Abstract

Recent development of active imaging system technology in the defense and security community have driven the need for a theoretical understanding of its operation and performance in military applications such as target acquisition. In this paper, the modeling of active imaging systems, developed at the U.S. Army RDECOM CERDEC Night Vision & Electronic Sensors Directorate, is presented with particular emphasis on the impact of coherent effects such as speckle and atmospheric scintillation. Experimental results from human perception tests are in good agreement with the model results, validating the modeling of coherent effects as additional noise sources. Example trade studies on the design of a conceptual active imaging system to mitigate deleterious coherent effects are shown.

© 2007 Optical Society of America

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  1. A. F. Milton, G. Klager, and T. Bowman, "Low cost sensors for UGVs," G. R. Gerhart, R. W. Gunderson, and C. M. Shoemaker, eds., Proc. SPIE 4024, 180-191 (2000).
    [CrossRef]
  2. O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, "Gated viewing for target detection and target recognition," G. W. Kamerman and C. Werner, eds., Proc. SPIE 3707, 432-448 (1999).
    [CrossRef]
  3. J. Busck, "Underwater 3-D optical imaging with a gated viewing laser radar," Opt. Eng. 44, 116,001 (2005).
    [CrossRef]
  4. P. Andersson, "Long-range three-dimensional imaging using range-gated laser radar images," Opt. Eng. 45, 034,301 (2006).
    [CrossRef]
  5. J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, Germany, 1975).
  6. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, WA, 2001).
    [CrossRef]
  7. E. L. Jacobs, R. H. Vollmerhausen, and C. E. Halford, "Modeling active imagers," G. C. Holst, ed., Proc. SPIE 5407, 201-210 (2004).
    [CrossRef]
  8. K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, "Atmospheric turbulence modulation transfer function for infrared target acquisition modeling," Opt. Eng. 40, 1906-1913 (2001).
    [CrossRef]
  9. J. W. Goodman, Statistical Optics (Wiley Interscience, New York, NY, 2000).
  10. R. Vollmerhausen, E. Jacobs, and R. Driggers, "New metric for predicting target acquisition performance," Opt. Eng. 43, 2806-2818 (2004).
    [CrossRef]
  11. P. G. J. Barten, "Evaluation of subjective image quality with the square-root integral method," J. Opt. Soc. Am. A 7, 2024-2031 (1990).
    [CrossRef]
  12. P. G. J. Barten, Contrast sensitivity of the human eye and its effects on image quality (SPIE Press Monograph, PM72, 1999).
    [CrossRef]
  13. W. Wolfe and G. Zissis, The Infrared Handbook (IRIA ERIM, Ann Arbor, MI, 1993).
  14. R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, "Impact of speckle on laser rangegated shortwave infrared imaging system target identification performance," Opt. Eng. 42, 738-746 (2003).
    [CrossRef]
  15. C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, "Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model," submitted to Opt. Eng. (2007).
    [CrossRef]
  16. R. J. Hill, "Models of the scalar spectrum for turbulent advection," J. Fluid Mech. 88, 541-562 (1978).
  17. J. R. Dunphy and J. Kerr, "Scintillation measurements for large integrated path turbulence," J. Opt. Soc. Am. 63, 981-986 (1973).
    [CrossRef]
  18. M. E. Gracheva, A. S. Gurvich, S. S. Kasharov, and V. V. Pokasov, "Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, NY, 1978).
  19. V. A. Banakh and V. L. Mironov, Lidar in a Turbulence Atmosphere (Artech House, Boston, MA, 1987).
  20. L. C. Andrews and R. L. Phillips, "I-K distribution as a universal propagation model of laser beams in atmospheric turbulence," J. Opt. Soc. Am. A 2, 160-163 (1985).
    [CrossRef]
  21. D. H. Tofsted and S. G. O’Brien, "Simulation of atmospheric turbulence image distortion and scintillation effects impacting short wave infrared (SWIR) active imaging systems," W. R.Watkins, D. Clement, and W. R. Reynolds, eds., Proc. SPIE 5432, 160-171 (2004).
    [CrossRef]
  22. J. A. Fleck, J. R. Morris, and M. J. Feit, "Time-dependent propagation of high-energy laser-beams through atmosphere," Appl. Phys. 10, 129-160 (1976).
    [CrossRef]
  23. A. E. Siegman, Lasers (Univ. Sci. Books, Mill Valley, CA, 1986).
  24. W. G. Tam and A. Zardecki, "Multiple scattering corrections to the Beer-Lambert Law. I: Open Detector," Appl. Opt. 21, 2405-2412 (1980).
    [CrossRef]
  25. D. H. Tofsted, "Turbulence Simulation: On Phase and Deflector Screen Generation," Tech. rep., U.S. Army Res. Lab. (2001).
  26. M. S. Belen’kii, "Effect of the inner scale of turbulence on the atmospheric modulation transfer function," J. Opt. Soc. Am. A 13, 1078-1082 (1996).
    [CrossRef]
  27. D. H. Tofsted, "Turbulence Simulation: Outer Scale Effects on the Refractive Index Spectrum," Tech. rep., U.S. Army Res. Lab. (2000).
  28. T. von Karman, "Progress in the statistical theory of turbulence," Proc. Natl. Acad. Sci. U.S. 34, 530-539 (1948).
    [CrossRef]
  29. J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, "Spectral characteristics of surface-layer turbulence," Q. J. Roy. Met. Soc. 98, 563-589 (1972).
    [CrossRef]
  30. E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, "Beam scintillation effects on identification performance with active imaging systems," R. G. Driggers and D. A. Huckridge, eds., Proc. SPIE 5987, 598,703-1-598,703-11 (2005).
  31. K. Weiss-Wrana, "Turbulence statistics applied to calculate expected turbulence-induced scintillation effects on electro-optical systems in different climatic regions," iS. M. Doss-Hammel and A. Kohnle, eds., Proc. SPIE 5891, 58,910D-1-58,910D-12 (2005).

