Abstract

A theory is developed for predicting a second-order hot-image formation in high-power laser systems. Light diffracted from a small optical scatterer interferes with an intense original wave in the nonlinear medium to produce a hologram like a Fresnel-zone plate. The theoretical model shows that the hologram produces a negative first-order diffractive wave focused to the traditional hot image and negative second-order diffraction that causes another intense image, namely, a second-order hot image. It is found by analysis that the location of the second-order hot image arises in a downstream plane with a half-distance from the medium to the scatterer. Results of the numerical calculations show that the peak intensity of the nonlinear image may reach a level high enough to damage optical components with the increase of the breakup integral (B integral), indicating that the image may also potentially damage expensive optical components in high-power laser systems.

© 2005 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science; Mill Valley, Calif., 1986), pp. 698–711.
  2. N. B. Baranova, N. E. Bykovskii, B. Ya. Zeldovich, Y.V. Senatskii, “Diffraction and self-focusing during amplification of high-power light pulses,” Sov. J. Quantum Electron. 4, 1362–1366 (1975).
    [CrossRef]
  3. J. T. Hunt, K. R. Manes, P. A. Renard, “Hot images from obscurations,” Appl. Opt, 32, 5973–5982 (1993).
    [CrossRef] [PubMed]
  4. W. H. Williams, K. R. Manes, J. T. Hunt, P. A. Renard, D. Milam, D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” 1CF Quart. Rep.6(1), 7–14, UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1995).
  5. C. C. Widmayer, D. Milam, S. P. deSzoeke, “Nonlinear formation of holographic images of obscurations in laser beams,” Appl. Opt. 36, 9342–9347 (1997).
    [CrossRef]
  6. C. C. Widmayer, M. R. Nickels, D. Milam, “Nonlinear holographic imaging of phase errors,” Appl. Opt. 37, 4801–4805 (1998).
    [CrossRef]
  7. C. C. Widmayer, L. R. Jones, D. Milam, “Measurement of the nonlinear coefficient of carbon disulfide using holographic self-focusing,” J. Nonlinear Opt. Phys. Mater. 7, 563–570 (1998).
    [CrossRef]
  8. L. P. Xie, J. L. Zhao, J. Q. Su, F. Jing, W. Y. Wang, H. S. Peng, “Theoretical analysis of the hot image effect from phase scatterer,” Acta Phys. Sin. 53, 2175–2179 (2004) (in Chinese).
  9. L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, H. S. Peng, “Nonlinear hot-image formation of an intense laser beam in media with gain and loss,” Opt. Commun. 236, 343–348 (2004).
    [CrossRef]
  10. M. W. Yu, Optical Holography and Its Applications (Beijing Institute of Technology Press, Beijing, 1996), pp. 139 (in Chinese).
  11. L. L. Zhao, Advanced Optics (National Defence Industry Press, Beijing, 2002), pp. 150–151 (in Chinese).
  12. V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” JETP Lett 3, 307–310 (1966).

2004

L. P. Xie, J. L. Zhao, J. Q. Su, F. Jing, W. Y. Wang, H. S. Peng, “Theoretical analysis of the hot image effect from phase scatterer,” Acta Phys. Sin. 53, 2175–2179 (2004) (in Chinese).

L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, H. S. Peng, “Nonlinear hot-image formation of an intense laser beam in media with gain and loss,” Opt. Commun. 236, 343–348 (2004).
[CrossRef]

1998

C. C. Widmayer, L. R. Jones, D. Milam, “Measurement of the nonlinear coefficient of carbon disulfide using holographic self-focusing,” J. Nonlinear Opt. Phys. Mater. 7, 563–570 (1998).
[CrossRef]

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

1997

1993

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

1975

N. B. Baranova, N. E. Bykovskii, B. Ya. Zeldovich, Y.V. Senatskii, “Diffraction and self-focusing during amplification of high-power light pulses,” Sov. J. Quantum Electron. 4, 1362–1366 (1975).
[CrossRef]

1966

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

Baranova, N. B.

N. B. Baranova, N. E. Bykovskii, B. Ya. Zeldovich, Y.V. Senatskii, “Diffraction and self-focusing during amplification of high-power light pulses,” Sov. J. Quantum Electron. 4, 1362–1366 (1975).
[CrossRef]

Bespalov, V. I.

