Abstract

20We describe a novel formulation of light beam propagation through any complex optical system that can be described by an ABCD ray-transfer matrix. Within the paraxial approximation, optical propagation can be formulated in terms of a Huygens principle expressed in terms of the ray-transfer ABCD matrix elements of the optical system. We extend and generalize previous treatments to include the effects of finite-sized limiting apertures (i.e., diffractive screens) in the optical train, tilt and random jitter of the optical elements, and distributed random inhomogeneities along the optical path (e.g., clear air turbulence and aerosols). In the presence of limiting apertures the ABCD matrix elements of the optical system are complex. For the case of laser beam propagation and Gaussian-shaped limiting apertures in the optical train, we obtain analytical expressions for both the spot radius and the wave-front radius of curvature at an arbitrary observation plane and give illustrative examples of practical concern. In particular, analytical expressions for the fringe visibility obtained in a coherent laser interferometric system are presented. An analytical expression for the mean spot radius of a laser beam propagating through an optical system in the presence of tilt and random jitter is obtained. We also consider the propagation of partially coherent light through optical systems. In particular, we derive a generalized van Cittert–Zernike theorem that is valid for an arbitrary optical system that can be characterized by an ABCD ray-transfer matrix. Finally, the propagation of laser beams through a general optical system in the presence of distributed random inhomogeneities is considered. An explicit expression for the mean irradiance distribution of a Gaussian-shaped beam is derived that is valid for an arbitrary optical system. In addition, we derive an expression for the mutual-coherence function for wave propagation through an arbitrary optical system. In all cases the results are expressed in terms of the ABCD matrix elements of the complete optical system. The formulation of optical propagation presented here is a rather simple and straightforward way of determining the effects of finite-sized optical elements, tilt and random jitter, and distributed random inhomogeneities along the optical path. It is merely necessary first to multiply the relevant ray matrices together to find the complete system matrix and then to substitute this matrix into the expressions given in this paper.

© 1987 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
    [CrossRef]
  2. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,”J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  3. J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics XI, E. Wolf, ed. (North-Holland, Amsterdam, 1973), pp. 247–304.
    [CrossRef]
  4. J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
  5. A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976); erratum QE-12, 315 (1976).
    [CrossRef]
  6. D. M. Walsh, L. V. Knight, “Transverse modes of a resonator with Gaussian mirrors,” Appl. Opt. 25, 2947–2954 (1986).
    [CrossRef] [PubMed]
  7. A. E. Siegman, Lasers (University Sciences, Mill Valley, Calif., 1986).
  8. J. P. Tache, “Ray matrices for tilted interfaces in laser resonators,” Appl. Opt. 26, 427–429 (1987).
    [CrossRef] [PubMed]
  9. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  10. P. Baues, “The connection of geometrical optics with propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  12. S. G. Hanson, “The laser gradient anemometer,” in Photon Correlation Techniques in Fluid Mechanics (Springer-Verlag, Berlin, 1983), pp. 212–220; A. S. Jensen, Risø National Laboratory, Roskilde, Denmark (personal communication, 1986).
    [CrossRef]
  13. R. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63. 1669–1685 (1975).
    [CrossRef]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).
  15. R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, New York, 1985).
    [CrossRef]
  16. L. W. Caspersen, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 13, 2243–2249 (1981).
    [CrossRef]
  17. A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).
  18. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.
  19. G. A. Massey, A. E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt. 8, 975–978 (1969).
    [CrossRef] [PubMed]

1987 (1)

1986 (1)

1981 (1)

L. W. Caspersen, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 13, 2243–2249 (1981).
[CrossRef]

1976 (1)

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976); erratum QE-12, 315 (1976).
[CrossRef]

1975 (1)

R. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63. 1669–1685 (1975).
[CrossRef]

1970 (2)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).

S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,”J. Opt. Soc. Am. 60, 1168–1177 (1970).
[CrossRef]

1969 (3)

P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
[CrossRef]

P. Baues, “The connection of geometrical optics with propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969).
[CrossRef]

G. A. Massey, A. E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt. 8, 975–978 (1969).
[CrossRef] [PubMed]

1966 (1)

Arnaud, J. A.

