Abstract

By expressing the solution to the nonlinear wage equation in operator form, we derive two methods for calculating a corrective phase to compensate for thermal blooming. The first method employs a form of glint return in which the uncorrected beam is first propagated through free space to the target. The resulting field is then propagated backward through the heated atmosphere to the aperture. The final phase of the beam represents the conjugate phase correction. The second method is based on the assumption that the atmospheric lens can be represented by a thin lens at the aperture. In this method one simply propagates the thermally bloomed beam through free space to the aperture. Again the final phase represents the conjugate phase correction. The second scheme performs as well as other predictive schemes and has the combined virtues of simplicity and versatility.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. T. Cathey, C. L. Hayes, W. C. Davis, V. F. Pizuro, Appl. Opt. 9, 701 (1970).
    [CrossRef] [PubMed]
  2. J. Hermann, J. Opt. Soc. Am. 67, 290 (1977).
    [CrossRef]
  3. See, for example, W. B. Bridges, P. T. Brunner, S. P. Lazzara, T. A. Nussmeier, T. R. O’Meara, J. A. Sanguinet, W. P. Brown, Appl. Opt. 13, 291 (1974).
    [CrossRef] [PubMed]
  4. W. P. Brown, Hughes Research Laboratory Report N60921-74-C-0249 (September1975).
  5. L. C. Bradley, J. Hermann, Appl. Opt. 13, 331 (1974).
    [CrossRef] [PubMed]
  6. J. Wallace, J. Pasciak, J. Opt. Soc. Am. 65, 1257 (1975).
    [CrossRef]
  7. J. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
    [CrossRef]
  8. P. J. Berger, P. B. Ulrich, J. T. Ulrich, F. O. Gebhardt, Appl. Opt. 16, 345 (1977).
    [CrossRef] [PubMed]
  9. J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 14, 99 (1977).
    [CrossRef]
  10. L. C. Bradley, J. Hermann, MIT Lincoln Laboratory internal report; unpublished.
  11. J. Hermann, MIT Lincoln Laboratory; private communication.

1977 (3)

1976 (1)

J. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

1975 (1)

1974 (2)

1970 (1)

Berger, P. J.

Bradley, L. C.

L. C. Bradley, J. Hermann, Appl. Opt. 13, 331 (1974).
[CrossRef] [PubMed]

L. C. Bradley, J. Hermann, MIT Lincoln Laboratory internal report; unpublished.

Bridges, W. B.

Brown, W. P.

Brunner, P. T.

Cathey, W. T.

Davis, W. C.

Feit, M. D.

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 14, 99 (1977).
[CrossRef]

J. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Fleck, J.

J. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Fleck, J. A.

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 14, 99 (1977).
[CrossRef]

Gebhardt, F. O.

Hayes, C. L.

Hermann, J.

J. Hermann, J. Opt. Soc. Am. 67, 290 (1977).
[CrossRef]

L. C. Bradley, J. Hermann, Appl. Opt. 13, 331 (1974).
[CrossRef] [PubMed]

L. C. Bradley, J. Hermann, MIT Lincoln Laboratory internal report; unpublished.

J. Hermann, MIT Lincoln Laboratory; private communication.

Lazzara, S. P.

Morris, J. R.

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 14, 99 (1977).
[CrossRef]

J. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

Nussmeier, T. A.

O’Meara, T. R.

Pasciak, J.

Pizuro, V. F.

Sanguinet, J. A.

Ulrich, J. T.

Ulrich, P. B.

Wallace, J.

Appl. Opt. (4)

Appl. Phys. (2)

J. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 10, 129 (1976).
[CrossRef]

J. A. Fleck, J. R. Morris, M. D. Feit, Appl. Phys. 14, 99 (1977).
[CrossRef]

J. Opt. Soc. Am. (2)

Other (3)

L. C. Bradley, J. Hermann, MIT Lincoln Laboratory internal report; unpublished.

J. Hermann, MIT Lincoln Laboratory; private communication.

W. P. Brown, Hughes Research Laboratory Report N60921-74-C-0249 (September1975).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Isointensity contours for cw beam in focal plane. Contours decrease from peak value in increments of 0.1. Problem parameters are NF = 18.3, NA = 0.4, Nω = 8, and ND = 107: (a) uncorrected beam, and (b) corrected beam using Eq. (11) and z T / R = 0.6. Correction decreases half-power area by a factor of 2.

Fig. 2
Fig. 2

Isointensity contours for multipulse beam in focal plane. Problem parameters are NF = 9.08, NA = 0.4, Nω = 0, and ND = 18.6. Initial beam shape is Gaussian: (a) uncorrected beam, and (b) beam corrected using Eq. (11) and z T / R = 1. For the uncorrected beam, the peak intensity is 0.53 × the peak for a diffraction-limited beam. Following correction, this rises to 0.95, the diffraction-limited peak.

Fig. 3
Fig. 3

Transverse wind velocity as a function of axial distance for noncoplanar scenario involving a stagnation zone. (a) Horizontal component vanishes 75 m from aperture, and (b) transverse velocity magnitude does not vanish at stagnation point due to 0.727 m/sec vertical component.

Fig. 4
Fig. 4

Isointensity contours at focus for stagnation zone scenario (a) uncorrected and (b) corrected. Correction increases average intensity by a factor of 2.

Fig. 5
Fig. 5

Isophase contours for corrected beam for stagnation zone scenario. Adjacent contours represent phase change increments of π/4.

Tables (2)

Tables Icon

Table I Comparison of Results Using Eqs. (11) and (12)

Tables Icon

Table II Comparison of Thermally Bloomed Beam Areas for Stagnation Zone Scenario

Equations (13)

Equations on this page are rendered with MathJax. Learn more.

2 i k ( E / z ) = 2 E + k 2 δ E ,
υ x δ ρ x + υ y δ ρ y = ( γ 1 ) c s 2 α I .
δ ρ t + υ x δ ρ x + υ y δ ρ y = γ 1 c 2 I ( x , y ) n δ ( t t n ) ,
E T = P NL E 0 ,
P NL = j exp ( i Δ z j 4 k 2 ) exp ( i k 2 δ ¯ j Δ z j ) exp ( i Δ z j 4 k 2 ) .
2 i k ( E / z ) = 2 E .
P NL exp ( i ψ ) E 0 = P L E 0 ,
P L = exp ( i z T 2 k 2 ) ,
exp ( i ψ ) E 0 = P N L ( P L E 0 ) .
P L exp ( i ψ ) E 0 = P N L E 0 .
exp ( i ψ ) E 0 = P N L P N L E 0 .
Δ ϕ = γ a 2 N D 2 N ω P ln ( 1 + N ω ) x | E ( z = 0 , x , y ) | 2 d x .
N A = 0.183 , N D ( z D ) = 851 , N F = 86.7 , N ω ( z D ) = | υ ( R ) | | υ ( z D ) | = 81.8 , and N D ( 0 ) = 450 , υ ( z D ) = 0.727 m / sec .

Metrics