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

Bykovskii, N. E.

N. B. Baranova, N. E. Bykovskii, B. Ya. Zeldovich, Y.V. Senatskii, “Diffraction and self-focusing during amplification of high-power light pulses,” Sov. J. Quantum Electron. 4, 1362–1366 (1975).
[CrossRef]

deSzoeke, S. P.

Eimerl, D.

W. H. Williams, K. R. Manes, J. T. Hunt, P. A. Renard, D. Milam, D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” 1CF Quart. Rep.6(1), 7–14, UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1995).

Hunt, J. T.

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

W. H. Williams, K. R. Manes, J. T. Hunt, P. A. Renard, D. Milam, D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” 1CF Quart. Rep.6(1), 7–14, UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1995).

Jing, F.

L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, H. S. Peng, “Nonlinear hot-image formation of an intense laser beam in media with gain and loss,” Opt. Commun. 236, 343–348 (2004).
[CrossRef]

L. P. Xie, J. L. Zhao, J. Q. Su, F. Jing, W. Y. Wang, H. S. Peng, “Theoretical analysis of the hot image effect from phase scatterer,” Acta Phys. Sin. 53, 2175–2179 (2004) (in Chinese).

Jones, L. R.

C. C. Widmayer, L. R. Jones, D. Milam, “Measurement of the nonlinear coefficient of carbon disulfide using holographic self-focusing,” J. Nonlinear Opt. Phys. Mater. 7, 563–570 (1998).
[CrossRef]

Manes, K. R.

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

W. H. Williams, K. R. Manes, J. T. Hunt, P. A. Renard, D. Milam, D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” 1CF Quart. Rep.6(1), 7–14, UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1995).

Milam, D.

C. C. Widmayer, L. R. Jones, D. Milam, “Measurement of the nonlinear coefficient of carbon disulfide using holographic self-focusing,” J. Nonlinear Opt. Phys. Mater. 7, 563–570 (1998).
[CrossRef]

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

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

W. H. Williams, K. R. Manes, J. T. Hunt, P. A. Renard, D. Milam, D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” 1CF Quart. Rep.6(1), 7–14, UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1995).

Nickels, M. R.

Peng, H. S.

L. P. Xie, J. L. Zhao, J. Q. Su, F. Jing, W. Y. Wang, H. S. Peng, “Theoretical analysis of the hot image effect from phase scatterer,” Acta Phys. Sin. 53, 2175–2179 (2004) (in Chinese).

L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, H. S. Peng, “Nonlinear hot-image formation of an intense laser beam in media with gain and loss,” Opt. Commun. 236, 343–348 (2004).
[CrossRef]

Renard, P. A.

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

W. H. Williams, K. R. Manes, J. T. Hunt, P. A. Renard, D. Milam, D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” 1CF Quart. Rep.6(1), 7–14, UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1995).

Senatskii, Y.V.

N. B. Baranova, N. E. Bykovskii, B. Ya. Zeldovich, Y.V. Senatskii, “Diffraction and self-focusing during amplification of high-power light pulses,” Sov. J. Quantum Electron. 4, 1362–1366 (1975).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science; Mill Valley, Calif., 1986), pp. 698–711.

Su, J. Q.

L. P. Xie, J. L. Zhao, J. Q. Su, F. Jing, W. Y. Wang, H. S. Peng, “Theoretical analysis of the hot image effect from phase scatterer,” Acta Phys. Sin. 53, 2175–2179 (2004) (in Chinese).

L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, H. S. Peng, “Nonlinear hot-image formation of an intense laser beam in media with gain and loss,” Opt. Commun. 236, 343–348 (2004).
[CrossRef]

Talanov, V. I.

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

Wang, W. Y.

L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, H. S. Peng, “Nonlinear hot-image formation of an intense laser beam in media with gain and loss,” Opt. Commun. 236, 343–348 (2004).
[CrossRef]

L. P. Xie, J. L. Zhao, J. Q. Su, F. Jing, W. Y. Wang, H. S. Peng, “Theoretical analysis of the hot image effect from phase scatterer,” Acta Phys. Sin. 53, 2175–2179 (2004) (in Chinese).

Widmayer, C. C.

