Abstract

Laser-beam propagation through atmospheric turbulence is analyzed theoretically and compared with measurements at λ = 0.63 and 10.6 μm. Calculations based either on irradiance statistics or mutual coherence function (MCF) are analyzed; a general expression for long-term-average beam spread based on the turbulence MCF is obtained. The spread of a laser beam focused over moderate distances can be separated into short- and long-term averages that differ by beam wander, which has been found to be essentially independent of wavelength and adequately described by geometric optics. However, a significant wavelength dependence of short-term-average beam spread is found experimentally. Measurements at 10.6 μm are nearly diffraction limited, whereas corresponding data for 0.63 μm are strongly influenced by variations of the refractive-index structure constant Cn. An empirical formula for long-term-average beam spread is at variance with the wave-number-dependent functional form predicted by MCF calculations. Both the irradiance and MCF approaches go over into the same asymptotic functional dependence for visible wavelengths and long ranges. Angular-beam-spread measurements at 0.63 μm support this conclusion.

© 1973 Optical Society of America

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References

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  1. V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  2. R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1966).
  3. A. Ishimaru, Proc. IEEE 57, 407 (1969).
    [Crossref]
  4. F. P. Carlson and A. Ishimaru, J. Opt. Soc. Am. 59, 319 (1969).
    [Crossref]
  5. F. G. Gebhardt and S. A. Collins, J. Opt. Soc. Am. 59, 1139 (1969).
    [Crossref]
  6. M. Greenebaum (private communication).
  7. A. L. Buck, Proc. IEEE 55, 448 (1967).
    [Crossref]
  8. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964).
  9. M. Beran, J. Opt. Soc. Am. 56, 1475 (1966).
    [Crossref]
  10. J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
    [Crossref]
  11. D. A. de Wolf, Radio Sci. 2, 1379 (1967); Radio Sci. 3, 308 (1968).
  12. W. P. Brown, IEEE Trans. Antennas Propag. 15, 81 (1967).
    [Crossref]
  13. A. D. Varvatsis and M. I. Sancer, Can. J. Phys. 49, 1233 (1971).
    [Crossref]
  14. H. T. Yura, Appl. Opt. 11, 1399 (1972).
    [Crossref] [PubMed]
  15. W. P. Brown, J. Opt. Soc. Am. 61, 1051 (1971).
    [Crossref]
  16. H. T. Yura, J. Opt. Soc. Am. 62, 889 (1972).
    [Crossref]
  17. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966); J. Opt. Soc. Am. 56, 1380 (1966).
    [Crossref]
  18. D. A. de Wolf (private communication).
  19. R. W. Harris, J. Opt. Soc. Am. 62, 722A (1972).
    [Crossref]
  20. G. R. Ochs, A Circuit for the Measurement of Normalized Crosscorrelations, ESSA Tech. Report No. ERL 63-WPL 2 (U.S. Government Printing Office, Washington, D.C., 1968).
  21. L. R. Zwang, Izv. Akad. Nauk SSSR Ser. Geofiz. 8, 1252 (1960).
  22. G. R. Ochs, A Resistance Thermometer for Measurement of Rapid Air Temperature Fluctuations, ESSA Tech. Report No. IER 47-ITSA 46 (U.S. Government Printing Office, Washington, D.C., 1967).
  23. G. R. Ochs, R. R. Bergman, and J. R. Snyder, J. Opt. Soc. Am. 59, 231 (1969).
    [Crossref]
  24. D. L. Fried and J. B. Seidman, J. Opt. Soc. Am. 57, 181 (1967).
    [Crossref]
  25. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
    [Crossref]
  26. M. A. Efroymson, in Multiple Regression Analysis in Mathematical Methods for Digital Computers, Volume 1, edited by Anthony Ralston and Herbert S. Wilf (Wiley, New York, 1960).
  27. C. A. Friehe, C. H. Gibson, and G. Dreyer, J. Opt. Soc. Am. 62, 1340 (1972).
  28. P. M. Livingston, Appl. Opt. 11, 684 (1972).
    [Crossref] [PubMed]
  29. E. C. Alcaraz and P. M. Livingston, in Proceedings Electro-Optical Design Conference, 1972–West (Indus, and Scientific Management Inc., Chicago, Ill., 1971), p. 76.
  30. T. Chiba, Appl. Opt. 10, 2456 (1971).
    [Crossref] [PubMed]

