Abstract

This paper considers the electromagnetic scattering problem of periodically corrugated surface with local imperfection of structural periodicity, and presents a formulation based on the coordinate transformation method (C-method). The C-method is originally developed to analyze the plane-wave scattering from perfectly periodic structures, and uses the pseudo-periodic property of the fields. The fields in imperfectly periodic structures are not pseudo-periodic and the C-method cannot be directly applied. This paper introduces the pseudo-periodic Fourier transform to convert the fields in imperfectly periodic structures to pseudo-periodic ones, and the C-method becomes then applicable.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. 46, 5435–5440 (2007).
    [CrossRef]
  2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton Univ. Press, Princeton, 1995).
  3. C. Yang, K. Shi, P. Edwards, and Z. Liu, “Demonstration of a PDMS based hybrid grating and Fresnel lens (G-Fresnel) device,” Opt. Express 18, 23529–23534 (2010).
    [CrossRef] [PubMed]
  4. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  5. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  6. K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagnetic Res. PIER 74, 241–271 (2007).
    [CrossRef]
  7. K. Watanabe, J. Pištora, and Y. Nakatake, “Rigorous coupled-wave analysis of electromagnetic scattering from lamellar grating with defects,” Opt. Express 19, 25799–25811 (2011).
    [CrossRef]
  8. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  9. M. G. Moharam and T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  10. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
    [CrossRef]
  11. K. Watanabe, “Numerical integration schemes used on the differential theory for anisotropic gratings,” J. Opt. Soc. Am. A 19, 2245–2252 (2002).
    [CrossRef]
  12. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  13. W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).
  14. H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974).
    [CrossRef]
  15. P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic Press, New York, 1984).

2011 (1)

2010 (1)

2007 (2)

T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. 46, 5435–5440 (2007).
[CrossRef]

K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagnetic Res. PIER 74, 241–271 (2007).
[CrossRef]

2002 (2)

1996 (1)

1982 (2)

1978 (1)

1974 (1)

H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974).
[CrossRef]

Chandezon, J.

Chew, W. C.

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

Cornet, G.

Davis, P. J.

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic Press, New York, 1984).

Dupuis, M. T.

Edwards, P.

Gaylord, T. K.

Gralak, B.

Itoh, K.

T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. 46, 5435–5440 (2007).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton Univ. Press, Princeton, 1995).

Knop, K.

Konishi, T.

T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. 46, 5435–5440 (2007).
[CrossRef]

Li, L.

Liu, Z.

Maystre, D.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton Univ. Press, Princeton, 1995).

Moharam, M. G.

Mori, M.

H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974).
[CrossRef]

Nakatake, Y.

Nevière, M.

Oonishi, T.

T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. 46, 5435–5440 (2007).
[CrossRef]

Pištora, J.

Popov, E.

Rabinowitz, P.

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic Press, New York, 1984).

Shi, K.

Takahasi, H.

H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974).
[CrossRef]

Tayeb, G.

Watanabe, K.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton Univ. Press, Princeton, 1995).

Yang, C.

Yasumoto, K.

K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagnetic Res. PIER 74, 241–271 (2007).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (3)

Jpn. J. Appl. Phys. (1)

T. Oonishi, T. Konishi, and K. Itoh, “Fabrication of phase only binary blazed grating with subwavelength structures designed by deterministic method based on electromagnetic analysis,” Jpn. J. Appl. Phys. 46, 5435–5440 (2007).
[CrossRef]

Opt. Express (2)

Prog. Electromagnetic Res. (1)

K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Prog. Electromagnetic Res. PIER 74, 241–271 (2007).
[CrossRef]

Publ. RIMS, Kyoto Univ. (1)

H. Takahasi and M. Mori, “Double exponential formulas for numerical integration,” Publ. RIMS, Kyoto Univ. 9, 721–741 (1974).
[CrossRef]

Other (4)

P. J. Davis and P. Rabinowitz, Methods of Numerical Integration, 2nd ed. (Academic Press, New York, 1984).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Modeling the Flow of Light (Princeton Univ. Press, Princeton, 1995).

