Abstract

A study of first-order Talbot resonators is presented. The general conditions for Talbot effect in ABCD systems are determined. These conditions are applied to the computation of the diffraction overlapping coefficients between array Gaussian emitters in a general first-order Talbot resonator. Relations on the ray-transfer matrix to generate the symmetric and the totally antisymmetric supermodes of the array are derived. These relations generalize the free-space, round-trip lengths of 1/2 and 1/4 of the Talbot distance. A new type of resonator based on a plano–convex gradient-index rod is proposed.

© 2003 Optical Society of America

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  1. S. Wang, J. Z. Wilcox, M. Jansen, J. J. Yang, “In-phase locking in diffraction-coupled phased-array diode lasers,” Appl. Phys. Lett. 48, 1770–1772 (1986).
    [CrossRef]
  2. J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
    [CrossRef]
  3. F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, “Coherent operation of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816–818 (1989).
    [CrossRef]
  4. A. M. Hornby, H. J. Baker, R. J. Morley, D. R. Hall, “Phase locking of linear arrays of CO2 waveguide lasers by the waveguide-confined Talbot effect,” Appl. Phys. Lett. 63, 2591–2593 (1993).
    [CrossRef]
  5. Y. Kono, M. Takeoka, K. Uto, A. Uchida, F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. 36, 607–614 (2000).
    [CrossRef]
  6. M. Wrage, P. Glas, D. Fischer, M. Leitner, D. V. Vysotsky, A. P. Napartovich, “Phase locking in a multicore fiber laser by means of a Talbot resonator,” Opt. Lett. 25, 1436–1438 (2000).
    [CrossRef]
  7. J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55, 334–336 (1989).
    [CrossRef]
  8. D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1165–1167 (1988).
    [CrossRef]
  9. D. Mehuys, W. Streifer, R. G. Waarts, D. F. Welch, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. 16, 823–825 (1991).
    [CrossRef] [PubMed]
  10. P. Peterson, A. Gavrielides, M. P. Sharma, “Extraction characteristics of a one-dimensional Talbot cavity with stochastic propagation phase,” Opt. Express OPEXFF 8, 670–681 (2001); http://www.osa.org/opticsexpress .
    [CrossRef] [PubMed]
  11. J. R. Leger, G. Mowry, Xu Li, “Modal properties of an external diode-laser-array cavity with diffractive mode-selecting mirrors,” Appl. Phys. Lett. 34, 4302–4310 (1995).
  12. M. T. Flores-Arias, C. Bao, M. V. Pérez, C. R. Fernández-Pousa, “Fractional Talbot effect in a Selfoc gradient-index lens,” Opt. Lett. 27, 2064–2066 (2002).
    [CrossRef]
  13. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  14. C Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin2002), Chap. 7.
  15. M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effect,” J. Mod. Opt. 43, 2139–2164 (1996).
    [CrossRef]
  16. T. Alieva, F. Agulló-López, “Imaging in first-order optical systems,” J. Opt. Soc. Am. A 13, 2375–2380 (1996).
    [CrossRef]
  17. L. M. Bernardo, “Talbot self-imaging in fractional Fourier planes of real and complex orders,” Opt. Commun. 140, 195–198 (1997).
    [CrossRef]
  18. A. W. Lohman, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
    [CrossRef]
  19. H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
    [CrossRef]

2002 (1)

2000 (2)

M. Wrage, P. Glas, D. Fischer, M. Leitner, D. V. Vysotsky, A. P. Napartovich, “Phase locking in a multicore fiber laser by means of a Talbot resonator,” Opt. Lett. 25, 1436–1438 (2000).
[CrossRef]

Y. Kono, M. Takeoka, K. Uto, A. Uchida, F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. 36, 607–614 (2000).
[CrossRef]

1997 (1)

L. M. Bernardo, “Talbot self-imaging in fractional Fourier planes of real and complex orders,” Opt. Commun. 140, 195–198 (1997).
[CrossRef]

1996 (2)

T. Alieva, F. Agulló-López, “Imaging in first-order optical systems,” J. Opt. Soc. Am. A 13, 2375–2380 (1996).
[CrossRef]

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effect,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

1995 (2)

J. R. Leger, G. Mowry, Xu Li, “Modal properties of an external diode-laser-array cavity with diffractive mode-selecting mirrors,” Appl. Phys. Lett. 34, 4302–4310 (1995).

