Abstract

We consider diffraction of an arbitrary two-dimensional electromagnetic field by a lamellar dielectric structure of arbitrary form, which fills a slit aperture in a conducting screen of finite thickness. The problem is treated rigorously, by expanding the field inside the index-modulated aperture in the form of a series of exact eigenfunctions. This discrete set of eigenfunctions is obtained by means of the theory of stratified media, combined with the appropriate boundary conditions at the perfectly conducting edges of the aperture. Then the eigenmode expansion is matched to the angular spectrum representations of the incident, forward-diffracted, and backward-diffracted fields to determine the total field everywhere in space. Numerical implementation and convergence properties of the method are discussed. Some illustrations of diffraction by nonperiodic structures are given.

© 1996 Optical Society of America

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References

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  1. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), pp. 1–121.
    [CrossRef]
  2. T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
    [CrossRef]
  3. F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
  4. D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
    [CrossRef]
  5. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981).
    [CrossRef]
  6. P. Sheng, R. S. Stepleman, 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. L. C. Botten, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, M. S. Craig, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  8. J. Y. Surrateau, M. Cadillac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).
  9. R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
    [CrossRef]
  10. J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
    [CrossRef]
  11. O. M. Mendez, M. Cadilhac, R. Petit, “Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen,” J. Opt. Soc. Am. 73, 328–331 (1983).
    [CrossRef]
  12. A. Roberts, “Electromagnetic theory of diffraction by a circular aperture in a thick, perfectly conducting screen,” J. Opt. Soc. Am. A 4, 1970–1983 (1987).
    [CrossRef]
  13. Y.-L. Kok, “Boundary-value solution to electromagnetic scattering by a rectangular groove in a ground plane,” J. Opt. Soc. Am. A 9, 302–311 (1992).
    [CrossRef]
  14. J. Huttunen, J. Turunen, “Phase images of grooves in a perfectly conducting surface,” Opt. Commun. 119, 485–490 (1995).
    [CrossRef]
  15. J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
    [CrossRef]
  16. Y.-L. Kok, “General solution to the multiple-metallic-grooves scattering problem: the fast-polarization case,” Appl. Opt. 32, 2573–2581 (1993).
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  17. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), pp. 229–337.
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  18. 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]

1995 (3)

J. Huttunen, J. Turunen, “Phase images of grooves in a perfectly conducting surface,” Opt. Commun. 119, 485–490 (1995).
[CrossRef]

J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[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]

1994 (1)

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

1993 (1)

1992 (1)

1991 (1)

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

1987 (1)

1985 (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

1983 (2)

O. M. Mendez, M. Cadilhac, R. Petit, “Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen,” J. Opt. Soc. Am. 73, 328–331 (1983).
[CrossRef]

J. Y. Surrateau, M. Cadillac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

1982 (1)

P. Sheng, R. S. Stepleman, 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 (2)

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

1972 (1)

D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

Adams, J. L.

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

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

Andrewartha, J. R.

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

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

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
[CrossRef]

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Cadilhac, M.

Cadillac, M.

J. Y. Surrateau, M. Cadillac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

Craig, M. S.

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

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

Derrick, G. H.

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
[CrossRef]

Friberg, A. T.

J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Huttunen, J.

J. Huttunen, J. Turunen, “Phase images of grooves in a perfectly conducting surface,” Opt. Commun. 119, 485–490 (1995).
[CrossRef]

J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Kok, Y.-L.

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), pp. 229–337.
[CrossRef]

Maystre, D.

D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

McPhedran, R. C.

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

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

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
[CrossRef]

Mendez, O. M.

Miller, J. M.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Moharam, M. G.

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Morf, R. H.

Noponen, E.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Petit, R.

J. Y. Surrateau, M. Cadillac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

O. M. Mendez, M. Cadilhac, R. Petit, “Diffraction of a two-dimensional electromagnetic beam wave by a thick slit pierced in a perfectly conducting screen,” J. Opt. Soc. Am. 73, 328–331 (1983).
[CrossRef]

D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

Roberts, A.

Sanda, P. N.

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

Sheng, P.

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

Surrateau, J. Y.

J. Y. Surrateau, M. Cadillac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

Taghizadeh, M. R.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Turunen, J.

