Abstract

A generalization of the method of random wave vectors [Appl. Opt. 36, 464 (1997)] that is suitable for a simulation of turbulence-induced anisoplanatic effects is proposed. A simulation of the cross-correlated phase fluctuations produced by two initially plane waves propagating through weak turbulence is considered. The variation of C n 2 along a propagation path and an effect of the finite outer scale of the turbulence are included in the simulation. The validity of the simulation method is verified by comparison of theoretical and simulated results. The simulation approach developed can be used in the problems related to adaptive optics, speckle inteferometry, guide stars, and imaging through turbulence.

© 1999 Optical Society of America

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References

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  1. J. H. Shapiro, “Propagation-medium limitations on phase-compensated atmospheric imaging,” J. Opt. Soc. Am. 66, 460–469 (1976).
    [CrossRef]
  2. D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
    [CrossRef]
  3. M. C. Roggermann, B. M. Welsh, D. Montera, T. A. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995).
    [CrossRef]
  4. M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
    [CrossRef]
  5. R. Foy, A. Laberye, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).
  6. M. Tallon, R. Foy, “Adaptive telescope with laser probe: isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).
  7. B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
    [CrossRef]
  8. D. L. Fried, “Focus anisoplanatism in the limit of infinitely many artificial-guide-star reference spots,” J. Opt. Soc. Am. A 12, 939–949 (1995).
    [CrossRef]
  9. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1981), Vol. 19, pp. 281–376.
    [CrossRef]
  10. J. C. Dainty, “The transfer function, signal-to-noise ratio, and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631–641 (1974).
  11. F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
    [CrossRef]
  12. M. C. Roggemann, E. L. Caudill, D. W. Tylper, M. J. Fox, M. A. Von Bokern, L. C. Matson, “Compensated speckle imaging: theory and experimental results,” Appl. Opt. 33, 3099–3110 (1994).
    [CrossRef] [PubMed]
  13. J. C. Ricklin, M. A. Vorontsov, G. W. Carhart, D. Gose, W. B. Miller, “Turbulent phase screen for study of imaging system performance,” J. Mod. Opt. 42, 13–17 (1995).
    [CrossRef]
  14. M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field,” J. Opt. Soc. Am. A 13, 1456–1466 (1996).
    [CrossRef]
  15. D. Kouznetsov, V. V. Voitsekhovich, R. Ortega-Martinez, “Simulation of turbulence-induced phase and log-amplitude distortions,” Appl. Opt. 36, 464–469 (1997).
    [CrossRef] [PubMed]
  16. B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulation anisoplanatic geometries and temporal evolution,” in Propagation and Imaging through the Atmosphere, L. R. Bissonnette, C. Dainty, eds., Proc. SPIE3125, 327–338 (1997).
    [CrossRef]
  17. J. A. Louthain, B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” in Airborne Laser Advanced Technology, P. H. Merritt, T. D. Steiner, eds., Proc. SPIE3381, 286–296 (1998).
    [CrossRef]
  18. D. Kouznetsov, V. V. Voitsekhovich, “Method of random wave-vectors in simulation of correlated random processes,” Meteorol. Z. N.F. 7, 230–236 (1998).
  19. V. I. Tatarski, “The Effects of the Turbulent Atmosphere on Wave Propagation,” (National Science Foundation, Washington, D.C., 1968).
  20. E. A. Vitrichenko, V. V. Voitsekhovich, M. I. Mishchenko, “Effect of atmospheric turbulence on the size of the field of view of adaptive systems,” USSR Acad. Sci. Atmos. Ocean. Phys. 20, 870–872 (1984).
  21. G. W. Reinhardt, S. A. Collins, “Outer scale effects in turbulence-degraded light-beam spectra,” J. Opt. Soc. Am. 62, 1526–1528 (1972).
    [CrossRef]
  22. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1995), Chap. 7.
  23. R. E. Hufnagel, “Variation of atmospheric turbulence,” in Digest of Topical Meeting on Optical Propagation through Turbulence, (Optical Society of America, Washington, D.C., 1974), pp. WA1-1–WA1-4.
  24. J. S. Accetta, D. L. Shumaker, eds., The Infrared and Electro-Optical Systems Handbook (SPIE Press, Bellingham, Wash., 1993), Vol. 2.

