Abstract

Self-focusing effects in large, high power laser amplifiers become manifest as small-scale beam instabilities and as large-scale phase aberrations. Spatial filtering has been shown to control instabilities; spatial filters constitute appropriate lens pair elements for image relaying as well. In this paper, image relaying is presented as a technique for preserving the transverse intensity profile of a high power beam as it propagates long distances through nonlinear elements. As a consequence, amplifier apertures can be filled more effectively, leading to a doubling of fixed-aperture system performance. A rationale for optimal selection of spatial filter bandpass is also presented. This selection, as might be expected, depends upon details of the beam's spatial structure as it enters any filter. A geometrical optics approach is used throughout; nevertheless, derived results remain valid when diffraction is included.

© 1978 Optical Society of America

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References

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  1. E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, IEEE J. Quantum Electron. QE-12, 402 (1976).
    [CrossRef]
  2. W. W. Simmons, S. Guch, F. Rainer, J. E. Murray, IEEE J. Quantum Electron. QE-11, 31D (1975).
  3. J. T. Hunt, P. A. Renard, W. W. Simmons, Appl. Opt. 16, 779 (1977).
    [PubMed]
  4. J. A. Glaze, Opt. Eng. 15, 136 (1976).
    [CrossRef]
  5. A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and Winston, New York, 1971), p. 20.
  6. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), p. 116.
  7. P. Renard, J. T. Hunt, “Artemis: A Diffraction Model for Light Propagation,” Lawrence Livermore Laboratory Internal Report UCID 17381 (February1977);A. E. Siegman, E. A. Sziklas, Appl. Opt. 13, 2775 (1974).
    [CrossRef] [PubMed]
  8. A. J. Campillo, J. E. Pearson, S. L. Shapiro, J. J. Terrell, Appl. Phys. Lett. 23, 85 (1973)
    [CrossRef]
  9. B. R. Suydam, IEEE J. Quantum Electron. QE-11, 225 (1975).
    [CrossRef]
  10. W. W. Simmons, D. R. Speck, J. T. Hunt, IEEE J. Quantum Electron. QE-13, 50 (1977).
  11. V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).
  12. E. S. Bliss, D. R. Speck, J. F. Holzrichter, J. H. Erkkila, A. J. Glass, Appl. Phys. Lett. 25, 448 (1974).
    [CrossRef]
  13. D. Auric, A. Labadens, J. Guyot, Opt. Commun. 18, 176 (1976).
    [CrossRef]
  14. ω is related to the more conventional waist w by ω = (2)1/2w.
  15. A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), p. 160.
  16. H. Sagan, Boundary Value and Eigenvalue Problems in Mathematical Physics (Wiley, New York, 1961), p. 107.
  17. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions, AMS 55 (U.S. Government Printing Office, Washington, D.C., 1965), p. 361.
  18. This result is similar to that presented by B. R. Suydam in Ref. 9.
  19. This approximation has been invoked by many authors (see, for example, Refs. 9 and 12). When it is not satisfied, the physical situation becomes difficult to model.
  20. J. B. Trenholme, Lawrence Livermore Laboratory; private communication.
  21. A. J. Campillo et al., Appl. Phys. Lett. 25, 408 (1974).
    [CrossRef]
  22. J. T. Hunt, P. A. Renard, R. G. Nelson, Appl. Opt. 15, 1458 (1976).
    [CrossRef] [PubMed]

1977 (2)

W. W. Simmons, D. R. Speck, J. T. Hunt, IEEE J. Quantum Electron. QE-13, 50 (1977).

J. T. Hunt, P. A. Renard, W. W. Simmons, Appl. Opt. 16, 779 (1977).
[PubMed]

1976 (4)

J. T. Hunt, P. A. Renard, R. G. Nelson, Appl. Opt. 15, 1458 (1976).
[CrossRef] [PubMed]

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, IEEE J. Quantum Electron. QE-12, 402 (1976).
[CrossRef]

J. A. Glaze, Opt. Eng. 15, 136 (1976).
[CrossRef]

D. Auric, A. Labadens, J. Guyot, Opt. Commun. 18, 176 (1976).
[CrossRef]

1975 (2)

