Abstract

Formulation of the Fourier modal method for multilevel structures with spatially adaptive resolution is presented for TE and TM polarizations using a slightly reformulated representation for the spatial coordinates. Projections to Fourier base in boundary value problem are used allowing extensions to multilayer profiles with differently placed transitions. We evade the eigenvalue problem in homogeneous regions demanded in the original formulation of the Fourier modal method with adaptive spatial resolution.

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References

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  1. K. Knop, "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves," J. Opt. Soc. Am. 68, 1206-1210 (1978).
    [CrossRef]
  2. P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  3. G. Granet and B. Guizal, "Efficient implementation for the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
    [CrossRef]
  4. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  5. G. Granet,"Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999).
    [CrossRef]
  6. R. Petit, ed., Electromagnetic theory of gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  7. J. Turunen, "Diffraction theory of microrelief gratings," Chap 2 in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, Cornwall, 1997)
  8. G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001).
    [CrossRef]
  9. 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]
  10. R. H. Morf, "Exponentially convergent and numerically efficient solution of Maxwell's equations for lamellar gratings," J. Opt. Soc. Am. A 12, 1043-1056 (1995).
    [CrossRef]
  11. D. Nyyssonen and C. P. Kirk, "Optical microscope imaging of lines patterned in thick layers with variable edge geometry," J. Opt. Soc. Am. A 5, 1270-1280 (1988).
    [CrossRef]

Other (11)

K. Knop, "Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves," J. Opt. Soc. Am. 68, 1206-1210 (1978).
[CrossRef]

P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

G. Granet and B. Guizal, "Efficient implementation for the coupled-wave method for metallic lamellar gratings in TM polarization," J. Opt. Soc. Am. A 13, 1019-1023 (1996).
[CrossRef]

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

G. Granet,"Reformulation of the lamellar grating problem through the concept of adaptive spatial resolution," J. Opt. Soc. Am. A 16, 2510-2516 (1999).
[CrossRef]

R. Petit, ed., Electromagnetic theory of gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

J. Turunen, "Diffraction theory of microrelief gratings," Chap 2 in Micro-Optics: Elements, Systems and Applications, H. P. Herzig, ed. (Taylor & Francis, Cornwall, 1997)

G. Granet, J. Chandezon, J.-P. Plumey, and K. Raniriharinosy, "Reformulation of the coordinate transformation method through the concept of adaptive spatial resolution. Application to trapezoidal gratings," J. Opt. Soc. Am. A 18, 2102-2108 (2001).
[CrossRef]

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]

R. H. Morf, "Exponentially convergent and numerically efficient solution of Maxwell's equations for lamellar gratings," J. Opt. Soc. Am. A 12, 1043-1056 (1995).
[CrossRef]

D. Nyyssonen and C. P. Kirk, "Optical microscope imaging of lines patterned in thick layers with variable edge geometry," J. Opt. Soc. Am. A 5, 1270-1280 (1988).
[CrossRef]

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Figures (11)

Fig. 1.
Fig. 1.

y-invariant grating geometry.

Fig. 2.
Fig. 2.

Convergence of the real part of the first eigenvalue in TE (a) and TM (b) polarization calculated by using the old FMM (---), the first parametric representation (⋯), and the new formulation (-).

Fig. 3.
Fig. 3.

Logarithmic decay of the error |γ 3 - γ exact| of the third eigenvalue in TE (a) and TM (b) polarization calculated by using the old FMM (- - -), the first parametric representation (⋯), and the new formulation (-).

Fig. 4.
Fig. 4.

The dependence between the coordinates x and u by using transition cx = 0.9d in the x space and in the u space cu = 0.5d which represents the new formulation (-) and cu = 0.9 which represents the old formulation (---).

Fig. 5.
Fig. 5.

A multilevel conducting grating with parameters d = 1, hj = 0.125j, na = 0.1217 + 3.2966i, and nb, = 1. Both n 1 and n 3 equal to 1.

Fig. 6.
Fig. 6.

