Abstract

This paper presents a series of studies on the effect of low-frequency fluctuations of the atmospheric refractive index on phase fluctuations of space-limited optical beams. Theoretical calculations have been made in the approximation of a smooth perturbation method. Numerous experimental data on statistical characteristics of optical wave phase fluctuations are given. These are used to verify the theoretical conclusions and to solve an inverse problem on the determination of the atmospheric turbulence characteristics.

© 1981 Optical Society of America

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1976 (5)

V. P. Lukin, V. V. Pokasov, V. L. Mironov, S. S. Khmelevtsov, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, No. 5, 550 (1976).

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O. N. Emaleev, V. P. Lukin, V. V. Pokasov, V. M. Sazanovich, S. S. Khmelevtsov, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 9, 100 (1976).

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V. P. Lukin, V. V. Pokasov, V. L. Mironov, S. S. Khmelevtsov, Radiotekh. Elektron. 20, 1164 (1975).

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A. S. Gurvitch, N. S. Time, Izv. Akad. Nauk SSSR Mekh. Zhidk. & Gaza 12, No. 6, (1973).

V. P. Lukin, V. V. Pokasov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, No. 11, 1726 (1973).

E. I. Gelfer, A. I. Kon, A. N. Cheremykhin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, No. 2, 723 (1973).

P. M. Livingston, Appl. Opt. 11, 684 (1973).
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1972 (10)

G. W. Reinhardt, S. A. Collins, J. Opt. Soc. Am. 62, 1526 (1972).
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A. I. Kon, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, No. 4, 533 (1972).

G. I. Arsenyan, F. F. Pashkov, A. A. Semenov, A. A. Tishchenko, N. N. Rimski, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, No. 8, 1228 (1972).

Yu. I. Bekhtin, Prib. Tekh. Eksp. No. 5, 146 (1972).

N. S. Time, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 8, No. 1, 90 (1972).

A. Consortini, L. Ronchi, Appl. Opt. 11, 1205 (1972).
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H. T. Yura, Appl. Opt. 11, 1399 (1972).
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Yu. I. Bekhtin, Prib. Tekh. Eksp. No. 5, 146 (1972).

V. P. Lukin, V. V. Pokasov, S. S. Khmelevtsov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, No. 12, 1861 (1972).

V. S. Klyatskin, V. I. Tatarskii, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, 1433 (1972).

1971 (8)

D. K. Barton, IEEE Trans. Antennas Propag. AP-19, 566 (1971).
[CrossRef]

V. I. Vaitsel, Radiotekh. Elektron. 16, 439 (1971).

Yu. I. Bekhtin, A. V. Shabelnikov, Prib. Tekh. Eksp. No. 4, 207 (1971).

V. P. Linnik, G. M. Bryanskaya, E. A. Sanotskaya, Opt. Mekh. Promst. No. 11, 27 (1971) [Sov. J. Opt. Tech. 38, 671 (1971)].

T. Miyamoto, K. Yasuura, Appl. Opt. 10, 161 (1971).
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M. Bertolotti, Appl. Opt. 10, 42 (1971).
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R. G. Buser, J. Opt. Soc. Am. 61, 488 (1971).
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S. F. Clifford, G. M. B. Bouricius, G. R. Ochs, M. H. Ackley, J. Opt. Soc. Am. 61, 1279 (1971).
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T. L. Ho, J. Opt. Soc. Am. 60, 667 (1970).
[CrossRef]

G. M. B. Bouricius, S. F. Clifford, J. Opt. Soc. Am. 60, 1484 (1970).
[CrossRef]

M. Bertolotti, L. Muzii, D. Sette, J. Opt. Soc. Am. 60, 1603 (1970).
[CrossRef]

M. Bertolotti, M. Carnevale, B. Daino, D. Sette, Appl. Opt. 9, 962 (1970).
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D. A. Gray, A. T. Waterman, J. Geophys. Res. 75, (1970).
[CrossRef]

