Abstract

Certain damage observed on the optics in NOVA is consistent with a phenomenon akin to holographic imaging. (NOVA is the Lawrence Livermore National Laboratory’s 10-beam Nd:glass laser used for inertial confinement fusion research.) The minimization of similar damage in next-generation laser systems is sought by first identifying the sources for these holographic images, specifying glass parameters appropriately, and staging the amplifier chain to circumvent the problem. The insight gained has lead to an explanation for a 20-year-old puzzle.

© 1993 Optical Society of America

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References

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  1. V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Zh. Eksp. Teor. Fiz. Pis’ma Red 3, 471–476 (1966) [JETP Lett. 3, 307 (1966)].
  2. H. W. Smith, Principles of Holography (Wiley, New York, 1969), Chap. 5.
  3. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 735.
  4. We follow J. B. Trenholme’s perturbation analysis, outlined in “1975 Laser annual report,” Rep. UCRL-50021-75 (Lawrence Livermore National Laboratory, Livermore, Calif., 1975), pp. 237–242.
  5. Ref. 3, pp. 698–711.
  6. N. B. Baranova, N. E. Bykovskii, B. Ya. Zel’dovich, Yu. V. Senatskii, “Diffraction and self-focusing during amplification of high-power light pulses,” Sov. J. Quantum Electron. 4, 1362–1366(1975).
    [Crossref]
  7. Sylvester’s theorem. See, for example, G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).
  8. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 16.
  9. NOVA is the Lawrence Livermore National Laboratory’s 10-beam Nd:glass laser used for inertial confinement fusion research.

1975 (1)

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

1966 (1)

V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Zh. Eksp. Teor. Fiz. Pis’ma Red 3, 471–476 (1966) [JETP Lett. 3, 307 (1966)].

Baranova, N. B.

N. B. Baranova, N. E. Bykovskii, B. Ya. Zel’dovich, Yu. 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,” Zh. Eksp. Teor. Fiz. Pis’ma Red 3, 471–476 (1966) [JETP Lett. 3, 307 (1966)].

Bykovskii, N. E.

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

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 16.

Korn, G. A.

Sylvester’s theorem. See, for example, G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

Korn, T. M.

Sylvester’s theorem. See, for example, G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

Senatskii, Yu. V.

N. B. Baranova, N. E. Bykovskii, B. Ya. Zel’dovich, Yu. 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), p. 735.

Smith, H. W.

H. W. Smith, Principles of Holography (Wiley, New York, 1969), Chap. 5.

Talanov, V. I.

V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Zh. Eksp. Teor. Fiz. Pis’ma Red 3, 471–476 (1966) [JETP Lett. 3, 307 (1966)].

Zel’dovich, B. Ya.

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

Sov. J. Quantum Electron. (1)

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

Zh. Eksp. Teor. Fiz. Pis’ma Red (1)

V. I. Bespalov, V. I. Talanov, “Filamentary structure of light beams in nonlinear liquids,” Zh. Eksp. Teor. Fiz. Pis’ma Red 3, 471–476 (1966) [JETP Lett. 3, 307 (1966)].

Other (7)

H. W. Smith, Principles of Holography (Wiley, New York, 1969), Chap. 5.

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

We follow J. B. Trenholme’s perturbation analysis, outlined in “1975 Laser annual report,” Rep. UCRL-50021-75 (Lawrence Livermore National Laboratory, Livermore, Calif., 1975), pp. 237–242.

Ref. 3, pp. 698–711.

Sylvester’s theorem. See, for example, G. A. Korn, T. M. Korn, Mathematical Handbook for Scientists and Engineers (McGraw-Hill, New York, 1961).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 16.

NOVA is the Lawrence Livermore National Laboratory’s 10-beam Nd:glass laser used for inertial confinement fusion research.

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Figures (8)

Fig. 1
Fig. 1

Planes of conjugate image and scatterer.

