Abstract

Coupling losses in a waveguide-laser resonator are considered for an arbitrary waveguide mode. For a rectangular waveguide the coupling efficiency is shown to be expressed in terms of a onefold integral, which can be evaluated in a closed form for three practical cases of interest corresponding to plane-parallel, half-concentric, and semiconfocal geometries. The results are used to discuss the extent of mode discrimination as well as the choice of optimum parameters for a compact waveguide-resonator design. The case of a phase-conjugate mirror is also considered. When an effective finite mirror size is used, a phase-conjugate waveguide resonator is shown to be capable of transverse-mode discrimination while providing low coupling losses for the fundamental waveguide mode.

© 1982 Optical Society of America

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References

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  1. P. W. Smith, “A waveguide gas laser,” Appl. Phys. Lett. 19, 132–134 (1971).
    [Crossref]
  2. J. J. Degnan, “The waveguide laser: a review,” Appl. Phys. 11, 1–33 (1976).
    [Crossref]
  3. R. L. Abrams, “Waveguide gas lasers,” in Laser Handbook, M. L. Stitch, ed. (North-Holland, Amsterdam, 1979), pp. 41–88.
  4. P. W. Smith, O. R. Wood, P. J. Maloney, and C. R. Adams, “Transversely excited waveguide gas lasers,” IEEE J. Quantum Electron. QE-17, 1166–1181 (1981).
    [Crossref]
  5. R. L. Abrams, “Coupling losses in hollow waveguide laser resonators,” IEEE J. Quantum Electron. QE-8, 838–843 (1972).
    [Crossref]
  6. J. J. Degnan and D. R. Hall, “Finite-aperture waveguide-laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
    [Crossref]
  7. R. L. Abrams and A. N. Chester, “Resonator theory for hollow waveguide lasers,” Appl. Opt. 13, 2117–2125 (1974).
    [Crossref] [PubMed]
  8. D. M. Henderson, “Waveguide lasers with intracavity electrooptic modulators: misalignment losses,” Appl. Opt. 15, 1066–1070 (1976).
    [Crossref] [PubMed]
  9. S. Avrillier and J. Verdonck, “Coupling losses in laser resonators containing a hollow rectangular dielectric waveguide,” J. Appl. Phys. 48, 4937–4941 (1977).
    [Crossref]
  10. K. D. Laakmann, “Recent developments in rf excited CO2waveguide lasers,” in Digest of Conference on Laser and Electro-Optical Systems (Optical Society of America, Washington, D.C., 1980), paper TUKK3.
  11. H. Krammer, “Field configuration and propagation constants of modes in hollow rectangular dielectric waveguides,” IEEE J. Quantum Electron. QE-12, 505–507 (1976).
    [Crossref]
  12. K. D. Laakman and W. H. Steier, “Waveguides: characteristic modes of hollow rectangular dielectric waveguides,” Appl. Opt. 15, 1334–1340 (1976).
    [Crossref]
  13. I. M. Bel’dyugin and E. M. Zenskov, “Theory of resonators with wavefront reversing mirrors,” Sov. J. Quantum Electron. 9, 1198–1199 (1979); Sov. J. Quantum Electron. 10, 120 (1980).
    [Crossref]
  14. J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theorical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
    [Crossref]
  15. J. F. Lam and W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 5, 61–63 (1980).
    [Crossref] [PubMed]
  16. P. A. Bélanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt. 19, 602–609 (1980).
    [Crossref] [PubMed]
  17. J. Feinberg and R. W. Hellwarth, “Phase-conjugate mirrors with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980).
    [Crossref]
  18. A. Hardy, “Sensitivity of phase-conjugate resonators to intracavity phase perturbations,” IEEE J. Quantum Electron. QE-17, 1581–1585 (1981).
    [Crossref]
  19. G. P. Agrawal and J. L. Boulnois, “Waveguide resonators with a phase-conjugate mirror,” Opt. Lett. 7, 159–161 (1982).
    [Crossref] [PubMed]
  20. See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  21. M. Abramowitz and I. A. Setgun, Handbook of Mathematical Functions (Dover, New York, 1964).
  22. M. Lyszik, F. Herlemont, and J. Lemaire, “A waveguide CO2laser operating in a Gaussian mode; experimental analysis,” Opt. Commun. 36, 327–330 (1981).
    [Crossref]
  23. Avrillier and Verdonck9 have given Γ11≃ 4.1/N2 for N 12.7. This expression appears to be incorrect in its magnitude as well as in its functional dependence on N.
  24. R. Trebino and A. E. Siegman, “Phase conjugate reflection at arbitrary angles using TEM00pump beams,” Opt. Commun. 32, 1–4 (1980).
    [Crossref]