J. Huttunen, J. Turunen, “Phase images of grooves in a perfectly conducting surface,” Opt. Commun. 119, 485–490 (1995).
[CrossRef]

J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Vasara, A.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), pp. 229–337.
[CrossRef]

Wyrowski, F.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Appl. Opt. (1)

J. Opt. (Paris) (1)

J. Y. Surrateau, M. Cadillac, R. Petit, “Sur la détermination numérique des efficacités de certains réseaux diélectriques profonds,” J. Opt. (Paris) 14, 273–288 (1983).

J. Opt. Soc. Am. (1)

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

Opt. Acta (2)

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

Opt. Commun. (3)

D. Maystre, R. Petit, “Diffraction par un réseau lamellaire infinement conducteur,” Opt. Commun. 5, 90–93 (1972).
[CrossRef]

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526–535 (1994).
[CrossRef]

J. Huttunen, J. Turunen, “Phase images of grooves in a perfectly conducting surface,” Opt. Commun. 119, 485–490 (1995).
[CrossRef]

Phys. Rev. B (1)

P. Sheng, R. S. Stepleman, 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. E (1)

J. Huttunen, A. T. Friberg, J. Turunen, “Scattering of partially coherent electromagnetic fields by microstructured media,” Phys. Rev. E 52, 3081–3092 (1995).
[CrossRef]

Proc. IEEE (1)

T. K. Gaylord, M. G. Moharam, “Analysis and applications of optical diffraction by gratings,” Proc. IEEE 73, 894–937 (1985).
[CrossRef]

Rep. Prog. Phys. (1)

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Other (3)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980), pp. 1–121.
[CrossRef]

R. C. McPhedran, G. H. Derrick, L. C. Botten, “Theory of crossed gratings,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 227–276.
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), pp. 229–337.
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Lamellar dielectric diffractive structure inside an aperture in a perfectly conducting screen, illuminated by an arbitrary two-dimensional electromagnetic wave.

Fig. 2
Fig. 2

(a) Example of the eigenvalue curves B(γ2) (TE polarization, thick solid curve) and C(γ2) (TM polarization, thin solid curve), showing the first few roots. In (b) and (c) we illustrate the closely spaced roots that appear in C(γ2).

Fig. 3
Fig. 3

Diffraction of a plane wave by a dielectric lamellar grating with (a) Q = 1, (b) Q = 2, and (c) Q = 4 periods inside the conducting aperture. Solid curves, TE polarization; dashed curves, TM polarization.

Fig. 4
Fig. 4

Radiant-intensity distribution generated by a ridge illuminated with a narrow Gaussian beam. Solid curve, TE polarization; dashed curve, TM polarization.

Tables (2)

Tables Icon

Table 1 Some Initial Eigenvalues for the Structure Considered in Fig. 2

Tables Icon

Table 2 Convergence of the Radiant Intensity J(θt) of the Transmitted Field When the Number of Waveguide Modes is Increaseda

Equations (38)