1998 (1)

D. Kouznetsov, V. V. Voitsekhovich, “Method of random wave-vectors in simulation of correlated random processes,” Meteorol. Z. N.F. 7, 230–236 (1998).

1997 (2)

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

D. Kouznetsov, V. V. Voitsekhovich, R. Ortega-Martinez, “Simulation of turbulence-induced phase and log-amplitude distortions,” Appl. Opt. 36, 464–469 (1997).
[CrossRef] [PubMed]

1996 (1)

1995 (3)

1994 (1)

1991 (1)

1990 (1)

M. Tallon, R. Foy, “Adaptive telescope with laser probe: isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

1985 (1)

R. Foy, A. Laberye, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

1984 (1)

E. A. Vitrichenko, V. V. Voitsekhovich, M. I. Mishchenko, “Effect of atmospheric turbulence on the size of the field of view of adaptive systems,” USSR Acad. Sci. Atmos. Ocean. Phys. 20, 870–872 (1984).

1982 (2)

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[CrossRef]

D. L. Fried, “Anisoplanatism in adaptive optics,” J. Opt. Soc. Am. 72, 52–61 (1982).
[CrossRef]

1976 (1)

1974 (1)

J. C. Dainty, “The transfer function, signal-to-noise ratio, and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631–641 (1974).

1972 (1)

Carhart, G. W.

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field,” J. Opt. Soc. Am. A 13, 1456–1466 (1996).
[CrossRef]

J. C. Ricklin, M. A. Vorontsov, G. W. Carhart, D. Gose, W. B. Miller, “Turbulent phase screen for study of imaging system performance,” J. Mod. Opt. 42, 13–17 (1995).
[CrossRef]

Caudill, E. L.

Collins, S. A.

Dainty, J. C.

J. C. Dainty, “The transfer function, signal-to-noise ratio, and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631–641 (1974).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1995), Chap. 7.

Fox, M. J.

Foy, R.

M. Tallon, R. Foy, “Adaptive telescope with laser probe: isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

R. Foy, A. Laberye, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

Fried, D. L.

Fugate, R. Q.

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

Gardner, C. S.

Gilli, J. M.

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[CrossRef]

Gose, D.

J. C. Ricklin, M. A. Vorontsov, G. W. Carhart, D. Gose, W. B. Miller, “Turbulent phase screen for study of imaging system performance,” J. Mod. Opt. 42, 13–17 (1995).
[CrossRef]

Hufnagel, R. E.

R. E. Hufnagel, “Variation of atmospheric turbulence,” in Digest of Topical Meeting on Optical Propagation through Turbulence, (Optical Society of America, Washington, D.C., 1974), pp. WA1-1–WA1-4.

Kouznetsov, D.

D. Kouznetsov, V. V. Voitsekhovich, “Method of random wave-vectors in simulation of correlated random processes,” Meteorol. Z. N.F. 7, 230–236 (1998).

D. Kouznetsov, V. V. Voitsekhovich, R. Ortega-Martinez, “Simulation of turbulence-induced phase and log-amplitude distortions,” Appl. Opt. 36, 464–469 (1997).
[CrossRef] [PubMed]

Laberye, A.

R. Foy, A. Laberye, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

Louthain, J. A.

J. A. Louthain, B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” in Airborne Laser Advanced Technology, P. H. Merritt, T. D. Steiner, eds., Proc. SPIE3381, 286–296 (1998).
[CrossRef]

Matson, L. C.

Miller, W. B.

J. C. Ricklin, M. A. Vorontsov, G. W. Carhart, D. Gose, W. B. Miller, “Turbulent phase screen for study of imaging system performance,” J. Mod. Opt. 42, 13–17 (1995).
[CrossRef]

Mishchenko, M. I.

