Abstract

The process of setting up problems of wave propagation through turbulence and reducing the expressions to integrals is typically lengthy. Furthermore, to yield useful results the integrals must be evaluated numerically, except for the simplest problems. Here procedures are given for quickly writing an integral expression and easily evaluating it analytically, yielding a series solution that requires only a few terms to yield accurate results. The solution can also be expressed as a finite sum of generalized hypergeometric functions. The approach uses the Rytov approximation and filter functions in the spatial domain to express quantities of interest such as Zernike modes and effects of anisoplanatism for single or counterpropagating or copropagating plane or spherical waves in integral form. The integrals are readily evaluated with Mellin transforms. We illustrate the technique by deriving the tilt jitter of a single wave and the jitter between two waves with outer-scale effects present. It is shown that outer scale has a significant effect on tilt even for large outer-scale sizes. The effect of outer scale on tilt anisoplanatism is less pronounced.

© 1993 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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1992 (1)

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

1991 (1)

1990 (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743, 1990.
[CrossRef]

1987 (3)

H.-M. Lee, “Rise and fall of directed transient; use of Mellin transformation in time domain problems,” Radio Sci. 22, 1102–1108 (1987).
[CrossRef]

D. Birmingham, S. Sen, “A Mellin summation technique,” J. Phys. A 20, 4557–4560 (1987).
[CrossRef]

D. Bessis, J. D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimensions of strange sets,” Phys. Rev. A 36, 920–928 (1987).
[CrossRef] [PubMed]

1985 (1)

1984 (1)

1983 (1)

P. Zwicke, I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern Recog. Machine Intell. PAMI-5, 191–199 (1983).
[CrossRef]

1982 (1)

1981 (1)

1980 (1)

1979 (2)

1978 (1)

R. A. Altes, “The Fourier-Mellin transform and mammalian hearing,” J. Acoust. Soc. Am. 63, 174–183 (1978).
[CrossRef] [PubMed]

1976 (4)

1975 (2)

W. Lipinski, P. Rolicz, R. Sikora, “Application of integral transforms to the analysis of the magnetic field of a spherical coil,” IEEE Trans. Magn. M-11, 1552–1554 (1975).
[CrossRef]

G. M. Cicuta, E. Montaldi, “Remarks on the full asymptotic expansion of Feynman parametrized integrals,” Lett. Nuovo Cimento Soc. Ital. Fis. 13, 310–312 (1975).
[CrossRef]

1972 (1)

1970 (1)

V. S. Banai, “Transient response of Euler–Cauchy-type time-varying nonlinear systems using multidimensional Mellin transforms,” Proc. IEEE 117, 1156–1160 (1970).

1969 (2)

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote sensing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 293–305 (1969).
[CrossRef]

Altes, R. A.

R. A. Altes, “The Fourier-Mellin transform and mammalian hearing,” J. Acoust. Soc. Am. 63, 174–183 (1978).
[CrossRef] [PubMed]

Banai, V. S.

V. S. Banai, “Transient response of Euler–Cauchy-type time-varying nonlinear systems using multidimensional Mellin transforms,” Proc. IEEE 117, 1156–1160 (1970).

Bessis, D.

D. Bessis, J. D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimensions of strange sets,” Phys. Rev. A 36, 920–928 (1987).
[CrossRef] [PubMed]

Birmingham, D.

D. Birmingham, S. Sen, “A Mellin summation technique,” J. Phys. A 20, 4557–4560 (1987).
[CrossRef]

Bufton, J. L.

Casasent, D.

Cicuta, G. M.

G. M. Cicuta, E. Montaldi, “Remarks on the full asymptotic expansion of Feynman parametrized integrals,” Lett. Nuovo Cimento Soc. Ital. Fis. 13, 310–312 (1975).
[CrossRef]

Ellerbroek, B. L.

B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors: expected values for the SOR-2 experiment,” OSC Rep. TR-613 (Optical Sciences Company, Placentia, Calif., 1984).

Fitzmaurice, M. W.

Fournier, J. D.

D. Bessis, J. D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimensions of strange sets,” Phys. Rev. A 36, 920–928 (1987).
[CrossRef] [PubMed]

Frehlich, R.

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

Fried, D.

Gardner, C. S.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743, 1990.
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Greenwood, D. P.

D. P. Greenwood, “Mutual coherence function of a wave front corrected by zonal adaptive optics,” J. Opt. Soc. Am. 69, 549–554 (1979).
[CrossRef]

D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Rep. RADC-TR-74-19 (ADA 776294/1GI) (Rome Air Development Center, Hanscom Air Force Base, Mass., 1974).

Harp, J. C.

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote sensing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

Herrmann, J.

Hufnagel, R. E.

R. E. Hufnagel, in Digest of Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

Ishimaru, A.