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H g I ( x , z ) = - A ( α ) exp { i [ α x + r ( α ) z ] } d α + - R ( α ) exp { i [ α x - r ( α ) z ] } d α ,
r ( α ) = { [ ( k n r ) 2 - α 2 ] 1 / 2 if α k n r i [ α 2 - ( k n r ) 2 ] 1 / 2 otherwise
H y III ( x , z ) = - T ( α ) exp { i [ α x + t ( α ) ( z - h ) ] } d α ,
t ( α ) = { [ ( k n t ) 2 - α 2 ] 1 / 2 if α k n t i [ α 2 - ( k n t ) 2 ] 1 / 2 otherwise .
{ x [ 1 n II 2 ( x ) x ] + 1 n II 2 ( x ) 2 z 2 + k 2 } H y II ( x , z ) = 0.
n II 2 ( x ) d d x [ 1 n II 2 ( x ) d d x ] X ( x ) + [ k 2 n II 2 ( x ) - γ 2 ] X ( x ) = 0 ,
d 2 d z 2 Z ( z ) + γ 2 Z ( z ) = 0.
X ( x ) = A l exp [ i β l ( x - x l ) ] + B l exp [ - i β l ( x - x l ) ] ,
β l = { [ ( k n l ) 2 - γ 2 ] 1 / 2 if γ k n l i [ γ 2 - ( k n l ) 2 ] 1 / 2 otherwise .
X ¯ l ( x ) = 1 n l 2 d d x X l ( x ) ,
X l ( x ) = X l ( x l ) cos [ β l ( x - x l ) ] + n l 2 β l X ¯ ( x l ) sin [ β l ( x - x l ) ] ,
X ¯ l ( x ) = X ¯ l ( x l ) cos [ β l ( x - x l ) ] - β l n l 2 X ( x l ) sin [ β l ( x - x l ) ] .
M l ( γ 2 ) = [ cos [ β l ( x l + 1 - x l ) ] ( n l 2 / β l ) sin [ β l ( x l + 1 - x l ) ] - ( β l / n l 2 ) sin [ β l ( x l + 1 - x l ) ] cos [ β l ( x l + 1 - x l ) ] ] ,
[ X l ( x l + 1 ) X ¯ l ( x l + 1 ) ] = M l ( γ 2 ) [ X l ( x l ) X ¯ l ( x l ) ] .
[ X l ( x l + 1 ) X ¯ l ( x l + 1 ) ] = M l ( γ 2 ) M l - 1 ( γ 2 ) M 1 ( γ 2 ) [ X 1 ( 0 ) X ¯ 1 ( 0 ) ] .
M ( γ 2 ) = l = 1 L M L - l + 1 ( γ 2 ) = [ A ( γ 2 ) B ( γ 2 ) C ( γ 2 ) D ( γ 2 ) ]
[ X L ( c ) X ¯ L ( c ) ] = M ( γ 2 ) [ X 1 ( 0 ) X ¯ 1 ( 0 ) ] ,
C ( γ 2 ) = 0
H y II ( x , z ) = m = 1 { a m exp ( i γ m z ) + b m exp [ - i γ m ( z - h ) ] } X m ( x ) ,
X l m ( x ) = X l m ( x l ) cos [ β l m ( x - x l ) ] + n l 2 β l m X ¯ l m ( x l ) sin [ β l m ( x - x l ) ] .
B ( γ 2 ) = 0
H y II ( x , 0 ) = H y II ( x , 0 )             when 0 < x < c ,
H y II ( x , h ) = H y II ( x , h )             when 0 < x < c ,
E x I ( x , 0 ) = { E x II ( x , 0 ) if 0 < x < c 0 otherwise ,
E x III ( x , h ) = { E y II ( x , h ) if 0 < x < c 0 otherwise .
m = 1 ( K m p γ m + δ p m ) a m + m = 1 ( δ m p - K m p γ m ) × exp ( i γ m h ) b m = 2 - I p * ( α ) A ( α ) d α ,
m = 1 ( δ m p - L m p γ m ) exp ( i γ m h ) a m + m = 1 ( L m p γ m + δ p m ) b m = 0 ,
K m p = n r 2 2 π - r - 1 ( α ) I m ( α ) I p * ( α ) d α ,
L m p = n t 2 2 π - t - 1 ( α ) I m ( α ) I p * ( α ) d α ,
I m ( α ) = 1 C m 0 c n II - 2 ( x ) X m ( x ) exp ( - i α x ) d x ,
C m = [ 0 c n II - 2 ( x ) X m ( x ) 2 d x ] 1 / 2 .
R ( α ) = A ( α ) - n r 2 2 π r ( α ) × m = 1 γ m [ a m - b m exp ( i γ m h ) ] I m ( α ) ,
T ( α ) = n t 2 2 π t ( α ) m = 1 γ m [ a m exp ( i γ m h ) - b m ] I m ( α ) .
K m p = 1 2 π - r ( α ) I m ( α ) I p * ( α ) d α ,
L m p = 1 2 π - t ( α ) I m ( α ) I p * ( α ) d α
J ( θ t ) = π ω t ( k n t ) 2 cos 2 θ t T ( k n t sin θ t ) 2
J ( θ r ) = π ω r ( k n r ) 2 cos 2 θ r R ( k n r sin θ r ) 2
A ( α ) = B w 2 π exp ( - 1 4 w 2 α 2 ) .

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