Abstract

We present a study of the formation of a hot image in an intense laser beam through a slab of Kerr medium with gain and loss, beyond the thin-medium approximation, to especially disclose the dependence of the hot image on the size of obscuration. Based on the angular spectrum description of light propagation and the mean-field approximation we obtain the expression for intensity of the hot image, which clearly shows the dependence of intensity of the hot image on the size of obscuration. It is shown that, as the size of obscuration increases, the intensity of the corresponding hot image first increases gradually, after reaching a maximum value, it decreases rapidly to a minimum value, meaning that there exists an optimum size of obscuration, which leads to the most intense hot image. Further analysis demonstrates that the optimum size of obscuration is approximately determined by the effective fastest growing spatial frequency for a given case. For the output light beam of a given intensity, with the gain coefficient of the Kerr medium slab increasing, or the loss coefficient decreasing, the optimum size of obscuration becomes bigger, while the hot image from the obscuration of a given size becomes weaker, suggesting that high gain and low loss can efficiently suppress the hot image from obscurations. The theoretical predictions are confirmed by numerical simulations.

© 2008 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2007 (2)

Y. W. Wang, Y. H. Hu, S. C. Wen, K. M. You, and X. Q. Fu, “Relationship between nonlinear hot image and dimensions of obscuration in high power lasers,” Acta Opt. Sin. 27, 1836-1841 (2007). (in Chinese).

T. Peng, J. Zhao, L. Xie, Z. Ye, H. W. J. Su, and J. Zhao, “Simulation analysis of the restraining effect of a spatial filter on a hot image,” Appl. Opt. 46, 3205-3209 (2007).
[CrossRef] [PubMed]

2005 (1)

2004 (4)

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. N. Honig, “The national ignition facility wavefront requirements and optical architecture,” Proc. SPIE 5341, 25-42 (2004).
[CrossRef]

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

B. Xu and M. Brandt-Pearce, “Analysis of noise amplification by a CW pump signal due to fiber nonlinearity,” IEEE Photon. Technol. Lett. 16, 1062-1064 (2004).
[CrossRef]

2000 (1)

S. C. Wen, and D. Y. Fan, “Theory of small-scale self-focusing of intense laser beams in media with gain and loss,” Acta Phys. Sin. 49, 1282-1286 (2000) (in Chinese).

1998 (1)

1997 (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]

R. Q. Hui, M. O'Sullivan, A. Robinson, and M. Taylor, “Modulation instability and its impact in multi-span optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071-1082 (1997).
[CrossRef]

R. Hui, D. Chowdhury, M. Newhouse, M. O'Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified system,” IEEE Photon. Technol. Lett. 9, 392-394 (1997).
[CrossRef]

1995 (1)

1993 (1)

1992 (1)

G. P. Agrawal, “Modulation instability in Erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 4, 562-564 (1992).
[CrossRef]

1989 (1)

1984 (1)

Acta Opt. Sin. (1)

Y. W. Wang, Y. H. Hu, S. C. Wen, K. M. You, and X. Q. Fu, “Relationship between nonlinear hot image and dimensions of obscuration in high power lasers,” Acta Opt. Sin. 27, 1836-1841 (2007). (in Chinese).

Acta Phys. Sin. (2)

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

S. C. Wen, and D. Y. Fan, “Theory of small-scale self-focusing of intense laser beams in media with gain and loss,” Acta Phys. Sin. 49, 1282-1286 (2000) (in Chinese).

Appl. Opt. (5)

IEEE Photon. Technol. Lett. (3)

G. P. Agrawal, “Modulation instability in Erbium-doped fiber amplifiers,” IEEE Photon. Technol. Lett. 4, 562-564 (1992).
[CrossRef]

R. Hui, D. Chowdhury, M. Newhouse, M. O'Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified system,” IEEE Photon. Technol. Lett. 9, 392-394 (1997).
[CrossRef]

B. Xu and M. Brandt-Pearce, “Analysis of noise amplification by a CW pump signal due to fiber nonlinearity,” IEEE Photon. Technol. Lett. 16, 1062-1064 (2004).
[CrossRef]

