;y L X
½hmnl;x þ expðig1 hÞymnl;x Cl ,
ðr00 ux a0 uz Þdm0 dn0 þ am Rmn;z þ rmn Rmn;x ð19Þ ¼
2
in which Zmnl,u is a further set of L independent amplitudes while ymnl,u is deduced from Zmnl,u by relations similar to (18a). fmnl(x) is the exponential (15a) and
L X
½hmnl;y þ expðig1 hÞymnl;y Cl
while at z ¼ h we have T mn;x ¼
L X
½emnl;x expðig1 hÞ þ Zmnl;x C1 ,
l¼1
T mn;y ¼
L X ½emnl;y expðig1 hÞ þ Zmnl;y C1 , l¼1
bn T mn;z tmn T mn;y ¼
3. Boundary conditions
L X ½hmnl;x expðig1 hÞ þ ymnl;x C1 , l¼1
We use the notations [3] for the incident (from zo0), reflected and transmitted fields (characterized by the superscripts i, r, t) in the semi-infinite media with the refractive indices n1 when zo0 and n2 for z4h.
a0 ¼ n1 k sin y cos j; b0 ¼ n1 k sin y sin j, r0;0 ¼ n1 k cos y; k ¼ 2p=l
ð24Þ
l¼1
(19a)
To sum up, the fieds (19) depend on 2L2 independent amplitudes to be determined by the boundary conditions on the faces z ¼ 0 and h of the modulated grazing region.
E i ðxÞ ¼ u exp½iða0 x þ b0 y þ r0;0 zÞ,
½emnl;y þ expðig1 hÞZmnl;y Cl ,
l¼1
l¼1
l¼1
fymnl ðxÞ ¼ exp½iðam x þ bn y gl fz hgÞ.
uy dm0 dn0 þ Rmn;y ¼
¼
l¼1
L X0 X ½hmnl;u fmn;l ðxÞ þ ymnl;u fymnl ðxÞCl H u ðxÞ ¼ mn
and
ux dm0 dn0 þ Rmn;x ¼
hmnl;y ¼ ðg1 emnl;x am emnl;z Þ=m0 o, hmnl;z ¼ ðam emnl;y bn emnl;x Þ=m0 o.
mn
513
(20)
ð20aÞ
tmn T mn;x am T mn;z ¼
L X ½hmnl;y expðig1 hÞ þ ymnl;y C1 . l¼1
ð25Þ
For m,n fixed, eliminating Rmn,u from (24) and Tmn,u from (25) and taking into account (16a) we get in each case L relations between the amplitudes emnl,x and Zmnl,x that is on the whole 2L2 relations needed to fully determine the expressions (19) of the electromagnetic field in the modulated grazing region and once this field known it is easy to get Rmn,u and Tmn,u.
ARTICLE IN PRESS 514
P. Hillion / Optik 119 (2008) 511–514
4. Conclusion
References
This variant of the method developed in [3] needs only L eigenvalues identified with the propagation constants gl instead of 2L for the same Rayleigh orders in the x and y directions. So, this variant could be computationally more efficient requiring, according to [3], approximatively an eight-fold reduction of computations to get the numerical solution. But, the L2 constraints imposed by the relations (17a) on the amplitudes emnl,u and the further L2 constraints on Zmnl,u have to be taken into account. I regret that the absence of computer resources available prevents me to present numerical results.
[1] M. Nevie`re, E. Popov, Light Propagation in Periodic Media, Marcel Dekker, Basel, 2003. [2] L. Li, New formulation of the Fourier modal method for crossed surface relief gratings, J. Opt. Soc. Am. A 14 (1997) 2758–2767. [3] E. Noponen, J. Turunen, Eigenmode method for electromagnetic synthesis of diffractive elements with threedimensional profiles, J. Opt. Soc. Am. A 11 (1994) 2494–2502.