1982 (1)

1981 (3)

A. Hardy, “Sensitivity of phase-conjugate resonators to intracavity phase perturbations,” IEEE J. Quantum Electron. QE-17, 1581–1585 (1981).
[Crossref]

M. Lyszik, F. Herlemont, and J. Lemaire, “A waveguide CO2laser operating in a Gaussian mode; experimental analysis,” Opt. Commun. 36, 327–330 (1981).
[Crossref]

P. W. Smith, O. R. Wood, P. J. Maloney, and C. R. Adams, “Transversely excited waveguide gas lasers,” IEEE J. Quantum Electron. QE-17, 1166–1181 (1981).
[Crossref]

1980 (4)

1979 (2)

I. M. Bel’dyugin and E. M. Zenskov, “Theory of resonators with wavefront reversing mirrors,” Sov. J. Quantum Electron. 9, 1198–1199 (1979); Sov. J. Quantum Electron. 10, 120 (1980).
[Crossref]

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theorical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

1977 (1)

S. Avrillier and J. Verdonck, “Coupling losses in laser resonators containing a hollow rectangular dielectric waveguide,” J. Appl. Phys. 48, 4937–4941 (1977).
[Crossref]

1976 (4)

H. Krammer, “Field configuration and propagation constants of modes in hollow rectangular dielectric waveguides,” IEEE J. Quantum Electron. QE-12, 505–507 (1976).
[Crossref]

J. J. Degnan, “The waveguide laser: a review,” Appl. Phys. 11, 1–33 (1976).
[Crossref]

D. M. Henderson, “Waveguide lasers with intracavity electrooptic modulators: misalignment losses,” Appl. Opt. 15, 1066–1070 (1976).
[Crossref] [PubMed]

K. D. Laakman and W. H. Steier, “Waveguides: characteristic modes of hollow rectangular dielectric waveguides,” Appl. Opt. 15, 1334–1340 (1976).
[Crossref]

1974 (1)

1973 (1)

J. J. Degnan and D. R. Hall, “Finite-aperture waveguide-laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
[Crossref]

1972 (1)

R. L. Abrams, “Coupling losses in hollow waveguide laser resonators,” IEEE J. Quantum Electron. QE-8, 838–843 (1972).
[Crossref]

1971 (1)

P. W. Smith, “A waveguide gas laser,” Appl. Phys. Lett. 19, 132–134 (1971).
[Crossref]

Abramowitz, M.

M. Abramowitz and I. A. Setgun, Handbook of Mathematical Functions (Dover, New York, 1964).

Abrams, R. L.

R. L. Abrams and A. N. Chester, “Resonator theory for hollow waveguide lasers,” Appl. Opt. 13, 2117–2125 (1974).
[Crossref] [PubMed]

R. L. Abrams, “Coupling losses in hollow waveguide laser resonators,” IEEE J. Quantum Electron. QE-8, 838–843 (1972).
[Crossref]

R. L. Abrams, “Waveguide gas lasers,” in Laser Handbook, M. L. Stitch, ed. (North-Holland, Amsterdam, 1979), pp. 41–88.

Adams, C. R.