H. M. Ozaktas, D. Mendlovic, “Fractional Fourier optics,” J. Opt. Soc. Am. A 12, 743–751 (1995).
[CrossRef]

1993 (2)

A. W. Lohman, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181–2186 (1993).
[CrossRef]

A. M. Hornby, H. J. Baker, R. J. Morley, D. R. Hall, “Phase locking of linear arrays of CO2 waveguide lasers by the waveguide-confined Talbot effect,” Appl. Phys. Lett. 63, 2591–2593 (1993).
[CrossRef]

1991 (1)

1989 (2)

F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, “Coherent operation of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816–818 (1989).
[CrossRef]

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55, 334–336 (1989).
[CrossRef]

1988 (2)

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1165–1167 (1988).
[CrossRef]

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[CrossRef]

1986 (1)

S. Wang, J. Z. Wilcox, M. Jansen, J. J. Yang, “In-phase locking in diffraction-coupled phased-array diode lasers,” Appl. Phys. Lett. 48, 1770–1772 (1986).
[CrossRef]

1970 (1)

Agulló-López, F.

Alieva, T.

Baker, H. J.

A. M. Hornby, H. J. Baker, R. J. Morley, D. R. Hall, “Phase locking of linear arrays of CO2 waveguide lasers by the waveguide-confined Talbot effect,” Appl. Phys. Lett. 63, 2591–2593 (1993).
[CrossRef]

Bao, C.

M. T. Flores-Arias, C. Bao, M. V. Pérez, C. R. Fernández-Pousa, “Fractional Talbot effect in a Selfoc gradient-index lens,” Opt. Lett. 27, 2064–2066 (2002).
[CrossRef]

C Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin2002), Chap. 7.

Bernardo, L. M.

L. M. Bernardo, “Talbot self-imaging in fractional Fourier planes of real and complex orders,” Opt. Commun. 140, 195–198 (1997).
[CrossRef]

Berry, M. V.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effect,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Collins, S. A.

D’Amato, F. X.

F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, “Coherent operation of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816–818 (1989).
[CrossRef]

Eng, L.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1165–1167 (1988).
[CrossRef]

Fernández-Pousa, C. R.

Fischer, D.

Flores-Arias, M. T.

Glas, P.

Gómez-Reino, C

C Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin2002), Chap. 7.

Hall, D. R.

A. M. Hornby, H. J. Baker, R. J. Morley, D. R. Hall, “Phase locking of linear arrays of CO2 waveguide lasers by the waveguide-confined Talbot effect,” Appl. Phys. Lett. 63, 2591–2593 (1993).
[CrossRef]

Hornby, A. M.

A. M. Hornby, H. J. Baker, R. J. Morley, D. R. Hall, “Phase locking of linear arrays of CO2 waveguide lasers by the waveguide-confined Talbot effect,” Appl. Phys. Lett. 63, 2591–2593 (1993).
[CrossRef]

Jansen, M.

S. Wang, J. Z. Wilcox, M. Jansen, J. J. Yang, “In-phase locking in diffraction-coupled phased-array diode lasers,” Appl. Phys. Lett. 48, 1770–1772 (1986).
[CrossRef]

Kannari, F.

Y. Kono, M. Takeoka, K. Uto, A. Uchida, F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. 36, 607–614 (2000).
[CrossRef]

Klein, S.

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effect,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

Kono, Y.

Y. Kono, M. Takeoka, K. Uto, A. Uchida, F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. 36, 607–614 (2000).
[CrossRef]

Leger, J. R.

J. R. Leger, G. Mowry, Xu Li, “Modal properties of an external diode-laser-array cavity with diffractive mode-selecting mirrors,” Appl. Phys. Lett. 34, 4302–4310 (1995).

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55, 334–336 (1989).
[CrossRef]

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[CrossRef]

Leitner, M.

