Abstract

The effects of turbulence-induced anisoplanatism cause the performance of an adaptive-optics system to be dependent on the separation angle between the imaged object and the source used as a reference. One method of quantifying this performance is through the optical transfer function (OTF). A new method is presented for calculating the upper bound on the OTF that is due to the residual or uncorrected phase and amplitude variations in an adaptive-optics system. The method includes diffraction effects, which in turn result in phase and amplitude effects. These results are compared with the geometric optics and are shown to yield a larger isoplanatic angle. A general expression for the OTF is obtained that permits evaluation of the effect of the inner- and outer-scale size of turbulence. A method is presented for layering the atmosphere and scaling the transfer function to different values of r0 and Cn2 profiles without the need to recompute the entire OTF. We show that with four layers placed at 200 m, 2 km, 10 km, and 18 km the calculated OTF’s are within 1% of the OTF’s obtained with a continuous atmosphere.

© 1994 Optical Society of America

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References

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1992

1991

1989

1983

1982

1978

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

1977

1976

1975

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1689 (1975).
[CrossRef]

1973

1970

S. F. Clifford, J. W. Strohbehn, “The theory of microwave line-of-sight propagation through a turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-18, 264–274 (1970).
[CrossRef]

1969

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

1966

Anderson, J. M.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Boeke, B. R.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Cleis, R. A.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Clifford, S. F.

S. F. Clifford, J. W. Strohbehn, “The theory of microwave line-of-sight propagation through a turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-18, 264–274 (1970).
[CrossRef]

Cochran, G.

G. Cochran, “The impact of scintillation on wavefront reconstruction in the SOR-3 experiments,” Tech. Rep. TR-766 (Optical Sciences Company, Placentia, Calif., 1986).

Coulman, C. E.

Cusumano, S. J.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Donovan, M. T.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Fante, R. L.

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1689 (1975).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1987).

Fried, D. L.

Fugate, R. Q.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Gardner, C. S.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Greenwood, D. P.

Hanson, D.

M. Miller, P. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” presented at the SPIE/SPSE Technical Symposium East on Imaging through the Atmosphere, March 22–23, 1976, Reston, Va.

Hardy, J. W.

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

Harp, J. C.

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

Higgins, C. H.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Hufnagel, R. E.

R. E. Hufnagel, “Variations of atmospheric turbulence,” in Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

Jelonek, M. P.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Korff, D.

Lange, W. J.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Lee, R. W.

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

McKechnie, T. S.

Merkle, F.

F. Merkle, “Adaptive optics,” Phys. World 4, 33–38 (1991).

Miller, M.

M. Miller, P. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” presented at the SPIE/SPSE Technical Symposium East on Imaging through the Atmosphere, March 22–23, 1976, Reston, Va.

Moroney, J. F.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Nickerson, K. S.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1987).

Roggemann, M. C.

Ruane, R. E.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Slavin, A. C.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Spinhirne, J. M.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Strohbehn, J. W.

S. F. Clifford, J. W. Strohbehn, “The theory of microwave line-of-sight propagation through a turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-18, 264–274 (1970).
[CrossRef]

Swindle, D. W.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Tatarski, V. I.

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

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1987).

Vernin, J.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1987).

Wallner, E. P.

Welsh, B. M.

Wild, W. J.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Wynia, J. M.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

Zieske, P.

M. Miller, P. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” presented at the SPIE/SPSE Technical Symposium East on Imaging through the Atmosphere, March 22–23, 1976, Reston, Va.

Appl. Opt.

Bull. Am. Astron. Soc.

R. Q. Fugate, C. H. Higgins, J. M. Wynia, W. J. Lange, A. C. Slavin, W. J. Wild, M. P. Jelonek, M. T. Donovan, S. J. Cusumano, J. M. Anderson, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, K. S. Nickerson, D. W. Swindle, R. A. Cleis, “Experimental demonstration of real time atmospheric compensation with adaptive optics employing laser guide stars,” Bull. Am. Astron. Soc. 23, 898 (1991).

IEEE Trans. Antennas Propag.

S. F. Clifford, J. W. Strohbehn, “The theory of microwave line-of-sight propagation through a turbulent atmosphere,” IEEE Trans. Antennas Propag. AP-18, 264–274 (1970).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Phys. World

F. Merkle, “Adaptive optics,” Phys. World 4, 33–38 (1991).

Proc. IEEE

J. W. Hardy, “Active optics: a new technology for the control of light,” Proc. IEEE 66, 651–697 (1978).
[CrossRef]

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

R. L. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63, 1669–1689 (1975).
[CrossRef]

Other

R. E. Hufnagel, “Variations of atmospheric turbulence,” in Topical Meeting on Optical Propagation through Turbulence (Optical Society of America, Washington, D.C., 1974).