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 293–305 (1969).
[CrossRef]

Kiss, I.

P. Zwicke, I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern Recog. Machine Intell. PAMI-5, 191–199 (1983).
[CrossRef]

Lee, H.-M.

H.-M. Lee, “Rise and fall of directed transient; use of Mellin transformation in time domain problems,” Radio Sci. 22, 1102–1108 (1987).
[CrossRef]

Lee, R. W.

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote sensing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

Lipinski, W.

W. Lipinski, P. Rolicz, R. Sikora, “Application of integral transforms to the analysis of the magnetic field of a spherical coil,” IEEE Trans. Magn. M-11, 1552–1554 (1975).
[CrossRef]

Marichev, O. I.

O. I. Marichev, Integral Transforms of Higher Transcendental Functions (Ellis Horwood, Chichester, England, 1983).

Minott, P. O.

Montaldi, E.

G. M. Cicuta, E. Montaldi, “Remarks on the full asymptotic expansion of Feynman parametrized integrals,” Lett. Nuovo Cimento Soc. Ital. Fis. 13, 310–312 (1975).
[CrossRef]

Noll, R. J.

Psaltis, D.

Roberts, P. H.

B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors: expected values for the SOR-2 experiment,” OSC Rep. TR-613 (Optical Sciences Company, Placentia, Calif., 1984).

Rolicz, P.

W. Lipinski, P. Rolicz, R. Sikora, “Application of integral transforms to the analysis of the magnetic field of a spherical coil,” IEEE Trans. Magn. M-11, 1552–1554 (1975).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

Sasiela, R. J.

R. J. Sasiela, “Unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” Rep. TR-807 ADA 198062 (Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Mass., 1988).

R. J. Sasiela, J. D. Shelton, “Mellin transform techniques applied to integral evaluation: Taylor series and asymptotic approximations,” submitted to J. Math. Phys.

Sen, S.

D. Birmingham, S. Sen, “A Mellin summation technique,” J. Phys. A 20, 4557–4560 (1987).
[CrossRef]

Servizi, G.

D. Bessis, J. D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimensions of strange sets,” Phys. Rev. A 36, 920–928 (1987).
[CrossRef] [PubMed]

Shelton, J. D.

R. J. Sasiela, J. D. Shelton, “Mellin transform techniques applied to integral evaluation: Taylor series and asymptotic approximations,” submitted to J. Math. Phys.

Sikora, R.

W. Lipinski, P. Rolicz, R. Sikora, “Application of integral transforms to the analysis of the magnetic field of a spherical coil,” IEEE Trans. Magn. M-11, 1552–1554 (1975).
[CrossRef]

Slater, L. J.

L. J. Slater, Generalized Hypergeometric Functions (Cambridge U. Press, London, 1966).

Strohbehn, J. W.

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, Berlin, 1978).
[CrossRef]

Tarazano, D. O.

D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Rep. RADC-TR-74-19 (ADA 776294/1GI) (Rome Air Development Center, Hanscom Air Force Base, Mass., 1974).

Tatarski, V. I.

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

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Washington, D.C., 1971).

Tavis, M. T.

Thompson, L. A.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743, 1990.
[CrossRef]

Titterton, P. J.

Turchetti, G.

D. Bessis, J. D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimensions of strange sets,” Phys. Rev. A 36, 920–928 (1987).
[CrossRef] [PubMed]

Tyler, G. A.

Vaienti, S.

D. Bessis, J. D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimensions of strange sets,” Phys. Rev. A 36, 920–928 (1987).
[CrossRef] [PubMed]

Valley, G. C.

Welsh, B. M.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743, 1990.
[CrossRef]

Winker, D. M.

Yura, H. T.

Zwicke, P.

P. Zwicke, I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern Recog. Machine Intell. PAMI-5, 191–199 (1983).
[CrossRef]

Appl. Opt. (4)

IEEE Trans. Magn. (1)

W. Lipinski, P. Rolicz, R. Sikora, “Application of integral transforms to the analysis of the magnetic field of a spherical coil,” IEEE Trans. Magn. M-11, 1552–1554 (1975).
[CrossRef]

IEEE Trans. Pattern Recog. Machine Intell. (1)

P. Zwicke, I. Kiss, “A new implementation of the Mellin transform and its application to radar classification of ships,” IEEE Trans. Pattern Recog. Machine Intell. PAMI-5, 191–199 (1983).
[CrossRef]

J. Acoust. Soc. Am. (1)

R. A. Altes, “The Fourier-Mellin transform and mammalian hearing,” J. Acoust. Soc. Am. 63, 174–183 (1978).
[CrossRef] [PubMed]

J. Atmos. Sci. (1)

R. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. 49, 1494–1509 (1992).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (3)

