Abstract

The modulation instability (MI) is one of the main factors responsible for the degradation of beam quality in high-power laser systems. The so-called B-integral restriction is commonly used as the criteria for MI control in passive optics devices. For amplifiers the adiabatic model, assuming locally the Bespalov-Talanov expression for MI growth, is commonly used to estimate the destructive impact of the instability. We present here the exact solution of MI development in amplifiers. We determine the parameters which control the effect of MI in amplifiers and calculate the MI growth rate as a function of those parameters. The safety range of operational parameters is presented. The results of the exact calculations are compared with the adiabatic model, and the range of validity of the latest is determined. We demonstrate that for practical situations the adiabatic approximation noticeably overestimates MI. The additional margin of laser system design is quantified.

© 2010 OSA

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References

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  1. V. I. Bespalov and V. I. Talanov, “Filamentary Structure of Light Beams in Nonlinear Media,” JETP Lett. 3, 307 (1966).
  2. C. A. Haynam, P. J. Wegner, J. M. Auerbach, M. W. Bowers, S. N. Dixit, G. V. Erbert, G. M. Heestand, M. A. Henesian, M. R. Hermann, K. S. Jancaitis, K. R. Manes, C. D. Marshall, N. C. Mehta, J. Menapace, E. Moses, J. R. Murray, M. C. Nostrand, C. D. Orth, R. Patterson, R. A. Sacks, M. J. Shaw, M. Spaeth, S. B. Sutton, W. H. Williams, C. C. Widmayer, R. K. White, S. T. Yang, and B. M. Van Wonterghem, “National Ignition Facility laser performance status,” Appl. Opt. 46(16), 3276–3303 (2007).
    [CrossRef] [PubMed]
  3. J. T. Hunt, J. Glaze, W. Simmons, and P. Renard, “Suppression of self-focusing through low-pass spatial filtering and relay imaging,” Appl. Opt. 17(13), 2053 (1978).
    [CrossRef] [PubMed]
  4. A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
    [CrossRef] [PubMed]
  5. B. R. Suydam, “Self-focusing of very powerful laser beam “ in Laser Induced Damage in Optical Materials, A.J.Glass and A.H.Guenther,Ed.Washington, D,C, NBS,1973,special publication 387,“Self-focusing of very powerful laser beam. II”, IEEE J. Quantum Electron. QE-10 837 (1974)
  6. J. Trenholme “LLNL annual report” (1974)
  7. M. Karlsson, “Modulational instability in lossy optical fibers,” J. Opt. Soc. Am. B 12, 2071 (1995).
  8. H. Segur and D. M. Henderson, “The modulation instability revisited,” Eur. Phys. J. Spec. Top. 147(1), 25–43 (2007).
    [CrossRef]
  9. V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
    [CrossRef]

2009 (2)

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[CrossRef] [PubMed]

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
[CrossRef]

2007 (2)

1995 (1)

1978 (1)

1966 (1)

V. I. Bespalov and V. I. Talanov, “Filamentary Structure of Light Beams in Nonlinear Media,” JETP Lett. 3, 307 (1966).

Auerbach, J. M.

Bespalov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary Structure of Light Beams in Nonlinear Media,” JETP Lett. 3, 307 (1966).

Bowers, M. W.

Dixit, S. N.

Erbert, G. V.

Fedoruk, M. P.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[CrossRef] [PubMed]

Glaze, J.

Haynam, C. A.

Heestand, G. M.

Henderson, D. M.

H. Segur and D. M. Henderson, “The modulation instability revisited,” Eur. Phys. J. Spec. Top. 147(1), 25–43 (2007).
[CrossRef]

Henesian, M. A.

Hermann, M. R.

Hunt, J. T.

Jancaitis, K. S.

Karlsson, M.

Manes, K. R.

Marshall, C. D.

Mehta, N. C.

Menapace, J.

Moses, E.

Murray, J. R.

Nostrand, M. C.

Orth, C. D.

Ostrovsky, L. A.

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
[CrossRef]

Patterson, R.

Renard, P.

Rubenchik, A. M.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[CrossRef] [PubMed]

Sacks, R. A.

Segur, H.

H. Segur and D. M. Henderson, “The modulation instability revisited,” Eur. Phys. J. Spec. Top. 147(1), 25–43 (2007).
[CrossRef]

Shaw, M. J.

Simmons, W.

Spaeth, M.

Sutton, S. B.

Talanov, V. I.

V. I. Bespalov and V. I. Talanov, “Filamentary Structure of Light Beams in Nonlinear Media,” JETP Lett. 3, 307 (1966).

Turitsyn, S. K.

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[CrossRef] [PubMed]

Van Wonterghem, B. M.

Wegner, P. J.

White, R. K.

Widmayer, C. C.

Williams, W. H.

Yang, S. T.

Zakharov, V. E.