E. A. Vitrichenko, V. V. Voitsekhovich, M. I. Mishchenko, “Effect of atmospheric turbulence on the size of the field of view of adaptive systems,” USSR Acad. Sci. Atmos. Ocean. Phys. 20, 870–872 (1984).

Montera, D.

Ortega-Martinez, R.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1995), Chap. 7.

Pruidze, D. V.

Reinhardt, G. W.

Rhoadarmer, T. A.

Ricklin, J. C.

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field,” J. Opt. Soc. Am. A 13, 1456–1466 (1996).
[CrossRef]

J. C. Ricklin, M. A. Vorontsov, G. W. Carhart, D. Gose, W. B. Miller, “Turbulent phase screen for study of imaging system performance,” J. Mod. Opt. 42, 13–17 (1995).
[CrossRef]

Roddier, F.

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[CrossRef]

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1981), Vol. 19, pp. 281–376.
[CrossRef]

Roggemann, M. C.

Roggermann, M. C.

Shapiro, J. H.

Tallon, M.

M. Tallon, R. Foy, “Adaptive telescope with laser probe: isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

Tatarski, V. I.

V. I. Tatarski, “The Effects of the Turbulent Atmosphere on Wave Propagation,” (National Science Foundation, Washington, D.C., 1968).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1995), Chap. 7.

Tylper, D. W.

Vernin, J.

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[CrossRef]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1995), Chap. 7.

Vitrichenko, E. A.

E. A. Vitrichenko, V. V. Voitsekhovich, M. I. Mishchenko, “Effect of atmospheric turbulence on the size of the field of view of adaptive systems,” USSR Acad. Sci. Atmos. Ocean. Phys. 20, 870–872 (1984).

Voelz, D. G.

Voitsekhovich, V. V.

D. Kouznetsov, V. V. Voitsekhovich, “Method of random wave-vectors in simulation of correlated random processes,” Meteorol. Z. N.F. 7, 230–236 (1998).

D. Kouznetsov, V. V. Voitsekhovich, R. Ortega-Martinez, “Simulation of turbulence-induced phase and log-amplitude distortions,” Appl. Opt. 36, 464–469 (1997).
[CrossRef] [PubMed]

E. A. Vitrichenko, V. V. Voitsekhovich, M. I. Mishchenko, “Effect of atmospheric turbulence on the size of the field of view of adaptive systems,” USSR Acad. Sci. Atmos. Ocean. Phys. 20, 870–872 (1984).

Von Bokern, M. A.

Vorontsov, M. A.

M. A. Vorontsov, G. W. Carhart, D. V. Pruidze, J. C. Ricklin, D. G. Voelz, “Image quality criteria for an adaptive imaging system based on statistical analysis of the speckle field,” J. Opt. Soc. Am. A 13, 1456–1466 (1996).
[CrossRef]

J. C. Ricklin, M. A. Vorontsov, G. W. Carhart, D. Gose, W. B. Miller, “Turbulent phase screen for study of imaging system performance,” J. Mod. Opt. 42, 13–17 (1995).
[CrossRef]

Welsh, B. M.

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

M. C. Roggermann, B. M. Welsh, D. Montera, T. A. Rhoadarmer, “Method for simulating atmospheric turbulence phase effects for multiple time slices and anisoplanatic conditions,” Appl. Opt. 34, 4037–4051 (1995).
[CrossRef]

B. M. Welsh, C. S. Gardner, “Effects of turbulence-induced anisoplanatism on the imaging performance of adaptive-astronomical telescopes using laser guide stars,” J. Opt. Soc. Am. A 8, 69–80 (1991).
[CrossRef]

J. A. Louthain, B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” in Airborne Laser Advanced Technology, P. H. Merritt, T. D. Steiner, eds., Proc. SPIE3381, 286–296 (1998).
[CrossRef]

B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulation anisoplanatic geometries and temporal evolution,” in Propagation and Imaging through the Atmosphere, L. R. Bissonnette, C. Dainty, eds., Proc. SPIE3125, 327–338 (1997).
[CrossRef]

Appl. Opt. (3)

Astron. Astrophys. (2)

R. Foy, A. Laberye, “Feasibility of adaptive telescope with laser probe,” Astron. Astrophys. 152, L29–L31 (1985).

M. Tallon, R. Foy, “Adaptive telescope with laser probe: isoplanatism and cone effect,” Astron. Astrophys. 235, 549–557 (1990).