J. Phys. A (1)

D. Birmingham, S. Sen, “A Mellin summation technique,” J. Phys. A 20, 4557–4560 (1987).
[CrossRef]

Lett. Nuovo Cimento Soc. Ital. Fis. (1)

G. M. Cicuta, E. Montaldi, “Remarks on the full asymptotic expansion of Feynman parametrized integrals,” Lett. Nuovo Cimento Soc. Ital. Fis. 13, 310–312 (1975).
[CrossRef]

Opt. Eng. (1)

D. Casasent, D. Psaltis, “Scale invariant optical transform,” Opt. Eng. 15, 258–261 (1976).
[CrossRef]

Phys. Rev. A (1)

D. Bessis, J. D. Fournier, G. Servizi, G. Turchetti, S. Vaienti, “Mellin transforms of correlation integrals and generalized dimensions of strange sets,” Phys. Rev. A 36, 920–928 (1987).
[CrossRef] [PubMed]

Proc. IEEE (3)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743, 1990.
[CrossRef]

V. S. Banai, “Transient response of Euler–Cauchy-type time-varying nonlinear systems using multidimensional Mellin transforms,” Proc. IEEE 117, 1156–1160 (1970).

R. W. Lee, J. C. Harp, “Weak scattering in random media, with applications to remote sensing,” Proc. IEEE 57, 375–406 (1969).
[CrossRef]

Radio Sci. (2)

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 293–305 (1969).
[CrossRef]

H.-M. Lee, “Rise and fall of directed transient; use of Mellin transformation in time domain problems,” Radio Sci. 22, 1102–1108 (1987).
[CrossRef]

Other (11)

R. J. Sasiela, J. D. Shelton, “Mellin transform techniques applied to integral evaluation: Taylor series and asymptotic approximations,” submitted to J. Math. Phys.

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, Berlin, 1978).
[CrossRef]

D. P. Greenwood, D. O. Tarazano, “A proposed form for the atmospheric microtemperature spatial spectrum in the input range,” Rep. RADC-TR-74-19 (ADA 776294/1GI) (Rome Air Development Center, Hanscom Air Force Base, Mass., 1974).

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

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Washington, D.C., 1971).

R. E. Hufnagel, in Digest of Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors: expected values for the SOR-2 experiment,” OSC Rep. TR-613 (Optical Sciences Company, Placentia, Calif., 1984).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980).

R. J. Sasiela, “Unified approach to electromagnetic wave propagation in turbulence and the evaluation of multiparameter integrals,” Rep. TR-807 ADA 198062 (Lincoln Laboratory, Massachusetts Institute of Technology, Lexington, Mass., 1988).

L. J. Slater, Generalized Hypergeometric Functions (Cambridge U. Press, London, 1966).

O. I. Marichev, Integral Transforms of Higher Transcendental Functions (Ellis Horwood, Chichester, England, 1983).

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Figures (7)

Fig. 1
Fig. 1

Geometry of focused and collimated beams of different diameters propagating in different directions with a separation that depends on the axial coordinate.

Fig. 2
Fig. 2

Tilt variance for the von Kármánn turbulence spectrum with outer scale, normalized to tilt variance with infinite outer scale. Notice that there is a significant reduction of tilt even if the outer scale is 100 times the diameter. This curve is independent of the turbulence distribution along the propagation path.

Fig. 3
Fig. 3

Tilt variance for the Greenwood turbulence spectrum with outer scale, normalized to tilt variance with infinite outer scale. Notice that there is a significant reduction of tilt even if the outer scale is 100 times the diameter. This curve is independent of the turbulence distribution along the propagation path.

Fig. 4
Fig. 4

Tilt anisoplanatism versus displacement, normalized to the tilt on the aperture. The tilt component parallel to the displacement is 1.73 times that of the perpendicular component for small displacements. For large displacements these components are equal. The tilt variance increases rapidly with displacement so that when the displacement is equal to the diameter, the tilt variance is one half that of a single wave. This curve is independent of the turbulence distribution along the propagation path.

Fig. 5
Fig. 5

Tilt anisoplanatism times the aperture diameter versus diameter for the HV-21 turbulence model. Notice that as the diameter gets larger the behavior of the tilt decay is the same for all displacements. In this region a one-term approximation for tilt anisoplanatism gives an accurate result.

Fig. 6
Fig. 6

Anisoplanatic tilt jitter parallel to the displacement normalized to the tilt on the aperture for various outer-scale sizes versus displacement.

Fig. 7
Fig. 7

Anisoplanatic tilt jitter perpendicular to the displacement normalized to the tilt on the aperture for various outer-scale sizes versus displacement.