2006

P. Andersson, "Long-range three-dimensional imaging using range-gated laser radar images," Opt. Eng. 45, 034,301 (2006).
[CrossRef]

2004

R. Vollmerhausen, E. Jacobs, and R. Driggers, "New metric for predicting target acquisition performance," Opt. Eng. 43, 2806-2818 (2004).
[CrossRef]

2003

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, "Impact of speckle on laser rangegated shortwave infrared imaging system target identification performance," Opt. Eng. 42, 738-746 (2003).
[CrossRef]

2001

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, "Atmospheric turbulence modulation transfer function for infrared target acquisition modeling," Opt. Eng. 40, 1906-1913 (2001).
[CrossRef]

1996

1990

1985

1980

1978

R. J. Hill, "Models of the scalar spectrum for turbulent advection," J. Fluid Mech. 88, 541-562 (1978).

1976

J. A. Fleck, J. R. Morris, and M. J. Feit, "Time-dependent propagation of high-energy laser-beams through atmosphere," Appl. Phys. 10, 129-160 (1976).
[CrossRef]

1973

1972

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, "Spectral characteristics of surface-layer turbulence," Q. J. Roy. Met. Soc. 98, 563-589 (1972).
[CrossRef]

1948

T. von Karman, "Progress in the statistical theory of turbulence," Proc. Natl. Acad. Sci. U.S. 34, 530-539 (1948).
[CrossRef]

Andersson, P.

P. Andersson, "Long-range three-dimensional imaging using range-gated laser radar images," Opt. Eng. 45, 034,301 (2006).
[CrossRef]

Andrews, L. C.

Barnard, K. J.

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, "Impact of speckle on laser rangegated shortwave infrared imaging system target identification performance," Opt. Eng. 42, 738-746 (2003).
[CrossRef]

Barten, P. G. J.

Belen’kii, M. S.

Busck, J.

J. Busck, "Underwater 3-D optical imaging with a gated viewing laser radar," Opt. Eng. 44, 116,001 (2005).
[CrossRef]

Cote, O. R.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, "Spectral characteristics of surface-layer turbulence," Q. J. Roy. Met. Soc. 98, 563-589 (1972).
[CrossRef]

Devitt, N.

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, "Impact of speckle on laser rangegated shortwave infrared imaging system target identification performance," Opt. Eng. 42, 738-746 (2003).
[CrossRef]

Driggers, R.

R. Vollmerhausen, E. Jacobs, and R. Driggers, "New metric for predicting target acquisition performance," Opt. Eng. 43, 2806-2818 (2004).
[CrossRef]

Driggers, R. G.

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, "Impact of speckle on laser rangegated shortwave infrared imaging system target identification performance," Opt. Eng. 42, 738-746 (2003).
[CrossRef]

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, "Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model," submitted to Opt. Eng. (2007).
[CrossRef]

Driggers, R. N.

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, "Atmospheric turbulence modulation transfer function for infrared target acquisition modeling," Opt. Eng. 40, 1906-1913 (2001).
[CrossRef]

Dunphy, J. R.

Feit, M. J.