J. Mod. Opt. (1)

J. C. Ricklin, M. A. Vorontsov, G. W. Carhart, D. Gose, W. B. Miller, “Turbulent phase screen for study of imaging system performance,” J. Mod. Opt. 42, 13–17 (1995).
[CrossRef]

J. Opt. (Paris) (1)

F. Roddier, J. M. Gilli, J. Vernin, “On the isoplanatic patch size in stellar speckle interferometry,” J. Opt. (Paris) 13, 63–70 (1982).
[CrossRef]

J. Opt. Soc. Am. (3)

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

Meteorol. Z. N.F. (1)

D. Kouznetsov, V. V. Voitsekhovich, “Method of random wave-vectors in simulation of correlated random processes,” Meteorol. Z. N.F. 7, 230–236 (1998).

Mon. Not. R. Astron. Soc. (1)

J. C. Dainty, “The transfer function, signal-to-noise ratio, and limiting magnitude in stellar speckle interferometry,” Mon. Not. R. Astron. Soc. 169, 631–641 (1974).

Rev. Mod. Phys. (1)

M. C. Roggemann, B. M. Welsh, R. Q. Fugate, “Improving the resolution of ground-based telescopes,” Rev. Mod. Phys. 69, 437–505 (1997).
[CrossRef]

USSR Acad. Sci. Atmos. Ocean. Phys. (1)

E. A. Vitrichenko, V. V. Voitsekhovich, M. I. Mishchenko, “Effect of atmospheric turbulence on the size of the field of view of adaptive systems,” USSR Acad. Sci. Atmos. Ocean. Phys. 20, 870–872 (1984).

Other (7)

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1995), Chap. 7.

R. E. Hufnagel, “Variation of atmospheric turbulence,” in Digest of Topical Meeting on Optical Propagation through Turbulence, (Optical Society of America, Washington, D.C., 1974), pp. WA1-1–WA1-4.

J. S. Accetta, D. L. Shumaker, eds., The Infrared and Electro-Optical Systems Handbook (SPIE Press, Bellingham, Wash., 1993), Vol. 2.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Pergamon, London, 1981), Vol. 19, pp. 281–376.
[CrossRef]

V. I. Tatarski, “The Effects of the Turbulent Atmosphere on Wave Propagation,” (National Science Foundation, Washington, D.C., 1968).

B. M. Welsh, “Fourier-series-based atmospheric phase screen generator for simulation anisoplanatic geometries and temporal evolution,” in Propagation and Imaging through the Atmosphere, L. R. Bissonnette, C. Dainty, eds., Proc. SPIE3125, 327–338 (1997).
[CrossRef]

J. A. Louthain, B. M. Welsh, “Fourier-series-based phase and amplitude optical field screen generator for weak atmospheric turbulence,” in Airborne Laser Advanced Technology, P. H. Merritt, T. D. Steiner, eds., Proc. SPIE3381, 286–296 (1998).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Typical samples of simulated phase fluctuations. The phase fluctuation S 1 is associated with the on-axis star, whereas the phase fluctuation S 2 is produced by the off-axis star.

Fig. 2
Fig. 2

Theoretical and simulated cross-structure functions for L 0 = 50 m. Theoretical functions were calculated with Eqs. (11) and (27).

Fig. 3
Fig. 3

Theoretical and simulated cross-structure functions for L 0 = 10 m. Theoretical functions were calculated with Eqs. (11) and (27).

Fig. 4
Fig. 4

Theoretical and simulated cross-structure functions for L 0 = 1 m. Theoretical functions were calculated with Eqs. (11) and (27).

Fig. 5
Fig. 5

Theoretical and simulated cross-structure functions for large separations between the observation points. Theoretical functions were calculated with Eqs. (11) and (27).