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics XI, E. Wolf, ed. (North-Holland, Amsterdam, 1973), pp. 247–304.
[CrossRef]

Baues, P.

P. Baues, “The connection of geometrical optics with propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969).
[CrossRef]

P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Burch, J. M.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Caspersen, L. W.

L. W. Caspersen, “Synthesis of Gaussian beam optical systems,” Appl. Opt. 13, 2243–2249 (1981).
[CrossRef]

Collins, S. A.

Fante, R.

R. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63. 1669–1685 (1975).
[CrossRef]

Fante, R. L.

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, New York, 1985).
[CrossRef]

Gerrard, A.

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Hanson, S. G.

S. G. Hanson, “The laser gradient anemometer,” in Photon Correlation Techniques in Fluid Mechanics (Springer-Verlag, Berlin, 1983), pp. 212–220; A. S. Jensen, Risø National Laboratory, Roskilde, Denmark (personal communication, 1986).
[CrossRef]

Knight, L. V.

Kogelnik, H.

Li, T.

Massey, G. A.

Siegman, A. E.

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976); erratum QE-12, 315 (1976).
[CrossRef]

G. A. Massey, A. E. Siegman, “Reflection and refraction of Gaussian light beams at tilted ellipsoidal surfaces,” Appl. Opt. 8, 975–978 (1969).
[CrossRef] [PubMed]

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.

A. E. Siegman, Lasers (University Sciences, Mill Valley, Calif., 1986).

Tache, J. P.

Walsh, D. M.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

Appl. Opt. (5)

Bell Syst. Tech. J. (1)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).

IEEE J. Quantum Electron. (1)

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976); erratum QE-12, 315 (1976).
[CrossRef]

J. Opt. Soc. Am. (1)

Opto-Electronics (2)

P. Baues, “Huygens principle in inhomogeneous isotropic media and a general integral equation applicable to optical resonators,” Opto-Electronics 1, 37–44 (1969).
[CrossRef]

P. Baues, “The connection of geometrical optics with propagation of Gaussian beams and the theory of optical resonators,” Opto-Electronics 1, 103–118 (1969).
[CrossRef]

Proc. IEEE (1)

R. Fante, “Electromagnetic beam propagation in turbulent media,” Proc. IEEE 63. 1669–1685 (1975).
[CrossRef]

Other (8)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1975).

R. L. Fante, “Wave propagation in random media: a systems approach,” in Progress in Optics XXII, E. Wolf, ed. (Elsevier, New York, 1985).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

S. G. Hanson, “The laser gradient anemometer,” in Photon Correlation Techniques in Fluid Mechanics (Springer-Verlag, Berlin, 1983), pp. 212–220; A. S. Jensen, Risø National Laboratory, Roskilde, Denmark (personal communication, 1986).
[CrossRef]

A. E. Siegman, Lasers (University Sciences, Mill Valley, Calif., 1986).

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” in Progress in Optics XI, E. Wolf, ed. (North-Holland, Amsterdam, 1973), pp. 247–304.
[CrossRef]

A. Gerrard, J. M. Burch, Introduction to Matrix Methods in Optics (Wiley, London, 1975).

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Chap. 8.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (17)

Fig. 1
Fig. 1

Schematic representation of beam wave propagation through a train of optical elements.

Fig. 2
Fig. 2

Schematic diagram of an optical system used to observe interference effects in the observation plane. Two beams are employed, one having its waist at a distance Δ from the left-hand focal plane and the other having its waist at the position of the left-hand focal plane.

Fig. 3
Fig. 3

Optical Fourier-transforming system with a limiting aperture (L.A.) of radius σ placed immediately to the right of the transform lens.

Fig. 4
Fig. 4

Normalized beam radius in the Fourier-transform plane as a function of σ/ωi for various values of γ = 2f/i2.

Fig. 5
Fig. 5

The fraction of the transmitted power impinging upon the Fourier-transform plane as a function of σωi for various values of γ = 2f/i2.

Fig. 6
Fig. 6

Normalized on-axis irradiance in the Fourier-transform plane as a function of σ/ωi for various values of ω = 2f/i2.

