Abstract

A simplified propagation model is presented that provides a useful balance between accuracy, flexibility, completeness and ease of computation. The atmosphere is modeled by a combination of aberrated lenses that includes the phase front distortions generated by the device, optical train, atmospheric turbulence, and thermal blooming. When atmospheric heating occurs away from the focal plane the method gives results that are virtually identical to the exact, distributed lens analysis. The method applies especially to vertical propagation through the atmosphere and horizontal propagation of repetitively pulsed lasers. Numerical results are presented that illustrate the analysis.

© 1978 Optical Society of America

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References

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  1. D. C. Smith, “High-power laser propagation: thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
    [CrossRef]
  2. J. N. Hayes, P. B. Ulrich, and A. H. Aitken, “Effects of the atmosphere on the propagation of 10.6 μ m laser beams,” Appl. Opt. 11, 257–260 (1972).
    [CrossRef] [PubMed]
  3. L. C. Bradley and J. Herrmann, “Numerical calculation of light propagation in a nonlinear medium,” J. Opt. Soc. Am. 61, 668 (1971).
  4. C. B. Hogge, “Propagation of high-energy laser beams in the atmosphere,” in High Energy Lasers and Their Applications, edited by S. Jacobs (Addison-Wesley, Reading, MA, 1974).
  5. H. J. Breaux, “An analysis of mathematical transformations and a comparison of numerical techniques for computation of high-energy cw laser propagation in an inhomogeneous medium,” (June1974).
  6. J. Wallace and J. Lilly, “Thermal blooming of repetitively pulsed laser beams,” J. Opt. Soc. Am. 64, 1651–1655 (1974).
    [CrossRef]
  7. W. P. Brown, “Computer simulation of adaptive optical systems,” (September1975).
  8. J. A. Fleck, J. R. Morris, and M. J. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” (June1975).
  9. E. H. Takken and D. M. Cordray, “Simplified analytical formulas for thermal blooming,” Appl. Opt. 13, 2753–2755 (1974).
    [CrossRef] [PubMed]
  10. F. G. Gebhardt, “High power laser propagation,” Appl. Opt. 15, 1479–1493 (1976).
    [CrossRef] [PubMed]
  11. J. A. Lilly, “Simplified calculation of laser beam propagation through the atmosphere,” (1976).
  12. R. Glauber, “High-energy collision theory” in Lectures in Theoretical Physics, Vol. 1, edited by W. Brittin and L. Dunham (Interscience, New York, 1959).
  13. J. B. Keller and B. R. Levy, “Scattering of short waves,” in Electromagnetic Scattering, edited by M. Kerker (Pergamon, New York, 1963).
  14. G. W. Zeiders, “A study of wave characteristics influences on laser selection for applications: propagation analysis,” (1974).
  15. H. J. Breaux, “A methodology for development of simple scaling laws for high-energy cw laser propagation,” (Jan.1978).
  16. G. S. S. Avila and J. B. Keller, “High-frequency asymptotic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363–381 (1963).
    [CrossRef]
  17. J. Wallace and J. Pasciak, “Theoretical aspects of thermal blooming compensation,” J. Opt. Soc. Am. 67, 1569–1575 (1977).
    [CrossRef]
  18. R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
    [CrossRef] [PubMed]

1977 (2)

D. C. Smith, “High-power laser propagation: thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
[CrossRef]

J. Wallace and J. Pasciak, “Theoretical aspects of thermal blooming compensation,” J. Opt. Soc. Am. 67, 1569–1575 (1977).
[CrossRef]

1976 (1)

1974 (2)

1972 (1)

1971 (2)

L. C. Bradley and J. Herrmann, “Numerical calculation of light propagation in a nonlinear medium,” J. Opt. Soc. Am. 61, 668 (1971).

R. F. Lutomirski and H. T. Yura, “Propagation of a finite optical beam in an inhomogeneous medium,” Appl. Opt. 10, 1652–1658 (1971).
[CrossRef] [PubMed]

1963 (1)

G. S. S. Avila and J. B. Keller, “High-frequency asymptotic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363–381 (1963).
[CrossRef]

Aitken, A. H.

Avila, G. S. S.

G. S. S. Avila and J. B. Keller, “High-frequency asymptotic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363–381 (1963).
[CrossRef]

Bradley, L. C.

L. C. Bradley and J. Herrmann, “Numerical calculation of light propagation in a nonlinear medium,” J. Opt. Soc. Am. 61, 668 (1971).

Breaux, H. J.