J. A. Fleck, J. R. Morris, and M. J. Feit, "Time-dependent propagation of high-energy laser-beams through atmosphere," Appl. Phys. 10, 129-160 (1976).
[CrossRef]

Fleck, J. A.

J. A. Fleck, J. R. Morris, and M. J. Feit, "Time-dependent propagation of high-energy laser-beams through atmosphere," Appl. Phys. 10, 129-160 (1976).
[CrossRef]

Halford, C.

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, "Impact of speckle on laser rangegated shortwave infrared imaging system target identification performance," Opt. Eng. 42, 738-746 (2003).
[CrossRef]

Halford, C. E.

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, "Atmospheric turbulence modulation transfer function for infrared target acquisition modeling," Opt. Eng. 40, 1906-1913 (2001).
[CrossRef]

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, "Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model," submitted to Opt. Eng. (2007).
[CrossRef]

Hill, R. J.

R. J. Hill, "Models of the scalar spectrum for turbulent advection," J. Fluid Mech. 88, 541-562 (1978).

Izumi, Y.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, "Spectral characteristics of surface-layer turbulence," Q. J. Roy. Met. Soc. 98, 563-589 (1972).
[CrossRef]

Jacobs, E.

R. Vollmerhausen, E. Jacobs, and R. Driggers, "New metric for predicting target acquisition performance," Opt. Eng. 43, 2806-2818 (2004).
[CrossRef]

Jacobs, E. L.

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, "Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model," submitted to Opt. Eng. (2007).
[CrossRef]

Kaimal, J. C.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, "Spectral characteristics of surface-layer turbulence," Q. J. Roy. Met. Soc. 98, 563-589 (1972).
[CrossRef]

Kerr, J.

Kopeika, N. S.

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, "Atmospheric turbulence modulation transfer function for infrared target acquisition modeling," Opt. Eng. 40, 1906-1913 (2001).
[CrossRef]

Krapels, K.

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, "Atmospheric turbulence modulation transfer function for infrared target acquisition modeling," Opt. Eng. 40, 1906-1913 (2001).
[CrossRef]

Morris, J. R.

J. A. Fleck, J. R. Morris, and M. J. Feit, "Time-dependent propagation of high-energy laser-beams through atmosphere," Appl. Phys. 10, 129-160 (1976).
[CrossRef]

Phillips, R. L.

Robinson, A. L.

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, "Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model," submitted to Opt. Eng. (2007).
[CrossRef]

Tam, W. G.

Vollmerhausen, R.

R. Vollmerhausen, E. Jacobs, and R. Driggers, "New metric for predicting target acquisition performance," Opt. Eng. 43, 2806-2818 (2004).
[CrossRef]

Vollmerhausen, R. H.

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, "Impact of speckle on laser rangegated shortwave infrared imaging system target identification performance," Opt. Eng. 42, 738-746 (2003).
[CrossRef]

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, "Atmospheric turbulence modulation transfer function for infrared target acquisition modeling," Opt. Eng. 40, 1906-1913 (2001).
[CrossRef]

von Karman, T.

T. von Karman, "Progress in the statistical theory of turbulence," Proc. Natl. Acad. Sci. U.S. 34, 530-539 (1948).
[CrossRef]

Wyngaard, J. C.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, "Spectral characteristics of surface-layer turbulence," Q. J. Roy. Met. Soc. 98, 563-589 (1972).
[CrossRef]

Zardecki, A.

Appl. Opt.

Appl. Phys.

J. A. Fleck, J. R. Morris, and M. J. Feit, "Time-dependent propagation of high-energy laser-beams through atmosphere," Appl. Phys. 10, 129-160 (1976).
[CrossRef]

J. Fluid Mech.

R. J. Hill, "Models of the scalar spectrum for turbulent advection," J. Fluid Mech. 88, 541-562 (1978).

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

R. Vollmerhausen, E. Jacobs, and R. Driggers, "New metric for predicting target acquisition performance," Opt. Eng. 43, 2806-2818 (2004).
[CrossRef]

R. G. Driggers, R. H. Vollmerhausen, N. Devitt, C. Halford, and K. J. Barnard, "Impact of speckle on laser rangegated shortwave infrared imaging system target identification performance," Opt. Eng. 42, 738-746 (2003).
[CrossRef]

P. Andersson, "Long-range three-dimensional imaging using range-gated laser radar images," Opt. Eng. 45, 034,301 (2006).
[CrossRef]

K. Krapels, R. N. Driggers, R. H. Vollmerhausen, N. S. Kopeika, and C. E. Halford, "Atmospheric turbulence modulation transfer function for infrared target acquisition modeling," Opt. Eng. 40, 1906-1913 (2001).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.