Fig. 2
Fig. 2

Propagation of a perturbed field, left to right, first through a vacuum following an obscuration, then through some nonlinear stuff, then through another vacuum region.

Fig. 3
Fig. 3

Scattered wave power gain versus spatial frequency when B = 3.

Fig. 4
Fig. 4

Series of N equally spaced slabs.

Fig. 5
Fig. 5

Mode gain for ten equally spaced slabs, total B = 2.

Fig. 6
Fig. 6

For ten equally spaced slabs, expected peak variation of axial intensity in the conjugate plane of a Gaussian scatterer.

Fig. 7
Fig. 7

For ten equally spaced slabs, expected peak variation of axial intensity in the conjugate plane of an opaque scatterer.

Fig. 8
Fig. 8

Axial intensity near the conjugate plane of a Gaussian scatterer (w0 = 0.1 cm). The nonlinear element is a single thin slab with B = 2.

Equations (68)

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ψ 0 = - w 0 w ( z ) exp [ - r 2 / w 2 ( z ) ] exp { i [ ϑ - k r 2 / 2 R ( z ) ] } ,
w ( z ) = w 0 [ 1 + ( z z R ) 2 ] 1 / 2 ,             z R = π w 0 2 λ , ϑ = tan - 1 ( z z R ) ,             R ( z ) = z + z R 2 z .
ψ ψ * 1 - 2 w 0 w ( z ) exp [ - r 2 / w 2 ( z ) ] cos [ ϑ - k r 2 2 R ( z ) ] .
U z = i 2 k 2 U + i k μ U ,
U ψ exp ( i k μ d ) = ψ exp ( i k d ψ ψ * n 2 / n ) .
θ ( r ) = - 2 B w 0 w ( z ) exp [ - r 2 / w 2 ( z ) ] cos [ ϑ - k r 2 2 R ( z ) ] .
ψ s ( r i ) = - i λ f exp ( i k r i 2 / 2 f ) × - exp [ i θ ( r ) + i k r 2 2 f - i k f r · r i ] d r .
ψ s ( r i ) = B λ f w 0 w exp ( i k r i 2 / 2 f ) - exp ( i k r 2 2 f - r 2 w 2 ) × [ exp ( i ϑ - i k r 2 2 R ) + exp ( - i ϑ + i k r 2 2 R ) ] × exp ( - i k f r · r i ) d r .
- exp [ - ( a r 2 + b · r ) ] d r = π a exp ( b 2 / 4 a ) ,             R ( a ) > 0.
ψ s ( r i ) = B z R exp ( i k r i 2 / 2 R ) { exp [ i ϑ - ( k w r i 2 R ) 2 ] converging wave + R R - i k w 2 exp [ - i ϑ - ( k w r i 2 R ) 2 R + i k w 2 R ( R 2 + k 2 w 4 ) ] diverging wave } .
w c = R λ π w ( R ) ,
I ( 0 ) = [ 1 + B exp ( i ϑ ) ] [ 1 + B exp ( i ϑ ) ] * = 1 + B 2 + 2 B cos ϑ .
I ( 0 ) = 1 + B 2 .
I ( 0 ) = ( 1 + B ) 2 .
