Abstract

We present a theoretical investigation on the formation of hot images in an intense laser beam through cascaded Kerr medium disks, to disclose the distribution and intensity of hot images in high-power disk amplifiers. It is shown that multiple hot images from an obscuration may be formed, instead of one hot image as reported previously in the literature. This gives a clear explanation for the curious damage pattern of hot images, namely, damage sites appearing on alternating optics in periodic trains. Further analysis demonstrates that the distribution and intensity of hot images depend closely on the number of Kerr medium disks, the distance from the obscuration to the front of the first disk downstream, the space between two neighboring disks, and the thickness and B integral of each disk. Moreover, we take two cascaded Kerr medium disks for example to detail multiple hot images from an obscuration and confirm the theoretical results by numerical simulations.

© 2008 Optical Society of America

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  1. J. T. Hunt, K. R. Manes, and P. A. Renard, “Hot images from obscurations,” Appl. Opt. 32, 5973-5982 (1993).
    [CrossRef] [PubMed]
  2. W. Williams, P. A. Renard, K. R. Manes, D. Milam, J. T. Hunt, and D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” Report UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1996), pp. 1-8.
  3. C. C. Widmayer, D. Milam, and S. P. Deszoeke, “Nonlinear formation of holographic images of obscurations in laser beams,” Appl. Opt. 36, 9342-9347 (1997).
    [CrossRef]
  4. C. C. Widmayer, M. R. Nickels, and D. Milam, “Nonlinear holographic imaging of phase errors,” Appl. Opt. 37, 4801-4805 (1998).
    [CrossRef]
  5. L. P. Xie, J. L. Zhao, and F. Jing, “Second-order hot image from a scatter in high-power laser systems,” Appl. Opt. 44, 2553-2557 (2005).
    [CrossRef] [PubMed]
  6. Y. Wang, S. Wen, L. Zhang, Y. Hu, and D. Fan, “Obscuration size dependence of hot image in laser beam through a Kerr medium slab with gain and loss,” Appl. Opt. 47, 1152-1163 (2008).
    [CrossRef] [PubMed]
  7. F. Y. Génin, M. D. Feit, M. R. Kozlowski, A. M. Rubenchik, A. Salleo, and J. Yoshiyama, “Rear-surface laser damage on 355 nm silica optics owing to Fresnel diffraction on front-surface,” Appl. Opt. 39, 3654-3663 (2000).
    [CrossRef]
  8. B. R. Suydam, “Effect of refractive-index nonlinearity on the optical quality of high-power laser beams,” IEEE J. Quantum Electron. 11, 225-230 (1975).
    [CrossRef]
  9. J. B. Trenholme, “Theory of irregularity growth of laser beams,” Report UCRL-50021-75 (Lawrence Livermore National Laboratory, Livermore, Calif., 1976), pp. 237-240.
  10. J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14, 353-363 (1978).
    [CrossRef]
  11. A. J. Kemp, G. J. Valentine, and D. Burns, “Progress towards high-power, high-brightness neodymium-based thin-disk lasers,” Prog. Quantum Electron. 28, 305-344 (2004).
    [CrossRef]
  12. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 424.
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 55-58.
  14. V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307-310(1966).
  15. C. J. Elliott and B. R. Suydam, “Self-focusing phenomena in air-glass laser structures,” IEEE J. Quantum Electron. 11, 863-866 (1975).
    [CrossRef]
  16. J. Murray, R. Sacks, J. Auerbach, J. Trenholme, J. Hunt, W. Williams, and K. Manes, “Laser requirements and performance,” Report UCRL-LR-105821-97-3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1997), pp. 99-105.
  17. Y. Chen and A. W. Snyder, “Four-photon parametric mixing in optical fibers: effect of pump depletion,”Opt. Lett. 14, 87-89(1989).
    [CrossRef] [PubMed]
  18. D. U. Martin and H. C. Yuen, “Quasi-recurring energy leakage in the two-space-dimensional nonlinear Schrödinger equation,” Phys. Fluids 23, 881-883 (1980).
    [CrossRef]

2008 (1)

2005 (1)

