Abstract

The modal method is well adapted for the modeling of deep-groove, high-contrast gratings of short period, possibly involving metal parts. Yet problems remain in the case of the TM polarization in the presence of metal parts in the corrugations: whereas most of the diffraction features are explained by the interplay of an astonishingly small number of true propagating and low-order evanescent modes, the exact solution of the diffraction problem requires the contribution of two types of evanescent modes that are usually overlooked. We investigate the nature and the role of these modes and show that metal gratings can be treated exactly by the modal method.

© 2006 Optical Society of America

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References

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  1. P. Lalanne and J.-P. Hugonin, "Numerical performance of finite-difference modal methods for the electromagnetic analysis of one-dimensional lamellar gratings," J. Opt. Soc. Am. A 17, 1033-1042 (2000).
    [CrossRef]
  2. T. Vallius, "Comparing the Fourier modal method with the C method: analysis of conducting multilevel gratings in TM polarization," J. Opt. Soc. Am. A 19, 1555-1562 (2002).
    [CrossRef]
  3. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
    [CrossRef]
  4. A. V. Tishchenko, "Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method," Opt. Quantum Electron. 37, 309-330 (2005).
    [CrossRef]
  5. J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).
    [CrossRef]
  6. P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
    [CrossRef]
  7. R. E. Collin, "Reflection and transmission at a slotted dielectric interface," Can. J. Phys. 34, 398-411 (1956).
    [CrossRef]
  8. S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).
  9. A.D.Boardman, ed., Electromagnetic Surface Modes (Wiley, 1982).
  10. D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
    [CrossRef]
  11. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
    [CrossRef]
  12. G. Tayeb and R. Petit, "On the numerical study of deep conducting lamellar diffraction gratings," Opt. Acta 31, 1361-1365 (1984).
    [CrossRef]
  13. See, for example, A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
    [CrossRef]

2005 (1)

A. V. Tishchenko, "Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method," Opt. Quantum Electron. 37, 309-330 (2005).
[CrossRef]

2002 (1)

2000 (1)

1984 (1)

G. Tayeb and R. Petit, "On the numerical study of deep conducting lamellar diffraction gratings," Opt. Acta 31, 1361-1365 (1984).
[CrossRef]

1983 (1)

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

1982 (1)

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

1981 (3)

D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

1973 (1)

See, for example, A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

1956 (2)

R. E. Collin, "Reflection and transmission at a slotted dielectric interface," Can. J. Phys. 34, 398-411 (1956).
[CrossRef]

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

Collin, R. E.

R. E. Collin, "Reflection and transmission at a slotted dielectric interface," Can. J. Phys. 34, 398-411 (1956).
[CrossRef]

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

Hugonin, J.-P.

Lalanne, P.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

Petit, R.

G. Tayeb and R. Petit, "On the numerical study of deep conducting lamellar diffraction gratings," Opt. Acta 31, 1361-1365 (1984).
[CrossRef]

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

Rytov, S. M.

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Sanda, P. N.

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Sarid, D.

D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
[CrossRef]

Sheng, P.

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

Tayeb, G.

G. Tayeb and R. Petit, "On the numerical study of deep conducting lamellar diffraction gratings," Opt. Acta 31, 1361-1365 (1984).
[CrossRef]

Tishchenko, A. V.

A. V. Tishchenko, "Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method," Opt. Quantum Electron. 37, 309-330 (2005).
[CrossRef]

Vallius, T.

Yariv, A.

See, for example, A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

Can. J. Phys. (1)

R. E. Collin, "Reflection and transmission at a slotted dielectric interface," Can. J. Phys. 34, 398-411 (1956).
[CrossRef]

IEEE J. Quantum Electron. (1)

See, for example, A. Yariv, "Coupled-mode theory for guided-wave optics," IEEE J. Quantum Electron. 9, 919-933 (1973).
[CrossRef]

J. Opt. (Paris) (1)

J. Y. Suratteau, M. Cadilhac, and R. Petit, "Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds," J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Acta (3)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

G. Tayeb and R. Petit, "On the numerical study of deep conducting lamellar diffraction gratings," Opt. Acta 31, 1361-1365 (1984).
[CrossRef]

Opt. Quantum Electron. (1)

A. V. Tishchenko, "Phenomenological representation of deep and high contrast lamellar gratings by means of the modal method," Opt. Quantum Electron. 37, 309-330 (2005).
[CrossRef]

Phys. Rev. B (1)

P. Sheng, R. S. Stepleman, and P. N. Sanda, "Exact eigenfunctions for square-wave gratings: Application to diffraction and surface-plasmon calculations," Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Phys. Rev. Lett. (1)

D. Sarid, "Long-range surface-plasma waves on very thin metal films," Phys. Rev. Lett. 47, 1927-1930 (1981).
[CrossRef]

Sov. Phys. JETP (1)

S. M. Rytov, "Electromagnetic properties of a finely stratified medium," Sov. Phys. JETP 2, 466-475 (1956).

Other (1)

A.D.Boardman, ed., Electromagnetic Surface Modes (Wiley, 1982).

