Abstract

Both a pulsed laser and a helium–neon laser have been used to examine the magnitude of scintillation as a function of range and turbulence strength over a near-earth, horizontal path. A photo-optical technique was utilized to record directly a 61-cm cross section of a received laser beam. Photographs of beam cross sections made at ranges of from 200 to 1500 m were used to compute log-irradiance variances. Simultaneous with the optical data, measurements of the index-structure coefficient Cn were made by use of a high-speed thermal technique. The data are used to test the spherical-wave equation that gives the log-amplitude variance as a function of range and Cn. The measurements indicate that the variance increases for ranges up to about 700 m, at which distance saturation occurs, i.e., no further growth of the variance is observed. The data are also compared with the saturation equations of Tatarski and deWolf. In addition, the effect of the transmitter-beam divergence on the magnitude of scintillation is examined. Finally, a modification of the inertial subrange of the Kolmogorov turbulence model is suggested to explain the occurrence of particular optical effects observed during temperature-inversion conditions.

© 1969 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 Book Co., New York, 1960), Ch. 9.
  2. V. V. Pisareva, Soviet Phys.—Acousti. Engl. Transl. 6, 81 (1960).
  3. Y. L. Feinberg, Propagation of Radiowaves along the Earth’s Surface (Akad. Nauk., Moscow, USSR, 1961).
  4. C. E. Coulman, J. Opt. Soc. Am. 56, 1232 (1966).
    [Crossref]
  5. D. L. Fried, J. Opt. Soc. Am. 57, 268 (1967).
    [Crossref]
  6. W. P. Brown, J. Opt. Soc. Am. 57, 1539 (1967).
    [Crossref]
  7. H. Siedentopf and F. Wisshak, Optik 3, 430 (1948).
  8. A. L. Buck, Appl. Opt. 6, 703 (1967).
    [Crossref] [PubMed]
  9. M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).
  10. L. R. Tsvang, IzvestiaANSSSR, Geophys. Ser. (1960), No. 8, p. 1252.
  11. Reference 1, p. 211.
  12. Reference 1, Eq. (9.43). The variance of the log amplitude, 〈χ2〉, is one fourth of σ2, the variance of the log irradiance.
  13. D. L. Fried, J. Opt. Soc. Am. 57, 175 (1967), Eq. (1.13).
    [Crossref]
  14. V. I. Tatarski, Radiofizika 10, 48 (1967).
  15. D. A. deWolf, J. Opt. Soc. Am. 58, 461 (1968), Eq. (23).
    [Crossref]
  16. A more complicated form of the saturation equation was fitted by Tatarski to the data of Gracheva and Gurvich. The shape of the modified curve differs from the general saturation equation in that it follows the plane-wave equation, Eq. (1), up to a value of about unity before falling off to saturate at a maximum value near two. deWolf incorrectly evaluated the general formulation [Ref. 15, Eq. (10), p. 463] by using the plane-wave equation to compute σ12.
  17. Reference 1, p. 55 ff.
  18. P. H. Deitz, in Modern Optics, J. Fox, Ed. (Polytechnic Press, New York, 1967).
  19. The XM–23 E2 Range Finder, supplied by Frankford Arsenal, Philadelphia, Pa.
  20. Cn= CT(−79×10−6p/T2), where p is pressure in millibars and T is absolute temperature. The apparatus used for the Cn measurements is manufactured by Flow Corporation and consists of two high-speed temperature probes, an anemometer bridge, model HWB-3, a secondary voltage supply, model HWI-3, and a random signal voltmeter, model 12A-1.
  21. Reference 15. deWolf has argued that I follows a Rice distribution, which, while tending to a log-normal distribution for weak fluctuations, tends to a Rayleigh distribution in the limit of strong fluctuations and/or long paths.
  22. Where P(I) = 1−exp(−I).
  23. Reference 1, Eq. (12.1).
  24. Equation (1) is the Rytov-approximated estimate of the logarithmic ratio of variance to square mean.
  25. Gracheva and Gurvich eliminated similar statistics from their graphs (Ref. 9).
  26. M. E. Gracheva, Radiofizika 10, 775 (1967), Fig. 3.
  27. The plane divergence of the cw laser was ~0.77 mrad and ~1.2 mrad for the pulsed laser.
  28. G. E. Mevers, private communication. In these experiments, an argon laser beam was propagated to a diffuse target board. The scattered return light was monitored temporally by a receiver near the transmitter.
  29. A system with a divergent beam has a complex equivalence to a diffraction-limited transmitter with an appropriately adjusted aperture and propagation path.
  30. Reference 18, Fig. 4, p. 760.
  31. The primary difficulty encountered in the spatial measurement arises from the variation in low-spatial-frequency content of the laser cross sections due both to profile effects and, in some cases, to beam breakup. Power-spectrum computations of irradiance scans show that the principal power is located at low frequencies. Statistical variation in low frequencies due to beam effects can contribute substantially to the spread in the variance plots, since the variance is given by the integral of the log-irradiance power spectrum. Variations in the saturation plot are also imposed by the reliability of the Cn measurement. Although the experiments were performed over level, open terrain, specially prepared for such experiments, there were still statistical variations of Cn along the optical path. In addition, experiment limitations permit only about a factor-or-two reliability under very weak-turbulence conditions. In spite of these constraints, we believe the phenomenon and general trend of the saturation effect to be accurately portrayed.
  32. F. P. Carlson, private communication. Unpublished results indicate that beam-wave theory may extend the region of applicability of the Rytov procedure.
  33. Reference 1, Fig. 11, p. 140.
  34. Reference 1, Eqs. (7.87) and (7.88), p. 151.
  35. In the limit of a large inner scale, the turbulence spectrum and the optical filter function would have little common area, and the irradiance fluctuations would approach zero (Ref. 1, Fig. 10, p. 139). If such variations exist in the turbulence spectrum, performance predictions of atmospheric-optical systems based on the single parameter Cn may prove to be inadequate.