M. Miller, P. Zieske, D. Hanson, “Characterization of atmospheric turbulence,” presented at the SPIE/SPSE Technical Symposium East on Imaging through the Atmosphere, March 22–23, 1976, Reston, Va.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, New York, 1987).

G. Cochran, “The impact of scintillation on wavefront reconstruction in the SOR-3 experiments,” Tech. Rep. TR-766 (Optical Sciences Company, Placentia, Calif., 1986).

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

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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Figures (9)

Fig. 1
Fig. 1

Geometry for the calculation of an angle-dependent optical transfer function (OTF).

Fig. 2
Fig. 2

Amplitude OTF calculated with a propagating wavelength of 0.5 μm, a Hufnagel–Valley turbulence profile with a 54-mph upper altitude wind,21 and a Von Karman power spectrum with Lm = 1 mm and Lo = 5 m.

Fig. 3
Fig. 3

Phase OTF calculated with a propagating wavelength of 0.5 μm, a separation angle of 2.4 μrad, a Hufnagel–Valley turbulence profile with a 54-mph high altitude wind,21 and a Von Karman refractive index power spectrum with Lm = 1 mm and Lo = 5 m.

Fig. 4
Fig. 4

Cn2(η) profiles. Model SLC-N represents a fit to the Advanced Research Projects Agency Maui Optical Station (AMOS) night data,22,23 and SLC-D represents a fit to the AMOS night data with the addition of an altered boundary layer to simulate daytime conditions.22,23 Model Greenwood is Greenwood’s good-seeing model,23 and TRW is a TRW high-turbulence model for the Capistrano Test Site environment. HV-21 and HV-54 represent Hufnagel–Valley atmospheres calculated with 21- and 54-mph winds, respectively.21

Fig. 5
Fig. 5

Measure of outer-scale significance on the OTF. The OTF is evaluated for d = 1 m, θ = 2 μrad, and a Hufnagel–Valley atmosphere21 with a 54-mph upper-atmosphere wind.

Fig. 6
Fig. 6

Hpi(d) for a turbulent layer located at (A) 200 m, (B) 2 km, (C) 10 km, and (D) 18 km. These OTF’s are calculated for λ = 0.5 μm, θ = 2.4 μrad, and r0i = 1 cm.

Fig. 7
Fig. 7

Function g i ( d ) for a turbulent layer located at (A) 200 m, (B) 2 km, (C) 10 km, and (D) 18 km. These functions are calculated for λ = 0.5 μm.

Fig. 8
Fig. 8

Comparison between diffraction theory and geometric-optics theory. The diffraction OTF’s are calculated with a four-layer atmosphere (at 200 m, 2 km, 10 km, and 18 km), λ = 0.5 μm, a Hufnagel–Valley turbulence profile with a 54-mph upper altitude wind,21 θ = 2.4 μrad, and a Von Karman refractive-index power spectrum with Lm = 1 mm and Lo = 5 m. The geometric OTF is calculated with a continuous atmosphere at the same wavelength and turbulence profile but with the use of a Kolmogorov refractive index power spectrum.

Fig. 9
Fig. 9

Comparison of the OTF high-frequency limit (d = ∞) between diffraction theory and geometric-optics theory as a function of separation angle θ. The diffraction OTF’s are calculated with a four-layer atmosphere (at 200 m, 2 km, 10 km, and 18 km), λ = 0.5 μm, a Hufnagel–Valley turbulence profile with a 54-mph upper altitude wind,21 and a Von Karman refractive-index power spectrum with Lm = 1 mm and Lo = 5 m. The geometric OTF is calculated with a continuous atmosphere at the same wavelength and turbulence profile but with the use of a Kolmogorov refractive index power spectrum. (a) Isoplanatic angle calculated for geometric optics. (b), (c) The isoplanatic angles computed with two different definitions for the diffraction method.

Tables (1)

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Table 1 Weights for a Four-Layer Atmospherea

Equations (42)