1972 (5)

H. T. Yura, Appl. Opt. 11, 1399 (1972).
[Crossref] [PubMed]

H. T. Yura, J. Opt. Soc. Am. 62, 889 (1972).
[Crossref]

R. W. Harris, J. Opt. Soc. Am. 62, 722A (1972).
[Crossref]

C. A. Friehe, C. H. Gibson, and G. Dreyer, J. Opt. Soc. Am. 62, 1340 (1972).

P. M. Livingston, Appl. Opt. 11, 684 (1972).
[Crossref] [PubMed]

1971 (3)

1969 (4)

1968 (1)

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

1967 (5)

D. A. de Wolf, Radio Sci. 2, 1379 (1967); Radio Sci. 3, 308 (1968).

W. P. Brown, IEEE Trans. Antennas Propag. 15, 81 (1967).
[Crossref]

A. L. Buck, Proc. IEEE 55, 448 (1967).
[Crossref]

D. L. Fried and J. B. Seidman, J. Opt. Soc. Am. 57, 181 (1967).
[Crossref]

D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967).
[Crossref]

1966 (3)

1960 (1)

L. R. Zwang, Izv. Akad. Nauk SSSR Ser. Geofiz. 8, 1252 (1960).

Alcaraz, E. C.

E. C. Alcaraz and P. M. Livingston, in Proceedings Electro-Optical Design Conference, 1972–West (Indus, and Scientific Management Inc., Chicago, Ill., 1971), p. 76.

Beran, M.

Bergman, R. R.

Born, M.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964).

Brown, W. P.

W. P. Brown, J. Opt. Soc. Am. 61, 1051 (1971).
[Crossref]

W. P. Brown, IEEE Trans. Antennas Propag. 15, 81 (1967).
[Crossref]

Buck, A. L.

A. L. Buck, Proc. IEEE 55, 448 (1967).
[Crossref]

Carlson, F. P.

Chiba, T.

Collins, S. A.

de Wolf, D. A.

D. A. de Wolf, Radio Sci. 2, 1379 (1967); Radio Sci. 3, 308 (1968).

D. A. de Wolf (private communication).

Dreyer, G.

C. A. Friehe, C. H. Gibson, and G. Dreyer, J. Opt. Soc. Am. 62, 1340 (1972).

Efroymson, M. A.

M. A. Efroymson, in Multiple Regression Analysis in Mathematical Methods for Digital Computers, Volume 1, edited by Anthony Ralston and Herbert S. Wilf (Wiley, New York, 1960).

Fried, D. L.

Friehe, C. A.

C. A. Friehe, C. H. Gibson, and G. Dreyer, J. Opt. Soc. Am. 62, 1340 (1972).

Gebhardt, F. G.

Gibson, C. H.

C. A. Friehe, C. H. Gibson, and G. Dreyer, J. Opt. Soc. Am. 62, 1340 (1972).

Greenebaum, M.

M. Greenebaum (private communication).

Harris, R. W.

R. W. Harris, J. Opt. Soc. Am. 62, 722A (1972).
[Crossref]

Ishimaru, A.

Livingston, P. M.

P. M. Livingston, Appl. Opt. 11, 684 (1972).
[Crossref] [PubMed]

E. C. Alcaraz and P. M. Livingston, in Proceedings Electro-Optical Design Conference, 1972–West (Indus, and Scientific Management Inc., Chicago, Ill., 1971), p. 76.

Ochs, G. R.

G. R. Ochs, R. R. Bergman, and J. R. Snyder, J. Opt. Soc. Am. 59, 231 (1969).
[Crossref]

G. R. Ochs, A Resistance Thermometer for Measurement of Rapid Air Temperature Fluctuations, ESSA Tech. Report No. IER 47-ITSA 46 (U.S. Government Printing Office, Washington, D.C., 1967).