1968 (1)

1967 (6)

1966 (1)

1965 (1)

M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).

1960 (2)

L. R. Tsvang, IzvestiaANSSSR, Geophys. Ser. (1960), No. 8, p. 1252.

V. V. Pisareva, Soviet Phys.—Acousti. Engl. Transl. 6, 81 (1960).

1948 (1)

H. Siedentopf and F. Wisshak, Optik 3, 430 (1948).

Brown, W. P.

Buck, A. L.

Carlson, F. P.

F. P. Carlson, private communication. Unpublished results indicate that beam-wave theory may extend the region of applicability of the Rytov procedure.

Coulman, C. E.

Deitz, P. H.

P. H. Deitz, in Modern Optics, J. Fox, Ed. (Polytechnic Press, New York, 1967).

deWolf, D. A.

Feinberg, Y. L.

Y. L. Feinberg, Propagation of Radiowaves along the Earth’s Surface (Akad. Nauk., Moscow, USSR, 1961).

Fried, D. L.

Gracheva, M. E.

M. E. Gracheva, Radiofizika 10, 775 (1967), Fig. 3.

M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).

Gurvich, A. S.

M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).

Mevers, G. E.

G. E. Mevers, private communication. In these experiments, an argon laser beam was propagated to a diffuse target board. The scattered return light was monitored temporally by a receiver near the transmitter.

Pisareva, V. V.

V. V. Pisareva, Soviet Phys.—Acousti. Engl. Transl. 6, 81 (1960).

Siedentopf, H.

H. Siedentopf and F. Wisshak, Optik 3, 430 (1948).

Tatarski, V. I.

V. I. Tatarski, Radiofizika 10, 48 (1967).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1960), Ch. 9.

Tsvang, L. R.

L. R. Tsvang, IzvestiaANSSSR, Geophys. Ser. (1960), No. 8, p. 1252.

Wisshak, F.

