Abstract

This paper proposes a spectral-domain approach to the electromagnetic scattering problem of lamellar grating with defects. The fields in imperfectly periodic structures have continuous spectra in the wavenumber space, and the main problem of the spectral-domain approach is connected to the discretization scheme on the wavenumber. The present approach introduces the pseudo-periodic Fourier transform to consider the discretization scheme in the Brillouin zone. This transformation also makes it possible to apply the conventional grating formulations to the problems of imperfectly periodic structures. The present formulation is based on the rigorous coupled-wave analysis with the help of pseudo-periodic Fourier transform.

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  1. O. Manyardo, R. Michaely, F. Schädelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. 29, 1437–1439 (2004).
    [CrossRef]
  2. Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. 18, 055016 (2008).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  14. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).
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    [CrossRef]
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2011 (1)

K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B 31, 219–237 (2011).

2010 (3)

2008 (3)

2007 (3)

W. T. Lu, Y. J. Huang, P. Vodo, R. K. Banyal, C. H. Perry, and S. Sridhar, “A new mechanism for negative refraction and focusing using selective diffraction from surface corrugation,” Opt. Express 15, 9166–9175 (2007).
[CrossRef] [PubMed]

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

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]

2005 (1)

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

2004 (2)

K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett. 325, 415–419 (2004).
[CrossRef]

O. Manyardo, R. Michaely, F. Schädelin, W. Noell, T. Overstolz, N. De Rooij, and H. P. Herzig, “Miniature lamellar grating interferometer based on silicon technology,” Opt. Lett. 29, 1437–1439 (2004).
[CrossRef]

2002 (1)

2001 (1)

1996 (1)

1982 (1)

1978 (1)

1974 (1)

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

Antoš, R.

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

Arnold, C.

Banyal, R. K.

Bonod, N.

Borghi, R.

Cai, Y. G.

B. G. Zhai, Y. G. Cai, and Y. M. Huang, “Transmission spectra of one-dimensional photonic crystal with a centered defect,” Mat. Sci. For. 663–665, 733–736 (2010).

Chen, Ch-H.

Chen, Y.

Davis, P. J.

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

De Rooij, N.

Enoch, S.

Fainman, Y.

Feiwen, L.

Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. 18, 055016 (2008).
[CrossRef]

Frezza, F.

Gaylord, T. K.

Greffet, J. J.

Guangya, Z.

Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. 18, 055016 (2008).
[CrossRef]

Herzig, H. P.

Hongbin, Y.

Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. 18, 055016 (2008).
[CrossRef]

Horie, M.

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

Huang, Y. J.

Huang, Y. M.

B. G. Zhai, Y. G. Cai, and Y. M. Huang, “Transmission spectra of one-dimensional photonic crystal with a centered defect,” Mat. Sci. For. 663–665, 733–736 (2010).

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]

Käpfe, T.

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]

Laroche, M.

Li, L.

Li, R.

K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett. 325, 415–419 (2004).
[CrossRef]

Liu, D.

K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett. 325, 415–419 (2004).
[CrossRef]

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

Lu, W. T.

Manyardo, O.

Marquier, F.

Maystre, D.

Michaely, R.

Mingsheng, Z.

Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. 18, 055016 (2008).
[CrossRef]

Mistrík, J.

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

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]

Nakagawa, W.

Nakatake, Y.

K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B 31, 219–237 (2011).

Noell, W.

Ohlídal, I.

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

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]

Overstolz, T.

Pajewski, L.

Parriaux, O.

Perry, C. H.

Pištora, J.

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

Popov, E.

Postava, K.

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

Rabinowitz, P.

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

Ren, K.

K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett. 325, 415–419 (2004).
[CrossRef]

Ren, X.

K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett. 325, 415–419 (2004).
[CrossRef]

Santarsiero, M.

Sato, A.

Schädelin, F.

Schettini, G.

Shouhua, W.

Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. 18, 055016 (2008).
[CrossRef]

Siong, Ch. F.

Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. 18, 055016 (2008).
[CrossRef]

Sridhar, S.

Sun, P.-Ch.

Takahasi, H.

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

Tayeb, G.

Višnovský, Š.

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

Vodo, P.

Watanabe, K.

K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B 31, 219–237 (2011).

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

Yamaguchi, T.

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

Yasumoto, K.

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

Zhai, B. G.

B. G. Zhai, Y. G. Cai, and Y. M. Huang, “Transmission spectra of one-dimensional photonic crystal with a centered defect,” Mat. Sci. For. 663–665, 733–736 (2010).