G. R. Ochs, A Circuit for the Measurement of Normalized Crosscorrelations, ESSA Tech. Report No. ERL 63-WPL 2 (U.S. Government Printing Office, Washington, D.C., 1968).

Sancer, M. I.

A. D. Varvatsis and M. I. Sancer, Can. J. Phys. 49, 1233 (1971).
[Crossref]

Schmeltzer, R. A.

R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1966).

Seidman, J. B.

Snyder, J. R.

Strohbehn, J. W.

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Varvatsis, A. D.

A. D. Varvatsis and M. I. Sancer, Can. J. Phys. 49, 1233 (1971).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964).

Yura, H. T.

Zwang, L. R.

L. R. Zwang, Izv. Akad. Nauk SSSR Ser. Geofiz. 8, 1252 (1960).

Appl. Opt. (3)

Can. J. Phys. (1)

A. D. Varvatsis and M. I. Sancer, Can. J. Phys. 49, 1233 (1971).
[Crossref]

IEEE Trans. Antennas Propag. (1)

W. P. Brown, IEEE Trans. Antennas Propag. 15, 81 (1967).
[Crossref]

Izv. Akad. Nauk SSSR Ser. Geofiz. (1)

L. R. Zwang, Izv. Akad. Nauk SSSR Ser. Geofiz. 8, 1252 (1960).

J. Opt. Soc. Am. (11)

Proc. IEEE (3)

J. W. Strohbehn, Proc. IEEE 56, 1301 (1968).
[Crossref]

A. L. Buck, Proc. IEEE 55, 448 (1967).
[Crossref]

A. Ishimaru, Proc. IEEE 57, 407 (1969).
[Crossref]

Q. Appl. Math. (1)

R. A. Schmeltzer, Q. Appl. Math. 24, 339 (1966).

Radio Sci. (1)

D. A. de Wolf, Radio Sci. 2, 1379 (1967); Radio Sci. 3, 308 (1968).

Other (8)

M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, Oxford, 1964).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

M. Greenebaum (private communication).

G. R. Ochs, A Circuit for the Measurement of Normalized Crosscorrelations, ESSA Tech. Report No. ERL 63-WPL 2 (U.S. Government Printing Office, Washington, D.C., 1968).

D. A. de Wolf (private communication).

M. A. Efroymson, in Multiple Regression Analysis in Mathematical Methods for Digital Computers, Volume 1, edited by Anthony Ralston and Herbert S. Wilf (Wiley, New York, 1960).

G. R. Ochs, A Resistance Thermometer for Measurement of Rapid Air Temperature Fluctuations, ESSA Tech. Report No. IER 47-ITSA 46 (U.S. Government Printing Office, Washington, D.C., 1967).

E. C. Alcaraz and P. M. Livingston, in Proceedings Electro-Optical Design Conference, 1972–West (Indus, and Scientific Management Inc., Chicago, Ill., 1971), p. 76.

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Figures (15)

Fig. 1
Fig. 1

Comparison of experimental focal-spot shapes (wavy line) with gaussian curve fits (dotted line) to the 1-σ width. Data were taken at 0.633-μm wavelength over a path of 1.75 km. Curve A, C n = 2.2 × 10 7 m 1 2; curve B, C n = 1.5 × 10 7 m 1 2; curve C, C n = 5 × 10 8 m 1 2.

Fig. 2
Fig. 2

Transmitter optical diagram. (B1) Ge beam splitter, (C1) visible beam expander and spatial filter, (C2) ir beam expander, (L1) 50-mW He–Ne laser, (L2) 5-W CO2 laser, (M1, M3, M5, M6) flat mirrors, (M4) 12.7-cm FL f/4 off-axis parabolic mirror, (7) 610-cm FL f/15 parabolic mirror.

Fig. 3
Fig. 3

Receiver optical configuration. (B) Incident focused laser beam, (M1) 91.5-cm diam, 12.2-m FL spherical mirror, (M2) flat folding mirror, (M3) rotating scanning mirror, (M4) 40-cm FL off-axis parabolic mirror, (D) dual-wavelength detector assembly.