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

W. C. Chew, Waves and Fields in Inhomogeneous Media (Van Nostrand Reinhold, New York, 1990).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

An example of imperfectly periodic surfaces (surface-relief grating with a defect).

Fig. 2
Fig. 2

Convergence of the field intensities at (x,y) = (0,d) for a sinusoidal profile grating with a defect.

Fig. 3
Fig. 3

Field intensities near a sinusoidal profile grating with a defect.

Fig. 4
Fig. 4

Numerical results of the reciprocity test for a sinusoidal profile grating with a defect.

Fig. 5
Fig. 5

Convergence of the field intensities at (x,y) = (0,d) for a sinusoidal profile grating with period modulation.

Fig. 6
Fig. 6

Field intensities near a sinusoidal profile grating with period modulation for the line-source excitation.

Fig. 7
Fig. 7

Numerical results of the reciprocity test for a sinusoidal profile grating with period modulation.

Equations (59)

Equations on this page are rendered with MathJax. Learn more.

g ( x ) = g ( p ) ( x ) + g ( a ) ( x )
x H y ( x , y ) y H x ( x , y ) = i ω ε r E z ( x , y )
y E z ( x , y ) = i ω μ r H x ( x , y )
x E z ( x , y ) = i ω μ r H y ( x , y )
[ u ( g ˙ ( p ) ( u ) + g ˙ ( a ) ( u ) ) v ] H y ( u , v ) v H x ( u , v ) = i ω ε r E z ( u , v )
v E z ( u , v ) = i ω μ r H x ( u , v )
[ u ( g ˙ ( p ) ( u ) + g ˙ ( a ) ( u ) ) v ] E z ( u , v ) = i ω μ r H y ( u , v )
f ¯ ( u ; ξ ) = m = f ( u m d ) e imd ξ
f ( u ) = 1 k d k d / 2 k d / 2 f ¯ ( u ; ξ ) d ξ
( u g ˙ ( p ) ( u ) v ) H ¯ y ( u ; ξ , v ) 1 k d k d / 2 k d / 2 g ˙ ¯ ( a ) ( u ; ξ ξ ) v H ¯ y ( u ; ξ , v ) d ξ v H ¯ x ( u ; ξ , v ) = i ω ε r E ¯ z ( u ; ξ , v )
v E ¯ z ( u ; ξ , v ) = i ω μ r H ¯ x ( u ; ξ , v )
( u g ˙ ( p ) ( u ) v ) E ¯ z ( u ; ξ , v ) 1 k d k d / 2 k d / 2 g ˙ ¯ ( a ) ( u ; ξ ξ ) v E ¯ z ( u ; ξ , v ) d ξ = i ω μ r H ¯ y ( u ; ξ , v ) .
E ¯ z ( u ; ξ , v ) = n = N N E ¯ z , n ( ξ , v ) e i α n ( ξ ) u
α n ( ξ ) = ξ + n k d