The diffraction efficiency of the zeroth transmitted order in TE polarization with normal incidence for the grating in fig. 5 as a function of the wavelength α according to the FMM (a) and the parametric representation (b) with different number of the modes: 5 modes (-) and 10 modes (◇ ◇ ◇). The dashed line is the accurate result calculated by the FMM with 240 modes.

Fig. 7.
Fig. 7.

Same as fig. 6 in TM polarization. The number of the modes: 12 modes (-) and 24 modes (◇ ◇ ◇). The dashed line: the FMM with 240 modes.

Fig. 8.
Fig. 8.

A checkerboard grating with parameters na = 5, nb , = 1.5, h 1 = 10, h 2 = 20, and d = 1. The refractive indices in the homogeneous regions are n 1 = 1 and n 3 = 1.

Fig. 9.
Fig. 9.

The diffraction efficiency of the zeroth transmitted diffraction order of the checkerboard grating in TM polarization as a function of the wavelength with normal incidence. The curves have been calculated by the FMM (a) and the parametric representation (b) with19 (-) and 38 (opex-10-1-24-i001) modes.

Fig. 10.
Fig. 10.

A cylindrical grating with parameters n 1 = n 3 = 1, na = 1, nb , = 5, h 1 = 0.5, h 2 = 1,d=1, and r = 0.25.

Fig. 11.
Fig. 11.

The diffraction efficiency of the zeroth transmitted diffraction order of the cylindrical grating in TE polarization as a function of the wavelength with normal incidence. The curves have been calculated by the FMM (a) and the parametric representation (b) with7 (-) and 14 (() modes.

Equations (28)

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E 1 ( x , z ) = exp [ i ( α 0 x + r 0 z ) ] + m R m exp [ i ( α m x r m z ) ] ,
E 3 ( x , z ) = m T m exp [ i ( α m x + t m z ) ] ,
r m = ( k n 1 ) 2 α m 2 ,
t m = ( k n 3 ) 2 α m 2
η R m = ( r m / r 0 ) R m 2
η T m = C ( t m / r 0 ) T m 2 ,
x E y x + z E y z + k 2 n 2 ( x ) E y = 0
n 2 ( x ) { x [ 1 n 2 x H y x ] + k 2 H y } + z H y z = o
x u x u ,
f ( u ) = x u .
a ( u ) = n 2 ( u ) f ( u ) ,
b ( u ) = f ( u ) n 2 ( u ) .
E y ( u , z ) = exp ( i γ z ) m E m exp ( i α m u ) ,
γ 2 E = f 1 [ k 2 a α f 1 α ] ,
γ 2 H = b 1 [ k 2 f α a 1 α ] ,
x l ( u ) = a 1 + a 2 u + a 3 2 π sin [ 2 π u u l 1 u l u l 1 ] ,
a 1 = u l x l 1 u l 1 x l u l u l 1
a 2 = x l x l 1 u l u l 1
a 3 = G ( u l u l 1 ) ( x l x l 1 )
E y ( u , z ) = v = 1 V { A v , j exp [ i γ v , j ( z h j ) ] + B v , j exp [ i γ v , j ( z h j + 1 ) ] }
× m = M M E m , v j exp [ i α m u j ( x ) ] ,
E y , j ( x , h j ) = E y , j + 1 ( x , h j )
z [ E y , j ( x , h j ) ] = z [ E y , j + 1 ( x , h j ) ]
H y , j ( x , h j ) = H y , j + 1 ( x , h j )
z [ 1 j ( x ) H y , j ( x , h j ) ] = z [ 1 j + 1 ( x ) H y , j + 1 ( x , h j ) ] .
K = [ K ] p , m 1 d 0 d f ( u ) exp [ i α p x ( u ) + i α m u ] d u ,
E j = K E u , H j = K H u , Q j = K Q u .
[ H j + 1 H j Q j + 1 Γ j + 1 Q j Γ j ] [ A j + 1 B j ] = [ H j + 1 X j H j X j + 1 Q j + 1 Γ j + 1 X j Q j Γ j X j + 1 ] [ A j B j + 1 ] ,

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