Abstract

This work presents an approach for constructing equations describing the effects of atmospheric turbulence on propagating light based on equations and concepts that will be familiar to those with a background in paraxial wave-optics modeling. The approach is developed and demonstrated by working through three examples of increasing complexity: the variance and power spectral density of the aperture-averaged phase gradient (G tilt) on a point-source beacon, the variance of the Zernike tilt difference between two physically separated point-source beacons, and the irradiance-weighted average phase gradient (centroid tilt) and target-plane jitter variance for a generic beam. The first two results are shown to be consistent with the existing literature; the third is novel, and it is shown to agree with wave optics and to be consistent with the literature in the special case of a Gaussian beam.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Corrections

Scot E. J. Shaw and Erin M. Tomlinson, "Analytic propagation variances and power spectral densities from a wave-optics perspective: publisher’s note," J. Opt. Soc. Am. A 36, 1333-1333 (2019)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-36-8-1333

5 July 2019: Typographical corrections were made to the math.

8 July 2019: Typographical corrections were made to the math.


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References

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  1. G. A. Tyler, “Bandwidth considerations for tracking through turbulence,” J. Opt. Soc. Am. A 11, 358–367 (1994).
    [Crossref]
  2. C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
    [Crossref]
  3. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
  4. R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, 2nd ed. (SPIE, 2007).
  5. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.
  6. V. I. Klyatskin and A. I. Kon, “On the displacement of spatially-bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
    [Crossref]
  7. J. W. Goodman, Introduction to Fourier Optics, 4th ed. (Roberts & Company, 2017).
  8. H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9(56), 401–407 (1836).
    [Crossref]
  9. L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
    [Crossref]
  10. M. Mansuripur, Classical Optics and its Applications, 2nd ed. (Cambridge University, 2009).
  11. B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representations of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
    [Crossref]
  12. J. W. Goodman, Statistical Optics (Wiley Classics, 2000).

1994 (1)

1990 (1)

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representations of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[Crossref]

1976 (1)

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
[Crossref]

1972 (1)

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially-bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[Crossref]

1881 (1)

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

1836 (1)

H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9(56), 401–407 (1836).
[Crossref]

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Butts, R.

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 4th ed. (Roberts & Company, 2017).

J. W. Goodman, Statistical Optics (Wiley Classics, 2000).

Herman, B. J.

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representations of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[Crossref]

Hogge, C.

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
[Crossref]

Klyatskin, V. I.

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially-bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[Crossref]

Kon, A. I.

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially-bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[Crossref]

Kravtsov, Y. A.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Mansuripur, M.

M. Mansuripur, Classical Optics and its Applications, 2nd ed. (Cambridge University, 2009).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

Rayleigh, L.

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

Rytov, S. M.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Sasiela, R. J.

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, 2nd ed. (SPIE, 2007).

Strugala, L. A.

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representations of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[Crossref]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9(56), 401–407 (1836).
[Crossref]

Tatarskii, V. I.

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

Tyler, G. A.

IEEE Trans. Antennas Propag. (1)

C. Hogge and R. Butts, “Frequency spectra for the geometric representation of wavefront distortions due to atmospheric turbulence,” IEEE Trans. Antennas Propag. 24, 144–154 (1976).
[Crossref]

J. Opt. Soc. Am. A (1)

Philos. Mag. (2)

H. F. Talbot, “Facts relating to optical science, No. IV,” Philos. Mag. 9(56), 401–407 (1836).
[Crossref]

L. Rayleigh, “On copying diffraction-gratings, and on some phenomena connected therewith,” Philos. Mag. 11(67), 196–205 (1881).
[Crossref]

Proc. SPIE (1)

B. J. Herman and L. A. Strugala, “Method for inclusion of low-frequency contributions in numerical representations of atmospheric turbulence,” Proc. SPIE 1221, 183–192 (1990).
[Crossref]

Radiophys. Quantum Electron. (1)

V. I. Klyatskin and A. I. Kon, “On the displacement of spatially-bounded light beams in a turbulent medium in the Markovian-random-process approximation,” Radiophys. Quantum Electron. 15, 1056–1061 (1972).
[Crossref]

Other (6)

J. W. Goodman, Introduction to Fourier Optics, 4th ed. (Roberts & Company, 2017).

M. Mansuripur, Classical Optics and its Applications, 2nd ed. (Cambridge University, 2009).

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).