P. W. Smith, O. R. Wood, P. J. Maloney, and C. R. Adams, “Transversely excited waveguide gas lasers,” IEEE J. Quantum Electron. QE-17, 1166–1181 (1981).
[Crossref]

Agrawal, G. P.

AuYeung, J.

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theorical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Avrillier, S.

S. Avrillier and J. Verdonck, “Coupling losses in laser resonators containing a hollow rectangular dielectric waveguide,” J. Appl. Phys. 48, 4937–4941 (1977).
[Crossref]

Bel’dyugin, I. M.

I. M. Bel’dyugin and E. M. Zenskov, “Theory of resonators with wavefront reversing mirrors,” Sov. J. Quantum Electron. 9, 1198–1199 (1979); Sov. J. Quantum Electron. 10, 120 (1980).
[Crossref]

Bélanger, P. A.

Boulnois, J. L.

Brown, W. P.

Chester, A. N.

Degnan, J. J.

J. J. Degnan, “The waveguide laser: a review,” Appl. Phys. 11, 1–33 (1976).
[Crossref]

J. J. Degnan and D. R. Hall, “Finite-aperture waveguide-laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
[Crossref]

Feinberg, J.

Fekete, D.

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theorical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Hall, D. R.

J. J. Degnan and D. R. Hall, “Finite-aperture waveguide-laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
[Crossref]

Hardy, A.

A. Hardy, “Sensitivity of phase-conjugate resonators to intracavity phase perturbations,” IEEE J. Quantum Electron. QE-17, 1581–1585 (1981).
[Crossref]

P. A. Bélanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt. 19, 602–609 (1980).
[Crossref] [PubMed]

Hellwarth, R. W.

Henderson, D. M.

Herlemont, F.

M. Lyszik, F. Herlemont, and J. Lemaire, “A waveguide CO2laser operating in a Gaussian mode; experimental analysis,” Opt. Commun. 36, 327–330 (1981).
[Crossref]

Jackson, J. D.

See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

Krammer, H.

H. Krammer, “Field configuration and propagation constants of modes in hollow rectangular dielectric waveguides,” IEEE J. Quantum Electron. QE-12, 505–507 (1976).
[Crossref]

Laakman, K. D.

Laakmann, K. D.

K. D. Laakmann, “Recent developments in rf excited CO2waveguide lasers,” in Digest of Conference on Laser and Electro-Optical Systems (Optical Society of America, Washington, D.C., 1980), paper TUKK3.

Lam, J. F.

Lemaire, J.

M. Lyszik, F. Herlemont, and J. Lemaire, “A waveguide CO2laser operating in a Gaussian mode; experimental analysis,” Opt. Commun. 36, 327–330 (1981).
[Crossref]

Lyszik, M.

M. Lyszik, F. Herlemont, and J. Lemaire, “A waveguide CO2laser operating in a Gaussian mode; experimental analysis,” Opt. Commun. 36, 327–330 (1981).
[Crossref]

Maloney, P. J.

P. W. Smith, O. R. Wood, P. J. Maloney, and C. R. Adams, “Transversely excited waveguide gas lasers,” IEEE J. Quantum Electron. QE-17, 1166–1181 (1981).
[Crossref]

Pepper, D. M.

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theorical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Setgun, I. A.

M. Abramowitz and I. A. Setgun, Handbook of Mathematical Functions (Dover, New York, 1964).

Siegman, A. E.

R. Trebino and A. E. Siegman, “Phase conjugate reflection at arbitrary angles using TEM00pump beams,” Opt. Commun. 32, 1–4 (1980).
[Crossref]

P. A. Bélanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt. 19, 602–609 (1980).
[Crossref] [PubMed]

Smith, P. W.

P. W. Smith, O. R. Wood, P. J. Maloney, and C. R. Adams, “Transversely excited waveguide gas lasers,” IEEE J. Quantum Electron. QE-17, 1166–1181 (1981).
[Crossref]

P. W. Smith, “A waveguide gas laser,” Appl. Phys. Lett. 19, 132–134 (1971).
[Crossref]

Steier, W. H.