A. I. Kon, Z. I. Feizulin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, No. 1, 71 (1970).

Z. I. Feizulin, Radiotekh. Elektron. 15, No. 7, 1390 (1970).

R. S. Lawrence, L. W. Strohbehn, IEEE Trans. AP-58, No. 10, 130 (1970).

V. I. Vaitsel, S. S. Khmelevtsov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 13, No. 7, 1072 (1970).

A. A. Semenov, A. N. Gordeev, G. I. Arsenyan, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 11, 103 (1970).

V. F. Turchin, V. P. Kozlov, M. S. Malkevich, Usp. Fiz. Nauk 102, No. 3, 345 (1970).

A. V. Shabelnikov, Radiotekh. Elektron. 15, 1077 (1970).

V. I. Vaitsel, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 5, 143 (1970).

S. S. Khmelevtsov, R. Sh. Tsvyk, Izv. Vyssh. Uchebn. Zaved. Fiz. No. 2, 52 (1970).

V. I. Tatarski, Rasprostranenje korot kikh voln v sluchaino neodnorodnoj srede v priblizhenii markovskogo sluchainogo protsessa, Preprint Otdeleniya Okeanografii, (“Short-Wave Propagation in a Randomly Inhomogeneous Medium in the Markovian Random Process Approximation,” Preprint of the Oceanography Department) Izv. Akad. Nauk SSSR Fiz. Atmos. Geogr. No. 1, 15 (1970).

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V. I. Klyatskin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 12, No. 5, 723 (1969).

D. L. Fried, IEEE Trans. AP-57, No. 4, 49 (1969).

R. W. Lee, L. C. Harp, IEEE Trans. 57, No. 4, 7 (1969).

F. P. Carlson, A. Ishimaru, J. Opt. Soc. Am. 59, 319 (1969).
[CrossRef]

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A. Peskoff, J. Opt. Soc. Am. 58, 1032 (1968).
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M. Bertolotti, M. Carnevale, L. Muzii, D. Sette, Appl. Opt. 7, 2246 (1968).
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V. I. Shishov, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, No. 6, 866 (1968).

A. S. Gurvich, M. A. Kalistratova, N. S. Time, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 11, No. 9 (1968).

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E. I. Gelfer, A. I. Kon, A. N. Cheremykhin, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 16, No. 2, 723 (1973).

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V. P. Lukin, V. V. Pokasov, S. S. Khmelevtsov, Variatsii vneshnego mashtaba atmosfernoj turbulentnosti v osadkakh (“Variations of the Outer Scale of Atmospheric Turbulence in Precipitation”) in Abstracts of Reports of Siberian Symposium on Laser Spectroscopy (Izd. Institute of Physics, Krasnoyarsk, 1973), pp. 79 and 80.

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V. P. Lukin, V. V. Pokasov, V. L. Mironov, V. M. Sazanovich, Izv. Akad. Nauk SSSR Fiz. Atmos. Okeana 12, No. 12, 1317 (1976).

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V. P. Lukin, V. V. Pokasov, S. S. Khmelevtsov, Variatsii vneshnego mashtaba atmosfernoj turbulentnosti v osadkakh (“Variations of the Outer Scale of Atmospheric Turbulence in Precipitation”) in Abstracts of Reports of Siberian Symposium on Laser Spectroscopy (Izd. Institute of Physics, Krasnoyarsk, 1973), pp. 79 and 80.

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[CrossRef]

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G. I. Arsenyan, F. F. Pashkov, A. A. Semenov, A. A. Tishchenko, N. N. Rimski, Izv. Vyssh. Uchebn. Zaved. Radiofiz. 15, No. 8, 1228 (1972).

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Yu. N. Barabanenkov, Yu. A. Kravtsov, S. M. Rytov, V. I. Tatarskii, Usp. Fiz. Nauk 102, 3 (1970) [Sov. Phys. Usp. 13, 551 (1971)].