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Figures (11)

Fig. 1
Fig. 1

Plane wave diffraction from a lamellar grating.

Fig. 2
Fig. 2

Modal field representation in an infinite grating.

Fig. 3
Fig. 3

TE dispersion curve in a metal–dielectric grating.

Fig. 4
Fig. 4

TM dispersion curve; the grating and the incident wave are the same as in Fig. 3.

Fig. 5
Fig. 5

TM dispersion curve in a case of metal–dielectric grating with line/period ratio of 0.95. All other parameters are the same as in Fig. 3.

Fig. 6
Fig. 6

TE mode field distribution, modes m = 17 (solid curve) and m = 18 (dashed curve). The modal fields are evenly distributed over the whole period of the grating.

Fig. 7
Fig. 7

TM mode field distribution, mode m = 25 ; solid curve—the power is located mostly in the dielectric layer—and m = 26 ; dashed curve—the power is located mostly in the metal layer.

Fig. 8
Fig. 8

Pair of hidden modes, m = 4 and m = 5 . The real parts of modal fields coincide (dashed curve), whereas the imaginary parts are in phase opposition (solid curve). Two grating periods are represented; the metal parts are shaded.

Fig. 9
Fig. 9

Pair of hidden modes, m = 19 and m = 20 . The real parts of modal fields coincide (dashed curve), whereas the imaginary parts are in phase opposition (solid curve). Two grating periods are represented; the metal parts are shaded.

Fig. 10
Fig. 10

Diagram representing the square of propagation constants ρ q of TE modes versus the line/period ratio d 1 d (ordinary case). The dispersion curves (dashed) are well separated and lie between the characteristic curves determined by condition (16) (solid curves). The grating and the incident wave are the same as in Fig. 3.

Fig. 11
Fig. 11

(a) Diagram representing the real part of ρ q of TM mode versus the line/period ratio d 1 d (the ordinary case). The dispersion curves can be shown by two sets of curves corresponding to metal-layer modes and to dielectric-layer modes. At the intersection points the modes become coupled with complex ρ q (hidden modes). (b) Dispersion curves determined by Eqs. (24, 25) give the location of metal-layer modes (thick curves) and dielectric-layer modes (thin curves).

Tables (5)

Tables Icon

Table 1 Zeroth-Order TM Diffraction on a Lossless Metal–Dielectric Grating, d 1 d = 0.5

Tables Icon

Table 2 Zeroth-Order TM Diffraction on a Lossless Metal–Dielectric Grating, d 1 d = 0.95

Tables Icon

Table 3 Square Propagation Constants, TE Modes, d 1 d = 0.5

Tables Icon

Table 4 Square Propagation Constants, TM Modes, d 1 d = 0.5

Tables Icon

Table 5 Square Propagation Constants, TM Modes, d 1 d = 0.95

Equations (26)