Fig. 6
Fig. 6

Anisotropy in the theoretical and simulated cross-structure functions. Theoretical functions were calculated with Eqs. (11) and (27).

Fig. 7
Fig. 7

Theoretical and simulated residual structure functions. Theoretical functions were calculated with Eqs. (11) and (28).

Equations (29)

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Ψ1r=χ1r+iS1r=ik 0Ldz  d2 κgnz, κexpiκ·r+κ2z2ik,
Ψ2r=χ2r+iS2r=ik 0Ldz  d2κgnz, κ×expiκ·r+κ2z2ik-iκ·nz,  n=sin γ cos α, sin γ cos β.
BS1S1ρ=BS2S2ρ=2π2k20Ldz  dκκΦnz, κ×J0κρcosκ2z/k+1,
BS1S2ρ=S1r1S2r2=2π2k20Ldz  dκκΦnz, κ×J0κ|ρ+nz|cosκ2z/k+1,
gnz1, κ1gn*z2, κ2=2πΦnz1, κ1δz1-z2×δκ1-κ2,  gnz1, κ1gnz2, κ2=2πΦnz1, κ1δz1-z2×δκ1+κ2,
Wκ=12π  d2ρ Bρexpiκ·ρ
WS1S1κ=WS2S2κ=2π2k20LdzΦnz, κcosκ2z/k+1,
WS1S2κ=2π2k20LdzΦnz, κexp-iκ·nz×cosκ2z/k+1.
Φnz, κ=0.033Cn2zκ2+1/L02-11/6,
n=sin γ, 0,
BS1S1ρ=BS2S2ρ=0.132π2k20LdzCn2z×0dκκκ2+1/L02-11/6J0κρ,
BS1S2ρ=0.132π2k20LdzCn2z0dκκκ2+1/L02-11/6J0κr+,  r+=ρ2+z2 sin2γ+2ρz sin γ cos φ1/2,
WS1S1κ=WS2S2κ=0.066π2k2κ2+1/L02-11/6×0LdzCn2zcosκ2z/k+1,
ReWS1S2κ=0.066π2k2κ2+1/L02-11/60LdzCn2z×cosκz sin γ cos θcosκ2z/k+1,  ImWS1S2κ=-0.066π2k2κ2+1/L02-11/60LdzCn2z×sinκz sin γ cos θcosκ2z/k+1,
S1r=m=1M Fpmcospm·r+φm,  S2r=m=1M Fpmcospm·r+φm+apmψm+bpm.
BS1S1ρ=BS2S2ρ=12m=1M F2pmcospm·ρ,
BS1S2ρ=12m=1M F2pmcospm·ρ-apmψm-bpm.
WS1S1κ=π2m=1M  d2pmF2pmΩpmδpm+κ+δpm-κ,  WS1S2κ=π2m=1M  dψmηψm  d2pmF2pmΩpm×(exp-iapmψm+bpmδpm+κ+expiapmψm+bpmδpm-κ),
Fκ=WS1S1κπMΩκ1/2,
12  dψηψexp-ia-κψ+b-κ+expiaκψ+bκ=WS1S2κWS1S1κ.
a-κ=-aκ,  b-κ=-bκ.
12-11dψ cosaκψ+bκ=sinaκcosbκaκ=wrκ,  12-11dψ sinaκψ+bκ=sinaκsinbκaκ=wiκ,  wrκ=ReWS1S2κWS1S1κ,  wiκ=ImWS1S2κWS1S1κ.
bκ=atan2wiκ, wrκ,  aκ=asincwrκ/cosbκsignsin θ,
Ωκκ=1Ω0κ2+1/L02,  Ω0=L0arctanK2L0-arctanK1L0,
Ωκ=12πΩ0κκ2+1/L02.
pm=1/L0tansmL0arctanK2L0-arctanK1L0.
Cn2z=C0r0-5/3k-2zz010 exp-zz1+exp-zz2,
DS1S2r1, r2=S1r1-S2r22=DS1S2r1-r2=DS1S2ρ, φ,
DSRr1, r2=SRr1-SRr22,  SRr=S1r-S2r

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