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S. M. Rytov, Vvedenje v statistiskuj fiziku (Introduction in Statistical Physics) (Nauka Press, Moscow, 1966).

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I. S. Gradshtein, I. M. Ryzhik, Tablitsi integralov, sum, ryadov i proizvedenij (Tables of Integrals, Sums, Series and Products) (Nauka Press, Moscow, 1971).

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P. A. Bakut, I. N. Troitski, A. A. Demin, A. N. Safronov, Zarubezhnaya Radioelektronika No. 11, 3 (1978).

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V. P. Linnik, G. M. Bryanskaya, E. A. Sanotskaya, Opt. Mekh. Promst. No. 11, 27 (1971) [Sov. J. Opt. Tech. 38, 671 (1971)].

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Prib. Tekh. Eksp. No. 5 (2)

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V. V. Pokasov, in Abstracts of the All-Union Symposium on Laser Radiation Propagation in the Atmosphere (Izd. Tomsk State U., 1971).

V. P. Lukin, V. V. Pokasov, Opredelenie spektra fluktuatsij pokazatelya prelomleniya iz fazovikh izmerenij (“Determination of the Refractive Index Fluctuation Spectrum Using Phase Measurements”) in Problems of Remote Sounding of the Atmosphere (Izd. Institute for Atmospheric Optics, Tomsk, 1975), pp. 160–170.

V. P. Lukin, Issledovanje opticheskikh voln v turbulentnoj atmosfere, (“Investigation of Phase Fluctuations of Optical Waves Propagating Through a Turbulent Atmosphere”) Thesis, Tomsk State U. (1975).

V. P. Lukin, V. V. Pokasov, O. L. Tuzov, Fazovie fluktuatsii opticheskikh voln v prizemnom sloe atmosferi (“Phase Fluctuations of an Optical Wave Propagating in an Atmospheric Boundary Layer”) in Abstracts of the Tenth All-Union Conference on Radio-Wave Propagation (Nauka Press, Moscow, 1972), Sec. 4, pp. 377–381.

V. P. Lukin, V. V. Pokasov, S. S. Khmelevtsov, Variatsii vneshnego mashtaba atmosfernoj turbulentnosti v osadkakh (“Variations of the Outer Scale of Atmospheric Turbulence in Precipitation”) in Abstracts of Reports of Siberian Symposium on Laser Spectroscopy (Izd. Institute of Physics, Krasnoyarsk, 1973), pp. 79 and 80.

V. P. Lukin, Vliyanie konechnosti vneshnego mashtaba turbulentnosty na fazovie fluktuatsii opticheskikh voln v prizemnom sloe atmosferi (“On the Influence of the Turbulence Outer-Scale Finiteness on Phase Fluctuations in an Optical Wave Propagating in a Boundary Atmospheric Layer”) in Abstracts of the Jubilee Scientific-Technical Conference of the Radiophysical Department of the Tomsk State U. (Izd. Tomsk State U., 1973), Part 2, pp. 100–104.

S. M. Rytov, Vvedenje v statistiskuj fiziku (Introduction in Statistical Physics) (Nauka Press, Moscow, 1966).

A. M. Obukhov, Izv. Akad. Nauk SSSR Ser. Geofiz., in press.

V. I. Tatarski, Rasprostranenje voln v turbulentnoj atmosfere (Wave Propagation in a Turbulent Atmosphere) (Nauka Press, Moscow, 1967).

L. Goldstein, P. A. Mailes, A. Chabot, IEEE Trans., in press.

A. S. Gurvich, A. I. Kon, V. L. Mironov, S. S. Khmelevtsov, Lazernoe izluchenie v turbulentnoj atmosfere, (Laser Radiation in a Turbulent Atmosphere) (Nauka Press, Moscow, 1976).

A. N. Tikhonov, V. Ya. Arsenin, Metody resheniya nekorrektnikh zadach (Nauka Press, Moscow, 1974).

A. S. Monin, A. M. Yaglom, Statisticheskaya gidromekhanika (Statistical Hydromechanics, Part 1) (Nauka Press, Moscow, 1965).