Fig. 4
Fig. 4

Dual-wavelength detector assembly. (D2) 30-mm × 2-mm TGS detector, (PA) TGS detector preamplifier, (BS) Ge beam splitter; (S) 30-mm × 0.1-mm slit, (M) flat mirror, (F) 0.63-μm bandpass filter, (PM) S-20 photomultiplier detector.

Fig. 5
Fig. 5

Four-trace oscilloscope display of focused beam profiles, 1.18-km path, C n = 3.57 × 10 7 m 1 3: upper traces, waveform coadder output, 0.63-μm top trace, 10.6-μm 2nd trace; lower traces, superposition of several individual video scans, 0.63-μm 3rd trace, 10.6-μm 4th trace; average beam spread, 122 μrad at 0.633 μm and 58.5 μrad at 10.6 μm; time axis, 0.5 ms/large div.

Fig. 6
Fig. 6

Beam-wander signal, 1.75-km path, C n = 1.50 × 10 7 m 1 3, upper trace 0.633 μm, lower trace 10.6 μm, time axis 0.2 s/div.

Fig. 7
Fig. 7

Optical Cn vs thermal Cn. Data for 0.4-, 1.18-, and 1.75-km paths. Fitted equation, Cn (opt) = 7.23 × 10−8 + 1.14Cn (therm); correlation coefficient for curve fit ρ is 0.845.

Fig. 8
Fig. 8

Beam spread and Cn vs time of day, 1.75-km path, beam-spread data: ○, 0.633 μm, □, 10.6 μm; Cn measurements, □, 200-m 0.633-μm scintillometer, ○, thermal Cn, Δ, long-path 10.6-μm scintillometer.

Fig. 9
Fig. 9

Short-term beam spread vs e−20.9Cn0.85z0.62k00.65. Data for 0.4-, 1.18-, 1.75-km paths. Fitted equation, θ2 = −1.7 × 10−9 + 1.44 × 10−9Cn0.85z0.62k00.65; multiple-regression coefficient for curve fit is 0.893.

Fig. 10
Fig. 10

Infrared vs visible beam wander. Data for 0.4-, 1.18-, 1.75-km paths. Fitted equation, Φir2 = 3.82 × 10−10 + 0.93Φvis3; correlation coefficient for curve fit ρ is 0.953.

Fig. 11
Fig. 11

Beam size vs beam wander. Data for 0.4-, 1.18-, 1.75-km paths and for λ = 0.633 μm. Fitted equation, θ2 = 8.8 × 10−10 + 2.47Φ3; correlation coefficient for curve fit ρ is 0.934.

Fig. 12
Fig. 12

Beam wander vs Cn2z. Data for 0.4-, 1.18-, 1.75-km paths. Fitted equation, Φ2 = 1.49 × 10−9 + 14.1Cn2z; correlation coefficient for curve fit ρ is 0.785.

Fig. 13
Fig. 13

Beam wander vs Cn2z5/6k01/6. Data for 0.4-, 1.18-, 1.75-km paths. Fitted equation, Φ2 = 5.64 × 10−9 + 3.56Cn2z5/6k01/6; correlation coefficient for curve fit ρ is 0.740.

Fig. 14
Fig. 14

Total beam broadening vs Cn2Az1.2k00.40. Data for 0.4-, 1.18-, 1.75-km paths. Fitted equation, θ′2 = 4.9 × 10−9 + 5.67Cn2.4z1.2k00.40; correlation coefficient for curve fit ρ is 0.823.

Fig. 15
Fig. 15

Total beam broadening vs e−12.9Cn1.2z1.04k00.33. Data for 0.4-, 1.18-, 1.75-km paths. Fitted equation, θ′2 = 4.08 × 10−10 + 2.81 × 10−6Cn1,2z1.04k00.33; multiple-regression coefficient for curve fit is 0.900.

Tables (3)

Tables Icon

Table I Summary of curve-fits for beam-spread and beam-wander data.

Tables Icon

Table II Residual correlation analysis.