J. Lightwave Technol. (1)

R. Q. Hui, M. O'Sullivan, A. Robinson, and M. Taylor, “Modulation instability and its impact in multi-span optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071-1082 (1997).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

L. P. Xie, F. Jing, J. L. Zhao, J. Q. Su, W. Y. Wang, and 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]

Opt. Lett. (2)

Proc. SPIE (1)

M. L. Spaeth, K. R. Manes, C. C. Widmayer, W. H. Williams, P. K. Whitman, M. A. Henesian, I. F. Stowers, and J. N. Honig, “The national ignition facility wavefront requirements and optical architecture,” Proc. SPIE 5341, 25-42 (2004).
[CrossRef]

Other (4)

W. Williams, P. A. Renard, K. R. Manes, D. Milam, J. T. Hunt, and D. EimerlModeling of Self-Focusing Experiments by Beam Propagation Codes, Report No. UCRL-LR-105821-96-1, (Lawrence Livermore National Laboratory, Livermore, Calif. 1996), pp. 1-8.

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

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

A. E. Siegman, Lasers (University Science, 1986), p. 735.

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Figures (7)

Fig. 1
Fig. 1

Schematic of nonlinear hot-image formation through the Kerr media with gain and loss.

Fig. 2
Fig. 2

Variation of the on-axis intensity of hot images with the width of obscurations, (a)  τ = 0 , of the pure amplitude modulation, and (b)  τ = 1 , θ = π , of the pure phase modulation, for four different gain and loss coefficients, the intensity of the input light beam is 1 GW / cm 2 , the thickness of the nonlinear medium d is 25 cm . Correspondingly, intensities of the output light beam for four sets of gain and loss coefficients from 0.02 / cm to 0.08 / cm are 1.649, 2.718, 4.482, and 7.389 GW / cm 2 , respectively. The corresponding B integrals are 0.529, 0.701, 0.947, and 1.303 rad , respectively. Curves denote analytical results and symbols denote simulative results.

Fig. 3
Fig. 3

Calculated variation of the on-axis intensity of the light field along the propagation axis for the nonlinear medium of thickness 25 cm , the gain and loss coefficient of the Kerr medium slab satisfying β α = 0.04 / cm . Correspondingly, the intensity of the output background field is 2.718 GW / cm 2 , and the corresponding B integral is 0.701 rad .

Fig. 4
Fig. 4

Variation of the on-axis intensity of hot images with the width of obscurations, (a)  τ = 0 , of the pure amplitude modulation, and (b)  τ = 1 , θ = π , of the pure phase modulation, for four slabs of Kerr medium of different thickness, the intensity of the input light beam is 1 GW / cm 2 , the gain and loss coefficient of the Kerr medium slab satisfying β α = 0.04 / cm . Correspondingly, intensities of the output background field for four varieties of thickness of the medium, d, from 20 to 35 cm are 2.26, 2.718, 3.320, and 4.055 GW / cm 2 , respectively. The corresponding B integrals are 0.5, 0.701, 0.947, and 1.247 rad , respectively. Curves denote analytical results and symbols denote simulative results.

Fig. 5
Fig. 5

Variation of the on-axis intensity of hot images with the width of obscurations, (a)  τ = 0 , of the pure amplitude modulation, and (b)  τ = 1 , θ = π , of the pure phase modulation, for four different gain and loss coefficients, the intensity of the output light beam is 3 GW / cm 2 , the thickness of the nonlinear medium d is 25 cm Correspondingly, intensities of the input background field for four sets of gain and loss coefficients from 0.02 / cm to 0.08 / cm are 1.820, 1.104, 0.669, and 0.406 GW / cm 2 , respectively. The corresponding B integrals are 0.963, 0.774, 0.634, and 0.529 rad , respectively. Curves denote analytical results and symbols denote simulative results.