Fig. 7
Fig. 7

The wave-front radius of curvature in the Fourier-transform plane divided by the focal length of the transform lens as a function of σ/ωi for various values of γ = 2f/i2.

Fig. 8
Fig. 8

Same as Fig. 2 except that a limiting aperture of radius σ is placed at the central focal plane.

Fig. 9
Fig. 9

Beam radius as a function of defocus for various values of α = (2f/iσ)2.

Fig. 10
Fig. 10

Wave-front radius of curvature versus normalized defocus for various values of α = (2f/iσ)2.

Fig. 11
Fig. 11

Fringe visibility versus normalized defocus for various values of α and ωR/ωi = 1.

Fig. 12
Fig. 12

Fringe visibility versus normalized defocus for various values of a and ωR/ωi = 2.

Fig. 13
Fig. 13

Fringe visibility versus normalized defocus for various values of α and ωR/ωi = ∞.

Fig. 14
Fig. 14

Fringe visibility versus normalized transverse displacement for various values of α and ωR/ωi = 0.5.

Fig. 15
Fig. 15

Fringe visibility versus normalized transverse displacement for various values of α and ωR/ωi → 1.

Fig. 16
Fig. 16

Fringe visibility versus normalized transverse displacement for various values of α and ωR/ωi →∞.

Fig. 17
Fig. 17

Schematic diagram used to illustrate the propagation of jitter through an optical system. Each lens is assumed to be characterized by a one-sigma, one-axis standard deviation jitter of σi.

Tables (2)

Tables Icon

Table 1 Ray Matrices for Various Simple Optical Elementsa

Tables Icon

Table 2 Ray-Matrix Elementsa

Equations (164)