Trebino, R.

R. Trebino and A. E. Siegman, “Phase conjugate reflection at arbitrary angles using TEM00pump beams,” Opt. Commun. 32, 1–4 (1980).
[Crossref]

Verdonck, J.

S. Avrillier and J. Verdonck, “Coupling losses in laser resonators containing a hollow rectangular dielectric waveguide,” J. Appl. Phys. 48, 4937–4941 (1977).
[Crossref]

Wood, O. R.

P. W. Smith, O. R. Wood, P. J. Maloney, and C. R. Adams, “Transversely excited waveguide gas lasers,” IEEE J. Quantum Electron. QE-17, 1166–1181 (1981).
[Crossref]

Yariv, A.

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theorical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

Zenskov, E. M.

I. M. Bel’dyugin and E. M. Zenskov, “Theory of resonators with wavefront reversing mirrors,” Sov. J. Quantum Electron. 9, 1198–1199 (1979); Sov. J. Quantum Electron. 10, 120 (1980).
[Crossref]

Appl. Opt. (4)

Appl. Phys. (1)

J. J. Degnan, “The waveguide laser: a review,” Appl. Phys. 11, 1–33 (1976).
[Crossref]

Appl. Phys. Lett. (1)

P. W. Smith, “A waveguide gas laser,” Appl. Phys. Lett. 19, 132–134 (1971).
[Crossref]

IEEE J. Quantum Electron. (6)

H. Krammer, “Field configuration and propagation constants of modes in hollow rectangular dielectric waveguides,” IEEE J. Quantum Electron. QE-12, 505–507 (1976).
[Crossref]

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theorical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[Crossref]

P. W. Smith, O. R. Wood, P. J. Maloney, and C. R. Adams, “Transversely excited waveguide gas lasers,” IEEE J. Quantum Electron. QE-17, 1166–1181 (1981).
[Crossref]

R. L. Abrams, “Coupling losses in hollow waveguide laser resonators,” IEEE J. Quantum Electron. QE-8, 838–843 (1972).
[Crossref]

J. J. Degnan and D. R. Hall, “Finite-aperture waveguide-laser resonators,” IEEE J. Quantum Electron. QE-9, 901–910 (1973).
[Crossref]

A. Hardy, “Sensitivity of phase-conjugate resonators to intracavity phase perturbations,” IEEE J. Quantum Electron. QE-17, 1581–1585 (1981).
[Crossref]

J. Appl. Phys. (1)

S. Avrillier and J. Verdonck, “Coupling losses in laser resonators containing a hollow rectangular dielectric waveguide,” J. Appl. Phys. 48, 4937–4941 (1977).
[Crossref]

Opt. Commun. (2)

M. Lyszik, F. Herlemont, and J. Lemaire, “A waveguide CO2laser operating in a Gaussian mode; experimental analysis,” Opt. Commun. 36, 327–330 (1981).
[Crossref]

R. Trebino and A. E. Siegman, “Phase conjugate reflection at arbitrary angles using TEM00pump beams,” Opt. Commun. 32, 1–4 (1980).
[Crossref]

Opt. Lett. (3)

Sov. J. Quantum Electron. (1)

I. M. Bel’dyugin and E. M. Zenskov, “Theory of resonators with wavefront reversing mirrors,” Sov. J. Quantum Electron. 9, 1198–1199 (1979); Sov. J. Quantum Electron. 10, 120 (1980).
[Crossref]

Other (5)

K. D. Laakmann, “Recent developments in rf excited CO2waveguide lasers,” in Digest of Conference on Laser and Electro-Optical Systems (Optical Society of America, Washington, D.C., 1980), paper TUKK3.

R. L. Abrams, “Waveguide gas lasers,” in Laser Handbook, M. L. Stitch, ed. (North-Holland, Amsterdam, 1979), pp. 41–88.

See, for example, J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

M. Abramowitz and I. A. Setgun, Handbook of Mathematical Functions (Dover, New York, 1964).

Avrillier and Verdonck9 have given Γ11≃ 4.1/N2 for N 12.7. This expression appears to be incorrect in its magnitude as well as in its functional dependence on N.