Zhou, J.

K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett. 325, 415–419 (2004).
[CrossRef]

Appl. Opt. (1)

J. Appl. Phys. (1)

R. Antoš, J. Pištora, I. Ohlídal, K. Postava, J. Mistrík, T. Yamaguchi, Š. Višňovský, and M. Horie, “Specular spectroscopic ellipsometry for the critical dimension monitoring of gratings fabricated on a thick transparent plate,” J. Appl. Phys. 97, 053107 (2005).
[CrossRef]

J. Micromech. Microeng. (1)

Y. Hongbin, Z. Guangya, Ch. F. Siong, L. Feiwen, W. Shouhua, and Z. Mingsheng, “An electromagnetically driven lamellar grating based Fourier transform microspectrometer,” J. Micromech. Microeng. 18, 055016 (2008).
[CrossRef]

J. Opt. Soc. Am. (2)

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]

Mat. Sci. For. (1)

B. G. Zhai, Y. G. Cai, and Y. M. Huang, “Transmission spectra of one-dimensional photonic crystal with a centered defect,” Mat. Sci. For. 663–665, 733–736 (2010).

Opt. Express (3)

Opt. Lett. (2)

Phys. Lett. (1)

K. Ren, X. Ren, R. Li, J. Zhou, and D. Liu, “Creating “defects” in photonic crystals by controlling polarizations,” Phys. Lett. 325, 415–419 (2004).
[CrossRef]

Prog. Electromagn. Res. B (1)

K. Watanabe and Y. Nakatake, “Spectral-domain formulation of electromagnetic scattering from circular cylinders located near periodic cylinder array,” Prog. Electromagn. Res. B 31, 219–237 (2011).

Progress in Electromagnetic Res. (1)

K. Watanabe and K. Yasumoto, “Two-dimensional electromagnetic scattering of non-plane incident waves by periodic structures,” Progress in 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 (3)

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

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel Dekker, New York, 1997).

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

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Figures (6)

Fig. 1
Fig. 1

Lamellar grating with defects.

Fig. 2
Fig. 2

Convergence of the field intensities at (x,y) = (0,d) for a lamellar grating with a defect for a line-source excitation.

Fig. 3
Fig. 3

Field intensities near a lamellar grating with a defect for the line-source excitation.

Fig. 4
Fig. 4

Convergence of the field intensities at (x,y) = (0,d) for a lamellar grating with five defects for a line-source excitation.

Fig. 5
Fig. 5

Field intensities near a lamellar grating with five defects for the line-source excitation.

Fig. 6
Fig. 6

Numerical results of the reciprocity test for a lamellar grating with five defects for the line-source excitation.

Equations (68)