2004 (1)

A. J. Kemp, G. J. Valentine, and D. Burns, “Progress towards high-power, high-brightness neodymium-based thin-disk lasers,” Prog. Quantum Electron. 28, 305-344 (2004).
[CrossRef]

2000 (1)

1998 (1)

1997 (1)

1993 (1)

1989 (1)

1980 (1)

D. U. Martin and H. C. Yuen, “Quasi-recurring energy leakage in the two-space-dimensional nonlinear Schrödinger equation,” Phys. Fluids 23, 881-883 (1980).
[CrossRef]

1978 (1)

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14, 353-363 (1978).
[CrossRef]

1975 (2)

B. R. Suydam, “Effect of refractive-index nonlinearity on the optical quality of high-power laser beams,” IEEE J. Quantum Electron. 11, 225-230 (1975).
[CrossRef]

C. J. Elliott and B. R. Suydam, “Self-focusing phenomena in air-glass laser structures,” IEEE J. Quantum Electron. 11, 863-866 (1975).
[CrossRef]

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307-310(1966).

Auerbach, J.

J. Murray, R. Sacks, J. Auerbach, J. Trenholme, J. Hunt, W. Williams, and K. Manes, “Laser requirements and performance,” Report UCRL-LR-105821-97-3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1997), pp. 99-105.

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307-310(1966).

Bliss, E. S.

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14, 353-363 (1978).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 424.

Burns, D.

A. J. Kemp, G. J. Valentine, and D. Burns, “Progress towards high-power, high-brightness neodymium-based thin-disk lasers,” Prog. Quantum Electron. 28, 305-344 (2004).
[CrossRef]

Chen, Y.

Deszoeke, S. P.

Eimerl, D.

W. Williams, P. A. Renard, K. R. Manes, D. Milam, J. T. Hunt, and D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” Report UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1996), pp. 1-8.

Elliott, C. J.

C. J. Elliott and B. R. Suydam, “Self-focusing phenomena in air-glass laser structures,” IEEE J. Quantum Electron. 11, 863-866 (1975).
[CrossRef]

Fan, D.

Feit, M. D.

Fleck, J. A.

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14, 353-363 (1978).
[CrossRef]

Génin, F. Y.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 55-58.

Hu, Y.

Hunt, J.

J. Murray, R. Sacks, J. Auerbach, J. Trenholme, J. Hunt, W. Williams, and K. Manes, “Laser requirements and performance,” Report UCRL-LR-105821-97-3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1997), pp. 99-105.

Hunt, J. T.

J. T. Hunt, K. R. Manes, and P. A. Renard, “Hot images from obscurations,” Appl. Opt. 32, 5973-5982 (1993).
[CrossRef] [PubMed]

W. Williams, P. A. Renard, K. R. Manes, D. Milam, J. T. Hunt, and D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” Report UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1996), pp. 1-8.

Jing, F.

Kemp, A. J.

A. J. Kemp, G. J. Valentine, and D. Burns, “Progress towards high-power, high-brightness neodymium-based thin-disk lasers,” Prog. Quantum Electron. 28, 305-344 (2004).
[CrossRef]

Kozlowski, M. R.

Manes, K.

J. Murray, R. Sacks, J. Auerbach, J. Trenholme, J. Hunt, W. Williams, and K. Manes, “Laser requirements and performance,” Report UCRL-LR-105821-97-3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1997), pp. 99-105.

Manes, K. R.

J. T. Hunt, K. R. Manes, and P. A. Renard, “Hot images from obscurations,” Appl. Opt. 32, 5973-5982 (1993).
[CrossRef] [PubMed]

W. Williams, P. A. Renard, K. R. Manes, D. Milam, J. T. Hunt, and D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” Report UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1996), pp. 1-8.

Martin, D. U.

D. U. Martin and H. C. Yuen, “Quasi-recurring energy leakage in the two-space-dimensional nonlinear Schrödinger equation,” Phys. Fluids 23, 881-883 (1980).
[CrossRef]

Milam, D.