H. J. Breaux, “An analysis of mathematical transformations and a comparison of numerical techniques for computation of high-energy cw laser propagation in an inhomogeneous medium,” (June1974).

H. J. Breaux, “A methodology for development of simple scaling laws for high-energy cw laser propagation,” (Jan.1978).

Brown, W. P.

W. P. Brown, “Computer simulation of adaptive optical systems,” (September1975).

Cordray, D. M.

Feit, M. J.

J. A. Fleck, J. R. Morris, and M. J. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” (June1975).

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. J. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” (June1975).

Gebhardt, F. G.

Glauber, R.

R. Glauber, “High-energy collision theory” in Lectures in Theoretical Physics, Vol. 1, edited by W. Brittin and L. Dunham (Interscience, New York, 1959).

Hayes, J. N.

Herrmann, J.

L. C. Bradley and J. Herrmann, “Numerical calculation of light propagation in a nonlinear medium,” J. Opt. Soc. Am. 61, 668 (1971).

Hogge, C. B.

C. B. Hogge, “Propagation of high-energy laser beams in the atmosphere,” in High Energy Lasers and Their Applications, edited by S. Jacobs (Addison-Wesley, Reading, MA, 1974).

Keller, J. B.

G. S. S. Avila and J. B. Keller, “High-frequency asymptotic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363–381 (1963).
[CrossRef]

J. B. Keller and B. R. Levy, “Scattering of short waves,” in Electromagnetic Scattering, edited by M. Kerker (Pergamon, New York, 1963).

Levy, B. R.

J. B. Keller and B. R. Levy, “Scattering of short waves,” in Electromagnetic Scattering, edited by M. Kerker (Pergamon, New York, 1963).

Lilly, J.

Lilly, J. A.

J. A. Lilly, “Simplified calculation of laser beam propagation through the atmosphere,” (1976).

Lutomirski, R. F.

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. J. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” (June1975).

Pasciak, J.

Smith, D. C.

D. C. Smith, “High-power laser propagation: thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
[CrossRef]

Takken, E. H.

Ulrich, P. B.

Wallace, J.

Yura, H. T.

Zeiders, G. W.

G. W. Zeiders, “A study of wave characteristics influences on laser selection for applications: propagation analysis,” (1974).

Appl. Opt. (4)

Commun. Pure Appl. Math. (1)

G. S. S. Avila and J. B. Keller, “High-frequency asymptotic field of a point source in an inhomogeneous medium,” Commun. Pure Appl. Math. 16, 363–381 (1963).
[CrossRef]

J. Opt. Soc. Am. (3)

Proc. IEEE (1)

D. C. Smith, “High-power laser propagation: thermal blooming,” Proc. IEEE 65, 1679–1714 (1977).
[CrossRef]

Other (9)

W. P. Brown, “Computer simulation of adaptive optical systems,” (September1975).

J. A. Fleck, J. R. Morris, and M. J. Feit, “Time dependent propagation of high energy laser beams through the atmosphere,” (June1975).

C. B. Hogge, “Propagation of high-energy laser beams in the atmosphere,” in High Energy Lasers and Their Applications, edited by S. Jacobs (Addison-Wesley, Reading, MA, 1974).

H. J. Breaux, “An analysis of mathematical transformations and a comparison of numerical techniques for computation of high-energy cw laser propagation in an inhomogeneous medium,” (June1974).

J. A. Lilly, “Simplified calculation of laser beam propagation through the atmosphere,” (1976).

R. Glauber, “High-energy collision theory” in Lectures in Theoretical Physics, Vol. 1, edited by W. Brittin and L. Dunham (Interscience, New York, 1959).

J. B. Keller and B. R. Levy, “Scattering of short waves,” in Electromagnetic Scattering, edited by M. Kerker (Pergamon, New York, 1963).

G. W. Zeiders, “A study of wave characteristics influences on laser selection for applications: propagation analysis,” (1974).

H. J. Breaux, “A methodology for development of simple scaling laws for high-energy cw laser propagation,” (Jan.1978).

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Figures (4)

FIG. 1
FIG. 1

Typical situation in atmospheric propagation for application of the simplified analysis. The laser is repetitively pulsed with a beam motion large enough to displace, near the focal plane, the column of air heated by the previous pulses.

FIG. 2
FIG. 2

Comparison of the Strehl ratios and the bloomed isoirradiance contours for a 10.6 μm, repetitively pulsed laser focused at 2.5 km. The contours determined by the analysis are shown as solid lines and the exact results by the dashed lines.