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

z i = z o i + a i ( z ) x i 2 + b i ( z ) y i 2 ,
U ( x , y ) = - i k 2 π B x B y exp ( - i k L ) d x o d y o U i ( x o , y o ) × exp [ - i k 2 B x ( D x x 2 - 2 x x 0 + A x y o 2 ) - i k 2 B y ( D y y 2 - 2 y y o - A y y o 2 ) ] ,
U ( r ) = - i k 2 π B exp ( - i k L ) d 2 r o U i ( r o ) × exp [ - i k 2 B ( D r 2 - 2 r · r o + A r o 2 ) ] .
U i ( x , y ) = A o exp [ - ( x / ω x i ) 2 ] H m ( 2 x ω x i ) × exp [ - ( y / ω y i ) 2 ] H n ( 2 y ω y i ) ,
U ( x , y ) = A o ( ω x i ω y i ω x ω y ) 1 / 2 exp { - i [ k L + ( 1 2 + m ) δ x + ( 1 2 + n ) δ y ] } exp [ - i ( π / λ q x ) x 2 ] H m ( 2 x ω x ) × exp [ - i ( π / λ q y ) y 2 ] H n ( 2 y ω y ) ,
q x , y = A x , y q i x , y + B x , y C x , y q i x , y + D x , y
1 q i x , y = - i λ π ω i x , y 2 .
ω x , y = ω i x , y [ A 2 + ( 2 B / k ω i x , y 2 ) 2 ] 1 / 2 ,
δ x , y = arctan ( - - λ B x , y π ω i x , y 2 A x , y ) .
1 q x , y = 1 R x , y - i λ π ω x , y 2 .
1 R x , y = ( ω i x , y ω x , y ) 2 [ A x , y C x , y + B x , y D x , y ( 2 k ω i 2 ) 2 ] .
M = [ - m - m Δ 0 - 1 m ] ,
ω = m ω i [ 1 + ( 2 Δ k ω i 2 ) 2 ] 1 / 2
R = m 2 Δ [ 1 + ( k ω i 2 2 Δ ) 2 ] ,
T = exp ( - r 2 / σ 2 ) ,
F - 1 = 2 i k σ 2 .
M L A = [ 1 0 - 1 F 1 ] = [ 1 0 - 2 i k σ 2 1 ] .
M = [ - 2 f i k σ 2 f ( 1 - 2 f i k σ 2 ) - 1 f ( 1 + 2 f i k σ 2 ) - 2 f i k σ 2 ] ,
exp ( - i k 2 q 2 r 2 ) = exp ( - i k 2 q 1 r 2 ) exp ( - r 2 σ 2 ) = exp [ - i k 2 ( 1 q 1 - 2 i k σ 2 ) ] ,
q 2 = q 1 1 - ( 2 i k σ 2 ) q 1 .
q o = A q i + B C q i + D ,
1 R = Re ( 1 / q ) = ( ω i ω 1 ) 2 { Re [ C A * + ( 2 k ω i 2 ) 2 D B * ] + ( 2 k ω i 2 ) Im ( D A * - C B * ) }
ω 2 = - 2 / k Im ( 1 / q ) = ω 1 2 Re ( A D * - B C * ) + 2 k ω i 2 { Im [ B D * + ( k ω i 2 2 ) 2 A C * ] } ,
ω 1 2 = 4 B 2 k 2 ω i 2 + ω i 2 A 2 + 4 k Im ( B A * ) .
I ( r ) = 2 P T π ω 1 2 exp ( - 2 r 2 ω 2 ) ,
P o = I ( r ) d 2 r = ( ω ω 1 ) 2 P T ,
ω 2 = ω 1 2 1 + ( 2 f k σ ω i ) 2 + ω i 2 σ 2
ω 1 2 = ( 2 f k ω i ) 2 [ ( 1 + ω i 2 σ 2 ) 2 + ( 2 f k σ 2 ) 2 ] .
P o P T = ( ω ω 1 ) 2 = 1 1 + ( 2 f k σ ω i ) 2 + ω i 2 σ 2 ,
I ( 0 ) = I [ 1 1 + ω i 2 σ 2 + ( 2 f k σ 2 ) 2 ] ,
R f = ( σ ω i ) 4 [ ( 1 + ω i 2 σ 2 ) 2 + ( 2 f k σ 2 ) 2 1 + 2 ω i 2 σ 2 + ( 2 f k σ 2 ) 2 ] .