C. C. Widmayer, M. R. Nickels, and D. Milam, “Nonlinear holographic imaging of phase errors,” Appl. Opt. 37, 4801-4805 (1998).
[CrossRef]

C. C. Widmayer, D. Milam, and S. P. Deszoeke, “Nonlinear formation of holographic images of obscurations in laser beams,” Appl. Opt. 36, 9342-9347 (1997).
[CrossRef]

W. Williams, P. A. Renard, K. R. Manes, D. Milam, J. T. Hunt, and D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” Report UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1996), pp. 1-8.

Morris, J. R.

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14, 353-363 (1978).
[CrossRef]

Murray, J.

J. Murray, R. Sacks, J. Auerbach, J. Trenholme, J. Hunt, W. Williams, and K. Manes, “Laser requirements and performance,” Report UCRL-LR-105821-97-3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1997), pp. 99-105.

Nickels, M. R.

Renard, P. A.

J. T. Hunt, K. R. Manes, and P. A. Renard, “Hot images from obscurations,” Appl. Opt. 32, 5973-5982 (1993).
[CrossRef] [PubMed]

W. Williams, P. A. Renard, K. R. Manes, D. Milam, J. T. Hunt, and D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” Report UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1996), pp. 1-8.

Rubenchik, A. M.

Sacks, R.

J. Murray, R. Sacks, J. Auerbach, J. Trenholme, J. Hunt, W. Williams, and K. Manes, “Laser requirements and performance,” Report UCRL-LR-105821-97-3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1997), pp. 99-105.

Salleo, A.

Snyder, A. W.

Suydam, B. R.

C. J. Elliott and B. R. Suydam, “Self-focusing phenomena in air-glass laser structures,” IEEE J. Quantum Electron. 11, 863-866 (1975).
[CrossRef]

B. R. Suydam, “Effect of refractive-index nonlinearity on the optical quality of high-power laser beams,” IEEE J. Quantum Electron. 11, 225-230 (1975).
[CrossRef]

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307-310(1966).

Trenholme, J.

J. Murray, R. Sacks, J. Auerbach, J. Trenholme, J. Hunt, W. Williams, and K. Manes, “Laser requirements and performance,” Report UCRL-LR-105821-97-3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1997), pp. 99-105.

Trenholme, J. B.

J. B. Trenholme, “Theory of irregularity growth of laser beams,” Report UCRL-50021-75 (Lawrence Livermore National Laboratory, Livermore, Calif., 1976), pp. 237-240.

Valentine, G. J.

A. J. Kemp, G. J. Valentine, and D. Burns, “Progress towards high-power, high-brightness neodymium-based thin-disk lasers,” Prog. Quantum Electron. 28, 305-344 (2004).
[CrossRef]

Wang, Y.

Wen, S.

Widmayer, C. C.

Williams, W.

J. Murray, R. Sacks, J. Auerbach, J. Trenholme, J. Hunt, W. Williams, and K. Manes, “Laser requirements and performance,” Report UCRL-LR-105821-97-3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1997), pp. 99-105.

W. Williams, P. A. Renard, K. R. Manes, D. Milam, J. T. Hunt, and D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” Report UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1996), pp. 1-8.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 424.

Xie, L. P.

Yoshiyama, J.

Yuen, H. C.

D. U. Martin and H. C. Yuen, “Quasi-recurring energy leakage in the two-space-dimensional nonlinear Schrödinger equation,” Phys. Fluids 23, 881-883 (1980).
[CrossRef]

Zhang, L.

Zhao, J. L.

Appl. Opt. (6)

IEEE J. Quantum Electron. (3)

J. A. Fleck, J. R. Morris, and E. S. Bliss, “Small-scale self-focusing effects in a high power glass laser amplifier,” IEEE J. Quantum Electron. 14, 353-363 (1978).
[CrossRef]

B. R. Suydam, “Effect of refractive-index nonlinearity on the optical quality of high-power laser beams,” IEEE J. Quantum Electron. 11, 225-230 (1975).
[CrossRef]

C. J. Elliott and B. R. Suydam, “Self-focusing phenomena in air-glass laser structures,” IEEE J. Quantum Electron. 11, 863-866 (1975).
[CrossRef]

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett. 3, 307-310(1966).