FIG. 3
FIG. 3

Effects of turbulence on the isoirradiance contours for a 10.6 μm, repetitively pulsed laser focused at 2.5 km. The lower contours are for thermal blooming alone and the upper contours are for moderate turbulence and thermal blooming.

FIG. 4
FIG. 4

Comparison of the bloomed isoirradiance contours for horizontal propagation of a cw laser beam focused at 2 and 3 km. Because heating occurs near the focal plane the results are less accurate than those in Fig. 2.

Equations (22)

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E = A ( x , y , z ) exp ( i k n z = α z / 2 ) ,
2 i F n A z + 2 A + 2 k 2 ( n - 1 ) R m 2 ρ A = 0 ,
A = A 0 ( x , y ) exp [ i F ( ψ 0 ( x , y ) - ½ ( x 2 + y 2 ) ) ] ,
A = F 2 π i A 0 ( x , y ) a ( ξ , η ; x 1 , y 1 ) × exp [ i F ( ψ ( ξ , η ; x 1 , y 1 ) + ψ 0 ( x , y ) + ½ ( x 1 2 + y 1 2 ) - x 1 x - y 1 y ) ] d x d y ,
ξ = x - x 1 ,             η = y - y 1 .
- ψ z + ½ [ ψ ξ 2 + ψ η 2 ] / ( 1 - z ) 2 = z f 2 ( n - 1 ) [ ρ ( x 1 + ξ ( 1 - z ) , y 1 + η ( 1 - z ) , z ) ] / R m 2 ,
- ( 1 - z ) 2 a z + ( a ) · ( ψ ) + ½ a ( 2 ψ ) - i ( 2 a ) / 2 F = 0 ,
ψ = N 0 1 ρ ( x 1 + ( x - x 1 ) ( 1 - z ) , y 1 + ( y - y 1 ) ( 1 - z ) , z ) d z ; a = 1.0 ,
N 0 1 ρ ( 0 , 0 , z ) d z < 1.0 .
I = ( F / 2 π ) 2 exp ( - α z f ) M t ( u , v ) M j ( u , v ) A 0 ( x , y ) A 0 ( x - u , y - v ) × exp [ i F ( N h 0 1 ρ h ( x 1 z + x ( 1 - z ) , y 1 z + y ( 1 - z ) , z ) d z + ψ q ( x , y ) - ψ q ( x - u , y - v ) - N h 0 1 ρ h ( x 1 z + ( x - u ) ( 1 - z ) , y 1 z + ( y - v ) ( 1 - z ) , z ) d z - x 1 u - y 1 v ) ] d u d v d x d y .
M t = exp [ - ( 1.45 k 2 z f R m 5 / 3 × 0 1 C n 2 ( z ) ( 1 - z ) 5 / 3 d z ) ( u 2 + v 2 ) 5 / 6 ] ,
M j = exp [ - ½ ( θ x 2 u 2 + θ y 2 v 2 ) / θ d 2 ] ,
N h 0 z h ρ h ( 0 , 0 , z ) d z < 1.0 and R s / R m < ( 1 - z h ) / z h ,
I = T - 1 ( M t ( u , v ) M j ( u , v ) H ( u , v ) / ( 2 π ) 2 ,
H ( u , v ) = T - 1 { | T ( A 0 ( x , y ) exp [ i F ( ψ q ( x , y ) + N h 0 z h ρ h ( x ( 1 - z ) , y ( 1 - z ) , z ) d z ) ] ) | 2 } ,
ρ h = - exp ( - α z ) ( γ - 1 ) α I a t s γ p × n = 1 I ( x - U ( z ) t s n / R m , y , z ) ,
I = I 0 ( x / ( 1 - z ) , y / ( 1 - z ) ) / ( 1 - z ) 2 ,
F ψ h = - k z f ( n - 1 ) exp ( - α z h / 2 ) × ( γ - 1 ) α I a R m γ p U t n = 1 1 n - x * I 0 ( u , y ) d u ,
x * = x - U w t s n / R m .
z h = ( 1 - U w t s / 2 R m ) / [ 1 + ( U t - U w ) t s / 2 R m ] < ( 1 + R s / R m ) - 1 .
ρ = - exp ( - α z ) ( γ - 1 ) α I a R m γ p U ( z ) - x I ( u , y , z ) d u ,
F ψ h = - k z f ( n - 1 ) ( γ - 1 ) α I a R m γ p × ( 0 z h exp ( - α z ) ( 1 - z ) U ( z ) d z ) - x I 0 ( u , y ) d u .