I I 0 ( 1 + B 2 )
E ¯ = p ^ 2 R [ U exp ( - i k z + w t ) ] ,
U z = - i 2 k 2 U - i k n 2 n U U * ,
U U exp ( - i k z C C * n 2 / n ) ,
U = C [ 1 + ψ s ( x , y , z ) ] = C [ 1 + j a j ( z ) e j ( x , y ) ] ,
C j a j z e j = - i 2 k C j a j 2 e j - i k n 2 n C C * j ( a j * + a j ) e j .
2 e j = - σ j 2 e j ,
a j z = i σ j 2 2 k a j - 2 i k C C * n 2 n R ( a j ) .
a j z = i σ j 2 2 k a j - 2 i k μ R ( a j ) .
a = [ u ( z ) v ( z ) ] = [ cos Θ S - sinh Θ S / S - S sinh Θ S cosh Θ S ] [ u ( 0 ) v ( 0 ) ] = M [ u ( 0 ) v ( 0 ) ] ,
Θ = σ 2 z 2 k = k ϕ 2 z 2 = π ϕ 2 z λ , S = ( 4 k 2 μ σ 2 - 1 ) 1 / 2 = ( 2 B Θ - 1 ) 1 / 2 .
| j a j ( z ) e j ( x , y ) | 1.
a = [ u ( z ) v ( z ) ] = [ cos Θ - sin Θ sin Θ cos Θ ] [ u ( 0 ) v ( 0 ) ] .
T ( Θ ) = [ cos Θ - sin Θ sin Θ cos Θ ] ,
T ( Θ 1 ) T ( Θ 2 ) T ( Θ 1 + Θ 2 ) T ( Θ 2 ) T ( Θ 1 ) .
M = [ m 11 m 12 m 21 m 22 ] ,
T = T ( Θ 2 ) M T ( Θ 1 ) .
[ T ( Θ ) , M ] = T ( Θ ) M - M T ( Θ ) sin Θ ( T ( π 2 ) M - M T ( π 2 ) ) = sin Θ [ T ( π 2 ) , M ] ,
sin Θ 2 T ( Θ 1 ) 1 2 T ( Θ 1 - Θ 2 ) T ( π 2 ) - 1 2 T ( Θ 1 + Θ 2 ) T ( π 2 . )
T = T ( Θ 2 ) M T ( Θ 1 ) = { M - 1 2 [ T ( π 2 ) , M ] T ( π 2 ) } T ( Θ 1 + Θ 2 ) + 1 2 [ T ( π 2 ) , M ] T ( π 2 ) T ( Θ 1 - Θ 2 ) ,
T = ( M - W ) T ( Θ 1 + Θ 2 ) + W T ( Θ 1 - Θ 2 ) ,
W = 1 2 [ T ( π 2 ) , M ] T ( π 2 ) .
M = [ cos Θ S - sinh Θ S / S - S sinh Θ S cosh Θ S ] [ 1 0 - 2 B 1 ] .
W = [ 0 - B - B 0 ] ,             M - W = [ 1 B - B 1 ] .
T = [ 1 B - B 1 ] [ cos 2 Θ 1 - sin 2 Θ 1 sin 2 Θ 1 cos 2 Θ 1 ] diverging wave + [ 0 - B - B 0 ] [ 1 0 0 1 ] converging wave .
a = W [ u ( 0 ) v ( 0 ) ] = [ 0 - B - B 0 ] [ u ( 0 ) v ( 0 ) ] = [ - B v ( 0 ) - B u ( 0 ) ] .
U ( x , y , z ) = [ 1 + j a j ( z ) e j ( x , y ) ] exp ( - i B ) = [ 1 - i B j u j ( 0 ) e j ( x , y ) ] exp ( - i B ) .
W = - 1 2 ( sinh Θ S S + S sinh Θ S ) [ 0 - 1 - 1 0 ] ,
G = 1 + B 2 sinh 2 ( 2 B Θ - Θ 2 ) 1 / 2 2 B Θ - Θ 2 .
a = W 2 [ u ( 0 ) v ( 0 ) ] = [ B 2 u ( 0 ) B 2 v ( 0 ) ] .
M = ( [ cos Θ v - sin Θ v sin Θ v cos Θ v ] [ 1 0 - 2 B / N 1 ] ) N .