Opt. Lett. (1)

Phys. Fluids (1)

D. U. Martin and H. C. Yuen, “Quasi-recurring energy leakage in the two-space-dimensional nonlinear Schrödinger equation,” Phys. Fluids 23, 881-883 (1980).
[CrossRef]

Prog. Quantum Electron. (1)

A. J. Kemp, G. J. Valentine, and D. Burns, “Progress towards high-power, high-brightness neodymium-based thin-disk lasers,” Prog. Quantum Electron. 28, 305-344 (2004).
[CrossRef]

Other (5)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), pp. 424.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), pp. 55-58.

J. Murray, R. Sacks, J. Auerbach, J. Trenholme, J. Hunt, W. Williams, and K. Manes, “Laser requirements and performance,” Report UCRL-LR-105821-97-3 (Lawrence Livermore National Laboratory, Livermore, Calif., 1997), pp. 99-105.

W. Williams, P. A. Renard, K. R. Manes, D. Milam, J. T. Hunt, and D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” Report UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1996), pp. 1-8.

J. B. Trenholme, “Theory of irregularity growth of laser beams,” Report UCRL-50021-75 (Lawrence Livermore National Laboratory, Livermore, Calif., 1976), pp. 237-240.

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Figures (10)

Fig. 1
Fig. 1

Schematic diagram of formation of multiple hot images from an obscuration in an intense laser beam through N cascaded Kerr medium disks. (a) The distance from the obscuration to the first disk, L o , is longer than the space between two neighboring disks, L. (b) The distance from the obscuration to the first disk, L o , is shorter than the space between two neighboring disks, L.

Fig. 2
Fig. 2

On-axis peak-to-mean intensity of the total light field along the propagation axis (a) for the obscuration of pure amplitude modulation, τ = 0 and (b) for the obscuration of pure phase modulation, τ = 1 , θ = π / 4 .

Fig. 3
Fig. 3

Distance of hot images from the rear of the second disk versus the distance from the obscuration to the front of the first disk for four different spaces between the two disks: (a) for the first hot image and (b) for the second hot image. Lines denote the analytical results and symbols denote the simulative results. They are the same in the other figures, so we will not describe them there.

Fig. 4
Fig. 4

Variation of the on-axis peak-to-mean intensity of hot images with B integral imposed by each of the disks: (a) for the obscuration of pure amplitude modulation, τ = 0 , and (b) for the obscuration of pure phase modulation, τ = 1 , θ = π / 4 .

Fig. 5
Fig. 5

Variation of the on-axis peak-to-mean intensity of hot images with the space between the two disks for two different distances from the obscuration to the front of the first disk: (a) for the first hot image and (b) for the second hot image. The other parameters are identical to these in Figs. 2, 3.

Fig. 6
Fig. 6

On-axis peak-to-mean intensity of the total light field along the propagation axis (a) for the obscuration of pure amplitude modulation, τ = 0 and (b) for the obscuration of pure phase modulation, τ = 1 , θ = π / 4 .

Fig. 7
Fig. 7

Distance of hot images from the rear of the second disk versus the distance from the obscuration to the front of the first disk for four different spaces between the two disks: (a) for the first hot image, (b) for the second hot image, and (c) for the third hot image.

Fig. 8
Fig. 8

Variation of the on-axis peak-to-mean intensity of hot images with B integral imposed by each of the disks: (a) for the obscuration of pure amplitude modulation, τ = 0 and (b) for the obscuration of pure phase modulation, τ = 1 , θ = π / 4 .

Fig. 9
Fig. 9

Variation of the on-axis peak-to-mean intensity of hot images with the space between the two disks: (a) for the first hot image, (b) for the second hot image, and (c) for the third hot image.

Fig. 10
Fig. 10

On-axes peak-to-mean intensity of the total light field along the propagation axis for N = 3 : (a) for the obscuration of pure amplitude modulation, τ = 0 , and (b) for the obscuration of pure phase modulation, τ = 1 , θ = π / 4 .