R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence, 2nd ed. (SPIE, 2007).

S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics (Springer-Verlag, 1989), Vol. 4.

J. W. Goodman, Statistical Optics (Wiley Classics, 2000).

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Figures (4)

Fig. 1.
Fig. 1. In this figure, light from a point-source beacon traces out a cone as it propagates a distance L through the atmosphere to an aperture. The atmospheric turbulence is indicated schematically as a series of phase screens as we would model it in wave-optics code.
Fig. 2.
Fig. 2. For Z-tilt anisoplanatism, we consider two beams that propagate along paths displaced from one another in space. In this illustration, we have two point sources propagating to a common aperture. Both beams follow cones (solid lines) with the same z -dependent diameter D ( z ) , with center lines (dashed lines) separated by the distance d ( z ) . Because the two cones do not completely overlap, the Z tilts picked up by the two beams on the way to the aperture will differ, with the difference being the Z-tilt anisoplanatism.
Fig. 3.
Fig. 3. Results for centroid jitter variance on target for focused beams that have propagated through different strengths of atmospheric turbulence. In all cases we took a 1 μm wavelength beam and propagated 3 km through Kolmogorov turbulence with constant C n 2 . For the top-hat beam we used a 20-cm aperture, and for the Gaussian we used w = 2 3 / 2 × 20 cm to match the top hat. The red and purple theory curves lie largely on top of their corresponding wave-optics results until we violate the weak-turbulence approximations in the theory for Rytov variance 0.3 .
Fig. 4.
Fig. 4. Illustration of our setup to calculate the effect of propagation on a sinusoidal phase profile. We start with a field of focal length f 0 , shown here as a diverging beam, with a sinusoidal phase profile on top of the parabolic phase. In propagating a distance z , our initial phase ϕ ( κ , r ) changes to ϕ ( κ , r , z 0 ; z 0 + z ) . We will derive an expression for the changes to this phase profile from propagation.

Equations (85)