N. Ts. Gomboev, E. V. Zubritski, G. F. Malygina, V. L. Mironov, S. S. Khmelevtsov, Opredelenje vnutrennego mashtaba turbulentnosti ir opticheskikh izmerenij (“Determination of a Turbulence Inner Scale Using Optical Measurements”) in Abstracts of the Third All-Union Symposium on Laser Radiation Propagation in the Atmosphere (Izd. Inst. Atmosph. Optics, 1975), pp. 183 and 184.

S. S. Zilitinkevich, Dinamika pogranichnogo sloya atmosfery (Dynamics of the Atmospheric Boundary Layer) (Gidrometeoizdat, Leningrad, 1970).

G. Jenkins, G. D. Watts, Spectral Analysis and Its Applications (Holden-Day, San Francisco, 1968; Mir, Moscow, 1971).

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Figures (18)

Fig. 1
Fig. 1

Normalized correlation functions of phase fluctuations;

Fig. 2
Fig. 2

Phase correlation coefficient bs(a) in a collimated beam for various parameter values [1,2,3 − bs(a) with parameter values d = 10−4, 10−3, 10−2].

Fig. 3
Fig. 3

Normalized correlation functions of the phase difference fluctuations for the base ρ − (1 − x 0 ρ = 0.2, 2 − x 0 ρ = 1.0, 3 − x 0 ρ = 3.0). The initial beams are collimated (q1 = 1, λ = 0.63 μm, L = 100 m). The dashed line shows the phase fluctuation correlation coefficient on the axis of one of the beams.

Fig. 4
Fig. 4

Spectral density of phase fluctuations as a function of dimensionless frequency a = 2πf/ x 0 v (1, the plane wave, q1 = 10−5; 2, the spherical wave, q1 = 105, x 0 2 L / k = 10 - 4 , x 0 2 / x m 2 = 10 - 6; 3, the limited collimated beam, q1 = 1).

Fig. 5
Fig. 5

Spectral density of phase difference fluctuations for the base ρ(q1 = 1, q2 = 0, x 0 2 L / k = 10 - 4 , x 0 / x m = 10 - 3 , x 0 ρ = π.

Fig. 6
Fig. 6

(a) Block diagram of the experimental apparatus for measuring phase spatial fluctuations for small separations. (b) Schemes of mixing blocks (M is the piezoceramic modulator, FM is the photoreceiver).

Fig. 7
Fig. 7

(a) Optical scheme of the apparatus for investigating spatial fluctuations of the optical wave phase for large separations of the observation points (1,4, semitransparent mirror; 2,3, reflecting mirror; 5, modulator). (b) An optical scheme of the many-beam experiment.

Fig. 8
Fig. 8

Measurements of D s (ρ) in a collimated beam (curves 1 and 2 are obtained in different turbulence conditions along the path).

Fig. 9
Fig. 9

Measurement of D s (ρ) in a spherical wave.

Fig. 10
Fig. 10

Spatial phase structure function for large observation point separations. In the logarithmic coordinates along the axes the law D s (ρ) ~ ρ5/3 is illustrated as a straight line.

Fig. 11
Fig. 11

Phase structure function in a turbid atmosphere (○, measurements in a pure atmosphere; △, measurements in turbid atmosphere conditions). Vertical sections indicate a confidential interval of the experimental values.

Fig. 12
Fig. 12

Phase structure function. Continuous lines show the asymptotic forms: I, D s (ρ) ~ ρ2; II, D s (ρ) ~ ρ5/3.

Fig. 13
Fig. 13

Synchronous measurements of the phase structure function for a 10-cm base with temperature gradients and average wind speed [△, D s (ρ = 10 cm); x, △v; ○, △T].

Fig. 14
Fig. 14

Normalized correlation functions of the phase difference fluctuations for variable bases.