Equations (99)

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

ϕ ( r , L ) χ ( r , L ) } = k 0 0 L d z d ν ( κ , z ) { cos [ ( γ , κ , z ) ] sin [ ( γ , κ , z ) ] } × exp ( i γ κ · r ) ,
P ( γ , κ , z ) = γ κ 2 ( L z ) 2 k 0 ,
P ( γ , κ , z ) = γ κ 2 z 2 k 0 .
G ( γ κ ) = d r g ( r ) exp ( i γ κ · r ) .
4 π D 2 d r g 2 ( r ) = 1 .
ϕ R ( r , L ) χ R ( r , L ) } = k 0 0 L d z d ν ( κ , z ) × ( G 1 ( γ 1 κ ) { cos [ P 1 ( γ 1 , κ , z ) ] sin [ P 1 ( γ 1 , κ , z ) ] } A ( κ , z ) G 2 ( γ 2 κ ) { cos [ P 2 ( γ 2 , κ , z ) ] sin [ P 2 ( γ 2 , κ , z ) ] } ) .
A ( κ , z ) = exp ( i γ κ · d ) .
σ ϕ R 2 σ χ R 2 } = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × ( G 1 ( γ 1 κ ) { cos [ P 1 ( γ 1 , κ , z ) ] sin [ P 1 ( γ 1 , κ , z ) ] } A ( κ , z ) G 2 ( γ 2 κ ) { cos [ P 2 ( γ 2 , κ , z ) ] sin [ P 2 ( γ 2 , κ , z ) ] } ) × ( G 1 * ( γ 1 κ ) { cos [ P 1 ( γ 1 , κ , z ) ] sin [ P 1 ( γ 1 , κ , z ) ] } A * ( κ , z ) G 2 * ( γ 2 κ ) { cos [ P 2 ( γ 2 , κ , z ) ] sin [ P 2 ( γ 2 , κ , z ) ] } ) ,
f ( κ ) = κ 11 / 3 .
f ( κ ) = ( κ 2 + κ 0 2 ) 11 / 6 ,
f ( κ ) = ( κ 2 + κ κ 0 ) 11 / 6 ,
σ ϕ R 2 σ χ R 2 σ ϕ χ R 2 } = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × { cos 2 [ P ( γ , κ , z ) ] sin 2 [ P ( γ , κ , z ) ] 0.5 sin [ 2 P ( γ , κ , z ) ] } F ( γ κ ) ,
F ( γ κ ) = G ( γ κ ) G * ( γ κ ) .
σ ϕ R 2 σ χ R 2 } = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × ( { cos [ P 1 ( γ , κ , z ) ] A ( κ , z ) cos [ P 2 ( γ , κ , z ) ] } 2 { sin [ P 1 ( γ , κ , z ) ] A ( κ , z ) sin [ P 2 ( γ , κ , z ) ] } 2 ) F ( γ κ ) .
σ ϕ R 2 σ χ R 2 } = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × { cos 2 [ P ( γ , κ , z ) ] sin 2 [ P ( γ , κ , z ) ] } | G 1 ( γ κ ) G 2 ( γ κ ) | 2 .
σ ϕ R 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × | G 1 ( γ 1 κ ) G 2 ( γ 2 κ ) | 2 .
σ ϕ R 2 σ χ R 2 } = [ D ϕ R ( d ) D χ R ( d ) ] = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × { cos 2 [ P ( γ , κ , z ) ] sin 2 [ P ( γ , κ , z ) ] } F ( γ κ ) 2 [ 1 cos ( γ κ · d ) ] .
σ ϕ R 2 σ ϕ R 2 } = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) { | cos ( κ 2 L / 2 k 0 ) cos ( κ 2 z / 2 k 0 ) exp ( i κ · d ) cos [ κ 2 ( L z ) / 2 k 0 ] | 2 | sin ( κ 2 L / 2 k 0 ) cos ( κ 2 z / 2 k 0 ) exp ( i κ · d ) sin [ κ 2 ( L z ) / 2 k 0 ] | 2 } F ( κ ) .