M = [ - m - m Δ - 2 f 1 f 2 i k σ 2 0 - 1 m ] .
q o = A q i + B C q i + D = m 2 ( q i + Δ ) + 2 f 2 2 i k σ 2 .
ω 2 ( Δ ) = ( m ω i ) 2 [ ( 1 + α ) 2 + δ 2 1 + α ]
R = k ( m ω i ) 2 2 [ ( 1 + α ) 2 + δ 2 δ ] ,
α = ( 2 f 1 k ω i 2 ) ( 2 f 1 k σ 2 )
δ = 2 Δ k ω i 2 .
R MIN = k ( m ω i ) 2 ( 1 + α ) ,
Δ MIN = k ω i 2 2 ( 1 + α )
V = 2 Re ( U 1 U 2 * ) W R d 2 r ( U 1 2 + U 2 2 ) W R d 2 r ,
W R = exp ( - r 2 / ω R 2 ) ,
V = ( 2 β γ 2 + β 2 ) 1 Ω 1 2 + Ω 2 2 ,
β = 1 ω 1 2 [ 2 D 2 + δ 2 D ( D 2 + δ 2 ) + 1 2 ] ,
γ = δ ω i 2 ( D 2 + δ 2 ) ,
= ω R ω i ,
D = 1 + α ,
1 Ω 1 2 = 2 ω 2 ( 0 ) + 1 ω R 2 ,
1 Ω 2 2 = 2 ω 2 ( Δ ) + 2 ω R 2 ,
V = 4 D 2 ( D 2 + δ 2 ) 4 D 4 + 5 D 2 δ 2 + δ 4 .
U 1 i = U o exp ( - r 2 / ω i 2 )
U 2 i = U o exp [ - ( r - r o ) 2 / ω i 2 ] .
U 1 ( r ) = U o ( ω i ω ) exp ( - i k r 2 / 2 q 1 ) = U o ( ω i ω ) exp ( - r 2 / ω 2 ) ,
ω = m ω i ( 1 + α ) 1 / 2 .
U 2 = U 1 exp [ - i k 2 ( A r o 2 - 2 r · r o q i A + B ) ] ,
q i = ( k ω i 2 2 ) i .
U 2 ( r ) = U o ( ω i ω ) exp [ - ( r + m r o ) 2 ω 2 ] .
V = 2 exp [ - r o 2 ω 2 ( 2 + D 2 2 + D ) ] 1 + exp [ - 2 r o 2 ω 2 ( D D + 2 2 ) ] ,
V = exp ( - r o 2 / 2 ω 2 ) .
θ j = θ j + δ θ j .
r j + = A j r j - + B j ( r j ) -
( r j ) + = C j r j - + D j ( r j ) - + θ j ,
R o = M R i + T ,
R = [ r r ] , M = [ A B C D ]
T = j = 1 n ( k = j n M k ) T j ,
T j = [ 0 θ j ] .
T = [ t t ] ,
t = j = 1 n B j θ j ,
t = j = 1 n D j θ j .
k = j n M k ,
U ( r 2 ) = - i k 2 π B exp ( - i k L ) d 2 r 1 U i ( r 1 ) × exp [ - i k 2 B ( D r 2 2 - 2 r 1 · r 2 + A r 1 2 ) ] × exp { - i k [ r 1 · t B + r 2 · ( t - D B t ) ] } ,
U ( r 2 ) = - i k 2 π z exp ( - i k z ) d 2 r 1 U i ( r 1 ) exp [ - i k 2 z ( r 2 - r 1 ) 2 ] × exp ( - i k r 1 · θ ) .
U i = A o g ( r ω i ) exp ( - i k 2 q i r 2 ) ,
g = ( 2 r ω i ) l L p l ( 2 r 2 ω i 2 ) .
x d 2 L p l d x 2 + ( l + 1 - x ) d L p l d x + p L p l = 0.
L o l ( x ) = 1 , L 1 l ( x ) = l + 1 - x , L 2 l ( x ) = ½ ( l + 1 ) ( l + 2 ) - ( l + 2 ) x + ½ x 2 .
I ( r 2 ) = U ( r 2 ) 2 = ( 2 P T π ω 2 ) g [ 2 ( r 2 - t ) ω ] 2 exp ( - 2 r 2 2 ω 2 ) ,
t = j = 1 n B j θ j ,
ω 2 = A 2 ω i 2 + 4 B 2 k 2 ω i 2 + 2 t 2 ,
t 2 = j = 1 n B j 2 σ j 2 ,
I ( r 2 ) d 2 r 2 = P T .
ω 2 2 = 4 z 1 2 k 2 ω i 2 + 2 σ 1 2 z 1 2 ,