B. R. Suydam, IEEE J. Quantum Electron. QE-11, 225 (1975).
[CrossRef]

W. W. Simmons, S. Guch, F. Rainer, J. E. Murray, IEEE J. Quantum Electron. QE-11, 31D (1975).

1974 (2)

E. S. Bliss, D. R. Speck, J. F. Holzrichter, J. H. Erkkila, A. J. Glass, Appl. Phys. Lett. 25, 448 (1974).
[CrossRef]

A. J. Campillo et al., Appl. Phys. Lett. 25, 408 (1974).
[CrossRef]

1973 (1)

A. J. Campillo, J. E. Pearson, S. L. Shapiro, J. J. Terrell, Appl. Phys. Lett. 23, 85 (1973)
[CrossRef]

1966 (1)

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Auric, D.

D. Auric, A. Labadens, J. Guyot, Opt. Commun. 18, 176 (1976).
[CrossRef]

Bespalov, V. I.

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Bliss, E. S.

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, IEEE J. Quantum Electron. QE-12, 402 (1976).
[CrossRef]

E. S. Bliss, D. R. Speck, J. F. Holzrichter, J. H. Erkkila, A. J. Glass, Appl. Phys. Lett. 25, 448 (1974).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), p. 116.

Campillo, A. J.

A. J. Campillo et al., Appl. Phys. Lett. 25, 408 (1974).
[CrossRef]

A. J. Campillo, J. E. Pearson, S. L. Shapiro, J. J. Terrell, Appl. Phys. Lett. 23, 85 (1973)
[CrossRef]

Dunn, M. H.

A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), p. 160.

Erkkila, J. H.

E. S. Bliss, D. R. Speck, J. F. Holzrichter, J. H. Erkkila, A. J. Glass, Appl. Phys. Lett. 25, 448 (1974).
[CrossRef]

Glass, A. J.

E. S. Bliss, D. R. Speck, J. F. Holzrichter, J. H. Erkkila, A. J. Glass, Appl. Phys. Lett. 25, 448 (1974).
[CrossRef]

Glaze, J. A.

J. A. Glaze, Opt. Eng. 15, 136 (1976).
[CrossRef]

Guch, S.

W. W. Simmons, S. Guch, F. Rainer, J. E. Murray, IEEE J. Quantum Electron. QE-11, 31D (1975).

Guyot, J.

D. Auric, A. Labadens, J. Guyot, Opt. Commun. 18, 176 (1976).
[CrossRef]

Holzrichter, J. F.

E. S. Bliss, D. R. Speck, J. F. Holzrichter, J. H. Erkkila, A. J. Glass, Appl. Phys. Lett. 25, 448 (1974).
[CrossRef]

Hunt, J. T.

W. W. Simmons, D. R. Speck, J. T. Hunt, IEEE J. Quantum Electron. QE-13, 50 (1977).

J. T. Hunt, P. A. Renard, W. W. Simmons, Appl. Opt. 16, 779 (1977).
[PubMed]

J. T. Hunt, P. A. Renard, R. G. Nelson, Appl. Opt. 15, 1458 (1976).
[CrossRef] [PubMed]

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, IEEE J. Quantum Electron. QE-12, 402 (1976).
[CrossRef]

P. Renard, J. T. Hunt, “Artemis: A Diffraction Model for Light Propagation,” Lawrence Livermore Laboratory Internal Report UCID 17381 (February1977);A. E. Siegman, E. A. Sziklas, Appl. Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

Labadens, A.

D. Auric, A. Labadens, J. Guyot, Opt. Commun. 18, 176 (1976).
[CrossRef]

Maitland, A.

A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), p. 160.

Murray, J. E.

W. W. Simmons, S. Guch, F. Rainer, J. E. Murray, IEEE J. Quantum Electron. QE-11, 31D (1975).

Nelson, R. G.

Pearson, J. E.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, J. J. Terrell, Appl. Phys. Lett. 23, 85 (1973)
[CrossRef]

Rainer, F.

W. W. Simmons, S. Guch, F. Rainer, J. E. Murray, IEEE J. Quantum Electron. QE-11, 31D (1975).

Renard, P.

P. Renard, J. T. Hunt, “Artemis: A Diffraction Model for Light Propagation,” Lawrence Livermore Laboratory Internal Report UCID 17381 (February1977);A. E. Siegman, E. A. Sziklas, Appl. Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

Renard, P. A.