( i U ¯ ( ξ ) g ˙ ( p ) v ) h ¯ y ( ξ , v ) 1 k d k d / 2 k d / 2 g ˙ ¯ ( a ) ( ξ ξ ) v h ¯ y ( ξ , v ) d ξ v h ¯ x ( ξ , v ) = i ω ε r e ¯ z ( ξ , v )
v e ¯ z ( ξ , v ) = i ω μ r h ¯ x ( ξ , v )
( i U ¯ ( ξ ) g ˙ ( p ) v ) e ¯ z ( ξ , v ) 1 k d k d / 2 k d / 2 g ˙ ¯ ( a ) ( ξ ξ ) v e ¯ z ( ξ , v ) d ξ = i ω μ r h ¯ y ( ξ , v )
( U ¯ ( ξ ) ) n , m = δ n , m α n ( ξ )
( g ˙ ( p ) ) n , m = 1 d d / 2 d / 2 g ˙ ( p ) ( u ) e i ( n m ) k d u d u
( g ˙ ¯ ( a ) ( ξ ) ) n , m = 1 d g ( a ) ( u ) e i α n m ( ξ ) u d u .
( i U ˜ G ˜ d d v ) h ˜ y ( v ) d d v h ˜ x ( v ) = i ω ε r e ˜ z ( v )
d d v e ˜ z ( v ) = i ω μ r h ˜ x ( v )
( i U ˜ G ˜ d d v ) e ˜ z ( v ) = i ω μ r h ˜ y ( v )
e ˜ z ( v ) = ( e ˜ z ( ξ 1 , v ) e ˜ z ( ξ L , v ) )
U ˜ = ( U ¯ ( ξ 1 ) 0 0 U ¯ ( ξ L ) )
G ˜ = ( G ¯ 1 , 1 G ¯ 1 , L G ¯ L , 1 G ¯ L , L )
G ¯ l , l = δ l , l g ˙ ( p ) + w l k d g ˙ ¯ ( a ) ( ξ l ξ l )
( e ˜ z ( v ) i d d v e ˜ z ( v ) ) = i M r d d v ( e ˜ z ( v ) i d d v e ˜ z ( v ) )
M r = ( ( k r 2 I U ˜ 2 ) 1 ( U ˜ G ˜ + G ˜ U ˜ ) ( k r 2 I U ˜ 2 ) 1 ( G ˜ 2 + I ) I 0 )
( P ˜ r , 11 P ˜ r , 12 P ˜ r , 21 P ˜ r , 22 ) = ( p r , 1 p r , 2 L ( 2 N + 1 ) )
( e ˜ z ( v ) i d d v e ˜ z ( v ) ) = ( P ˜ r , 11 P ˜ r , 12 P ˜ r , 21 P ˜ r , 22 ) ( a ˜ e , r ( ) ( v ) a ˜ e , r ( + ) ( v ) )
( a ˜ e , r ( ) ( v ) a ˜ e , r ( + ) ( v ) ) = V r ( v v ) ( a ˜ e , r ( ) ( v ) a ˜ e , r ( + ) ( v ) )
( V r ( v ) ) n , m = δ n , m e i η r , n v .
( e ˜ z ( v ) h ˜ t ( v ) ) = ( P ˜ r , 11 P ˜ r , 12 Q ˜ e , r , 1 Q ˜ e , r , 2 ) ( a ˜ e , r ( ) ( v ) a ˜ e , r ( + ) ( v ) )
Q ˜ e , r , q = 1 ω μ r [ G ˜ U ˜ P ˜ r , 1 q ( G ˜ 2 + I ) P ˜ r , 2 q ]
( h ˜ z ( v ) e ˜ t ( v ) ) = ( P ˜ r , 11 P ˜ r , 12 Q ˜ h , r , 1 Q ˜ h , r , 2 ) ( a ˜ h , r ( ) ( v ) a ˜ h , r ( + ) ( v ) )
Q ˜ h , r , q = 1 ω ε r [ G ˜ U ˜ P ˜ r , 1 q ( G ˜ 2 + I ) P ˜ r , 2 q ]
( a ˜ f , s ( + ) ( + 0 ) a ˜ f , c ( ) ( 0 ) ) = ( S ˜ f , 11 S ˜ f , 12 S ˜ f , 21 S ˜ f , 22 ) ( a ˜ f , s ( ) ( + 0 ) a ˜ f , c ( + ) ( 0 ) )