Li, Xu

J. R. Leger, G. Mowry, Xu Li, “Modal properties of an external diode-laser-array cavity with diffractive mode-selecting mirrors,” Appl. Phys. Lett. 34, 4302–4310 (1995).

Lohman, A. W.

Marshall, W. K.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1165–1167 (1988).
[CrossRef]

Mehuys, D.

D. Mehuys, W. Streifer, R. G. Waarts, D. F. Welch, “Modal analysis of linear Talbot-cavity semiconductor lasers,” Opt. Lett. 16, 823–825 (1991).
[CrossRef] [PubMed]

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1165–1167 (1988).
[CrossRef]

Mendlovic, D.

Mitsunaga, K.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1165–1167 (1988).
[CrossRef]

Morley, R. J.

A. M. Hornby, H. J. Baker, R. J. Morley, D. R. Hall, “Phase locking of linear arrays of CO2 waveguide lasers by the waveguide-confined Talbot effect,” Appl. Phys. Lett. 63, 2591–2593 (1993).
[CrossRef]

Mowry, G.

J. R. Leger, G. Mowry, Xu Li, “Modal properties of an external diode-laser-array cavity with diffractive mode-selecting mirrors,” Appl. Phys. Lett. 34, 4302–4310 (1995).

Napartovich, A. P.

Ozaktas, H. M.

Pérez, M. V.

M. T. Flores-Arias, C. Bao, M. V. Pérez, C. R. Fernández-Pousa, “Fractional Talbot effect in a Selfoc gradient-index lens,” Opt. Lett. 27, 2064–2066 (2002).
[CrossRef]

C Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin2002), Chap. 7.

Roychoudhuri, C.

F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, “Coherent operation of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816–818 (1989).
[CrossRef]

Scott, M. L.

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[CrossRef]

Siebert, E. T.

F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, “Coherent operation of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816–818 (1989).
[CrossRef]

Streifer, W.

Takeoka, M.

Y. Kono, M. Takeoka, K. Uto, A. Uchida, F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. 36, 607–614 (2000).
[CrossRef]

Uchida, A.

Y. Kono, M. Takeoka, K. Uto, A. Uchida, F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. 36, 607–614 (2000).
[CrossRef]

Uto, K.

Y. Kono, M. Takeoka, K. Uto, A. Uchida, F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. 36, 607–614 (2000).
[CrossRef]

Veldkamp, W. B.

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[CrossRef]

Vysotsky, D. V.

Waarts, R. G.

Wang, S.

S. Wang, J. Z. Wilcox, M. Jansen, J. J. Yang, “In-phase locking in diffraction-coupled phased-array diode lasers,” Appl. Phys. Lett. 48, 1770–1772 (1986).
[CrossRef]

Welch, D. F.

Wilcox, J. Z.

S. Wang, J. Z. Wilcox, M. Jansen, J. J. Yang, “In-phase locking in diffraction-coupled phased-array diode lasers,” Appl. Phys. Lett. 48, 1770–1772 (1986).
[CrossRef]

Wrage, M.

Yang, J. J.

S. Wang, J. Z. Wilcox, M. Jansen, J. J. Yang, “In-phase locking in diffraction-coupled phased-array diode lasers,” Appl. Phys. Lett. 48, 1770–1772 (1986).
[CrossRef]

Yariv, A.

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1165–1167 (1988).
[CrossRef]

Appl. Phys. Lett. (7)

S. Wang, J. Z. Wilcox, M. Jansen, J. J. Yang, “In-phase locking in diffraction-coupled phased-array diode lasers,” Appl. Phys. Lett. 48, 1770–1772 (1986).
[CrossRef]

J. R. Leger, M. L. Scott, W. B. Veldkamp, “Coherent addition of AlGaAs lasers using microlenses and diffractive coupling,” Appl. Phys. Lett. 52, 1771–1773 (1988).
[CrossRef]

F. X. D’Amato, E. T. Siebert, C. Roychoudhuri, “Coherent operation of diode lasers using a spatial filter in a Talbot cavity,” Appl. Phys. Lett. 55, 816–818 (1989).
[CrossRef]