Fig. 15
Fig. 15

Normalized phase-difference fluctuation spectrum for 5-and 3-cm bases. The vertical section shows an 80% confidential interval. The straight line 1 corresponds to the power dependence of the spectrum f2, line 2, f−8/3.

Fig. 16
Fig. 16

Normalized phase structure functions: 1,3, calculated according to Eq. (31) (1, x 0 1 = 6.5 m−1; 3, x 0 1 = 3.2m−1); 2,4, calculated according to Eq. (32) (2, x 0 2 = 26.5 m−1; 4, x 0 2 = 11.8 m−1); 5, function f(y) = ay5/3, where a is chosen using its best fit with the experimental data, 6, calculated according to Eq. (31) with x 0 1 = 4.3 m−1.

Fig. 17
Fig. 17

A 1-D phase fluctuation spectrum: ○, △ x, ●, experimental data corresponding to Eq. (29) with x 0 1 = 4.3 m−1 and a simulated spectrum asymptotic form.

Fig. 18
Fig. 18

Agreement of meteorological and optical measurements of the inner scale.

Equations (65)

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D s ( r 1 , r 2 , τ ) = 2 π 2 0 L d x 0 d x x [ [ J 0 ( 2 i γ 2 x ρ 1 ) + J 0 ( 2 i γ 2 x ρ 2 ) - J 0 ( x p ) - J 0 ( x p * ) ] H 2 - [ 1 - J 0 ( x g ) ] H 2 - [ 1 - J 0 ( x g * ) ] H 2 * ] ϕ n ( x ) ,
B s ( r 1 , r 2 , τ ) = π 2 0 L d x 0 d x x [ { J 0 ( x p ) + J 0 ( x p * ) } H 2 - J 0 ( x g ) H 2 - J 0 ( x g * ) H 2 * ] ϕ n ( x ) ,
p 2 = ( γ 1 y d - 2 i γ 2 y c + V y τ ) 2 + ( γ 1 z d - 2 i γ 2 z c + V z τ ) 2 , g 2 = ( γ y d - V y τ ) 2 + ( γ z d - V z τ ) 2 , y d = ( y 1 - y 2 ) ,             z d = ( z 1 - z 2 ) , y c = 1 2 ( y 1 + y 2 ) ,             z c = 1 2 ( z 1 + z 2 ) , H 2 = k 2 exp ( - γ 2 ( L - x ) k x 2 ) , H 2 = - k 2 exp ( - i γ ( L - x ) k x 2 ) , ρ 1 = ( y 1 2 + z 1 2 ) 1 / 2 , ρ 2 = ( y 2 2 + z 2 2 ) 1 / 2 , γ = 1 - i α x 1 + i α x = γ 1 - i γ 2 .
u 0 ( y , z ) = exp [ - ( 1 a 0 2 + i k f 0 ) ( y 2 + z 2 ) ] ,
a 2 = a 0 2 [ ( 1 - α 2 L ) 2 + ( α 1 L ) 2 ] ,
ϕ n ( x ) = 0.033 C n 2 x - 1 / 3 exp ( - x 2 / x m 2 ) ,
D s ( ρ ) 3.28 k 2 C n 2 L l 0 - 1 / 3 ρ 2 ,             ρ l 0 ,