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Figures (8)

Fig. 1
Fig. 1

Schematic of the waveguide and mirror geometry used for calculation of the coupling efficiency.

Fig. 2
Fig. 2

EH11 mode-coupling efficiency as a function of the Fresnel number N for several values of the curvature parameter β.

Fig. 3
Fig. 3

EH22 mode-coupling efficiency as a function of the Fresnel number N for several values of the curvature parameter β.

Fig. 4
Fig. 4

Coupling efficiency for various waveguide modes EHpq and a fixed number N = 0.5 as a function of the curvature parameter β.

Fig. 5
Fig. 5

Coupling efficiency for various waveguide modes EHpq and a fixed number N = 1 as a function of the curvature parameter β.

Fig. 6
Fig. 6

Coupling efficiency for various waveguide modes EHpq and a fixed number N = 2 as a function of the curvature parameter β.

Fig. 7
Fig. 7

EH11 mode-coupling efficiency of a phase-conjugate mirror (PCM) as a function of the Fresnel number N for several values of the effective PCM size parameter γ.

Fig. 8
Fig. 8

PCM coupling efficiency for various waveguide modes and a fixed effective PCM size parameter γ = 5 as a function of the Fresnel number N.

Tables (1)

Tables Icon

Table 1 Optimum Parameters β and N for which C11 is Maximuma

Equations (44)