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ϕ ( κ , r , z ) = sin [ κ · r + α ( κ , z ) ] ,
ϕ ( κ , r , z ; L ) = sin [ D ( z ) D ( L ) κ · r + α ( κ , z ) ] cos ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) ,
θ G , m = 1 k R d r ( m ^ · ) ϕ ( r ) / R d r .
θ G , m ( κ , z ; L ) = 4 π k D 2 ( L ) r D ( L ) / 2 d r ( m ^ · ) sin [ D ( z ) D ( L ) κ · r + α ( κ , z ) ] cos ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) .
θ G , m ( κ , z ; L ) = 4 k D ( L ) cos [ α ( κ , z ) ] cos ( φ m φ κ ) × cos ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) J 1 ( D ( z ) κ 2 ) .
σ G , m 2 ( κ , z ; L ) = 8 k 2 D 2 ( L ) cos 2 ( φ m φ κ ) cos 2 ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) × J 1 2 ( D ( z ) κ 2 ) .
d σ G , m 2 ( z ; L ) = 4 π k 2 d z C n 2 ( z ) d κ Φ n 0 ( κ ) σ G , m 2 ( κ , z ; L ) ,
Φ n 0 ( κ ) = 5 18 π Γ ( 1 3 ) ( κ 2 + k o 2 ) 11 / 6 e κ 2 / k i 2 .
σ G , m 2 = 4 π k 2 0 L d z C n 2 ( z ) 0 d κ κ Φ n 0 ( κ ) 0 2 π d φ κ σ G , m 2 ( κ , z ; L ) .
σ G , m 2 = 80 9 Γ ( 1 3 ) D 2 ( L ) 0 L d z C n 2 ( z ) 0 d κ κ ( κ 2 + k o 2 ) 11 / 6 × e κ 2 / k i 2 0 2 π d φ κ cos 2 ( φ m φ κ ) cos 2 ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) × J 1 2 ( D ( z ) κ 2 ) .
σ G , m 2 = 80 π 9 Γ ( 1 3 ) D 2 ( L ) 0 L d z C n 2 ( z ) 0 d κ κ ( κ 2 + k o 2 ) 11 / 6 × e κ 2 / k i 2 cos 2 ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) J 1 2 ( D ( z ) κ 2 ) .
σ G , m 2 = 20 π Γ ( 4 3 ) 9 Γ ( 2 3 ) Γ ( 11 6 ) Γ ( 17 6 ) 2.838 D 2 ( L ) 0 L d z C n 2 ( z ) D 5 / 3 ( z ) .
ϕ ( κ , r , z ; L ) = sin [ D ( z ) D ( L ) κ · r κ · v ( z ) t + α ( κ , z ) ] × cos ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) .
f = κ · v ( z ) 2 π ,
κ = 2 π f v ( z ) cos [ φ κ φ v ( z ) ] .
φ f = φ κ φ v ( z ) .
0 2 π d φ κ 0 d κ κ 2 π / 2 π / 2 d φ f 0 d f f ( 2 π v ( z ) cos ( φ f ) ) 2 .
σ G , m 2 = 8 π k 2 0 L d z C n 2 ( z ) π / 2 π / 2 d φ f 0 d f f ( 2 π v ( z ) cos ( φ f ) ) 2 × Φ n 0 ( 2 π f v ( z ) cos ( φ f ) ) × σ G , m 2 [ 2 π f v ( z ) cos ( φ f ) , φ f + φ v ( z ) , z ; L ] ,
σ 2 0 d f PSD ( f )