D s ( ρ ) 2.91 k 2 C n 2 L ρ 5 / 3 - 0.616 C n 2 k 7 / 6 L 11 / 6 [ 1 - b χ ( ρ ) ] , ρ l 0 , λ L l 0 ,
ϕ n ( x ) = 0.033 C n 2 ( x 2 + x 0 1 2 ) - 11 / 6 ,
ϕ n ( x ) = 0.033 C n 2 x - 11 / 6 [ 1 - exp ( - x 2 / x 0 2 2 ) ] ,
p 2 = v 2 τ 2 ,             g 2 = v 2 τ 2 ,             v 2 = V y 2 + V z 2 .
b s ( a ) = 0 1 d s 0 d x x ( x 2 + 1 ) - 11 / 6 J 0 ( a x ) exp [ - b 1 x 0 1 2 L k × ( 1 - s ) 2 x 2 ] { 1 + cos [ γ 1 x 0 1 2 k ( 1 - s ) x 2 ] } × ( 0 1 d s 0 d x x ( x 2 + 1 ) - 11 / 6 × exp [ - b 1 x 0 1 2 L k ( 1 - s ) 2 x 2 ] × { 1 + cos [ γ 1 x 0 1 2 L k ( 1 - s ) x 2 ] } ) - 1 ,
b 1 = q 1 ( 1 - q 2 ) 2 + q 1 2 ,             γ 1 = ( 1 - q 2 ) + ( q 1 2 + q 2 2 - q 2 ) s ( 1 - q 2 ) 2 + q 1 2 , α 1 L = q 1 ,             α 2 L = q 2 ;
u 0 ( ρ ) = exp { - ( ρ - ρ 0 j ) 2 2 a 0 j 2 - i k ( ρ - ρ 0 j ) 2 2 f 0 j } ,
ψ 1 ( ρ 0 1 , ρ 1 ) ψ 1 * ( ρ 0 2 , ρ 2 ) = 4 π 2 k 2 0 L d x 0 d x x ϕ n ( x ) × exp { - i x 2 2 k ( L - x ) [ φ 1 ( x ) - φ 2 * ( x ) ] } J 0 ( x { φ 1 ( x ) ρ 1 - φ 2 * ( x ) ρ 2 + [ 1 - φ 1 ( x ) ] ρ 0 1 - [ 1 - φ 2 * ( x ) ] ρ 0 2 } ) ,
ψ 1 ( ρ 0 1 , ρ 1 ) ψ 1 ( ρ 0 2 , ρ 2 ) = - 4 π 2 k 2 0 L d x 0 d x x ϕ n ( x ) × exp { - i x 2 2 k ( L - x ) [ φ 1 ( x ) + φ 2 ( x ) ] } J 0 ( x { φ 1 ( x ) ρ 1 - φ 2 ( x ) ρ 2 + [ 1 - φ 1 ( x ) ] ρ 0 1 - [ 1 - φ 2 ( x ) ] ρ 0 2 } ) ,
ρ j = ρ j ,             ρ 0 j = ρ 0 j ,             φ j = 1 + i α j x 1 + i α j L , 1 α j 2 = 1 a 0 j 2 + i k f 0 j ,             j = 1 , 2.
ψ 1 ( ρ 0 1 ) ψ 1 * ( ρ 0 2 ) = 4 π 2 k 2 0 L d x 0 d x x ϕ n ( x ) J 0 ( x ρ ) × exp { - i x 2 2 k ( L - x ) [ φ 1 ( x ) - φ 2 * ( x ) ] } ,
ψ 1 ( ρ 0 1 ) ψ 1 ( ρ 0 2 ) = - 4 π 2 k 2 0 L d x 0 d x x ϕ n ( x ) J 0 ( x ρ ) × exp { - i x 2 2 k ( L - x ) [ φ 1 ( x ) + φ 2 ( x ) ] } , ρ = ρ .
B s ( ρ ) = 1 2 Re [ ψ 1 ( ρ 0 1 ) ψ 1 * ( ρ 0 2 ) - ψ 1 ( ρ 0 1 ) ψ 1 ( ρ 0 2 ) ] .
( φ 1 ( x ) = φ 2 ( x ) = γ ,             φ 1 ( x ) - φ 2 * ( x ) = - 2 i γ 2 , φ 1 ( x ) + φ 2 ( x ) = 2 γ ,             γ 2 = - J m γ ,