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
[CrossRef]

Appl. Opt. (2)

Eur. Phys. J. Spec. Top. (1)

H. Segur and D. M. Henderson, “The modulation instability revisited,” Eur. Phys. J. Spec. Top. 147(1), 25–43 (2007).
[CrossRef]

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

JETP Lett. (1)

V. I. Bespalov and V. I. Talanov, “Filamentary Structure of Light Beams in Nonlinear Media,” JETP Lett. 3, 307 (1966).

Phys. Rev. Lett. (1)

A. M. Rubenchik, M. P. Fedoruk, and S. K. Turitsyn, “Laser beam self-focusing in the atmosphere,” Phys. Rev. Lett. 102(23), 233902 (2009).
[CrossRef] [PubMed]

Physica D (1)

V. E. Zakharov and L. A. Ostrovsky, “Modulation instability: the beginning,” Physica D 238(5), 540–548 (2009).
[CrossRef]

Other (2)

B. R. Suydam, “Self-focusing of very powerful laser beam “ in Laser Induced Damage in Optical Materials, A.J.Glass and A.H.Guenther,Ed.Washington, D,C, NBS,1973,special publication 387,“Self-focusing of very powerful laser beam. II”, IEEE J. Quantum Electron. QE-10 837 (1974)

J. Trenholme “LLNL annual report” (1974)

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Figures (6)

Fig. 1
Fig. 1

γas function of transversal perturbation wave-number ω. Red line q = 1 blue q = 2 green q = 3 , black adiabatic approximation and q = 1 , g 0 L = 3 .

Fig. 2
Fig. 2

γ as function of transversal perturbation wave-number ω . Red line g 0 L = 1 , blue g 0 L = 2 , green g 0 L = 3 , black adiabatic approximation g 0 L = 1 , q = 1.

Fig. 3
Fig. 3

γ as function q for ω = 0.5 (red line –exact solution, blue – adiabatic approximation) and for ω = 1 (green line – exact solution, black – adiabatic approximation), g 0 L = 3.

Fig. 4
Fig. 4

ω max as function q for different values of g 0 L . Red line g 0 L = 3 , blue g 0 L = 2 , green g 0 L = 1. The dashed lines are results of adiabatic approximation.

Fig. 5
Fig. 5

Maximum γ (solid line) and B-integral (dashed line) as function of q for different values of g 0 L . Red line g 0 L = 3 , blue g 0 L = 2 , green, g 0 L = 1. Dotted lines correspond the adiabatic approximation.

Fig. 6
Fig. 6

Counterplot ln γ max as function q and g 0 L .

Equations (17)

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i ψ z + 1 2 n 0 k 0 Δ ψ + k 0 n 2 | ψ | 2 ψ = i g 0 2 ψ .
i U z + 1 2 n 0 k 0 Δ U + k 0 n 2 ( z ) | U | 2 U = 0.
U ( z , r ) = ( U 0 + a + i b ) × exp [ i k 0 | U 0 | 2 n 2 ( z ' ) d z ' , a z + 1 2 n 0 k 0 Δ b = 0 , b z + 1 2 n 0 k 0 Δ a + 2 k 0 n 2 ( z ) | U 0 | 2 a = 0.
2 a z 2 + 1 2 n 0 k 0 Δ { 1 2 n 0 k 0 Δ + 2 k 0 n 2 ( z ) | U 0 | 2 } a = 0.
k z 2 = k 2 2 n 0 k 0 [ k 2 2 n 0 k 0 2 k 0 n 2 | U 0 | 2 ] .
k 2 2 n 0 k 0 = k 0 n 2 | A | 2
k 2 2 n 0 k 0 = 2 k 0 n 2 | A 0 | 2 .
a ( 0 ) , b ( 0 ) exp [ k z L ] = a ( 0 ) , b ( 0 ) exp [ k 0 n 2 | A 0 | 2 L ] = a ( 0 ) , b ( 0 ) exp [ B ] .
a z = k 2 2 n 0 k 0 b , b z = k 2 2 n 0 k 0 a + 2 k 0 n 2 ( z ) | U 0 | 2 a .
B ' ( L ) = k 0 n 2 ( 0 ) 0 L | U 0 | 2 exp [ g 0 z ] d z = k 0 n 2 ( 0 ) | U 0 | 2 exp [ g 0 L ] 1 g 0 = q exp [ g 0 L ] 1 4 , q = 4 k 0 n 2 ( 0 ) | U 0 | 2 / g 0 .
γ a = 0 L Im k z d z = 0.5 q f ( ω , g 0 L ) , ω 2 = k 2 4 n 0 k 0 2 n 2 ( 0 ) | U 0 | 2 ,
f ( ω , g 0 L ) = 0 g 0 L [ ( e x ω 2 ) ω 2 ] 1 / 2 d x : ω 2 < 1 f ( ω , g 0 L ) = ln ω 2 g 0 L [ ( e x ω 2 ) ω 2 ] 1 / 2 d x : ω 2 > 1
f ( ω , g 0 L ) = 2 ω { ( e g 0 L ω 2 ) 1 / 2 1 ω 2 ω tan 1 e g 0 L ω 2 ω + ω tan 1 1 ω 2 ω } ; ω 2 < 1 f ( ω , g 0 L ) = 2 ω { ( e g 0 L ω 2 ) 1 / 2 ω tan 1 e g 0 L ω 2 ω } ; ω 2 > 1
d 2 a d z 2 + k 2 2 n 0 k 0 [ k 2 2 n 0 k 0 2 k 0 n 2 ( z ) | U 0 | 2 ] a = 0.
d 2 a d z 2 + q 2 ω 2 g 0 2 4 ( ω 2 exp [ g 0 z ] ) a = 0.
x 2 d 2 a d x 2 + x d a d x + ( ν 2 x 2 ) a = 0.
a ( z ) = i f 0 q ω π 2 sinh [ ν π ] { I i ν ' ( q ω ) I i ν ( q ω e g 0 z / 2 ) I i ν ' ( q ω ) I i ν ( q ω e g 0 z / 2 ) } + i f 1 π g 0 sinh [ ν π ] { I i ν ( q ω ) I i ν ( q ω e g 0 z / 2 ) I i ν ( q ω ) I i ν ( q ω e g 0 z / 2 ) } .

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