σ ϕ R 2 σ χ R 2 } = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × [ sin 2 ( κ 2 L / 2 k 0 ) sin 2 ( κ 2 z / 2 k 0 ) cos 2 ( κ 2 L / 2 k 0 ) sin 2 ( κ 2 z / 2 k 0 ) ] F ( κ ) .
σ ϕ R 2 σ χ R 2 } = 1 π 0 d ω { S ϕ ( ω ) S χ ( ω ) .
κ x = ω / υ ( z ) ,
c 2 = κ y 2 υ 2 ( z ) ω 2 + 1 ,
S ϕ ( ω ) S χ ( ω ) } = 2.606 k 0 2 ω F servo ( ω ) 0 L d z C n 2 ( z ) υ 2 ( z ) 0 c d c U ( c 1 ) c 2 1 × f ( ω c υ ( z ) ) { cos 2 [ γ ω 2 c 2 ( L z ) 2 υ 2 ( z ) k 0 ] sin 2 [ γ ω 2 c 2 ( L z ) 2 υ 2 ( z ) k 0 ] } F ( γ ω c υ ( z ) ) .
U ( x ) = 1 for x > 0 , U ( x ) = 0 for x < 0 .
S R = 1 2 π d α K ( α ) exp ( i k 0 D ρ · α / z D ( α D ) / 2 | ρ = 0 = 1 2 π d α K ( α ) exp [ D ( α ) / 2 ] .
K ( α ) = 16 π [ cos 1 ( α ) α ( 1 α 2 ) 1 / 2 ] U ( 1 α ) .
S R = 0 d α α K ( α ) exp [ D ( α ) / 2 ] .
h ( x ) H ( s ) [ h ( x ) ] 0 d x x h ( x ) x s ,
h ( x ) = 1 2 π i C d s H ( s ) x s .
h ( x ) = 0 d t t h 0 ( t ) h 1 ( x / t ) [ h ( x ) ] H ( s ) = H 0 ( s ) H 1 ( s ) .
h ( x ) = 1 2 π i C d s H 0 ( s ) H 1 ( s ) x s .
h ( x 1 , , x N ) = 0 d y y h 0 ( y ) j = 1 N h j ( x j / y ) [ h ( x 1 , , x N ) ] = H 0 ( s 1 + s 2 + + s N ) j = 1 N H j ( s j ) = H ( s 1 , s 2 , s N ) .
h ( x 1 , , x N ) = 1 ( 2 π i ) N c 1 c N d s 1 d s N H ( s 1 , s 2 , , s N ) x 1 s 1 x N s N .
Γ [ α 1 , , α m β 1 , , β n ] = Γ [ α 1 ] Γ [ α 2 ] Γ [ α m ] Γ [ β 1 ] Γ [ β 2 ] Γ [ β n ] ·
T Z 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ κ 11 / 3 ( 16 k 0 D ) 2 × [ J 2 ( γ κ D / 2 ) γ κ D / 2 ] 2 .
T Z 2 = 105.1 D 1 / 3 0 L d z C n 2 ( z ) γ 5 / 3 0 d x x x 11 / 3 J 2 2 ( x ) .
T Z 2 = 105.1 2 π D 1 / 3 0 L d z C n 2 ( z ) γ 5 / 3 × Γ [ s / 2 + 2 , s / 2 + ½ s / 2 + 3 , s / 2 + 1 ] | s = 11 / 3 = 105.1 2 π D 1 / 3 0 L d z C n 2 ( z ) γ 5 / 3 Γ [ 1 / 6 , 7 / 3 29 / 6 , 17 / 6 ] .
T Z 2 = 6.08 μ 0 D 1 / 3 = 0.3641 ( D / r o ) 5 / 3 ( λ / D ) 2 ,
T G 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ κ 11 / 3 ( 4 / k 0 D ) 2 J 1 2 ( γ κ D / 2 ) .
T G 2 = 6.564 μ 0 D 1 / 3 2 π Γ [ 1 / 6 , 4 / 3 17 / 6 , 11 / 6 ] = 5.675 μ 0 D 1 / 3 = 0.3399 ( D / r o ) 5 / 3 ( λ / D ) 2 .
T G Z 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ κ 11 / 3 ( 4 / k 0 D ) 2 × { [ 4 J 2 ( κ D / 2 ) κ D / 2 ] J 1 ( κ D / 2 ) } 2 ·
T G Z 2 = 0.102 μ 0 D 1 / 3 = 0.0061 ( D / r o ) 5 / 3 ( λ / D ) 2 .
T Z 2 = 6.08 C n 2 L D 1 / 3 .
T Z 2 = 6.08 μ 5 / 3 D 1 / 3 L 5 / 3 = 0.169 λ 2 D 1 / 3 L 5 / 3 θ o 5 / 3 ·