B s ( ρ ) = 2 π 2 k 2 0 L d x 0 d x x ϕ n ( x ) J 0 ( x ρ ) exp [ - γ 2 x 2 ( L - x ) k ] × { 1 + cos [ γ 1 x 2 ( L - x ) k ] } .
Γ s ( τ ) = [ s ( ρ , t ) - s ( O , t ) ] [ s ( ρ , t + τ ) - s ( O , t + τ ) ] = 2 B s ( v τ ) - B s ( ρ + v τ ) - B s ( ρ - v τ ) .
γ s ( τ ) = Γ s ( τ ) Γ s ( O ) = b s ( v τ ) - 1 2 b s ( ρ - v τ ) - 1 2 b s ( ρ + v τ ) 1 - b s ( ρ ) ,
W s ( f ) = 4 0 d τ cos ( 2 π f τ ) B s ( v τ ) .
w s ( f ) = 0 L d x v 2 π f d x x ϕ n ( x ) f s ( x , x ) ( x 2 - 4 π 2 f 2 v 2 ) - 1 / 2 × [ 0 L d x v 0 d x x ϕ n ( x ) f s ( x , x ) ] - 1 ,
f s ( x , x ) = exp [ - γ 2 x 2 ( L - x ) k ] { 1 + cos [ γ 1 x 2 ( L - x ) k ] } .
ϕ n ( x ) = 0.033 C r 2 ( x 2 + x 0 2 ) - 11 / 6 exp ( - x 2 / x m 2 ) ,
( 2 π f x m v 1 ) .
w s ( f ) = Γ ( 11 6 ) 2 Γ ( 4 3 ) 0 L d x v C n 2 ( a 2 + 1 ) - 4 / 3 x 0 - 8 / 3 { [ exp ( - β 1 , x 0 2 a 2 ) ] × ψ 1 2 , - 1 3 ; β 1 x 0 2 ( a 2 + 1 ) + exp ( - β 2 x 0 2 a 2 ) 2 × ψ [ 1 2 , - 1 3 ; β 2 x 0 2 ( a 2 + 1 ) ] + exp ( - β 3 x 2 a 2 ) 2 ψ [ 1 2 , - 1 3 ; β 3 x 2 ( a 2 + 1 ) ] } × ( 0 L d x v C n 2 x 0 - 8 / 3 ) - 1 ,
β 1 = x m - 2 + q 1 ( 1 - s ) 2 ( 1 + q 1 2 ) · L k ,             β 2 = β 1 - i ( 1 - s ) ( 1 + q 1 2 s ) ( 1 + q 1 2 ) · L k , β 3 = β 2 * ,
I , plane wave :             q 1 = 10 - 5 ; II , spherical wave :             q 1 = 10 5 ; III , limited collimated beam :             q 1 = 1 ; x 0 2 L k = 10 - 4 , x m x 0 = 10 3 .
σ s 2 = 2 π 2 k 2 0 L d x 0 d x x ϕ n ( x ) f s ( x , x )
( 2 π f x m v 1 )
( 2 π f v L k 1 ,             2 π f x m v 1 )
w s ( f ) = [ 1 + ( 2 π f x 0 v ) 2 ] - 4 / 3 ,
ψ ( α , β ; z ) z - α [ 1 + 0 ( 1 z ) ] ,             z α , β ,
w s ( f ) = Γ ( 11 6 ) Γ ( 4 3 ) ( 2 π f x 0 v ) - 1 / 3 ( x m x 0 ) exp [ - ( 2 π f x m v ) 2 ] .
( 2 π f x m v 2 π f v L k 2 π f x 0 v ) .
w s ( f ) = 1 2 ( 2 π f x 0 v ) - 8 / 3 ,
w s = Γ ( 11 6 ) 2 Γ ( 4 3 ) ( 2 π f x 0 v ) - 8 / 3 ( 2 π f x m v ) - 1 exp [ - ( 2 π f x m v ) 2 ] .