A. M. Hornby, H. J. Baker, R. J. Morley, D. R. Hall, “Phase locking of linear arrays of CO2 waveguide lasers by the waveguide-confined Talbot effect,” Appl. Phys. Lett. 63, 2591–2593 (1993).
[CrossRef]

J. R. Leger, “Lateral mode control of an AlGaAs laser array in a Talbot cavity,” Appl. Phys. Lett. 55, 334–336 (1989).
[CrossRef]

D. Mehuys, K. Mitsunaga, L. Eng, W. K. Marshall, A. Yariv, “Supermode control in diffraction-coupled semiconductor laser arrays,” Appl. Phys. Lett. 53, 1165–1167 (1988).
[CrossRef]

J. R. Leger, G. Mowry, Xu Li, “Modal properties of an external diode-laser-array cavity with diffractive mode-selecting mirrors,” Appl. Phys. Lett. 34, 4302–4310 (1995).

IEEE J. Quantum Electron. (1)

Y. Kono, M. Takeoka, K. Uto, A. Uchida, F. Kannari, “A coherent all-solid-state laser array using the Talbot effect in a three-mirror cavity,” IEEE J. Quantum Electron. 36, 607–614 (2000).
[CrossRef]

J. Mod. Opt. (1)

M. V. Berry, S. Klein, “Integer, fractional and fractal Talbot effect,” J. Mod. Opt. 43, 2139–2164 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

L. M. Bernardo, “Talbot self-imaging in fractional Fourier planes of real and complex orders,” Opt. Commun. 140, 195–198 (1997).
[CrossRef]

Opt. Lett. (3)

Other (2)

C Gómez-Reino, M. V. Pérez, C. Bao, Gradient-Index Optics: Fundamentals and Applications (Springer-Verlag, Berlin2002), Chap. 7.

P. Peterson, A. Gavrielides, M. P. Sharma, “Extraction characteristics of a one-dimensional Talbot cavity with stochastic propagation phase,” Opt. Express OPEXFF 8, 670–681 (2001); http://www.osa.org/opticsexpress .
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Illumination of a periodic transparency in the input plane of a first-order system.

Fig. 2
Fig. 2

Paraxial relation between plane and curved surfaces.

Fig. 3
Fig. 3

Notation for the paraxial propagation between curved surfaces.

Fig. 4
Fig. 4

Geometry of a general plano–convex Talbot resonator surrounding a first-order system.

Fig. 5
Fig. 5

Threshold gains for the symmetric (continuous line) and totally antisymmetric supermodes (dotted–dashed line) as a function of the cavity length. The arrows mark the corresponding values of B, the equivalent free-space, round-trip length.  

Equations (39)