Equations (51)

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T 0 ( x , y ) = { τ e i θ  inside the obscuration 1 outside the obscuration ,
T ( x , y ) = 1 T 0 ( x , y ) = { ( 1 τ e i θ ) inside the obscuration 0 outside the obscuration .
E ( x , y , 0 , t ) = A 0 ( 0 ) T 0 ( x , y ) exp [ i ( k 0 z ω 0 t ) ] = A 0 ( 0 ) [ 1 T ( x , y ) ] exp [ i ( k 0 z ω 0 t ) ] ,
G ( q x , q y , 0 ) = I [ A 2 ( 0 ) ] = u ˜ ( q x , q y , 0 ) + i v ˜ ( q x , q y , 0 ) ,
G ( q x , q y , L ) = G ( q x , q y , 0 ) exp ( i q 2 L / 2 k 0 ) = [ u ˜ ( q x , q y , 0 ) + i v ˜ ( q x , q y , 0 ) ] exp ( i q 2 L / 2 k 0 ) = u ˜ ( q x , q y , L ) + i v ˜ ( q x , q y , L ) ,
[ u ˜ ( q x , q y , L ) v ˜ ( q x , q y , L ) ] = [ cos Ω L sin Ω L sin Ω L cos Ω L ] [ u ˜ ( q x , q y , 0 ) v ˜ ( q x , q y , 0 ) ] = M [ u ˜ ( q x , q y , 0 ) v ˜ ( q x , q y , 0 ) ] ,
A z = i 2 k 2 A + i k 2 n 2 n | A | 2 A ,
[ u ˜ ( q x , q y , d + L ) v ˜ ( q x , q y , d + L ) ] = M [ u ˜ ( q x , q y , L ) v ˜ ( q x , q y , L ) ] ,
M = [ cosh ( g d ) Ω g sinh ( g d ) g Ω sinh ( g d ) cosh ( g d ) ] .
g = Ω ( q c 2 / 2 k Ω ) ,
[ u ˜ ( q x , q y , z ) v ˜ ( q x , q y , z ) ] = M L I M d ( M L M d ) N 1 M L O [ u ˜ ( q x , q y , 0 ) v ˜ ( q x , q y , 0 ) ] ,
M " = M d ( M L M d ) N 1 = 1 sin φ [ cosh ( g d ) Ω g sinh ( g d ) g Ω sinh ( g d ) cosh ( g d ) ] [ M 11 M 12 M 21 M 22 ] ,
M 11 = [ cos ( Ω L ) cosh ( g d ) + sin ( Ω L ) g Ω sinh ( g d ) ] sin ( N 1 ) φ sin ( N 2 ) φ ,
M 12 = [ cos ( Ω L ) Ω g sinh ( g d ) + sin ( Ω L ) cosh ( g d ) ] sin ( N 1 ) φ ,
M 21 = [ sin ( Ω L ) cosh ( g d ) + cos ( Ω L ) g Ω sinh ( g d ) ] sin ( N 1 ) φ ,
M 22 = [ sin ( Ω L ) Ω g sinh ( g d ) + cos ( Ω L ) cosh ( g d ) ] sin ( N 1 ) φ sin ( N - 2 ) φ ,
cos φ = cos ( Ω L ) cosh ( g d ) + ( q c 2 / 4 k Ω ) sin ( Ω L ) sinh ( g d ) / g .
M L I M M L O = ( M W ) M L I + L O + W M L I L O ,
W = [ W 11 W 12 W 21 W 22 ] = B sin ( N φ ) sin φ sinh ( g d ) g d [ 0 1 1 0 ] ,
M = [ M 11 M 12 M 21 M 22 ] ,
M 11 = 1 sin φ [ cosh ( g d ) sin N φ cos ( Ω L ) sin ( N 1 ) φ ] ,
M 12 = 1 sin φ [ Ω g sinh ( g d ) sin N φ + sin ( Ω L ) sin ( N 1 ) φ ] ,