W δ s ( f ) = 4 sin 2 ( π ρ f v ) W s ( f ) .
a W δ s ( a ) σ Δ s 2 ,
( a 1 , π ρ f v 1 )             a W δ s ( a ) σ Δ s 2 = 4 π ρ 2 v 2 f 3
( a 1 , π ρ f v 1 )
a W δ s ( a ) σ Δ s 2 = 4 ( π ρ f v ) 2 ( 2 π f x 0 v ) - 5 / 3
( π ρ f v 1 ) a W δ s ( a ) σ Δ s 2 = 4 sin 2 ( π ρ f v ) ( 2 π f x 0 v ) - 8 / 3 f
A ( r , t ) exp [ i Δ ϕ ( r , t ) + A 0 exp ( i k a cos Ω t ) ,
2 A 0 A ( r , t ) cos [ k a cos Ω t + Δ ϕ ( r , t ) ] ;
i Ω A ( r , t ) J 1 2 ( k a ) sin Δ ϕ ( r , t ) sin Ω t , i 2 Ω A ( r , t ) J 2 2 ( k a ) cos Δ ϕ ( r , t ) cos 2 Ω t ,
ϕ ( Δ , t ) = s ( Δ , t ) - ϕ ( - Δ , t ) ,
b ^ s ( y ) = { 1 - D s ( y ) / D s ( y m ) , y y m , 0 , y > y m ,
B s ( ρ ) = - + d 2 x F s ( x ) exp ( i x ρ ) , F ( x ) = ϕ r ( O , x ) f s ( x ) , f s ( x ) = π k 2 Re 0 L d x { exp [ - γ 2 x 2 ( L - x ) k ] + exp [ - i γ x 2 ( L - x ) k ] } ,
B s ( y ) = - + d x 2 V s ( x 2 ) exp ( i x 2 y ) ,
V s ( x 2 ) = - + d x 3 f s ( x 2 , x 3 ) ϕ n ( O , x 2 , x 3 ) .
V r ( O , x 2 ) = - + d x 3 ϕ r ( O , x 2 , x 3 ) .
ϕ n 1 ( O , x 2 , x 3 ) = 0.033 C n 2 ( x 01 2 + x 2 2 + x 3 2 ) - 11 / 6 ,
ϕ n 2 ( O , x 2 , x 3 ) = 0.033 C n 2 ( x 2 2 + x 3 2 ) - 11 / 6 { 1 - exp [ - ( x 2 2 + x 3 2 ) / x 02 2 ] } .
b s 1 ( p ˜ 1 ) = Γ ( 1 6 ) π - 1 ( ρ ˜ 1 / 2 ) 5 / 6 K - 5 , 6 ( ρ ˜ 1 ) ,
b s 2 ( ρ ˜ 2 ) = F 1 1 [ - 5 6 , 1 ; ( ρ ˜ 2 / 2 ) 2 ] - ( ρ ˜ 2 / 2 ) 5 / 3 Γ - 1 ( 11 6 ) ,
v ¯ s ( x 2 ) = Δ y π [ 1 + 2 k = 1 L - 1 b s ( y k ) w ( y k ) cos ( x 2 y k ) ] , w ( u ) = { 1 - 6 ( u / μ ) 2 + 6 ( u / μ ) 3 , u μ 2 , 2 ( 1 - u / μ ) 3 , μ 2 < u μ , 0 , u > μ ,
ϕ n ( x ) = 0.0197 C r 2 ( α z k ) 11 / 3 ( x α z k ) - 11 / 6 K - ( 11 / 6 ) ( x α z k )
D s ( ρ ) = 2.91 k 2 C r 2 L { [ ρ 2 + ( α z k ) 2 ] 5 / 6 - ( α z k ) 5 / 3 }             L k 2 α 2 z k 2 ,
D s ( ρ 2 ) / D s ( ρ 1 ) = ( α z k ) 1 / 3 ρ 2 5 / 3 ρ 1 - 2 ,
D s ( ρ 2 ) / D s ( ρ 1 ) = ( 1 + a 2 b 2 ) 5 / 6 - 1 ( 1 + a 2 ) 5 / 6 - 1 ,

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