H. Siedentopf and F. Wisshak, Optik 3, 430 (1948).

Appl. Opt. (1)

Izv. Vuzov, Radiofizika (1)

M. E. Gracheva and A. S. Gurvich, Izv. Vuzov, Radiofizika 8, 717 (1965).

Izvestia (1)

L. R. Tsvang, IzvestiaANSSSR, Geophys. Ser. (1960), No. 8, p. 1252.

J. Opt. Soc. Am. (5)

Optik (1)

H. Siedentopf and F. Wisshak, Optik 3, 430 (1948).

Radiofizika (2)

V. I. Tatarski, Radiofizika 10, 48 (1967).

M. E. Gracheva, Radiofizika 10, 775 (1967), Fig. 3.

Soviet Phys.—Acousti. Engl. Transl. (1)

V. V. Pisareva, Soviet Phys.—Acousti. Engl. Transl. 6, 81 (1960).

Other (23)

Y. L. Feinberg, Propagation of Radiowaves along the Earth’s Surface (Akad. Nauk., Moscow, USSR, 1961).

V. I. Tatarski, Wave Propagation in a Turbulent Medium (McGraw–Hill Book Co., New York, 1960), Ch. 9.

The plane divergence of the cw laser was ~0.77 mrad and ~1.2 mrad for the pulsed laser.

G. E. Mevers, private communication. In these experiments, an argon laser beam was propagated to a diffuse target board. The scattered return light was monitored temporally by a receiver near the transmitter.

A system with a divergent beam has a complex equivalence to a diffraction-limited transmitter with an appropriately adjusted aperture and propagation path.

Reference 18, Fig. 4, p. 760.

The primary difficulty encountered in the spatial measurement arises from the variation in low-spatial-frequency content of the laser cross sections due both to profile effects and, in some cases, to beam breakup. Power-spectrum computations of irradiance scans show that the principal power is located at low frequencies. Statistical variation in low frequencies due to beam effects can contribute substantially to the spread in the variance plots, since the variance is given by the integral of the log-irradiance power spectrum. Variations in the saturation plot are also imposed by the reliability of the Cn measurement. Although the experiments were performed over level, open terrain, specially prepared for such experiments, there were still statistical variations of Cn along the optical path. In addition, experiment limitations permit only about a factor-or-two reliability under very weak-turbulence conditions. In spite of these constraints, we believe the phenomenon and general trend of the saturation effect to be accurately portrayed.

F. P. Carlson, private communication. Unpublished results indicate that beam-wave theory may extend the region of applicability of the Rytov procedure.

Reference 1, Fig. 11, p. 140.

Reference 1, Eqs. (7.87) and (7.88), p. 151.

In the limit of a large inner scale, the turbulence spectrum and the optical filter function would have little common area, and the irradiance fluctuations would approach zero (Ref. 1, Fig. 10, p. 139). If such variations exist in the turbulence spectrum, performance predictions of atmospheric-optical systems based on the single parameter Cn may prove to be inadequate.

Reference 1, p. 211.

Reference 1, Eq. (9.43). The variance of the log amplitude, 〈χ2〉, is one fourth of σ2, the variance of the log irradiance.

A more complicated form of the saturation equation was fitted by Tatarski to the data of Gracheva and Gurvich. The shape of the modified curve differs from the general saturation equation in that it follows the plane-wave equation, Eq. (1), up to a value of about unity before falling off to saturate at a maximum value near two. deWolf incorrectly evaluated the general formulation [Ref. 15, Eq. (10), p. 463] by using the plane-wave equation to compute σ12.

Reference 1, p. 55 ff.

P. H. Deitz, in Modern Optics, J. Fox, Ed. (Polytechnic Press, New York, 1967).

The XM–23 E2 Range Finder, supplied by Frankford Arsenal, Philadelphia, Pa.

Cn= CT(−79×10−6p/T2), where p is pressure in millibars and T is absolute temperature. The apparatus used for the Cn measurements is manufactured by Flow Corporation and consists of two high-speed temperature probes, an anemometer bridge, model HWB-3, a secondary voltage supply, model HWI-3, and a random signal voltmeter, model 12A-1.

Reference 15. deWolf has argued that I follows a Rice distribution, which, while tending to a log-normal distribution for weak fluctuations, tends to a Rayleigh distribution in the limit of strong fluctuations and/or long paths.

Where P(I) = 1−exp(−I).

Reference 1, Eq. (12.1).

Equation (1) is the Rytov-approximated estimate of the logarithmic ratio of variance to square mean.

Gracheva and Gurvich eliminated similar statistics from their graphs (Ref. 9).