Fig. 6
Fig. 6

Variation of the on-axis intensity of hot images with the width of obscurations, (a)  τ = 0 , of the pure amplitude modulation, and (b)  τ = 1 , θ = π , of the pure phase modulation, for four slabs of Kerr medium of different thickness, the intensity of the output light beam is 3 GW / cm 2 , the gain and loss coefficient of the Kerr medium slab satisfying β α = 0.04 / cm . Correspondingly, intensities of the input light beam for four varieties of thickness of the medium, d, from 20 to 35 cm are 1.348, 1.104, 0.904, and 0.740 GW / cm 2 , respectively. The corresponding B integrals are 0.674, 0.774, 0.855, and 0.922 rad , respectively. Curves denote analytical results and symbols denote simulative results.

Fig. 7
Fig. 7

Variation of the on-axis intensity of hot images with the width of obscurations for three different distances from the obscuration to the nonlinear medium, (a) the case of the input light beam of a given intensity of 1 GW / cm 2 , and (b) the case of the output light beam of a given intensity of 3 GW / cm 2 . The thickness of the Kerr medium slab is 25 cm , gain and loss coefficients of the Kerr medium slab satisfying β α = 0.04 / cm , τ = 0 . (a) The intensity of the output background field is 2.718 GW / cm 2 , the corresponding B integral is 0.701 rad , and for (b) the intensity of the input background field is 1.104 GW / cm 2 , the corresponding B integral is 0.744 rad .

Tables (4)

Tables Icon

Table 1 Comparison of the Size of the Obscuration Corresponding to the Hot Image of Maximum Intensity a max with Inverse the Effective Fastest Growing Spatial Frequency π / q ¯ max , the Intensity of the Input Light Beam is 1 GW / cm 2 , the Thickness of the Nonlinear Medium d is 25 cm

Tables Icon

Table 2 Comparison of the Size of the Obscuration Corresponding to the Hot Image of Maximum Intensity a max with Inverse the Effective Fastest Growing Spatial Frequency π / q ¯ max , the Intensity of the Input Light Beam is 1 GW / cm 2 the Gain and Loss Coefficients of the Kerr Medium Slab Satisfying β α = 0.04 / cm

Tables Icon

Table 3 Comparison of the Size of the Obscuration Corresponding to the Hot Image of Maximum Intensity with Inverse the Effective Fastest Growing Spatial Frequency π / q ¯ max , the Intensity of the Output Light Beam is 3 GW / cm 2 , the Thickness of the Nonlinear Medium d is 25 cm

Tables Icon

Table 4 Comparison of the Size of the Obscuration Corresponding to the Hot Image of Maximum Intensity with Inverse the Effective Fastest Growing Spatial Frequency π / q ¯ max , the Intensity of the Output Light Beam is 3 GW / cm 2 , the Gain and Loss Coefficient of the Kerr Medium Slab Satisfying β α = 0.04 / cm

Equations (22)