M 21 = 1 sin φ [ g Ω sinh ( g d ) sin N φ sin ( Ω L ) sin ( N 1 ) φ ] ,
M 22 = 1 sin φ [ cosh ( g d ) sin N φ cos ( Ω L ) sin ( N 1 ) φ ] .
M W = [ M 11 W 11 M 12 W 12 M 21 W 21 M 22 W 22 ] .
[ u ˜ ( q x , q y , z ) v ˜ ( q x , q y , z ) ] = [ ( M W ) M L I + L O + W M L I L O ] [ u ˜ ( q x , q y , 0 ) v ˜ ( q x , q y , 0 ) ] .
u ˜ ( q x , q y , z ) + i v ˜ ( q x , q y , z ) = { [ cosh ( g d ) + i ( q c 2 / 4 k Ω ) sinh ( g d ) g ] sin N φ sin φ exp ( i Ω L ) sin ( N 1 ) φ sin φ } × G ( q x , q y , 0 ) exp [ i Ω ( L I + L O ) ] + i B sinh ( g d ) g d sin ( N φ ) sin φ G * ( q x , q y , 0 ) exp [ i Ω ( L O L I ) ] .
u ˜ ( q x , q y , z ) + i v ˜ ( q x , q y , z ) = G ( q x , q y , 0 ) sin φ { ( 1 + i B ) sin N φ exp ( i Ω L ) sin ( N 1 ) φ } exp [ i Ω ( L I + L O ) ] + i B sin ( N φ ) sin φ G * ( q x , q y , 0 ) exp [ i Ω ( L O L I ) ] .
cos φ = cosh 2 ( g d ) + [ ( q c 2 / 4 k Ω ) sinh ( g d ) / g ] 2 cos ( Ω L ϕ ) ,
exp ( i φ ) + exp ( i φ ) = cosh 2 ( g d ) + [ ( q c 2 / 4 k Ω ) sinh ( g d ) / g ] 2 { exp [ i ( Ω L φ ) ] + exp [ i ( Ω L φ ) ] } .
sin ( N φ ) sin φ = exp [ i ( N 1 ) φ ] + exp [ i ( N 3 ) φ ] + + exp [ i ( N 3 ) φ ] + exp [ i ( N 1 ) φ ] .
exp [ i ( N 1 ) φ ] + exp [ i ( N 1 ) φ ] = { cosh 2 ( g d ) + [ ( q c 2 / 4 k Ω ) sinh ( g d ) / g ] 2 } N / 2 × { exp [ i ( N 1 ) ψ ] + C N 1 1 exp [ i ( N 3 ) ψ ] + C N 1 2 exp [ i ( N 5 ) ψ ] + + C N 1 N 3 exp [ i ( N 5 ) ψ ] + C N 1 N 2 exp [ i ( N 3 ) ψ ] + exp [ i ( N 1 ) ψ ] } { C N 1 1 exp [ i ( N 3 ) φ ] + C N 1 2 exp [ i ( N 5 ) φ ] + + C N N 3 exp [ i ( N 5 ) φ ] + C N 1 N 2 exp [ i ( N 3 ) φ ] } .
exp [ i ( N 1 ) φ ] + exp [ i ( N 3 ) φ ] + + exp [ i ( N 3 ) φ ] + exp [ i ( N 1 ) φ ] = { exp [ i ( N 1 ) ψ ] + a 1 exp [ i ( N 3 ) ψ ] + a 2 exp [ i ( N 5 ) ψ ] + + a 2 exp [ i ( N 5 ) ψ ] + a 1 exp [ i ( N 3 ) ψ ] + exp [ i ( N 1 ) ψ ] } .
i B sinh ( g d ) g d sin ( N φ ) sin φ G * ( q x , q y , 0 ) exp [ i Ω ( L O L I ) ] = i B sinh ( g d ) g d G * ( q x , q y , 0 ) exp [ i Ω ( L O L I ) ] × { exp [ i ( N 1 ) ψ ] + a 1 exp [ i ( N 3 ) ψ ] + a 2 exp [ i ( N 5 ) ψ ] + + a 2 exp [ i ( N 5 ) ψ ] + a 1 exp [ i ( N 3 ) ψ ] + exp [ i ( N 1 ) ψ ] } .