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P a ( η , κ , x ) = A ( η , κ ) sin ( η α ) cos { η β [ κ · x + ϕ 0 ( η , κ ) ] } ,
P p ( η , κ , x ) = A ( η , κ ) cos ( η α ) cos { η β [ κ · x + ϕ 0 ( η , κ ) ] } ,
( k 2 κ 2 ) 1 / 2 k = k [ ( 1 κ 2 k 2 ) 1 / 2 1 ] κ 2 2 k ,
Δ P p ( η , κ , x ) = A ( η , κ ) cos ( η α ) × cos { η β [ κ · x + ϕ 0 ( η , κ ) ] } + A ( η , κ ) cos ( η α ) cos [ κ · x + ϕ 0 ( η , κ ) ] .
Δ P p ( x ) = d η d κ ( A ( η , κ ) cos ( η α ) × cos { η β [ κ · x + ϕ 0 ( η , κ ) ] } A ( η , κ ) cos ( η α ) cos [ κ · x + ϕ 0 ( η , κ ) ] ) .
P a ( x ) = { d η d κ Ã ( η , κ ) × exp [ j ( κ · x ) ] sin ( η α ) exp ( j η β ) } ,
Δ P p ( x ) = { d η d κ Ã ( η , κ ) × exp [ j ( κ · x ) ] cos ( η α ) [ 1 exp ( j η β ) ] } ,
P a _ ( x ) = d η d κ Ã ( η , κ ) exp [ j ( κ · x ) ] sin ( η α ) × exp ( j η β ) ,
Δ P p _ ( x ) = d η d κ Ã ( η , κ ) exp [ j ( κ · x ) ] × cos ( η α ) [ 1 exp ( j η β ) ] ,
t s ( x ) = t 0 + P a ( x ) ,
H a ( d ) = 1 + Γ P a ( d ) 1 + Γ P a ( 0 ) ,
Γ P a ( d ) _ = P a ( x ) _ · P a * ( x d ) _ = 2 Γ P a ( d ) .
Γ P a ( d ) _ = d η 1 d κ 1 d η 2 d κ 2 × Ã ( η 1 , κ 1 ) Ã * ( η 2 , κ 2 ) × exp { j [ κ 1 · x κ 2 · ( x d ) ] } × sin ( η 1 α 1 ) sin ( η 2 α 2 ) exp [ j ( β η 1 β η 2 ) ] ,
à ( η 1 , κ 1 ) à * ( η 2 , κ 2 ) = 8 π k 2 δ ( κ 1 κ 2 ) δ ( η 1 η 2 ) × Φ ( κ 1 , η 1 ) ,
Γ P a ( d ) _ = 8 π k 2 d η d κ Φ ( κ , η ) exp [ j ( κ · d ) ] sin 2 ( η α ) .
Γ P a ( d ) = 4 π k 2 d η d κ Φ ( κ , η ) sin 2 ( η α ) cos ( κ · d ) .
H a ( d ) = 1 + 4 π k 2 d η d κ Φ ( κ , η ) sin 2 ( η α ) cos ( κ · d ) 1 + 4 π k 2 d η d κ Φ ( κ , η ) sin 2 ( η α ) .
H p ( d ) = Γ Δ P P ( d ) Γ Δ P P ( 0 ) = exp [ j Δ P P ( x ) ] exp [ j Δ P P ( x d ) ] exp [ j Δ P P ( x ) ] exp [ j Δ P P ( x 0 ) ] = exp { j [ Δ P P ( x ) Δ P P ( x d ) ] } 1 .
H p ( d ) = exp { 1 2 [ Δ P P ( x ) Δ P P ( x d ) ] 2 } .
H p ( d ) = exp [ Γ Δ P P ( d ) Γ Δ P P ( 0 ) ] .
Γ Δ P P _ ( d ) = Δ P P _ ( x ) · Δ P P * _ ( x d ) .
Γ Δ P P _ ( d ) = d η 1 d κ 1 d η 2 d κ 2 Ã ( η 1 , κ 1 ) Ã * ( η 2 , κ 2 ) × exp { j [ κ 1 · x κ 2 · ( x d ) ] } × cos ( η 1 α 1 ) [ 1 exp ( j β η 1 ) ] × cos ( η 2 α ) [ 1 exp ( j β η 2 ) ] .
Γ Δ P P _ ( d ) = 8 π k 2 d η d κ Φ ( κ , η ) exp [ j ( κ · d ) ] cos 2 ( η α ) × [ 1 exp ( j β η ) ] [ 1 exp ( j β η ) ] = 8 π k 2 d η d κ Φ ( κ , η ) cos 2 ( η α ) × | 1 exp ( j β η ) | 2 exp ( j κ · d ) .