T. von Karman, "Progress in the statistical theory of turbulence," Proc. Natl. Acad. Sci. U.S. 34, 530-539 (1948).
[CrossRef]

Q. J. Roy. Met. Soc.

J. C. Kaimal, J. C. Wyngaard, Y. Izumi, and O. R. Cote, "Spectral characteristics of surface-layer turbulence," Q. J. Roy. Met. Soc. 98, 563-589 (1972).
[CrossRef]

Other

E. Jacobs, R. L. Espinola, C. Halford, and D. Tofsted, "Beam scintillation effects on identification performance with active imaging systems," R. G. Driggers and D. A. Huckridge, eds., Proc. SPIE 5987, 598,703-1-598,703-11 (2005).

K. Weiss-Wrana, "Turbulence statistics applied to calculate expected turbulence-induced scintillation effects on electro-optical systems in different climatic regions," iS. M. Doss-Hammel and A. Kohnle, eds., Proc. SPIE 5891, 58,910D-1-58,910D-12 (2005).

D. H. Tofsted and S. G. O’Brien, "Simulation of atmospheric turbulence image distortion and scintillation effects impacting short wave infrared (SWIR) active imaging systems," W. R.Watkins, D. Clement, and W. R. Reynolds, eds., Proc. SPIE 5432, 160-171 (2004).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley Interscience, New York, NY, 2000).

A. F. Milton, G. Klager, and T. Bowman, "Low cost sensors for UGVs," G. R. Gerhart, R. W. Gunderson, and C. M. Shoemaker, eds., Proc. SPIE 4024, 180-191 (2000).
[CrossRef]

O. K. Steinvall, H. Olsson, G. Bolander, C. A. Groenwall, and D. Letalick, "Gated viewing for target detection and target recognition," G. W. Kamerman and C. Werner, eds., Proc. SPIE 3707, 432-448 (1999).
[CrossRef]

J. Busck, "Underwater 3-D optical imaging with a gated viewing laser radar," Opt. Eng. 44, 116,001 (2005).
[CrossRef]

J. C. Dainty, Laser Speckle and Related Phenomena (Springer-Verlag, Heidelberg, Germany, 1975).

L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications (SPIE Press, Bellingham, WA, 2001).
[CrossRef]

E. L. Jacobs, R. H. Vollmerhausen, and C. E. Halford, "Modeling active imagers," G. C. Holst, ed., Proc. SPIE 5407, 201-210 (2004).
[CrossRef]

C. E. Halford, A. L. Robinson, R. G. Driggers, and E. L. Jacobs, "Tilted surfaces in SWIR imagery: speckle simulation and a simple contrast model," submitted to Opt. Eng. (2007).
[CrossRef]

P. G. J. Barten, Contrast sensitivity of the human eye and its effects on image quality (SPIE Press Monograph, PM72, 1999).
[CrossRef]

W. Wolfe and G. Zissis, The Infrared Handbook (IRIA ERIM, Ann Arbor, MI, 1993).

A. E. Siegman, Lasers (Univ. Sci. Books, Mill Valley, CA, 1986).

M. E. Gracheva, A. S. Gurvich, S. S. Kasharov, and V. V. Pokasov, "Similarity relations and their experimental verification for strong intensity fluctuations of laser radiation," in Laser Beam Propagation in the Atmosphere, J. W. Strohbehn, ed. (Springer, New York, NY, 1978).

V. A. Banakh and V. L. Mironov, Lidar in a Turbulence Atmosphere (Artech House, Boston, MA, 1987).

D. H. Tofsted, "Turbulence Simulation: On Phase and Deflector Screen Generation," Tech. rep., U.S. Army Res. Lab. (2001).

D. H. Tofsted, "Turbulence Simulation: Outer Scale Effects on the Refractive Index Spectrum," Tech. rep., U.S. Army Res. Lab. (2000).

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Figures (11)

Fig. 1.
Fig. 1.

Operational concept and implementation of an active imaging system.

Fig. 2.
Fig. 2.

Relationship between system CTF, limiting frequency, and supra-threshold contrast.

Fig. 3.
Fig. 3.

Illustration of the radiometric laser model.

Fig. 4.
Fig. 4.

Speckle from a target board at 1 kilometer as imaged by a laser range gated imager under conditions with virtually no turbulence.

Fig. 5.
Fig. 5.

Propagation through turbulence.

Fig. 6.
Fig. 6.

Test setup.

Fig. 7.
Fig. 7.