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f ¯ ( x ; ξ ) = m = f ( x m d ) e i m d ξ ,
f ( x ) = 1 k d k d / 2 k d / 2 f ¯ ( x ; ξ ) d ξ
η ( x ) = q g ( η ) ( x ) + q d ( η ) ( x )
1 η ( x ) = r g ( η ) ( x ) + r d ( η ) ( x )
q g ( η ) ( x ) = η s + ( η c η s ) l = u ( x l d )
q d ( η ) ( x ) = ( η c η s ) l 𝒟 u ( x l d )
r g ( η ) ( x ) = 1 η s + ( 1 η c 1 η s ) l = u ( x l d )
r d ( η ) ( x ) = ( 1 η c 1 η s ) l 𝒟 u ( x l d )
u ( x ) = { 1 for | x | a / 2 0 for | x | > a / 2 .
x H y ( x , y ) y H x ( x , y ) = i ω D z ( x , y )
y E z ( x , y ) = i ω B x ( x , y )
x E z ( x , y ) = i ω B y ( x , y )
D z ( x , y ) = ( q g ( ɛ ) ( x ) + q d ( ɛ ) ( x ) ) E z ( x , y )
H x ( x , y ) = ( r g ( μ ) ( x ) + r d ( μ ) ( x ) ) B x ( x , y )
B y ( x , y ) = ( q g ( μ ) ( x ) + q d ( μ ) ( x ) ) H y ( x , y ) .
x H ¯ y ( x ; ξ , y ) y H ¯ x ( x ; ξ , y ) = i ω D ¯ z ( x ; ξ , y )
y E ¯ z ( x ; ξ , y ) = i ω B ¯ x ( x ; ξ , y )
x E ¯ z ( x ; ξ , y ) = i ω B ¯ y ( x ; ξ , y )
D ¯ z ( x ; ξ , y ) = q g ( ɛ ) ( x ) E ¯ z ( x ; ξ , y ) + 1 k d k d / 2 k d / 2 q ¯ d ( ɛ ) ( x ; ξ ξ ) E ¯ z ( x ; ξ , y ) d ξ
H ¯ x ( x ; ξ , y ) = r g ( μ ) ( x ) B ¯ x ( x ; ξ , y ) + 1 k d k d / 2 k d / 2 r ¯ d ( μ ) ( x ; ξ ξ ) B ¯ x ( x ; ξ , y ) d ξ
B ¯ y ( x ; ξ , y ) = q g ( μ ) ( x ) H ¯ y ( x ; ξ , y ) + 1 k d k d / 2 k d / 2 q ¯ d ( μ ) ( x ; ξ ξ ) H ¯ y ( x ; ξ , y ) d ξ .
E ¯ z ( x ; ξ , y ) = n = N N E ¯ z , n ( ξ , y ) e i α n ( ξ ) x
α n ( ξ ) = ξ + n k d
( [ [ f ] ] ) n , m = 1 d d / 2 d / 2 f ( x ) e i ( n m ) k d x d x .
( [ [ g ¯ ] ] ( ξ ) ) n , m = 1 d d / 2 d / 2 g ¯ ( x ; ξ ) e i α n m ( ξ ) x d x .
i X ¯ ( ξ ) h ¯ y ( ξ , y ) y h ¯ x ( ξ , y ) = i ω d ¯ z ( ξ , y )
y e ¯ z ( ξ , y ) = i ω b ¯ x ( ξ , y )
i X ¯ ( ξ ) e ¯ z ( ξ , y ) = i ω b ¯ y ( ξ , y )
d ¯ z ( ξ , y ) = [ [ q g ( ɛ ) ] ] e ¯ z ( ξ , y ) + 1 k d k d / 2 k d / 2 [ [ q ¯ d ( ɛ ) ] ] ( ξ ξ ) e ¯ z ( ξ , y ) d ξ
h ¯ x ( ξ , y ) = [ [ r g ( μ ) ] ] b ¯ x ( ξ , y ) + 1 k d k d / 2 k d / 2 [ [ r ¯ d ( μ ) ] ] ( ξ ξ ) b ¯ x ( ξ , y ) d ξ
b ¯ y ( ξ , y ) = [ [ q g ( μ ) ] ] h ¯ y ( ξ , y ) + 1 k d k d / 2 k d / 2 [ [ q ¯ d ( μ ) ] ] ( ξ ξ ) h ¯ y ( ξ , y ) d ξ
i X ˜ h ˜ y ( y ) d d y h ˜ x ( y ) = i ω d ˜ z ( y )
d d y e ˜ z ( y ) = i ω b ˜ x ( y )
i X ˜ e ˜ z ( y ) = i ω b ˜ y ( y )
d ˜ z ( y ) = Q ˜ ( ɛ ) e ˜ z ( y )
h ˜ x ( y ) = R ˜ ( μ ) b ˜ x ( y )
b ˜ y ( y ) = Q ˜ ( μ ) h ˜ y ( y )
e ˜ z ( y ) = ( e ¯ z ( ξ 1 , y ) e ¯ z ( ξ L , y ) )
X ˜ = ( X ¯ ( ξ 1 ) 0 0 X ¯ ( ξ L ) )
Q ˜ ( υ ) = ( Q 1 , 1 ( υ ) Q 1 , L ( υ ) Q L , 1 ( υ ) Q L , L ( υ ) )