Tables Icon

Table III Confidence limits for empirical curve fits of the form θ 2 θ 2 } = a 0 X + b 0, with X = αCnβzγk0δ.

Equations (36)

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I ( w ) = ( k 0 2 π z ) 2 d r F ( r , z ) e 1 2 D ( r , z ) e i k w · r / z .
F ( r , z ) = d R U ( R + 1 2 r ; 0 ) U * ( R 1 2 r ; 0 ) e i k 0 R · r / z ,
D ( r , z ) = 4 π k 0 2 z 0 1 d κ Φ n ( κ ) × ( 1 exp [ i κ · r ( 1 y ) ] ) .
Φ n ( κ ) = 0.033 C n 2 exp ( κ 2 / κ 2 ) κ 0 11 / 3 ( 1 + κ 2 / κ 0 2 ) 11 / 6 ,
D n ( r ) = C n 2 r 2 3 ; l 0 < r < L 0 ,
D n ( r ) = ( n ( R ) n ( R + r ) ) 2 AV ,
w 2 = N w 2 I ( w ) d w ,
N 1 = I ( w ) d w .
w 2 = ( z k 0 ) 2 [ 2 ρ 0 2 + ( d 2 d r 2 e 1 2 D + 1 r d e 1 2 D d r ) r = 0 ]
θ t 2 = θ 0 2 + 8 k 0 2 ( d 2 d r 2 e 1 2 D ( r ) + 1 r d e 1 2 D ( r ) d r ) r = 0 ,
θ 0 2 = 16 k 2 ρ 0 2
θ t 2 = θ 0 2 + 32 π 2 z 3 L 1 ,
L 1 [ 1 2 × 0.033 C n 2 κ m 1 3 ] 1 .
θ t 2 = θ 0 2 + 32 π 2 z 3 L 1 ( 1 f ) ,
Φ 2 = 32 π 2 z f 3 L 1 ,
θ = ( θ t 2 θ 0 2 ) 1 2 ,
C n = 77.6 × 10 6 ( P / T 2 ) ( 1 + 0.00752 λ 2 ) × Δ T 2 1 2 r 1 2 ,
C l s ( 0 ) = 0.124 k 0 7 / 6 z 11 / 6 C n 2 ,
C n ( optical ) = 7.23 × 10 8 [ m 1 3 ] + 1.14 C n ( thermal ) .
X = α C n β z γ k 0 δ ,
Q = a 0 X + b 0 .
U = Y a 0 X + b 0
x θ 2 C a 0 + A b 0 B a 0 A B X , x θ 2 1.67 × 10 8 + 1.22 X ;
x 0 θ 2 C b 0 + ( C a 0 + A b 0 ) X , x 0 θ 2 4.47 × 10 19 + 1.31 X .
Φ ir 2 = 3.82 × 10 10 + 0.930 Φ vis 2 ,
Φ 2 C n 2 z 5 / 6 k 0 1 / 6 .
θ = ( θ 2 + Φ 2 ) 1 2 ,
θ 2 = 2.9 × 10 5 C n 6 / 5 z k 0 1 2 .
θ 2 = 1.7 × 10 9 + 1.44 × 10 9 C n 0.85 z 0.62 k 0 0.65 .
θ 2 = 4.9 × 10 9 + 5.67 C n 2.4 z 1.04 k 0 0.33 .
θ 2 = 4.08 × 10 10 + 2.81 × 10 6 C n 1.2 z 1.04 k 0 0.33 .
θ T 2 = θ 0 2 + 18.3 C n 2 κ m 1 3 z .
θ 2 C n 2 z k 1 1 3
k 1 = { κ m k 0 2 z κ m 5 / 3 C n 2 1 ( π 2 C n 2 k 2 z ) 3 / 5 k 0 2 z κ m 5 / 3 C n 2 1 .
1 2 D ( 0 , z ) C n 2 z k 0 2 κ m 1 3 N f 2 F 3 × ( 1 6 , 3 2 ; 5 2 , 2 , 2 , κ m 2 ρ 0 2 ) ,
N f 2 π ρ 0 2 / λ z > 1 ,