M = ( [ cos Θ v - sin Θ v sin Θ v cos Θ v ] [ 1 0 - 2 B / N 1 ] ) N = 1 sin φ [ ( cos Θ v + 2 B N sin Θ v ) sin N φ - sin ( N - 1 ) φ - sin Θ v sin N φ ( sin Θ v - 2 B N cos Θ v ) sin N φ cos Θ v sin N φ - sin ( N - 1 ) φ ] ,
W = B sin N φ N sin φ [ sin Θ v - cos Θ v - cos Θ v - sin Θ v ] = [ 0 - B sin N φ N sin φ - B sin N φ N sin φ 0 ] T ( - Θ v ) .
T W T ( Θ 1 - Θ 2 ) = [ 0 - B sin N φ N sin φ - B sin N φ N sin φ 0 ] T ( - Θ v ) T ( Θ 1 - Θ 2 ) = [ 0 - B sin N φ N sin φ - B sin N φ N sin φ 0 ] T [ ( Θ 1 - Θ v ) - Θ 2 ] ,
a = W [ u ( 0 ) v ( 0 ) ] = [ 0 - B sin N φ N sin φ - B sin N φ N sin φ 0 ] [ u ( 0 ) v ( 0 ) ] = B sin N φ N sin φ [ - v ( 0 ) - u ( 0 ) ] .
U ( x , y , z ) = exp ( - i B ) [ 1 - i B N j sin N φ sin φ u j ( 0 ) e j ( x , y ) ] .
G = B sin N φ N sin φ ,
cos φ = cos Θ v + B N sin Θ v .
G = B sinh [ N ln ( u + u 2 - 1 ) ] N sinh [ ln ( u + u 2 - 1 ) ] ,
u = cos Θ v + B N sin Θ v .
G max sinh B ,
Θ v = B / N .
Θ 0 , 1 = tan - 1 B N + cos - 1 { cos ( π / N ) [ 1 + ( B / N ) 2 ] 1 / 2 } ,
Θ 0 , 1 = B + π N .
a ( z = 0 ) = 0 exp ( - r 2 / w 0 2 ) J 0 ( k ϕ r ) r d r = w 0 2 2 exp ( - k 2 ϕ 2 w 0 2 / 4 ) .
ψ s = 0 k ϕ 0 , 1 G w 0 2 2 exp ( - k 2 ϕ 2 w 0 2 / 4 ) J 0 ( k ϕ r ) k ϕ d ( k ϕ ) .
k ϕ 0 , 1 = ( 2 k Θ 0 , 1 L v ) 1 / 2 .
ψ s 0 k ϕ 0 , 1 G max w 0 2 2 exp ( - k 2 ϕ 2 w 0 2 / 4 ) J 0 ( 0 ) k ϕ d ( k ϕ ) = 0 k ϕ 0 , 1 sinh B 0.01 2 exp ( - 0.01 k 2 ϕ 2 / 4 ) k ϕ d ( k ϕ ) = sinh B [ 1 - exp ( - 0.01 k 2 ϕ 0 , 1 2 / 4 ) ] = sinh ( 2 ) [ 1 - exp ( - 0.01 π Θ 0 , 1 / 60 λ ) ] 3.4.
a ( z = 0 ) = 0 α 1 × J 0 ( k ϕ r ) r d r = α J 1 ( k ϕ α ) k ϕ .
ψ s sinh B [ 1 - J 0 ( 0.1 π Θ 0 , 1 / 60 λ ) ] = 3.81.
U ( r , z ) = 1 + ψ s ( r , z ) 1 - i B ( - w 0 w ( z ) exp [ - r 2 / w 2 ( z ) ] × exp { - i [ k r 2 / 2 R ( z ) - ϑ ( z ) ] } ) .
I = ψ ψ * = 1 + B w 0 w ( z ) exp [ - r 2 / w 2 ( z ) ] × { B w 0 w ( z ) exp [ - r 2 / w 2 ( z ) ] - 2 sin [ ϑ ( z ) + k r 2 / 2 R ( z ) ] shift from conjugate plane } ,
z ± = B ± B 2 + 4 2 z R ,
B tan [ 2 ϑ ( z ± ) ] = - 2.

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