L I = L o + m L ( m = N 1 , N 3 , , 1 or 0 ) ,
L I = L o m L ( m = N 1 , N 3 , , 2 or 1 ) ,
[ cosh ( g d ) + i ( q c 2 4 k Ω ) sinh ( g d ) g ] sin N φ sin φ exp ( i Ω L ) sin ( N 1 ) φ sin φ = c 0 exp [ i ( N 1 ) ψ ] + c 1 exp [ i ( N 3 ) ψ ] + c 2 exp [ i ( N 5 ) ψ ] + + c N 3 exp [ i ( N 5 ) ψ ] + c N 2 exp [ i ( N 3 ) ψ ] + c N 1 exp [ i ( N 1 ) ψ ]
L I = m L L O ( m = N 1 , N 3 , ... , 2 or 1 ) .
u ˜ ( q x , q y , z ) + i v ˜ ( q x , q y , z ) = G ( q x , q y , 0 ) { { [ cosh 2 ( g d ) + ( q c 2 4 k Ω ) 2 sinh 2 ( g d ) g 2 ] 1 } exp [ i Ω ( L I + L O L ) ] + [ cosh ( g d ) + i ( q c 2 4 k Ω ) sinh ( g d ) g ] 2 exp [ i Ω ( L I + L O + L ) ] } + i B sinh ( g d ) g d G * ( q x , q y , 0 ) { [ cosh ( g d ) i ( q c 2 4 k Ω ) sinh ( g d ) g ] exp [ i Ω ( L O L I + L ) ] + [ cosh ( g d ) + i ( q c 2 4 k Ω ) sinh ( g d ) g ] exp [ i Ω ( L O L I L ) ] } ,
u ˜ ( q x , q y , z ) + i v ˜ ( q x , q y , z ) = G ( q x , q y , 0 ) { B 2 exp [ i Ω ( L I + L O L ) ] + ( 1 + i B ) 2 exp [ i Ω ( L I + L O + L ) ] } + i B G * ( q x , q y , 0 ) { ( 1 i B ) exp [ i Ω ( L O L I + L ) ] + ( 1 + i B ) exp [ i Ω ( L O L I L ) ] } .
I = | A 0 + I 1 { i B sinh ( g d ) g d [ cosh ( g d ) i ( q c 2 / 4 k Ω ) sinh ( g d ) / g ] G * ( q x , q y , 0 ) } | 2 ,
I = A 0 2 [ 1 2 B 2 + 2 B 2 τ cos θ + 2 B τ sin θ + ( B 4 + B 2 ) ( 1 + τ 2 2 τ cos θ ) ] .
I = | A 0 + I 1 { i B sinh ( g d ) g d [ cosh ( g d ) + i ( q c 2 / 4 k Ω ) sinh ( g d ) / g ] G * ( q x , q y , 0 ) } | 2 ,
I = A 0 2 [ 1 + 2 B 2 2 B 2 τ cos θ + 2 B τ sin θ + ( B 4 + B 2 ) ( 1 + τ 2 2 τ cos θ ) ] .
I = | A 0 + I 1 { { cosh 2 ( g d ) + [ ( q c 2 / 4 k Ω ) sinh ( g d ) / g ] 2 1 } G ( q x , q y , 0 ) } | 2 ,
I = A 0 2 [ ( 1 B 2 ) 2 + 2 B 2 ( 1 B 2 ) τ cos θ + B 4 τ 2 ] .
T 0 ( x , y ) = { τ e i θ x | a / 2 1 otherwise .
T ( x , y ) = 1 T 0 ( x , y ) = ( 1 τ e i θ ) rect ( x / a ) ,
G ( q x , q y , 0 ) = I [ A 0 ( τ e i θ 1 ) rect ( x / a ) ] = A 0 ( τ e i θ 1 ) 2 π a sin ( a q x / 2 ) a q x / 2 δ ( q y ) ,
A 2 ( z ) = I 1 [ u ˜ ( q x , q y , z ) + i v ˜ ( q x , q y , z ) ] .
I ( x , y , z ) = | A 0 + A 2 ( z ) | 2 .

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