Γ Δ P P _ ( d ) = 4 π k 2 d η d κ Φ ( κ , η ) cos ( κ · d ) cos 2 ( η α ) × [ 2 2 cos ( η β ) ] ,
Γ Δ P P ( d ) = 8 π k 2 Φ ( κ , η ) cos 2 ( η | κ | 2 2 k ) cos ( κ · d ) × [ 1 cos ( η κ · θ ) ] d η d κ .
H P ( θ , d ) = exp { 8 π k 2 Φ ( κ , η ) cos 2 ( η | κ | 2 2 k ) × [ 1 cos ( κ · d ) ] [ 1 cos ( η κ · θ ) ] d η d κ } .
Φ ( κ , η ) = 0.033 C n 2 ( η ) Φ 0 ( κ ) .
[ 1 1 1 h 1 h 2 h n h 1 2 h 2 2 h n 2 h 1 n 1 h 2 n 1 h n n 1 ] [ W 1 W 2 W n ] = [ 1 m 1 m 2 m n 1 ] ,
H P ( θ , d ) = exp { 8 π 0.033 k 2 i = 1 m C n 2 ( η i ) Φ 0 ( κ ) × cos 2 ( η i κ 2 2 k ) [ 1 cos ( κ · d ) ] × [ 1 cos ( η i κ · θ ) ] d κ } = i = 1 m exp { 8 π 0.033 k 2 C n 2 ( η i ) Φ 0 ( κ ) × cos 2 ( η i κ 2 2 k ) [ 1 cos ( κ · d ) ] × [ 1 cos ( η i κ · θ ) ] d κ } ,
C n 2 ( η i ) = W i C n 2 ( η ) d η = W i Ĉ n 2 .
H P ( θ , d ) = i = 1 m exp { 8 π 0.033 k 2 W i Ĉ n 2 Φ 0 ( κ ) × cos 2 ( η i κ 2 2 k ) [ 1 cos ( κ · d ) ] × [ 1 cos ( η i κ · θ ) ] d κ } .
H a ( d ) = 1 + 4 π 0.033 k 2 Ĉ n 2 i = 1 m W i Φ 0 ( κ ) sin 2 ( η i α ) cos ( κ · d ) d κ 1 + 4 π 0.033 k 2 C n 2 i = 1 m W i Φ 0 ( κ ) sin 2 ( η i α ) d κ .
r 0 = 0.185 ( λ 2 Ĉ n 2 ) 3 / 5 or Ĉ n 2 λ 2 = ( 0.185 ) 5 / 3 r 0 5 / 3 = k 2 Ĉ n 2 4 π 2 .
H P ( θ , d ) = i = 1 m exp { 32 π 3 0.033 ( 0.185 ) 5 / 3 W i r 0 5 / 3 × Φ 0 ( κ ) cos 2 ( η i κ 2 2 k ) [ 1 cos ( κ · d ) ] × [ 1 cos ( η i κ · θ ) ] d κ } = i = 1 m ( exp { 1.967 Φ 0 ( κ ) cos 2 ( η i κ 2 2 k ) × [ 1 cos ( d · κ ) ] [ 1 cos ( η i κ · θ ) ] d κ } ) W i / r 0 5 / 3 = i = 1 m H p i ( θ , d ) 1 / r 0 i 5 / 3 ,
H p i ( θ , d ) = exp { 1.967 Φ 0 ( κ ) cos 2 ( η i κ 2 2 k ) × [ 1 cos ( d · κ ) ] [ 1 cos ( η i κ · θ ) ] d κ }
r 0 5 / 3 = i = 1 m r 0 i 5 / 3 .
H a ( d ) = 1 + i = 1 m 1 r 0 i 5 / 3 g i ( d ) 1 + i = 1 m 1 r 0 i 5 / 3 g i ( 0 ) ,
g i ( d ) = 16 π 3 0.033 ( 0.185 ) 5 / 3 Φ 0 ( κ ) sin 2 ( η i α ) cos ( κ · d ) d κ = 0.983 Φ 0 ( κ ) sin 2 ( η i α ) cos ( κ · d ) d κ .
Φ 0 ( κ ) = exp ( | κ | 2 L m 2 4 π 2 ) ( | κ | 2 + 4 π 2 L o 2 ) 11 / 6 ,
H P ( d ) = i = 1 4 H p i ( d ) 1 / r 0 i 5 / 3 = [ H p 1 ( d ) ] ( 5.258 ) 5 / 3 [ H p 2 ( d ) ] ( 43.40 ) 5 / 3 × [ H p 3 ( d ) ] ( 9.526 ) 5 / 3 [ H p 4 ( d ) ] ( 30.61 ) 5 / 3 .
H a ( d ) = 1 1 + i = 1 m 1 r 0 i 5 / 3 g i ( 0 ) .
H a ( c ) = 1 1 + i = 1 m 1 r 0 i 5 / 3 g i ( 0 ) = 1 1 + ( 5.258 ) 5 / 3 g 1 ( 0 ) + ( 43.40 ) 5 / 3 g 2 ( 0 ) + ( 9.526 ) 5 / 3 g 3 ( 0 ) + ( 30.61 ) 5 / 3 g 4 ( 0 ) = 0.8082 .

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