Williams, W. H.

W. H. Williams, K. R. Manes, J. T. Hunt, P. A. Renard, D. Milam, D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” 1CF Quart. Rep.6(1), 7–14, UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1995).

Xie, L. P.

L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, H. S. Peng, “Nonlinear hot-image formation of an intense laser beam in media with gain and loss,” Opt. Commun. 236, 343–348 (2004).
[CrossRef]

L. P. Xie, J. L. Zhao, J. Q. Su, F. Jing, W. Y. Wang, H. S. Peng, “Theoretical analysis of the hot image effect from phase scatterer,” Acta Phys. Sin. 53, 2175–2179 (2004) (in Chinese).

Yu, M. W.

M. W. Yu, Optical Holography and Its Applications (Beijing Institute of Technology Press, Beijing, 1996), pp. 139 (in Chinese).

Zeldovich, B. Ya.

N. B. Baranova, N. E. Bykovskii, B. Ya. Zeldovich, Y.V. Senatskii, “Diffraction and self-focusing during amplification of high-power light pulses,” Sov. J. Quantum Electron. 4, 1362–1366 (1975).
[CrossRef]

Zhao, J. L.

L. P. Xie, J. L. Zhao, J. Q. Su, F. Jing, W. Y. Wang, H. S. Peng, “Theoretical analysis of the hot image effect from phase scatterer,” Acta Phys. Sin. 53, 2175–2179 (2004) (in Chinese).

L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, H. S. Peng, “Nonlinear hot-image formation of an intense laser beam in media with gain and loss,” Opt. Commun. 236, 343–348 (2004).
[CrossRef]

Zhao, L. L.

L. L. Zhao, Advanced Optics (National Defence Industry Press, Beijing, 2002), pp. 150–151 (in Chinese).

Acta Phys. Sin.

L. P. Xie, J. L. Zhao, J. Q. Su, F. Jing, W. Y. Wang, H. S. Peng, “Theoretical analysis of the hot image effect from phase scatterer,” Acta Phys. Sin. 53, 2175–2179 (2004) (in Chinese).

Appl. Opt

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

Appl. Opt.

J. Nonlinear Opt. Phys. Mater.

C. C. Widmayer, L. R. Jones, D. Milam, “Measurement of the nonlinear coefficient of carbon disulfide using holographic self-focusing,” J. Nonlinear Opt. Phys. Mater. 7, 563–570 (1998).
[CrossRef]

JETP Lett

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

Opt. Commun.

L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, H. S. Peng, “Nonlinear hot-image formation of an intense laser beam in media with gain and loss,” Opt. Commun. 236, 343–348 (2004).
[CrossRef]

Sov. J. Quantum Electron.

N. B. Baranova, N. E. Bykovskii, B. Ya. Zeldovich, Y.V. Senatskii, “Diffraction and self-focusing during amplification of high-power light pulses,” Sov. J. Quantum Electron. 4, 1362–1366 (1975).
[CrossRef]

Other

A. E. Siegman, Lasers (University Science; Mill Valley, Calif., 1986), pp. 698–711.

W. H. Williams, K. R. Manes, J. T. Hunt, P. A. Renard, D. Milam, D. Eimerl, “Modeling of self-focusing experiments by beam propagation codes,” 1CF Quart. Rep.6(1), 7–14, UCRL-LR-105821-96-1 (Lawrence Livermore National Laboratory, Livermore, Calif., 1995).

M. W. Yu, Optical Holography and Its Applications (Beijing Institute of Technology Press, Beijing, 1996), pp. 139 (in Chinese).

L. L. Zhao, Advanced Optics (National Defence Industry Press, Beijing, 2002), pp. 150–151 (in Chinese).

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Figures (8)

Fig. 1
Fig. 1

Hologram like a Fresnel-zone plate, induced by the sum of the wave diffracted from an optical scatterer and an intense background wave, producing a −1-order diffracted wave (conjugate wave) that develops into a traditional hot image and a −2-order diffracted wave focused to another intense image, namely, the second-order hot image.

Fig. 2
Fig. 2

Theoretical predictions and numerical results of the second-order hot-image distance d1 versus the scatterer distance d0. Parameters τ, θ, the B integral, and the input intensity are 1, π, 2, 1.13 GW/cm2, respectively.