J. T. Hunt, P. A. Renard, W. W. Simmons, Appl. Opt. 16, 779 (1977).
[PubMed]

J. T. Hunt, P. A. Renard, R. G. Nelson, Appl. Opt. 15, 1458 (1976).
[CrossRef] [PubMed]

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, IEEE J. Quantum Electron. QE-12, 402 (1976).
[CrossRef]

Sagan, H.

H. Sagan, Boundary Value and Eigenvalue Problems in Mathematical Physics (Wiley, New York, 1961), p. 107.

Shapiro, S. L.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, J. J. Terrell, Appl. Phys. Lett. 23, 85 (1973)
[CrossRef]

Simmons, W. W.

W. W. Simmons, D. R. Speck, J. T. Hunt, IEEE J. Quantum Electron. QE-13, 50 (1977).

J. T. Hunt, P. A. Renard, W. W. Simmons, Appl. Opt. 16, 779 (1977).
[PubMed]

W. W. Simmons, S. Guch, F. Rainer, J. E. Murray, IEEE J. Quantum Electron. QE-11, 31D (1975).

Sommargren, G. E.

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, IEEE J. Quantum Electron. QE-12, 402 (1976).
[CrossRef]

Speck, D. R.

W. W. Simmons, D. R. Speck, J. T. Hunt, IEEE J. Quantum Electron. QE-13, 50 (1977).

E. S. Bliss, D. R. Speck, J. F. Holzrichter, J. H. Erkkila, A. J. Glass, Appl. Phys. Lett. 25, 448 (1974).
[CrossRef]

Suydam, B. R.

B. R. Suydam, IEEE J. Quantum Electron. QE-11, 225 (1975).
[CrossRef]

Talanov, V. I.

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Terrell, J. J.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, J. J. Terrell, Appl. Phys. Lett. 23, 85 (1973)
[CrossRef]

Trenholme, J. B.

J. B. Trenholme, Lawrence Livermore Laboratory; private communication.

Weaver, H. J.

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, IEEE J. Quantum Electron. QE-12, 402 (1976).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), p. 116.

Yariv, A.

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and Winston, New York, 1971), p. 20.

Appl. Opt. (2)

Appl. Phys. Lett. (3)

A. J. Campillo et al., Appl. Phys. Lett. 25, 408 (1974).
[CrossRef]

A. J. Campillo, J. E. Pearson, S. L. Shapiro, J. J. Terrell, Appl. Phys. Lett. 23, 85 (1973)
[CrossRef]

E. S. Bliss, D. R. Speck, J. F. Holzrichter, J. H. Erkkila, A. J. Glass, Appl. Phys. Lett. 25, 448 (1974).
[CrossRef]

IEEE J. Quantum Electron. (4)

B. R. Suydam, IEEE J. Quantum Electron. QE-11, 225 (1975).
[CrossRef]

W. W. Simmons, D. R. Speck, J. T. Hunt, IEEE J. Quantum Electron. QE-13, 50 (1977).

E. S. Bliss, J. T. Hunt, P. A. Renard, G. E. Sommargren, H. J. Weaver, IEEE J. Quantum Electron. QE-12, 402 (1976).
[CrossRef]

W. W. Simmons, S. Guch, F. Rainer, J. E. Murray, IEEE J. Quantum Electron. QE-11, 31D (1975).

JETP Lett. (1)

V. I. Bespalov, V. I. Talanov, JETP Lett. 3, 307 (1966).

Opt. Commun. (1)

D. Auric, A. Labadens, J. Guyot, Opt. Commun. 18, 176 (1976).
[CrossRef]

Opt. Eng. (1)

J. A. Glaze, Opt. Eng. 15, 136 (1976).
[CrossRef]

Other (10)

A. Yariv, Introduction to Optical Electronics (Holt, Rinehart, and Winston, New York, 1971), p. 20.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), p. 116.

P. Renard, J. T. Hunt, “Artemis: A Diffraction Model for Light Propagation,” Lawrence Livermore Laboratory Internal Report UCID 17381 (February1977);A. E. Siegman, E. A. Sziklas, Appl. Opt. 13, 2775 (1974).
[CrossRef] [PubMed]

ω is related to the more conventional waist w by ω = (2)1/2w.

A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), p. 160.