R ˜ ( υ ) = ( R 1 , 1 ( υ ) R 1 , L ( υ ) R L , 1 ( υ ) R L , L ( υ ) )
Q l , l ( υ ) = δ l , l [ [ q g ( υ ) ] ] + w l k d [ [ q ¯ d ( υ ) ] ] ( ξ l ξ l )
R l , l ( υ ) = δ l , l [ [ r g ( υ ) ] ] + w l k d [ [ r ¯ d ( υ ) ] ] ( ξ l ξ l )
d 2 d y 2 e ˜ z ( y ) = C ˜ g ( e ) e ˜ z ( y )
C ˜ g ( e ) = R ˜ ( μ ) 1 ( ω 2 Q ˜ ( ɛ ) X ˜ Q ˜ ( μ ) 1 X ˜ ) .
( e ˜ z ( y ) h ˜ x ( y ) ) = Ξ g ( e ) ( a ˜ g ( e , ) ( y ) a ˜ g ( e , + ) ( y ) )
Ξ g ( e ) = ( P ˜ g ( e ) P ˜ g ( e ) 1 ω R ˜ ( μ ) P ˜ g ( e ) Y ˜ g ( e ) 1 ω R ˜ ( μ ) P ˜ g ( e ) Y ˜ g ( e ) )
P ˜ g ( e ) = ( p g , 1 ( e ) p g , L ( 2 N + 1 ) ( e ) )
( Y ˜ g ( e ) ) n , m = δ n , m β g , n ( e ) .
a ˜ g ( e , ± ) ( y ) = A ˜ g ( e ) ( ± ( y y ) ) a ˜ g ( e , ± ) ( y )
( A ˜ g ( e ) ( y ) ) n , m = δ n , m e i β g , n ( e ) y
( e ˜ z ( y ) h ˜ x ( y ) ) = Ξ r ( e ) ( a ˜ r ( e , ) ( y ) a ˜ r ( e , + ) ( y ) )
Ξ r ( e ) = ( I I 1 ω μ r Y ˜ r 1 ω μ r Y ˜ r )
( Y ˜ r ) n , m = δ n , m k r 2 ( X ˜ ) n , n
( a ˜ g ( f , ) ( h r ) a ˜ g ( f , + ) ( h r ) ) = ( G ˜ r ( f , + ) G ˜ r ( f , ) G ˜ r ( f , ) G ˜ r ( f , + ) ) ( a ˜ r ( f , ) ( h r ) a ˜ r ( f , + ) ( h r ) )
G ˜ r ( e , ± ) = 1 2 ( P ˜ g ( e ) 1 ± 1 μ r Y ˜ g ( e ) 1 P ˜ g ( e ) 1 R ˜ ( μ ) 1 Y ˜ r )
G ˜ r ( h , ± ) = 1 2 ( P ˜ g ( h ) 1 ± 1 ɛ r Y ˜ g ( h ) 1 P ˜ g ( h ) 1 R ˜ ( ɛ ) 1 Y ˜ r )
( a ˜ s ( f , + ) ( h s ) a ˜ c ( f , ) ( h c ) a ˜ g ( f , ) ( h s ) a ˜ g ( f , + ) ( h c ) ) = ( S ˜ 11 ( f ) S ˜ 12 ( f ) S ˜ 21 ( f ) S ˜ 22 ( f ) S ˜ 31 ( f ) S ˜ 32 ( f ) S ˜ 41 ( f ) S ˜ 42 ( f ) ) ( a ˜ s ( f , ) ( h s ) a ˜ c ( f , + ) ( h c ) )
S ˜ 11 ( f ) = ( G ˜ s ( f , + ) A ˜ g ( f ) ( t ) G ˜ c ( f , ) G ˜ c ( f , + ) 1 A ˜ g ( f ) ( t ) G ˜ s ( f , ) ) 1 ( G ˜ s ( f , ) A ˜ g ( f ) ( t ) G ˜ c ( f , ) G ˜ c ( f , + ) 1 A ˜ g ( f ) ( t ) G ˜ s ( f , + ) )
S ˜ 12 ( f ) = ( G ˜ s ( f , + ) A ˜ g ( f ) ( t ) G ˜ c ( f , ) G ˜ c ( f , + ) 1 A ˜ g ( f ) ( t ) G ˜ s ( f , ) ) 1 A ˜ g ( f ) ( t ) ( G ˜ c ( f , + ) G ˜ c ( f , ) G ˜ c ( f , + ) 1 G ˜ c ( f , ) )
S ˜ 21 ( f ) = G ˜ c ( f , + ) 1 A ˜ g ( f ) ( t ) ( G ˜ s ( f , + ) + G ˜ s ( f , ) S ˜ 11 ( f ) )
S ˜ 22 ( f ) = G ˜ c ( f , + ) 1 ( G ˜ c ( f , ) A ˜ g ( f ) ( t ) G ˜ s ( f , ) S ˜ 12 ( f ) )
S ˜ 31 ( f ) = G ˜ s ( f , + ) + G ˜ s ( f , ) S ˜ 11 ( f )
S ˜ 32 ( f ) = G ˜ s ( f , ) S ˜ 12 ( f )
S ˜ 41 ( f ) = G ˜ c ( f , ) S ˜ 21 ( f )
S ˜ 42 ( f ) = G ˜ c ( f , + ) + G ˜ c ( f , ) S ˜ 22 ( f ) .
ψ ( i ) ( x , y ) = H 0 ( 1 ) ( k s ρ ( x x 0 , y y 0 ) )
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 ) |

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