ω 3 2 = 4 z 2 2 k 2 ω i 2 + ( z 2 z 1 ) 2 ω i 2 + 2 z 2 2 ( σ 1 2 + σ 2 2 ) ,
ω 4 2 = 4 [ z 3 ( 1 - z 1 z 2 ) ] 2 k 2 ω i 2 + ( z 3 z 1 ) 2 ω i 2 + 2 [ z 3 ( 1 - z 1 z 2 ) 2 ] 2 σ 1 2 + 2 z 3 2 ( σ 2 2 + σ 3 2 ) .
ω 2 2 = 4 z 1 2 k 2 ω i 2 + 2 z 1 2 σ 1 2 ,
ω 3 2 = 4 z 2 2 k 2 ω i 2 + ( z 2 z 1 ) 2 ω i 2 + 2 z 2 2 ( σ 1 2 + σ 2 2 ) ,
ω 4 2 = 4 ( z 1 z 3 / z 2 ) 2 k 2 ω i 2 + 2 ( z 1 z 3 z 2 ) 2 σ 1 2 + 2 z 3 2 σ 3 2 .
ω 2 2 = 4 z 1 2 k 2 ω i 2 + 2 z 1 2 σ 1 2 ,
ω 3 2 = ( z 2 z 1 ) 2 ω i 2 + 2 z 2 2 σ 2 2 ,
ω 4 2 = 4 ( z 1 z 3 / z 2 ) 2 k 2 ω i 2 + 2 ( z 1 z 3 z 2 ) 2 σ 1 2 + 2 z 3 2 σ 3 2 .
ω 4 2 = ( z 3 / z 2 ) 2 ω 2 2 + 2 z 3 2 σ 3 2 .
γ ( r , r ) = U ( r ) U * ( r ) [ U ( r ) 2 ] 1 / 2 [ U ( r ) 2 ] 1 / 2 ,
γ = Γ ( r , r ) [ I ( r ) I ( r ) ] 1 / 2 ,
Γ ( r , r ) = | k 2 π B | 2 d 2 r 1 d 2 r 1 U i ( r 1 ) U i * ( r 1 ) × exp { - i k B [ D R · ρ + A R 1 · ρ 1 - ( ρ · R 1 + ρ 1 · R ) ] } ,
I ( r ) = Γ ( r , r ) ,
ρ = r - r ,
R = ½ ( r + r ) ,
ρ 1 = r 1 - r 1 ,
R 1 = ½ ( r 1 + r 1 ) .
U i ( r 1 ) U i * ( r 1 ) = S ( R 1 ) δ ( ρ 1 ) ,
γ ( ρ , R ) = exp ( - i k B D ρ · R ) d 2 r S ( r ) exp ( i k B ρ · r ) d 2 r S ( r ) ,
S ( r ) = exp ( - 2 r 2 / r 0 2 ) ,
γ = exp ( - i k B D ρ · R ) exp ( - ρ 2 / ρ o 2 ) ,
ρ o = 2 2 B k r o .
γ = exp ( - ρ 2 / ρ o 2 ) ,
S ( r ) = { 1 r a 0 otherwise .
γ = exp ( - i k B D ρ · R ) [ 2 J 1 ( v ) v ] 2 ,
v = k a ρ B
I ( r 1 ) I ( r 2 ) = I ( r 1 ) I ( r 2 ) ( 1 + γ 2 )
Cov I = γ 2 ,
U ( p ) = ( - i k 2 π B ) d 2 r U i ( r ) exp [ ψ ( p , r ) ] × exp [ - i k 2 B ( D p 2 - 2 p · r + A r 2 ) ] .
U ( p ) = exp ψ U o ,
exp ψ = exp [ χ + i ϕ + ½ ( χ 2 + 2 i ϕ χ - ϕ 2 ) ] ,
exp ψ exp ( - ½ ϕ 2 ) ;
ϕ 2 = 2 π 0 L d z d 2 Q Φ n ( Q ; z ) ,
Φ n ( Q ; z ) = 0.033 C n 2 ( z ) exp ( - Q 2 / Q m 2 ) ( Q 2 + 1 / L o 2 ) 11 / 6 ,
Φ n 0.033 C n 2 ( z ) Q - 11 / 3 .
( Φ n ) aerosols = ( 1 2 π k 4 ) σ v ( θ ) ,
ϕ 2 0.78 k 2 L o 5 / 3 0 L C n 2 ( z ) d z ,
U = exp [ - 0.39 k 2 L o 5 / 3 0 L C n 2 ( z ) d z ] U o .
I ( p ) = U ( p ) 2 .
I ( p ) = | k 2 π B | 2 d 2 r exp ( i k B p · r ) K ( r ) M ( r ) ,
K ( r ) = d 2 R U i ( R + ½ r ) u i * ( R - ½ r ) exp ( - i k A B r · R )
M ( r ) = exp [ ψ ( r 1 , p ) + ψ * ( r 2 , p ) ]
M ( r ) = exp [ - ½ D w ( r ) ] ,