H. Sagan, Boundary Value and Eigenvalue Problems in Mathematical Physics (Wiley, New York, 1961), p. 107.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions, AMS 55 (U.S. Government Printing Office, Washington, D.C., 1965), p. 361.

This result is similar to that presented by B. R. Suydam in Ref. 9.

This approximation has been invoked by many authors (see, for example, Refs. 9 and 12). When it is not satisfied, the physical situation becomes difficult to model.

J. B. Trenholme, Lawrence Livermore Laboratory; private communication.

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Figures (3)

Fig. 1
Fig. 1

A typical optical relay system for suppressing self-focusing effects in high power Nd:glass laser systems.

Fig. 2
Fig. 2

Intensity profile distortion caused by self-focusing in a high power laser system.

Fig. 3
Fig. 3

Power spectrum of a high power beam with a harmonic perturbation.

Equations (39)

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t k = [ m k m k d k 1 ( m k ) 1 d k 2 + f k 1 + f k 2 0 1 / m k ] ,
m k = ( f k 2 ) / ( f k 1 )
T N = k = 1 N t k .
m k d k 1 + ( m k ) 1 d k 2 f k 1 f k 2 = 0 .
I k ( r , z ) = I k ( 0 , z ) F k ( r ) ,
Δ ϕ k = Δ B k F k ( r ) ,
Δ B k = 2 π λ γ 0 t I k ( 0 , z ) d z ,
r N = r 1 + d k = 1 N 1 ( N k ) Δ θ k ,
θ N = k = 1 N Δ θ k ,
Δ θ k = Δ B k F k ( r ) r | r = r k ,
r ¯ N = ( 1 ) N [ r 1 ( 4 f d ) l = 1 N ( N l ) Δ θ ¯ l + 1 ] ,
θ ¯ N = ( 1 ) N l = 1 N Δ θ ¯ l ,
Δ θ l ¯ = Δ B l F l ( r ) ¯ r .
d ¯ = 4 f d
r ¯ N = r 1 ,
θ ¯ N = l Δ θ l ¯ .
ψ = Ψ o Ψ s ,
Ψ o = A 0 F ( r ) 1 / 2 ,
Ψ s = A 0 exp { ½ [ r / ω ( z 1 ) ] 2 } exp [ i k r 2 2 R ( z 1 ) ] .
Ψ s ( f ) = A 0 ω 0 ω ( f ) exp { ½ [ r / ω ( f ) ] 2 } exp [ i k r 2 / 2 R ( f ) ] ,
ω ( f ) = f k ω 0 , R ( f ) = f 1 z 1 / f .
ψ s = A 0 2 ω 0 2 ( k D 2 f ) 2 [ 2 J 1 ( X ) X ] ,
X = [ ( k D ) / ( 2 f ) ] r .
K c = ( k D ) / ( 2 f ) .
Δ ψ rms = ( 1 T ) 1 / 2 Ψ i ,
Δ ψ rms = 1 π ( 40 π k D ) 1 / 2 Ψ i = 0.07 Ψ i .
ψ i ( r ) = A 0 [ F ( r ) ] 1 / 2 [ 1 + δ cos κ · r ] .
ψ f ( r ) = A 0 [ F ( r ) ] 1 / 2 [ 1 + δ cos κ · r ] × exp [ i B F ( r ) ( 1 + δ cos κ · r ) 2 ] ,
ψ f ( r ) = A 0 [ F ( r ) ] 1 / 2 exp [ i B F ( r ) ] j α j cos ( j κ · r ) ,
x = 2 B δ F ( r ) .
α 0 = J 0 ( x ) + i δ J 1 ( x ) ,
α 1 = δ + 2 i J 1 ( x ) δ J 2 ( x ) ,
α 2 = i δ J 1 ( x ) 2 J 2 ( x ) i J 3 ( x ) .
| α 1 | 2 / | α 0 | 2 = ( 1 + 2 B 2 ) δ 2 .
ϕ D = B F ( r ) .
Δ z = λ 2 π B f 2 1 r d F ( r ) d r
Δ z ¯ = ½ Δ z | max
ɛ = [ ( 2 r 0 ) / f ] Δ z ¯ ,
F ( r ) = ( 1 0 r r 0 1 r / a 1 r 0 / a r 0 r a )

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