Fig. 3
Fig. 3

On-axis peak-to-average intensity ratio in the second-order image plane plotted versus the B integral. Parameters d0, τ, and θ are 100 cm, 0, and 0 for curve (1); 100 cm, 1, and π/3 for curve (2); 100 cm, 1, and π for curve (3).

Fig. 4
Fig. 4

Plot of the on-axis peak-to-average intensity ratio in this image plane versus the phase modulation θ. Parameters d0, τ, the B integral, and the input intensity are 100 cm, 1, 0.5, and 0.28 GW/cm2 for curve (1); 100 cm, 1, 1.5, and 0.84 GW/cm2 for curve (2); 100 cm, 1, 2, 1.13 GW/cm2 for curve (3).

Fig. 5
Fig. 5

Some numerical results of the on-axis relative intensity versus the propagation distance when d0 = 100 cm, τ = 1, θ = π, B = 0.5 for curve (1); d0 = 100 cm, τ = 1, θ = π, B = 1.5 for curve (2); d0 = 100 cm, τ = 1, θ = π, B = 2 for curve (3).

Fig. 6
Fig. 6

Calculated spatial fluence profile showing a sharp image of the optical scatterer in the plane 48 cm behind the nonlinear medium. Parameters d0, τ, θ, the B integral, and the input intensity are 100 cm, 1, π, 2, and 1.13 GW/cm2.

Fig. 7
Fig. 7

Calculated spatial fluence profile in the plane 48 cm behind the nonlinear medium. Parameters d0, τ, θ; the B integral, and the input intensity are 100 cm, 1, π, 0.5, and 0.28 GW/cm2.

Fig. 8
Fig. 8

Plot of some numerical results of the axial peak-to-average intensity ratio in the image plane of a pure phase scatterer (phase shift, π). Parameters d0, τ, the B integral, and the input intensity are 100 cm, 1, 0.5, and 0.28 GW/cm2 for curve (1); 100 cm, 1, 1.5, and 0.84 GW/cm2 for curve (2); 100 cm, 1, 2, and 1.13 GW/cm2 for curve (3).

Equations (11)

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k d 0 ( x 1 2 + y 1 2 ) 1
U ( x 2 , y 2 ) = A exp ( i k d 0 ) A i λ d 0 exp ( i k d 0 ) × exp [ i k 2 d 0 ( x 2 2 + y 2 2 ) ] T ( x 2 λ d 0 , y 2 λ d 0 ) = U R + U O ,
U R = A exp ( i k d 0 ) ,
U O = A i λ d 0 exp ( i k d 0 ) exp [ i k 2 d 0 ( x 2 2 + y 2 2 ) ] × T ( x 2 λ d 0 , y 2 λ d 0 ) ,
t 0 ( x 1 , y 1 ) = 1 t ( x 1 , y 1 ) = { τ exp ( i θ ) inside the scatterer 1 outside the scatterer ,
| U O | | U R | .
1 2 Φ + 2 i k Φ z = k 2 n 2 | Φ | 2 n 0 Φ ;
Φ = ( U R + U O ) exp [ i k ( n 2 2 n 0 ) | U R + U O | 2 d ] .
Φ = ( U R + U O ) exp ( i B ) { 1 + i B ( | U O | 2 | U R | 2 + U O U R * | U R | 2 + U O * U R | U R | 2 ) 1 2 B 2 [ U O 2 ( U R * ) 2 | U R | 4 + ( U O * ) 2 U R 2 | U R | 4 + 2 | U O | 2 | U R | 2 | U R | 4 ] + 0 ( | U O U R | 2 ) } ,
U 2 ( x 3 , y 3 ) = A B 2 2 i λ d 1 exp [ i ( k d 1 + B ) ] + + × ( U O * ) 2 U R 2 | U R | 4 exp { i k 2 d 1 [ ( x 3 x 2 ) 2 + ( y 3 y 2 ) 2 ] } d x 2 d y 2 .
U 2 ( x 3 , y 3 ) = A B 2 i λ d 0 exp [ i ( k d 0 2 + B ) ] × [ t ( 2 x 3 , 2 y 3 ) t ( 2 x 3 , 2 y 3 ) ] * ,

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