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( E i ( x , z , t ) H i ( x , z , t ) ) = ( E 0 I + H 0 I + ) exp ( i k x x ) exp ( i k 0 I z ) exp ( i ω t ) ,
k x = k 0 n I sin θ , k 0 I = k 0 n I cos θ ,
ϵ I = n I 2 ϵ 0 , k 0 = ω ϵ 0 μ 0 .
( E d ( x , z ) H d ( x , z ) ) = q = 0 ( E q ( x ) H q ( x ) ) [ a q + exp ( i β q z ) + a q exp ( i β q z ) ] ,
h 2 < z < h 2 ,
E q ( x , y , z , t ) = n y exp ( i k x m d ) exp ( i β q z ) exp ( i ω t ) × { b q 1 + exp ( i k 1 x ) + b q 1 ( i k 1 x ) , m d x m d + d 1 b q 2 + exp ( i k 2 x ) + b q 2 exp ( i k 2 x ) , m d + d 1 x ( m + 1 ) d } ,
k j = k 0 2 n j g 2 β q 2 , j = 1 , 2 ,
cos k 1 d 1 cos k 2 d 2 1 2 ( k 1 k 2 + k 2 k 1 ) sin k 1 d 1 sin k 2 d 2 = cos k x d .
H q ( x , y , z , t ) = n y exp ( i k x m d ) exp ( i β q z ) exp ( i ω t ) × { b q 1 + exp ( i k 1 x ) + b q 1 ( i k 1 x ) , m d x m d + d 1 b q 2 + exp ( i k 2 x ) + b q 2 exp ( i k 2 x ) , m d + d 1 x ( m + 1 ) d } .
cos k 1 d 1 cos k 2 d 2 1 2 ( n 2 g 2 k 1 n 1 g 2 k 2 + n 1 g 2 k 2 n 2 g 2 k 1 ) sin k 1 d 1 sin k 2 d 2 = cos k x d .
f TE ( ρ ) = cos ( k 0 d 1 n 1 g 2 ρ ) cos ( k 0 d 2 n 2 g 2 ρ ) 1 2 ( n 1 g 2 ρ n 2 g 2 ρ + n 2 g 2 ρ n 1 g 2 ρ ) sin ( k 0 d 1 n 1 g 2 ρ ) sin ( k 0 d 2 n 2 g 2 ρ ) ,
f TM ( ρ ) = cos ( k 0 d 1 n 1 g 2 ρ ) cos ( k 0 d 2 n 2 g 2 ρ ) 1 2 ( n 2 g 2 n 1 g 2 ρ n 1 g 2 n 2 g 2 ρ + n 1 g 2 n 2 g 2 ρ n 2 g 2 n 1 g 2 ρ ) sin ( k 0 d 1 n 1 g 2 ρ ) sin ( k 0 d 2 n 2 g 2 ρ ) ,
ρ q = ( β q k 0 ) 2 .
f ( ρ ) = cos k x d .
ρ pl = n 1 g 2 n 2 g 2 n 1 g 2 + n 2 g 2 .
f TE ( ρ ) = 1 4 ( n 1 g 2 ρ n 2 g 2 ρ + n 2 g 2 ρ n 1 g 2 ρ ) 2 cos ( k 0 d 1 n 1 g 2 ρ + k 0 d 2 n 2 g 2 ρ ) 1 4 ( n 1 g 2 ρ n 2 g 2 ρ n 2 g 2 ρ n 1 g 2 ρ ) 2 cos ( k 0 d 1 n 1 g 2 ρ k 0 d 2 n 2 g 2 ρ ) .
k 0 d 1 n 1 g 2 ρ + k 0 d 2 n 2 g 2 ρ = m π
1 4 ( n 1 g 2 ρ n 2 g 2 ρ + n 2 g 2 ρ n 1 g 2 ρ ) 2 = 1 4 ( n 1 g 2 ρ n 2 g 2 ρ n 2 g 2 ρ n 1 g 2 ρ ) 2 + 1 .
f TM ( ρ ) = 1 4 ( n 2 g 2 n 1 g 2 ρ n 1 g 2 n 2 g 2 ρ + n 1 g 2 n 2 g 2 ρ n 2 g 2 n 1 g 2 ρ ) 2 cos ( k 0 d 1 n 1 g 2 ρ k 0 d 2 n 2 g 2 ρ ) 1 4 ( n 2 g 2 n 1 g 2 ρ n 1 g 2 n 2 g 2 ρ n 1 g 2 n 2 g 2 ρ n 2 g 2 n 1 g 2 ρ ) 2 cos ( k 0 d 1 n 1 g 2 ρ + k 0 d 2 n 2 g 2 ρ ) .
k 0 d 1 n 1 g 2 ρ k 0 d 2 n 2 g 2 ρ = m π .
f TM ( ρ ) = 1 + 1 4 ( n 2 g 2 n 1 g 2 ρ n 1 g 2 n 2 g 2 ρ n 1 g 2 n 2 g 2 ρ n 2 g 2 n 1 g 2 ρ ) 2 sin 2 ( k 0 d 1 n 1 g 2 ρ ) cos { k 0 d 2 n 2 g 2 ρ + arctan [ 1 2 ( n 2 g 2 n 1 g 2 ρ n 1 g 2 n 2 g 2 ρ + n 1 g 2 n 2 g 2 ρ n 2 g 2 n 1 g 2 ρ ) tan ( k 0 d 1 n 1 g 2 ρ ) ] } .
k 0 d 2 n 2 g 2 ρ arctan [ 1 2 ( n 2 g 2 n 1 g 2 ρ n 1 g 2 n 2 g 2 ρ + n 1 g 2 n 2 g 2 ρ n 2 g 2 n 1 g 2 ρ ) tan ( k 0 d 1 n 1 g 2 ρ ) ] = m π .
1 + 1 8 ( n 2 g 2 n 1 g 2 ρ n 1 g 2 n 2 g 2 ρ n 1 g 2 n 2 g 2 ρ n 2 g 2 n 1 g 2 ρ ) 2 ( k 0 d 1 n 1 g 2 ρ m π ) 2 cos ( k x d ) cos ( k 0 d 2 n 2 g 2 ρ ) ,
k 1 d 1 m π ± j 8 1 cos ( k x d ) cos ( k 0 d 2 n 2 g 2 ρ ) n 2 g 2 n 1 g 2 ρ n 1 g 2 n 2 g 2 ρ n 1 g 2 n 2 g 2 ρ n 2 g 2 n 1 g 2 ρ .
k 0 d 1 n 1 g 2 ρ = π m 1
k 0 d 2 n 2 g 2 ρ = π m 2 .

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