T o 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ ( κ 2 + κ o 2 ) 11 / 6 ( 16 / k 0 D ) 2 × [ J 2 ( κ D / 2 ) κ D / 2 ] 2 .
T o 2 = 1334 μ 0 κ o 11 / 3 D 4 0 d x x J 2 2 ( x ) [ ( 2 x / κ o D ) 2 + 1 ] 11 / 6 .
T o 2 = 400 μ 0 κ o 11 / 3 D 4 1 2 π i c d s ( κ o D 2 ) 2 s × Γ [ s + 2 , s + ½ , s , s + 11 / 6 s + 3 , s + 1 ] ,
T o 2 = 6.08 μ 0 D 1 / 3 { F 2 3 [ 11 6 , 7 3 ; 5 6 , 29 6 , 17 6 ; ( π D L o ) 2 ] 1.4234 ( D L o ) 1 / 3 F 2 3 [ 2 , 5 2 ; 7 6 , 5 , 3 ; ( π D L o ) 2 ] } .
T o 2 6.08 μ 0 D 1 / 3 { 1 1.42 ( D L o ) 1 / 3 + 3.70 ( D L o ) 2 4.01 ( D L o ) 7 / 3 + 4.21 ( D L o ) 4 4.00 ( D L o ) 13 / 3 } .
T o 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ ( κ 2 + κ κ o ) 11 / 6 ( 16 / k 0 D ) 2 × [ J 2 ( κ D / 2 ) κ D / 2 ] 2 .
T o 2 = 6.08 μ 0 D 1 / 3 { F 3 4 [ 11 12 , 7 3 , 17 12 ; 5 6 , 29 6 , 17 6 , 1 2 ; ( π D L o ) 2 ] + 1.764 ( D L o ) F 3 4 [ 17 12 , 23 12 , 17 6 ; 7 6 , 4 3 , 3 2 , 16 3 , 10 3 , ( π D L o ) 2 ] 1.848 ( D L o ) 1 / 3 F 3 4 [ 5 2 , 13 12 , 19 12 ; 7 6 , 5 , 3 7 6 , 2 3 , ( π D L o ) 2 ] } .
T o 2 6.08 μ 0 D 1 / 3 [ 1 1.85 ( D L o ) 1 / 3 + 1.76 ( D L o ) 5.24 ( D L o ) 2 + 6.70 ( D L o ) 7 / 3 3.77 ( D L o ) 3 ] .
σ x 2 σ y 2 σ 2 } = 0.2073 0 L d z C n 2 ( z ) d κ { cos 2 ( φ ) sin 2 ( φ ) 1 } κ 11 / 3 ( 16 / D ) 2 × [ J 2 ( κ D / 2 ) κ D / 2 ] 2 2 { 1 cos [ κ d cos ( φ ) ] } ,
0 π d x exp [ i β cos ( x ) ] sin 2 ν ( x ) = π ( 2 / β ) ν Γ [ ν + 1 2 ] J ν ( β ) , Re ν > 1 2 .
cos 2 ( φ ) = 1 sin 2 ( φ )
J 0 ( r ) = 1 2 π 0 2 π d φ exp [ i r cos ( φ ) ] = 1 2 π 0 2 π d φ cos [ r cos ( φ ) ] ,
σ x 2 σ y 2 σ 2 } = 667 D 2 0 L d z C n 2 ( z ) 0 d κ κ 8 / 3 [ J 2 ( κ D / 2 ) κ D / 2 ] 2 × { 1 2 + J 1 ( κ d ) κ d J 0 ( κ d ) 1 2 J 1 ( κ d ) κ d 1 J 0 ( κ d )
I 1 = 0 d κ κ 8 / 3 [ J 2 ( κ D / 2 ) κ D / 2 ] 2 [ J 1 ( κ d ) κ d 1 2 ] ,
I T = 0 d κ κ 8 / 3 [ J 2 ( κ D / 2 ) κ D / 2 ] 2 [ J 0 ( κ d ) 1 ] .
σ x 2 σ y 2 σ 2 } = 667 D 2 0 L d z C n 2 ( z ) { I T I 1 I 1 I T .
σ x 2 σ y 2 } = 6.08 μ 0 D 1 / 3 × ( { 1 F 4 3 [ 1 6 , 23 6 , 11 6 , 3 2 ; 4 3 , 2 , 1 2 ; ( d D ) 2 ] 1 F 3 2 [ 1 6 , 23 6 , 11 6 ; 4 3 , 2 ; ( d D ) 2 ] } + { 2.19 ( d D ) 14 / 3 F 4 3 [ 5 2 , 3 2 , 1 2 , 23 6 ; 17 6 , 13 3 , 10 3 ; ( d D ) 2 ] 0.0388 ( d D ) 14 / 3 F 3 2 [ 5 2 , 3 2 , 1 2 ; 13 3 , 10 3 ; ( d D ) 2 ] } ) .