S ˜ f , 11 = ( Q ˜ f , c , 1 P ˜ c , 11 1 P ˜ s , 12 Q ˜ f , s , 2 ) 1 ( Q ˜ f , c , 1 P ˜ c , 11 1 P ˜ s , 11 Q ˜ f , s , 1 )
S ˜ f , 12 = ( Q ˜ f , c , 1 P ˜ c , 11 1 P ˜ s , 12 Q ˜ f , s , 2 ) 1 ( Q ˜ f , c , 1 P ˜ c , 11 1 P ˜ c , 12 Q ˜ f , c , 2 )
S ˜ f , 21 = P ˜ c , 11 1 ( P ˜ s , 12 S ˜ f , 11 + P ˜ s , 11 )
S ˜ f , 22 = P ˜ c , 11 1 ( P ˜ s , 12 S ˜ f , 12 P ˜ c , 12 )
( a ˜ f , r ( ) ( v ) a ˜ f , r ( + ) ( v ) ) = ( P ˜ r , 11 P ˜ r , 12 P ˜ r , 21 P ˜ r , 22 ) 1 ( ψ ˜ ( v ) i d d v ψ ˜ ( v ) )
ψ ¯ n ( ξ , v ) = 1 d ψ ( u , v + g ( u ) ) e i α n ( ξ ) u d u
i v ψ ¯ n ( ξ , v ) = i d v ψ ( u , v + g ( u ) ) e i α n ( ξ ) u d u
g ( p ) ( x ) = h 1 + ( h 2 h 1 ) sin 2 ( π d x )
ψ ( i ) ( x , y ) = H 0 ( 1 ) ( k s ( x x 0 ) 2 + ( y y 0 ) 2 )
ψ ¯ n ( i ) ( ξ , v ) = 1 d ψ ( i ) ( u , v + g ( p ) ( u ) ) e i α n ( ξ ) u d u + 1 d a 1 a 2 ( ψ ( i ) ( u , v + g ( u ) ) ψ ( i ) ( u , v + g ( p ) ( u ) ) ) e i α n ( ξ ) u d u
i v ψ ¯ n ( i ) ( ξ , v ) = i d v ψ ( i ) ( u , v + g ( p ) ( u ) ) e i α n ( ξ ) u d u i d a 1 a 2 v ( ψ ( i ) ( u , v + g ( u ) ) ψ ( i ) ( u , v + g ( p ) ( u ) ) ) e i α n ( ξ ) u d u .
H 0 ( 1 ) ( k s ρ ( x , y ) ) = 1 π 1 k s 2 ζ 2 e i ( ζ x + k s 2 ζ 2 | y | ) d ζ
e i β ( h 2 g ( p ) ( u ) ) = m = c m ( β ) e i m k d u .
1 d ψ ( i ) ( u , v + g ( p ) ( u ) ) e i α n ( ξ ) u d u = m = c n m ( β m ( ξ ) ) 2 d β m ( ξ ) e i [ α m ( ξ ) x 0 β m ( ξ ) ( y 0 h 2 ) ] e i β m ( ξ ) v
i d v ψ ( i ) ( u , v + g ( p ) ( u ) ) e i α n ( ξ ) u d u = 2 d m = c n m ( β m ( ξ ) ) e i [ α m ( ξ ) x 0 β m ( ξ ) ( y 0 h 2 ) ] e i β m ( ξ ) v
β m ( ξ ) = k s 2 α m ( ξ ) 2
c m ( β ) = i m J m ( β h 1 h 1 2 ) e i β h 2 h 1 2
g ( a ) ( x ) = { ( h 2 h 1 ) cos 2 ( π d x ) for  | x | d 2 0 for  | x | > d 2
σ ( x p , y p ; x q , y q ) = | ψ ( x p , y p ; x q , y q ) ψ ( x q , y q ; x p , u p ) | | ψ ( x p , y p ; x q , y q ) | .
g ( a ) ( x ) = { g ( p ) ( x ) + g ( p ) ( 0 x d d + Δ ( ζ ) d ζ ) for  | x | a 0 for  | x | > a
Δ ( x ) = d min d 2 ( 1 + cos ( π a x ) ) .

Metrics