PSD G , m ( f ) = 32 π 3 k 2 f 0 L d z C n 2 ( z ) v 2 ( z ) π / 2 π / 2 d φ f sec 2 ( φ f ) × Φ n 0 ( 2 π f v ( z ) cos ( φ f ) ) × σ G , m 2 [ 2 π f v ( z ) cos ( φ f ) , φ f + φ v ( z ) , z ; L ] .
PSD G , m ( f ) = 80 2 2 / 3 9 π 5 / 3 Γ ( 1 3 ) 0.3102 D 2 ( L ) f 8 / 3 0 L d z C n 2 ( z ) v 5 / 3 ( z ) × π / 2 π / 2 d φ f cos 5 / 3 ( φ f ) cos 2 [ φ f + φ v ( z ) φ m ] × J 1 2 ( π f D ( z ) v ( z ) cos ( φ f ) ) .
σ 2 = 4 π k 2 0 L d z C n 2 ( z ) 0 d κ κ Φ n 0 ( κ ) 0 2 π d φ κ σ 2 ( κ , z ; L ) .
PSD ( f ) = 32 π 3 k 2 f 0 L d z C n 2 ( z ) v 2 ( z ) π / 2 π / 2 d φ f sec 2 ( φ f ) × Φ n 0 ( 2 π f v ( z ) cos ( φ f ) ) × σ 2 [ 2 π f v ( z ) cos ( φ f ) , φ f + φ v ( z ) , z ; L ] ,
θ Z , m = r D / 2 d r k ( r · m ^ ) ϕ ( r ) / r D / 2 d r .
θ Z , m ( κ , z ; L ) = 4 π D 2 ( L ) 0 D ( L ) / 2 d r r 0 2 π d φ ( 4 k D ( L ) ) 2 × k ( r · m ^ ) ϕ ( κ , r , z ; L ) ,
= 32 k κ D ( z ) D ( L ) cos [ α ( κ , z ) ] cos ( φ m φ κ ) × cos ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) J 2 ( D ( z ) κ 2 ) ,
cos [ α ( κ , z ) ] θ ˜ Z , m ( κ , z ; L ) .
σ Z , m 2 ( κ , z ; L ) = 1 2 ( 32 k κ D ( z ) D ( L ) ) 2 cos 2 ( φ m φ κ ) × cos 2 ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) J 2 2 ( D ( z ) κ 2 ) ,
= 1 2 θ ˜ Z , m 2 ( κ , z ; L ) .
θ Z , m ( κ , d , z ; L ) = cos { α ( κ , z ) + κ d ( z ) cos [ φ κ φ d ( z ) ] } × θ ˜ Z , m ( κ , z ; L ) .
θ Δ Z , m ( κ , d , z ; L ) = θ Z , m ( κ , 0 , z ; L ) θ Z , m ( κ , d , z ; L ) .
σ Δ Z , m 2 ( κ , d , z ; L ) = ( cos [ α ( κ , z ) ] cos { α ( κ , z ) + κ d ( z ) × cos [ φ κ φ d ( z ) ] } ) 2 θ ˜ Z , m 2 ( κ , z ; L ) .
σ Δ Z , m 2 ( κ , d , z ; L ) = ( 1 cos { κ d ( z ) cos [ φ κ φ d ( z ) ] } ) × θ ˜ Z , m 2 ( κ , z ; L ) .
σ Δ Z , m 2 = 4 π k 2 5 18 π Γ ( 1 3 ) ( 32 k D ( L ) ) 2 0 L d z C n 2 ( z ) D 2 ( z ) 0 d κ κ × ( κ 2 + k o 2 ) 11 / 6 e κ 2 / k i 2 1 κ 2 cos 2 ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) × J 2 2 ( D ( z ) κ 2 ) 0 2 π d φ κ cos 2 ( φ κ φ m ) × ( 1 cos { κ d ( z ) cos [ φ κ φ d ( z ) ] } ) ,
σ Δ Z , m 2 = 10240 π 9 Γ ( 1 3 ) D 2 ( L ) 0 L d z C n 2 ( z ) D 2 ( z ) 0 d κ κ 1 × ( κ 2 + k o 2 ) 11 / 6 e κ 2 / k i 2 cos 2 ( κ 2 D ( z ) ( L z ) 2 k D ( L ) ) × J 2 2 ( D ( z ) κ 2 ) { 1 J 0 [ κ d ( z ) ] + J 2 [ κ d ( z ) ] cos [ 2 φ d ( z ) 2 φ m ] } .