σ x 2 σ y 2 } = 6.08 μ 0 D 1 / 3 ( { 1 0.532 ( D d ) 1 / 3 F 4 3 [ 1 6 , 5 2 , 1 6 , 2 3 ; 5 , 3 , 1 3 ; ( D d ) 2 ] 1 0.798 ( D d ) 1 / 3 F 3 2 [ 1 6 , 5 2 , 1 6 ; 5 , 3 ; ( D d ) 2 ] } ) .
σ x 2 σ y 2 } 2.67 μ 0 D 1 / 3 ( d D ) 2 { 3 1 , d D .
T 3.27 μ 0 1 / 2 D 1 / 6 d D , d D .
σ x 2 σ y 2 } = { σ x 2 σ y 2 } L + { σ x 2 σ y 2 } U ·
σ x 2 σ y 2 } L = 6.08 D 1 / 3 0 H c d z C n 2 ( z ) × ( { 1 F 4 3 [ 1 6 , 23 6 , 11 6 , 3 2 ; 4 3 , 2 , 1 2 ; ( θ z D ) 2 ] 1 F 3 2 [ 1 6 , 23 6 , 11 6 ; 4 3 , 2 ; ( θ z D ) 2 ] } + { 2.19 ( θ z D ) 14 / 3 F 4 3 [ 5 2 , 3 2 , 1 2 , 23 6 ; 17 6 , 13 3 , 10 3 ; ( θ z D ) 2 ] 0.388 ( θ z D ) 14 / 3 F 3 2 [ 5 2 , 3 2 , 1 2 ; 13 3 , 10 3 ; ( θ z D ) 2 ] } ) .
σ x 2 σ y 2 } U = 6.08 D 1 / 3 ( μ 0 + ( H c ) { 1 1 } H c L d z C n 2 ( z ) × { 0.532 ( D θ z ) 1 / 3 F 4 3 [ 1 6 , 5 2 , 1 6 , 2 3 ; 5 , 3 , 1 3 ; ( D θ z ) 2 ] 0.798 ( D θ z ) 1 / 3 F 3 2 [ 1 6 , 5 2 , 1 6 ; 5 , 3 ; ( D θ z ) 2 ] } ) .
σ x 2 σ y 2 } = 2.67 μ 2 ( H c ) D 1 / 3 ( θ D ) 2 { 3 1 } 3.68 μ 4 ( H c ) D 1 / 3 ( θ D ) 4 { 5 1 } + 2.35 μ 14 / 3 ( H c ) D 1 / 3 ( θ D ) 14 / 3 { 17 / 3 1 } + .
θ T c = 0.184 λ D 1 / 6 μ 2 1 / 2 .
σ x 2 σ y 2 } 1 D 1 / 3 [ 6.08 μ 0 + ( H c ) { 1 1 } 4.85 μ 1 / 3 + ( H c ) ( D θ ) 1 / 3 × { 2 / 3 1 } + 0.0205 μ 7 / 3 + ( H c ) ( D θ ) 7 / 3 { 4 / 3 1 ] .
σ x 2 σ y 2 σ 2 } = 0.4146 ( 16 D ) 2 0 L d z C n 2 ( z ) d κ { cos 2 ( φ ) sin 2 ( φ ) 1 } × ( κ 2 + κ o 2 ) 11 / 6 [ J 2 ( κ D / 2 ) κ D / 2 ] 2 { 1 cos [ κ d cos ( φ ) ] } .
I 1 I T } = 4 d 11 / 3 D 2 0 d t t { t 14 / 3 [ J 1 ( t ) ( t / 2 ) ] t 11 / 3 [ J 0 ( t ) 1 ] } J 2 2 ( t / x ) × [ 1 + ( y / t ) 2 ] 11 / 6 .
I 1 I T } = 0.0945 d 11 / 3 D 2 1 ( 2 π i ) 2 c 1 c 2 d s 1 d s 2 ( π d / L o ) 2 s 1 ( d / D ) 2 s 2 × Γ [ s 1 + s 2 11 6 * , s 2 + 2 , s 2 + 1 2 , s 1 , s 1 + 11 6 s 2 + 3 , s 2 + 1 ] × { 1 2 Γ [ s 1 s 2 + 23 6 ] 1 Γ [ s 1 s 2 + 17 6 ] .
σ x 2 σ y 2 } = 31.5 μ 0 d 1 / 3 { n = 0 m = 0 ( 1 ) n + m n ! m ! ( π d L o ) 2 m ( d D ) 2 n + 5 × Γ [ m n 7 3 , n + 5 2 , m + 11 6 n + 5 2 , n + 1 2 , n + m + 13 3 ] × { 2 n + 2 m + 17 3 1 } + n = 1 m = 0 ( 1 ) n + m n ! m ! ( π D L o ) 2 m × ( d D ) 2 n + 1 / 3 Γ [ n m + 1 6 , m n + 7 3 , m + 11 6 n + m + 29 6 , n + m + 17 6 , n + 2 ] × { 2 n + 1 1 } + n = 0 m = 1 ( 1 ) n + m n ! m ! ( π d L o ) 2 m + 1 / 3 ( π D L o ) 2 n × Γ [ n + 5 2 , m n 1 6 , m + n + 2 n + 5 , n + 3 , m + 2 ] { 2 m + 1 1 } } .