Example imagery showing scintillation patterns varying with C2 n . The images were taken at measured C2 n values of (Left) 1.22E-12, (Center) 1.41E-13, and (Right) 1.11E-14 m2/3.

Fig. 8.
Fig. 8.

Scintillated beam profiles generated by simulation: no scintillation, C2 n =1E-14,C2 n =3.16E-14, and C2 n =1E-13.

Fig. 9.
Fig. 9.

9(a) Speckled images: incoherent, 1 shot, 2 shot average, and 8 shot average. 9(b) Comparison of model predictions and experimental results. The TTPF explains approximately 94% of the variance in the measured data.

Fig. 10.
Fig. 10.

10(a) Scintillated images: unscintillated, C2 n =1E-14, C2 n =3.16E-14, and C2 n =1E-13. 10(b) Comparison of model predictions and experimental results. The TTPF explains approximately 89% of the variance in the measured data.

Fig. 11.
Fig. 11.

11(a) Trade study comparing aperture size vs. C2 n on range performance. 11(b) Effect of frame averaging vs. turbulence strength on range performance.

Equations (31)

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

CTF sys ( f ) = CTF Eye ( f ) H sys ( f ) 1 + κ 2 σ 2 ( f ) L 2
H atmospheric ( f ) short = e 57.4 a C n 2 R λ 1 3 f 5 3 ( 1 u ( f λ D 0 ) 1 3 )
σ 2 ( f ) = S n ( ξ ) H Post ( ξ ) H Per ( ξ , f ) 2 d ξ
H Per ( ξ , f c ) = exp [ 2.2 log 2 ( ξ f c ) ]
TTP = f lo f limit C t CTF sys ( f ) d f [ cyc mrad ]
CTF sys ( f ) f lo , f limit = C t f limit > f lo
V ( R ) = 1000 ( A t R ) TTP [ cyc ]
P task ( R ) = ( V ( R ) V 50 ( task ) ) E 1 + ( V ( R ) V 50 ( task ) ) E
E = 1.51 + 0.24 ( V ( R ) V 50 )
P R x = ( P T x π R 2 ( tan ( ϕ las 2 ) ) 2 ) ( A det f 2 ) ( 1 π ) ( π D 0 2 4 ) τ atm 2 τ R x
σ speckle 2 ( f ) = 1 N S speckle ( ξ ) H Post ( ξ ) H Det ( ξ ) H Per ( ξ , f ) 2 d ξ
S speckle ( f ) = L H Optics ( f ) 0 H Optics ( ξ ) d ξ
C speckle = σ I I
K = imager pixel size l c / sin α
B ( ρ ) = exp [ B ln x ( ρ ) + B ln y ( ρ ) ] 1 ,
B ln x ( ρ ) = K 0 1 0 η 11 6 e η / η x J 0 ( ρ ξ k η L ) { 1 cos [ η ξ ( 1 ξ ) ] } , d η d ξ
B ln x ( ρ ) = K 0 1 0 ( η + η y ) 11 6 J 0 ( ρ ξ k η L ) { 1 cos [ η ξ ( 1 ξ ) ] } , d η d ξ
K = 2.65 β 0 2
β 0 2 = 0.4 σ 1 2
σ 1 2 = 1.23 C n 2 k 7 6 L 11 6
η x = 8.56 1 + 0.19 σ 1 12 5
η y = 9 ( 1 + 0.23 σ 1 12 5 )
σ scintillation 2 ( f ) = S scintillation ( ξ ) H Post ( ξ ) H Per ( ξ , f ) 2 d ξ .
p 0 ( r ) = A p exp ( r 2 w 0 2 ) = exp ( θ 2 θ 0 2 ) n 0 ( r θ ) d θ
n 0 ( r , θ ) = A n exp ( r 2 v 0 2 ) exp ( ik r θ )
ϕ ( r ) = ( kc / n 0 ) δτ ( r )
T lm = ( 2 π ) 3 Δ XYc 2 G lm Φ n ( 2 π l / X , 2 π m / Y , 0 )
Φ n ( κ x , κ y , κ z ) = Φ n ( κ ) = Φ n ( κ ) = Φ I ( κ , l o ) + Φ X ( κ , L o ) Φ K ( κ )
Φ I ( κ , l o ) = 0.033 C n 2 κ 11 3 F ( κ l o / 1.412 )
F ( x ) = 2.2 J ( x / 7.5 ) 1.2 J ( x / 2.5 )
Φ X ( κ , L o ) = 0.033 C n 2 i = 1,2 a i X ( κ , i )

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