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Figures (14)

Fig. 1
Fig. 1

Ray diagram of optical receiver.

Fig. 2
Fig. 2

Cross section of pulsed-laser beam at 200 m. Calibrated step tablet is also shown.

Fig. 3
Fig. 3

Cross section of pulsed-laser beam at 1000 m. Measurement field is 61 cm in diameter.

Fig. 4
Fig. 4

Irradiance scan taken from image shown in Fig. 3.

Fig. 5
Fig. 5

Cumulative distribution curve for irradiance scan shown in Fig. 4. The cumulative distribution on a probability scale is plotted (●—●) against the natural log of the normalized irradiance. Thus, a cumulative log-normal distribution plots as a straight line. A cumulative Rayleigh curve (○—○) is also plotted for comparison [P(I) = 1−exp(−I)].

Fig. 6
Fig. 6

Pulsed-laser measurements in which range was varied from 200 to 1500 m under high-Cn conditions ((● 200 m, △ 600 m, ○ 1000 m, × 1500 m). Measured natural-log standard deviations (σ) are plotted against G = ( 0.496 ) 1 2 κ 7 / 12 L 11 / 12 C n. Spherical-wave line (S) indicates a slope of unity and conformity to theory. Plane-wave line is designated P. Geometrical-optics saturation curves of Tatarski [Eq. (2)] are also shown (200-, 600-, 1000-, and 1500-m curves, lowest to highest).

Fig. 7
Fig. 7

Cn vs hour of day (EDST), 18 August 1967, 10-min-average values (λ = 6328 Å).

Fig. 8
Fig. 8

Beam-cross-section of cw laser taken at 0750 h, 18 August 1967, 650-m range, 2-msec exposure. Low-contrast patterns are due to operation of receiving system on highly coherent optical signal. Standard deviations of optical data are low; observed coherence effects are high.

Fig. 9
Fig. 9

Beam-cross-section photograph of cw laser taken at 1306 h, 18 August 1967, 650-m range, 2-msec exposure. Standard deviations of optical data are high; observed coherence effects are low.

Fig. 10
Fig. 10

Beam-cross-section photograph of cw laser taken at 2046 h, 18 August 1967, 650-m range, 2-msec exposure. Standard deviations of optical data are high; observed coherence effects are high.

Fig. 11
Fig. 11

cw- (○) and pulsed-laser (●) measurements in which range was held constant at 650 m while Cn varied over two orders of magnitude. Axis units are identical to Fig. 6. Tatarski’s saturation equation (T) is plotted for a range of 650 m.

Fig. 12
Fig. 12

Combined pulsed-laser measurements from Figs. 6 and 11. Tatarski’s saturation equation (T) is plotted for a range of 650 m. deWolf’s equation [Eq. (4)] is also shown (d). Axis units are identical to Fig. 6.

Fig. 13
Fig. 13

Laser measurements taken under temperature inversion conditions. Axis units are identical to Fig. 6. Figure 13(a) shows pulsed (●) and cw (○) data from 18 August 1967. Figure 13(b) shows cw data from 27 March 1968.

Fig. 14
Fig. 14

Beam-cross-section photograph of cw laser taken at 2050 h on 27 March 1968, 650 m, 2-msec exposure. This unusual optical effect is seen only during temperature-inversion conditions.

Tables (2)

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Table I Divergence test No. 1.

Tables Icon

Table II Divergence test No. 2.

Equations (7)

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σ T 2 = 1.23 C n 2 k 7 / 6 L 11 / 6 ,
σ 2 = 4 [ 1 - ( 1 + 6 σ 1 2 ) - 1 / 6 ] ,
σ 1 2 = π 2 0.033 Γ ( 7 / 6 ) 24 C n 2 L 3 κ m 7 / 3 .
σ 2 = ln [ 2 - exp ( - σ T 2 ) ] ,
( T r 1 - T r 2 ) 2 = D T ( r ) = C T 2 r 2 3 ,             l 0 r L 0 .
( n r 1 - n r 2 ) 2 = D n ( r ) = C n 2 r 2 3 ,             l 0 r L 0 ,
σ 2 = ln { 1 + [ σ N 2 / ( Ī ) 2 ] } ,