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T 0 ( x , y ) = { τ exp ( i θ ) inside     the     obscuration 1 outside     the     obscuration ,
T ( x , y ) = 1 T 0 ( x , y ) .
E ( x , y , 0 ) = A 0 T 0 ( x , y ) exp ( i k z ) = A 0 [ 1 T ( x , y ) ] exp ( i k z ) ,
G ( q x , q y , 0 ) = I [ A 2 ( 0 ) ] = u ˜ 0 ( q x , q y ) + i v ˜ 0 ( q x , q y ) ,
G ( q x , q y , L ) = G ( q x , q y , 0 ) exp ( i q 2 L 2 k ) = [ u ˜ 0 ( q x , q y ) + i v ˜ 0 ( q x , q y ) ] exp ( i q 2 L 2 k ) = u ˜ ( q x , q y , L ) + i v ˜ ( q x , q y , L ) ,
A z = i 2 k 0 2 A + i k 0 2 n 2 n 0 | A | 2 A + ( β α 2 ) A ,
A 0 = A 0 exp [ 0 z ( β α 2 ) d z ] exp { i A 0 2 0 z k 0 n 2 2 n 0 exp [ 0 z ( β α ) d z ] d z } .
A = A 0 ( 1 + σ ) exp [ 0 z ( β α 2 ) d z ] exp { i A 0 2 0 z k 0 n 2 2 n 0 exp [ 0 z ( β α ) d z ] d z } ,
σ z = i 2 k 0 Δ 2 σ + i k 0 n 2 2 n 0 | A 0 | 2 exp [ 0 z ( β α ) d z ] ( σ + σ * ) .
u ˜ = u ( x , y , z ) exp [ i ( q x x + q y y ) ] d x d y v ˜ = v ( x , y , z ) exp [ i ( q x x + q y y ) ] d x d y ,
u ˜ z = q 2 2 k 0 v ˜ v ˜ z = q 2 2 k 0 u ˜ + k 0 n 2 n 0 | A 0 | 2 exp [ 0 z ( β α ) d z ] u ˜ .
| A | 2 ¯ = 1 d 0 d { | A 0 | 2 exp [ 0 z ( β α ) d z ] } d z ,
u ˜ z = q 2 2 k 0 v ˜ v ˜ z = q 2 2 k 0 u ˜ + k 0 n 2 n 0 | A | 2 ¯ u ˜ .
u ˜ ( q x , q y , z 2 ) = u ˜ ( q x , q y , z 1 ) cosh ( g d ) + v ˜ ( q x , q y , z 1 ) q 2 2 k 0 g sinh ( g d ) , v ˜ ( q x , q y , z 2 ) = u ˜ ( q x , q y , z 1 ) 2 k 0 g q 2 sinh ( g d ) + v ˜ ( q x , q y , z 1 ) cosh ( g d ) ,
u ˜ ( q x , q y , z 1 ) = u ˜ 0 ( q x , q y ) cos ( q 2 d 1 2 k ) + v ˜ 0 ( q x , q y ) sin ( q 2 d 1 2 k ) , v ˜ ( q x , q y , z 1 ) = u ˜ 0 ( q x , q y ) sin ( q 2 d 1 2 k ) + v ˜ 0 ( q x , q y ) cos ( q 2 d 1 2 k ) .
A 1 ( z ) = A 0 exp [ 0 d ( β α 2 ) d z ] ,
G ( q x , q y , z ) = [ u ˜ ( q x , q y , z 2 ) + i v ˜ ( q x , q y , z 2 ) ] e i ( q 2 d 2 / 2 k )
G ( q x , q y , z ) = ( u ˜ 0 + i v ˜ 0 ) [ cosh ( 2 B ¯ χ 1 χ 2 ) + i 1 2 χ 2 2 χ 1 χ 2 sinh ( 2 B ¯ χ 1 χ 2 ) ] e i 2 B ¯ ( κ + γ ) χ 2 + ( u ˜ 0 + i v ˜ 0 ) * i sinh ( 2 B ¯ χ 1 χ 2 ) 2 χ 1 χ 2 e i 2 B ¯ ( κ γ ) χ 2 ,
A 2 ( z ) = I 1 [ G ( q x , q y , z ) ] = A 2 ( z ) + A 2 ( z )
A 2 ( z ) = I 1 { ( u ˜ 0 + i v ˜ 0 ) [ cosh ( 2 B ¯ χ 1 χ 2 ) + i 1 2 χ 2 2 χ 1 χ 2 sinh ( 2 B ¯ χ 1 χ 2 ) ] e i 2 B ¯ ( κ + γ ) χ 2 } ,
A 2 ( z ) = I 1 { ( u ˜ 0 + i v ˜ 0 ) * i sinh ( 2 B ¯ χ 1 χ 2 ) 2 χ 1 χ 2 e i 2 B ¯ ( κ γ ) χ 2 } ,
I = exp [ 0 d ( β α ) d z ] | { A 0 + I 1 [ ( u ˜ 0 + i v ˜ 0 ) * i sinh ( 2 B ¯ χ 1 χ 2 ) 2 χ 1 χ 2 ] } | 2 = A 0 2 exp [ 0 d ( β α ) d z ] | { 1 + [ T * ( x , y ) ] I 1 [ i sinh ( 2 B ¯ χ 1 χ 2 ) 2 χ 1 χ 2 ] } | 2 ,

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