σ Δ Z , m 2 = 2560 π 9 Γ ( 1 3 ) 333.6 D 2 ( L ) 0 L d z C n 2 ( z ) 0 d κ κ 8 / 3 ( J 2 [ D ( z ) κ / 2 ] D ( z ) κ / 2 ) 2 × { 1 J 0 [ κ d ( z ) ] + J 2 [ κ d ( z ) ] cos [ 2 φ d ( z ) 2 φ m ] } .
θ C , m = 1 k d r a 2 ( r ) ( m ^ · ) ϕ ( r ) / d r a 2 ( r )
θ C , m ( κ , z ) = κ k cos ( φ m φ κ ) d r a 2 ( r , z ) P cos [ κ · r + α ( κ , z ) ] .
θ C , m ( κ , z ) = κ k cos ( φ m φ κ ) Re { exp [ i α ( κ , z ) ] F [ a 2 ( r , z ) P ] } ,
σ C , m 2 ( κ , z ) = κ 2 2 k 2 cos 2 ( φ m φ κ ) | F [ a 2 ( r , z ) P ] | 2 .
X m ( κ , z ; L ) = ( L z ) θ C , m ( κ , z ) ,
σ X , m 2 ( κ , z ; L ) = ( L z ) 2 σ C , m 2 ( κ , z ) ,
= ( L z ) 2 κ 2 2 k 2 cos 2 ( φ m φ κ ) | F [ a 2 ( r , z ) P ] | 2 .
σ X , m 2 = 4 π k 2 0 L d z C n 2 ( z ) 0 d κ κ [ 5 18 π Γ ( 1 3 ) ( κ 2 + k o 2 ) 11 / 6 × e κ 2 / k i 2 ] 0 2 π d φ κ { ( L z ) 2 κ 2 2 k 2 cos 2 ( φ m φ κ ) × | F [ a 2 ( r , z ) P ] | 2 } .
σ X , m 2 = 5 π 9 Γ ( 1 3 ) 0 L d z C n 2 ( z ) ( L z ) 2 0 d κ κ 3 ( κ 2 + k o 2 ) 11 / 6 × e κ 2 / k i 2 { F [ a 2 ( r , z ) P ] } 2 .
a 2 ( r , z ) P = 2 π w 2 ( z ) e 2 r 2 / w 2 ( z ) ,
σ X , m 2 = 5 π 9 Γ ( 1 3 ) 0 L d z C n 2 ( z ) ( L z ) 2 0 d κ κ 3 ( κ 2 + k o 2 ) 11 / 6 × e κ 2 / k i 2 e κ 2 w 2 ( z ) / 4 .
σ X , m 2 = 5 π Γ ( 1 6 ) 2 2 / 3 9 Γ ( 1 3 ) 2.285 0 L d z C n 2 ( z ) ( L z ) 2 w 1 / 3 ( z ) .
ψ ( r , z 0 ) = exp ( i k 2 f 0 r 2 ) exp [ i ϵ sin ( κ · r + α ) ] ,
ψ ( r , z 0 ) = exp ( i k 2 f 0 r 2 ) n = J n ( ϵ ) e i n κ · r e i n α .
ψ ( r , z 0 + z ) = e i k z i λ z n = J n ( ϵ ) e i n α d r 0 e i n κ · r 0 exp ( i k 2 f 0 r 0 2 ) × exp [ i k 2 z ( r 0 r ) 2 ] .
k 2 f 0 r 0 2 + k 2 z ( r 0 r ) 2 = k 2 f 1 r 2 + k s 2 z ( r 0 r s ) 2 .
ψ ( r , z 0 + z ) = e i k z i λ z exp ( i k 2 f 1 r 2 ) n = J n ( ϵ ) e i n α × d r 0 e i n κ · r 0 exp [ i k s 2 z ( r 0 r s ) 2 ] .
ψ ( r , z 0 + z ) = e i k z s exp ( i k 2 f 1 r 2 ) n = J n ( ϵ ) e i n κ · r / s e i n α × exp ( i n 2 κ 2 z 2 k s ) .
ψ ( r , z 0 + z ) e i k z s exp ( i k 2 f 1 r 2 ) × exp [ i ϵ cos ( κ 2 z 2 k s ) sin ( κ · r s + α ) ] × exp [ ϵ sin ( κ 2 z 2 k s ) sin ( κ · r s + α ) ] .