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E p q ( x , y ) = 1 a { cos ( π p x / 2 a ) sin ( π p x / 2 a ) } { cos ( π q y / 2 a ) sin ( π q y / 2 a ) } ,
- a a - a a E p q * ( s ) E p q ( s ) d 2 s = δ p p δ q q ,
E ˜ p q ( s ) = - a a - a a E p q ( s 2 ) K ( s , s 2 ) d 2 s 2 ,
K ( s , s 2 ) = - b b - b b G ( s , s 1 ) G ( s 1 , s 2 ) d 2 s 1 ,
G ( s , s 1 ) = 1 i λ d exp [ i k ( d + s - s 1 2 2 d - β s 1 2 2 d ) ] ,
β = d / R c
χ p q p q = - a a - a a E p q * ( s ) E ˜ p q ( s ) d 2 s .
C p q = χ p q p q 2 ,
K ( s , s 2 ) = exp ( 2 i k d ) 2 i λ d ( 1 - β ) × exp [ i k 4 d ( s - s 2 2 - β 1 - β s + s 2 2 ) ] .
K ( s , s 2 ) = exp [ i k ( 2 d + s 2 / d ) ] δ ( x + x 2 ) δ ( y + y 2 ) .
K ( s , s 2 ) = exp ( 2 i k d ) i λ d exp ( - i k d s · s 2 ) .
C p q = ( C p p C q q ) 1 / 2 .
u = ( x + x 2 ) / 2 a ,             v = ( x - x 2 ) / 2 a .
C p p = ( 2 1 - β ) 1 / 2 0 1 exp [ - 2 i π N β ( 1 - β ) u 2 ] G p ( u ) d u | 4 ,
G p ( u ) = 2 N 0 1 - u exp ( 2 i π N v 2 ) × [ cos ( π p u ) - ( - 1 ) p cos ( π p v ) ] d v
N = a 2 / λ d .
F ( x ) = C ( x ) + i S ( x ) = 0 x exp ( i π 2 t 2 ) d t .
G p ( u ) = cos ( π p u ) F ( η - η u ) - ( - 1 ) p 1 2 exp ( - i π p 2 2 η 2 ) × [ F ( η + p η - η u ) + F ( η - p η - η u ) ] ,
η = 2 N = ( 4 a 2 / λ d ) 1 / 2 .
C p p = | 1 η F ( η ) - ( - 1 ) p 1 2 η exp ( - i π p 2 2 η 2 ) × [ F ( η + p / η ) + F ( η - p / η ) ] | 4             ( β = 1 ) .
C p p = | 8 N 0 1 exp ( 2 i π N u 2 ) × [ ( 1 - u ) cos ( π p u ) + sin ( π p u ) / π p ] d u | 4 .
C p p = 1 4 | F ( η + p / η ) + F ( η - p / η ) + 1 i π p ( 1 + i π p 2 η 2 + η η ) × [ F ( η + p / η ) - F ( η - p / η ) - 2 F ( p / η ) ] | 4 ( β = 0 ) .
C p p = | 2 π p N 0 1 cos ( π p ξ / 2 ) cos ( 2 π N ξ ) p 2 π 2 4 - 4 π 2 N 2 ξ 2 d ξ | 4
C p p = 1 π 4 η 4 | cos ( π 2 p 2 η 2 ) Σ p ( η ) sin ( π 2 p 2 η 2 ) Σ p ± ( η ) | 4             ( β = 0.5 ) ,
Σ p + ( η ) = 2 C i [ π 2 ( p 2 η 2 - η 2 ) ] - C i [ π 2 ( p η + η ) ] - C i [ π 2 ( p η - η ) ] ,
Σ p - ( η ) = 2 S i [ π 2 ( p 2 η 2 - η 2 ) ] - S i [ π 2 ( p η + η ) ] - S i [ π 2 ( p η - η ) ] .
C p p = ( 2 π ) 8 ( 8 1 - β ) 2 N 2 p 8 1 - ( - 1 ) p 2             ( N 1 ; β 1 ) .
C p p = 1 β 2 η 4 | F ( η ) - ( - 1 ) p 1 2 exp ( - i π 2 p 2 η 2 ) × [ F ( η + p η ) + F ( η - p η ) ] | 4             ( N 1 ; β 0 )
C p p 1 ( 2 β N ) 2             ( p - odd ; N 1 ; β 0 ) ,
C p p ( π p 2 16 ) 4 1 ( 2 β N 3 ) 2 ( p - even ; N 1 , β 0 ) .
C p p [ 1 - p 2 6 N 3 / 2 - π 240 p 4 N 5 / 2 + 1 72 p 4 N 3 + ] 2 .
Γ p q ( p 2 + q 2 ) 6 N - 3 / 2 .
R c = B ( d / B + B / d ) ,
β ( N ) = 1 1 + σ 11 2 N 2 ,
R ( s 1 ) = exp ( - s 1 2 / 2 b 2 ) ,
E ˜ p q ( s ) = - a a - a a E p q * ( s 2 ) K c ( s , s 2 ) d 2 s 2 ,
K c ( s , s 2 ) = - - R ( s 1 ) G ( s , s 1 ) G * ( s 1 , s 2 ) d 2 s 1 .
K c ( s , s 2 ) = 2 π b 2 λ 2 d 2 exp [ i k 2 d ( s 2 - s 2 2 ) - k 2 b 2 2 d 2 s - s 2 2 ] .
K c ( s , s 2 ) = δ ( x - x 2 ) δ ( y - y 2 ) .
C p p = [ γ η 2 ( π 2 ) 1 / 2 0 1 exp [ - ½ ( π γ η 2 u ) 2 ] H p ( u ) d u ] 4 ,
H p ( u ) = 2 0 1 - u [ cos ( π p u ) - ( - 1 ) p cos ( π p v ) ] × cos ( π η 2 u v ) d v
γ = b / a .
H p ( u ) = 1 π { 2 cos ( π p u ) sin [ π η 2 u ( 1 - u ) ] η 2 u - ( - 1 ) p ( sin [ π ( p + η 2 u ) ( 1 - u ) ] p + η 2 u ) + sin [ π ( p - η 2 u ) ( 1 - u ) ] p - η 2 u } .
C p p [ 1 - 1 2 γ 2 ( p 2 16 N 2 + 1 3 - 2 π 2 p 2 ) ] 4 .