D w ( r ) = 4 π k 2 0 L d z d 2 Q Φ n ( Q ; z ) { 1 - exp [ i Q · r ( z ) ] } ,
r ( z ) = [ B ( z ) B ] r ,
U i = U o exp ( - r 2 / ω i 2 )
I ( p ) = 2 P T π ω 2 exp ( - 2 p 2 ω 2 ) ,
ω 2 = A 2 ω i 2 + 4 B 2 k 2 ω i 2 + 2 t 2 + 8 k 2 q T 2 ,
q T [ 1.46 k 2 0 L d z C n 2 ( z ) B ( z ) 5 / 3 ] - 3 / 5 .
exp [ - 0 L d z α a ( z ) ] ,
8 k 2 q a 2 ,
q a 2 = [ k 2 θ a 2 4 ( 1 l a ) 0 L B 2 ( z ) d z ] - 1 / 2
γ = Γ ( r , r ) [ I ( r ) I ( r ) ] 1 / 2 ,
Γ ( r , r ) = | k 2 π B | 2 exp [ - i k 2 ( D B r 2 - D * B * r 2 ) ] d 2 r 1 S ( r 1 ) × exp [ i k r 1 · ( r B - r B * ) ] exp [ k Im ( A / B ) r 1 2 ]
I ( r ) = Γ ( r , r ) .
γ = exp ( - ρ 2 / ρ 0 2 ) ,
ρ = r - r
ρ 0 2 = 8 B 2 k 2 r o 2 + 4 k Im ( B A * ) .
Γ ( r 1 , r 2 ) = S o π ρ o 2 exp [ - ( r 1 - r 2 ) 2 ρ o 2 ] exp [ - ( r 1 2 + r 2 2 ) ω 2 ] ,
ω = ρ o { B 2 [ 2 ( Im B ) 2 + Im B D * Im B A * ] } 1 / 2 .
I ( r ) = Γ ( r , r ) = S o π ρ o 2 exp ( - 2 r 2 / ω 2 ) ,
Γ ( r 1 , r 2 ; ρ 1 , ρ 2 ) = exp [ ψ ( r 1 , p 1 ) + ψ * ( r 2 , p 2 ) ] ,
Γ = exp [ - ½ D w ( r 1 - r 2 ; ρ 1 - ρ 2 ) ] ,
D w = 4 π k 2 0 L d z d 2 Q Φ n ( Q ; z ) { 1 - exp [ i Q · r ( z ) ] } ,
r 1 ( z ) = A ( z ) p 1 + B ( z ) p 1
r 2 ( z ) = A ( z ) p 2 + B ( z ) p 2 ,
p 1 - p 2 = r - p A ( L ) B ( L ) ,
r = r 1 - r 2 ,
p = p 1 - p 2 .
A ( L ) = D ,             B ( L ) = B ,
r ( z ) = r 1 ( z ) - r 2 ( z ) = A ( z ) p + B ( z ) B ( r - D p ) .
D w = 4 π k 2 0 L d z d 2 Q Φ n ( Q ; z ) × ( 1 - exp { i Q · [ A ( z ) p + B ( z ) B ( r - D p ) ] } ) .
D w = 8 π 2 k 2 0 L d z 0 d Q Q Φ n ( Q ; z ) × { 1 - J o [ Q | A ( z ) p + B ( z ) B ( r - D p ) | ] } ,
r ( z ) = B ( z ) B r
D w = 4 π k 2 0 L d z d 2 Q Φ n ( Q ; z ) { 1 - exp [ i Q · r B ( z ) B ] } ,
D w = 8 π 2 k 2 0 L d z 0 d Q Q Φ n ( Q ; z ) { 1 - J o [ B ( z ) B Q r ] } .
D w ( r ) 2.92 k 2 r 5 / 3 B 5 / 3 0 L d z C n 2 ( z ) B ( z ) 5 / 3
M = exp ( - ½ D w ) = exp [ - ( r / r o ) 5 / 3 ] ,
r o = [ 1.46 k 2 B 5 / 3 0 L d z C n 2 ( z ) B ( z ) 5 / 3 ] - 3 / 5
B χ ( r ) = 2 π 2 k 2 0 L d z 0 d Q Q Φ n ( Q ; z ) J o [ Q r ( z ) ] [ 1 - cos ( β Q 2 / k ) ] ,
r ( z ) = B ( 0 , z ) B ( 0 , L ) r ,
β ( z ) = B ( 0 , z ) B ( z , L ) B ( 0 , L ) .
χ 2 0.563 k 7 / 6 0 L d z C n 2 ( z ) β ( z ) 5 / 6 .

Metrics