σ x 2 σ y 2 } = 2.67 μ 0 D 1 / 3 ( d D ) 2 { 3 1 } × [ 1 20.6 ( D L o ) 2 + 27.4 ( D L o ) 7 / 3 + ] .
σ x 2 σ y 2 } = 2.67 μ 2 D 1 / 3 ( θ D ) 2 { 3 1 } × [ 1 20.6 ( D L o ) 2 + 27.4 ( D L o ) 7 / 3 + ] .
F m , n ( κ ) x F m , n ( κ ) y F 0 , n ( κ ) } = ( n + 1 ) [ 2 J n + 1 ( κ D / 2 ) κ D / 2 ] 2 { 2 cos 2 ( m φ ) 2 sin 2 ( m φ ) 1 ( m = 0 ) .
F ( κ ) = [ 2 J 1 ( κ D / 2 ) κ D / 2 ] 2 .
F x ( κ ) F y ( κ ) F ( κ ) } = [ 4 J 2 ( κ D / 2 ) κ D / 2 ] 2 { cos 2 ( φ ) sin 2 ( φ ) 1 .
F ( κ ) = 1 [ 2 J 1 ( κ D / 2 ) κ D / 2 ] 2 [ 4 J 2 ( κ D / 2 ) κ D / 2 ] 2 .
F x ( κ ) F y ( κ ) F ( κ ) } = J 1 2 ( κ D / 2 ) { cos 2 ( φ ) sin 2 ( φ ) 1 .
F x ( κ ) F y ( κ ) F ( κ ) } = { 16 k 0 D [ 1 ( D i / D ) 4 ] } 2 × [ J 2 ( κ D / 2 ) κ D / 2 ( D i D ) 3 J 2 ( κ D i / 2 ) κ D i / 2 ] 2 { cos 2 ( ϕ ) sin 2 ( ϕ ) 1 .
F x ( κ ) F y ( κ ) F ( κ ) } = { 4 k 0 D [ 1 ( D i / D ) 2 ] } 2 × [ J 1 ( κ D / 2 ) D i D J 1 ( κ D / 2 ) ] 2 { cos 2 ( ϕ ) sin 2 ( ϕ ) 1 .
F ( κ ) = { 2 J 1 [ κ D ( 1 z / L ) / 2 ] κ D ( 1 z / L ) / 2 } 2 .
F ( κ ) = [ 2 J 1 ( κ D / 2 ) κ D / 2 ] 2 .
F ( κ ) = [ 2 J 1 ( κ D s z / 2 L ) κ D s z / 2 L ] 2 .
F ( κ ) = [ 2 J 1 ( κ D ( 1 z / L ) / 2 ) κ D ( 1 z / L ) / 2 ] 2 [ 2 J 1 ( κ D s z / 2 L ) κ D s z / 2 L ] 2 .
μ m = 0 d z C n 2 ( z ) z m = sec m + 1 ( ξ ) 0 d h C n 2 ( h ) h m ,
L = sec ( ξ ) H ,
μ m + ( L ) = L d z C n 2 ( z ) z m = sec m + 1 ( ξ ) H d h C n 2 ( h ) h m ,
μ m ( L ) = 0 L d z C n 2 ( z ) z m = sec m + 1 ( ξ ) 0 H d h C n 2 ( h ) h m .
C n 2 ( h ) = 0.00594 ( W / 27 ) 2 ( 10 5 h ) 10 exp ( h / 1000 ) + 2.7 × 10 16 exp ( h / 1500 ) + A exp ( h / 100 ) ,
μ m = 0 d z C n 2 ( z ) z m = sec m + 1 ( ξ ) × [ 5.94 × 10 20 + 3 m ( W / 27 ) 2 Γ ( m + 11 ) + 4.05 × 10 13 Γ ( m + 1 ) ( 1500 ) n + A × 100 m + 1 Γ ( m + 1 ) ] .
μ m + ( L ) = L d z C n 2 ( z ) z m = sec m + 1 ( ξ ) × [ 5.94 × 10 20 + 3 m ( W / 27 ) 2 Γ ( m + 11 , H / 1000 ) + 4.05 × 10 13 Γ ( m + 1 , H / 1500 ) ( 1500 ) m + A × 100 m + 1 Γ ( m + 1 , H / 100 ) ] ,
μ m ( L ) = 0 L d z C n 2 ( z ) z m = sec m + 1 ( ξ ) × [ 5.94 × 10 20 + 3 m ( W / 27 ) 2 γ ( m + 11 , H / 1000 ) + 4.05 × 10 13 γ ( m + 1 , H / 1500 ) ( 1500 ) m + A × 100 m + 1 γ ( m + 1 , H / 100 ) ] .
γ [ b + 1 , x ] = 0 x d y y b exp ( y ) ,
Γ [ b + 1 , x ] = x d y y b exp ( y ) .
r o 5 / 3 = 0.423 k 0 2 μ 0 ,
θ o 5 / 3 = 2.91 k 0 2 μ 5 / 3 .

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