ϕ ( κ , r , z 0 ; z 0 + z ) = sin ( D ( z 0 ) D ( z 0 + z ) κ · r + α ) × cos ( κ 2 z 2 k D ( z 0 ) D ( z 0 + z ) )
cos ( κ 2 z 2 k D ( z 0 ) D ( z 0 + z ) ) = cos ( 2 π z z T D ( z 0 ) D ( z 0 + z ) ) ,
Δ ϕ ( r , z ) = d κ g ( κ , z ) q 1 / 2 [ F Δ ϕ ( κ , z ) ] 1 / 2 e i κ · r ,
Δ ϕ ( r , z ) = 2 d κ x 0 d κ y B ( κ , z ) [ F Δ ϕ ( κ , z ) ] 1 / 2 × sin [ κ · r + α ( κ , z ) ] .
θ G , m ( z ; L ) = 2 d κ x 0 d κ y B ( κ , z ) [ F Δ ϕ ( κ , z ) ] 1 / 2 × cos [ α ( κ , z ) ] θ ˜ G , m ( κ , z ; L ) ,
σ G , m 2 ( z ; L ) = 4 d κ x 0 d κ y F Δ ϕ ( κ , z ) σ G , m 2 ( κ , z ; L ) ,
σ G , m 2 ( z ; L ) = 2 d κ F Δ ϕ ( κ , z ) σ G , m 2 ( κ , z ; L ) .
F Δ ϕ ( κ , z ) = 2 π k 2 Φ n 0 ( κ ) z z + Δ z d z C n 2 ( z )
d σ G , m 2 ( z ; L ) = 4 π k 2 d z C n 2 ( z ) d κ Φ n 0 ( κ ) σ G , m 2 ( κ , z ; L ) .
σ G , m 2 = 4 π k 2 0 L d z C n 2 ( z ) 0 d κ κ Φ n 0 ( κ ) 0 2 π d φ κ σ G , m 2 ( κ , z ; L ) ,
P = d r | ψ ( r ) | 2 = 1 4 π 2 d κ | F [ ψ ( r ) ] | 2 ,
X κ = 1 4 π 2 P d κ κ | F [ ψ ( r ) ] | 2 .
θ C = X κ k = 1 i 4 π 2 P k d κ F [ ψ ( r ) ] { F [ ψ ( r ) ] } * .
θ C = 1 i P k d r [ ψ ( r ) ] ψ * ( r ) .
θ C = 1 2 i P k d r [ a 2 ( r ) ] + 1 P k d r a 2 ( r ) ϕ ( r ) .
θ C = 1 P k d r a 2 ( r ) ϕ ( r ) ,
F [ ψ ( r , z ) ] = F [ ψ ( r , 0 ) ] H ( κ , z ) .
H ( κ , z ) = exp [ i 2 π λ z i λ z 4 π ( κ x 2 + κ y 2 ) ] .
| F [ ψ ( r , z ) ] | 2 = | F [ ψ ( r , 0 ) ] | 2 .
ψ ( r , z ) = e i k z i λ z exp ( i k 2 z r 2 ) F [ ψ ( r 0 , 0 ) exp ( i k 2 z r 0 2 ) ] ,
X ( z ) = 1 P d r r | e i k z i λ z exp ( i k 2 z r 2 ) F [ ψ ( r 0 , 0 ) exp ( i k 2 z r 0 2 ) ] | 2 .
X ( z ) = λ z 8 π 3 P d κ κ F [ ψ ( r 0 , 0 ) exp ( i k 2 z r 0 2 ) ] × F * [ ψ ( r 1 , 0 ) exp ( i k 2 z r 1 2 ) ] .
X ( z ) = λ z i 8 π 3 P d κ F { [ ψ ( r 0 , 0 ) exp ( i k 2 z r 0 2 ) ] } × F * [ ψ ( r 1 , 0 ) exp ( i k 2 z r 1 2 ) ] .
X ( z ) = λ z i 2 π P d r [ ψ ( r , 0 ) exp ( i k 2 z r 2 ) ] × ψ * ( r , 0 ) exp ( i k 2 z r 2 )
= 1 P d r r | ψ ( r , 0 ) | 2 + λ z i 2 π P d r ψ * ( r , 0 ) ψ ( r , 0 ) .
X ( z ) = X ( 0 ) + z θ C ( 0 ) .
θ C ( z ) = 0 z d z d θ C d z | z = z .
d X d z = θ C ( z ) ,
X ( L ) = 0 L d z θ C ( z ) = 0 L d z 0 z d z d θ C d z | z = z .
X ( L ) = 0 L d z ( L z ) d θ C d z .