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ψ ( x 1 ,   y 1 ;   z ) = k   exp ( ikz ) 2 π i ( B x B y ) 1 / 2   d x 0 d y 0   × exp ik 2 B x   ( A x x 0 2 - 2 x 0 x 1 + C x x 1 2 ) × exp ik 2 B y   ( A y y 0 2 - 2 y 0 y 1 + C y y 1 2 ) × ψ ( x 0 ,   y 0 ;   0 ) ,
K ( x 0 ,   x 1 ;   z ) = exp ( ikz ) k 2 π iB 1 / 2   × exp ik 2 B   ( Ax 0 2 - 2 x 0 x 1 + Cx 1 2 )
ψ ( x 0 ,   0 ) = t ( x 0 ) ϕ g ( x 0 ) ,
q ( 0 ) q ˙ ( 0 ) = 1 z ¯ 0 1   q 0 q ˙ 0 .
ϕ g ( x 0 ,   z = 0 ) = q 0 q ( 0 ) 1 / 2   exp ikx 0 2 2   U ( 0 ) ,
U ( 0 ) = q ˙ ( 0 ) q ( 0 ) = 1 R ( 0 ) + i   λ π w 2 ( 0 ) ,
t ( x 0 ) = l = - + a l   exp ( - 2 π ilx 0 / p ) .
ψ ( x 1 ,   z )
= exp ( ikz ) k 2 π iB 1 / 2 d x 0
× exp ik 2 B   ( Ax 0 2 - 2 x 0 x 1 + Dx 1 2 ) ϕ g ( x 0 ) t ( x 0 ) .
ψ ( x 1 ,   z ) = exp ( ikz ) q 0 q ( 0 ) 1 / 2   exp ikx 1 2 2 U ( z ) × l a l   exp - 2 π il 2   q ( 0 ) q ( z )   B z T × exp - 2 π ilx 1 p   q ( 0 ) q ( z ) ,
U ( z ) = q ˙ ( z ) q ( z ) = 1 R ( z ) + i   λ π w 2 ( z ) ,
q ( z ) q ˙ ( z ) = A B C D   q ( 0 ) q ˙ ( 0 ) .
B   Re q ( 0 ) q ( z ) = β α   p 2 λ = β α   z T 2
B   A + B / R ( 0 ) [ A + B / R ( 0 ) ] 2 + [ B λ / π w ( 0 ) 2 ] 2 = β α   p 2 λ ,
M T = [ A + B / R ( 0 ) ] 2 + [ B λ / π w ( 0 ) 2 ] 2 A + B / R ( 0 ) .
ψ C ( x ) = exp ( - ikx 2 / 2 R ) ψ C ( x ) ,
K ( x 0 ,   x 1 ;   z ) = exp ( ikz ) k 2 π iB 1 / 2   exp - ikx 0 2 2 R 0 + ikx 1 2 2 R 1 × exp ik 2 B   ( A x 0 2 - 2 x 0 x 1 + D x 1 2 ) ,
A B C D = 1 0 1 / R 1 1   A B C D   1 0 - 1 / R 0 1 .
ψ C 0 ( x 0 ) = l = - a l   exp ( - 2 π ilx 0 / p ) .
ψ C 1 ( x 1 ) = exp ( ikz ) k 2 π iB 1 / 2   d x 0   × exp ik 2 B   ( Ax 0 2 - 2 x 0 x 1 + Dx 1 2 ) ψ C 0 ( x 0 ) ,
B A = β α   p 2 λ ,
φ gm ( x 0 ,   0 ) = q 0 q ( 0 ) 1 / 2   exp ik 2   ( x 0 - mp ) 2 U ( 0 ) ,
A B C D = A S B S C S D S   1 0 - 2 / R 1   A S B S C S D S ,
φ gm ( x 1 ) = exp ( ikd ) q 0 q ( 0 ) 1 / 2 × exp ikx 1 2 2   U × exp - ikmpx 1   q ˙ ( 0 ) q exp ikm 2 p 2 2   A   q ˙ ( 0 ) q ,
q q ˙ = A B C D   q ( 0 ) q ˙ ( 0 ) ,
R nm = φ gn * ( x ) φ gm ( x ) d x | φ gm ( x ) | 2 d x .
R nm = exp ( ikd ) 4 i k Δ 1 / 2 × exp - ikp 2 q ˙ q ˙ ( 0 ) * 2 Δ   n 2 - 2 nm A - C U + m 2 A - C U A - C U ( 0 ) * ,
R nm = exp ( ikd ) 1 + iB 2 z R - 1 / 2   exp i   π 2   z T   ( n - m ) 2 B - 2 iz R .
B = 1 N   p 2 λ = z T 2 N .
R nm = exp ( ikd ) 1 + i   z T 4 Nz R - 1 / 2   × exp π Ni   ( n - m ) 2 1 - 4 Niz R / z T ,
A S B S C S D S = cos ( gz ) 1 g   sin ( gz ) g   sin ( gz ) cos ( gz ) ,
A = D = cos ( gd ) + 1 Rg   sin ( gd ) ,
B = 1 g   sin ( gd ) + 2 Rg 2   sin 2 gd 2 ,
C = - g   sin ( gd ) + 2 R   cos 2 gd 2 .
R = 1 g   cot gd 2 ,
B = 2 g   tan gd 2 ,
d N = 2 z N = 2 g   arctan gp 2 n 0 2 N λ 0 , R N = 2 N λ 0 n 0   1 ( gp ) 2 ,
[ r   exp ( i 2 σ L ) R - 1 ] E = 0 ,

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