Abstract

Adaptive optics has been used in a cooperative mode to measure the phase distortion of the light from a star and to correct its image with a deformable mirror. A wave-front sensor in the adaptive-optics system that measures the phase aberrations requires that an object be fairly bright for accurate performance of the measurement. The use of synthetic beacons provides a means of correcting the images of objects that are too dim to allow one to use their light to provide correction in a cooperative mode. Synthetic beacons at a finite distance do not provide a perfect correction in imaging an object at a greater distance. The error in making a correction with one or more beacons is analyzed. Analytical expressions that can be used to determine performance in a variety of geometries, with various beacon altitudes and numbers, are derived. This analysis is applied to 60-cm and 4-m systems.

© 1994 Optical Society of America

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References

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  1. L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
    [Crossref]
  2. R. Foy, M. Tallon, “ATLAS experiment to test the laser probe technique for wavefront measurements,” in Active Telescope Systems, J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 174–183 (1989).
    [Crossref]
  3. L. A. Thompson, C. S. Gardner, “Excimer laser guide star techniques for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 184–190 (1989).
    [Crossref]
  4. C. S. Gardner, L. A. Thompson, B. M. Welsh, “Sodium laser guide star technique for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 191–202 (1989).
    [Crossref]
  5. C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
    [Crossref]
  6. B. M. Welsh, C. S. Gardner, L. A. Thomspon, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
    [Crossref]
  7. B. M. Welsh, C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4153 (1989).
    [Crossref] [PubMed]
  8. B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
    [Crossref]
  9. B. M. Welsh, L. A. Thompson, “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]
  10. O. I. Marichev, Integral Transforms of Higher Transcendental Functions (Horwood, Chichester, UK, 1983).
  11. R. J. Sasiela, J. D. Shelton, “Mellin transform techniques applied to integral evaluation: Taylor series and asymptotic approximations,”J. Math. Phys. 34, 2572–2617 (1993).
    [Crossref]
  12. C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London)141–143 (1991).
    [Crossref]
  13. D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 15, 1797–1799 (1991).
    [Crossref]
  14. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961), Chap. 7, pp. 122–163.
  15. V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Washington, D.C., 1971), Chap. 3, pp. 218–258.
  16. A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 293–305 (1969).
    [Crossref]
  17. R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1992).
    [Crossref]
  18. J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, Berlin, 1978).
    [Crossref]
  19. R. J. Noll, “Zernike polynomials and atmospheric turbulence,”J. Opt. Soc. Am. 66, 207–211 (1976).
    [Crossref]
  20. R. A. Humphreys, C. A. Primmerman, L. C. Bradley, J. Herrmann, “Atmospheric-turbulence measurements using a synthetic beacon in the mesospheric sodium layer,” Opt. Lett. 16, 1367–1369 (1991).
    [Crossref] [PubMed]
  21. R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
    [Crossref]
  22. R. J. Sasiela, “Strehl ratios with various types of anisoplanatism,” J. Opt. Soc. Am. A 9, 1398–1405 (1992).
    [Crossref]
  23. J. Herrmann, “Least-squares wave front errors of minimum norm,”J. Opt. Soc. Am. 70, 28–35 (1980).
    [Crossref]
  24. R. Parenti, R. J. Sasiela, SWAT System Performance Predictions, SWP-1 (MIT Lincoln Laboratory, Lexington, Mass., 1993).
  25. B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors; expected values for the SOR-2 experiment,” (Optical Sciences Company, Placentia, Calif., 1984).
  26. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Chap. 3, p. 482.

1993 (1)

R. J. Sasiela, J. D. Shelton, “Mellin transform techniques applied to integral evaluation: Taylor series and asymptotic approximations,”J. Math. Phys. 34, 2572–2617 (1993).
[Crossref]

1992 (2)

1991 (5)

R. A. Humphreys, C. A. Primmerman, L. C. Bradley, J. Herrmann, “Atmospheric-turbulence measurements using a synthetic beacon in the mesospheric sodium layer,” Opt. Lett. 16, 1367–1369 (1991).
[Crossref] [PubMed]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London)141–143 (1991).
[Crossref]

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 15, 1797–1799 (1991).
[Crossref]

B. M. Welsh, L. A. Thompson, “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]

1990 (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

1989 (2)

1987 (1)

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[Crossref]

1980 (1)

1976 (1)

1969 (1)

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 293–305 (1969).
[Crossref]

Ameer, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Barclay, H. T.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London)141–143 (1991).
[Crossref]

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 15, 1797–1799 (1991).
[Crossref]

Boeke, B. R.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Bradley, L. C.

Browne, S. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Ellerbroek, B. L.

B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors; expected values for the SOR-2 experiment,” (Optical Sciences Company, Placentia, Calif., 1984).

Foy, R.

R. Foy, M. Tallon, “ATLAS experiment to test the laser probe technique for wavefront measurements,” in Active Telescope Systems, J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 174–183 (1989).
[Crossref]

Fried, D. L.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Fugate, R. Q.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Gardner, C. S.

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[Crossref]

B. M. Welsh, C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4153 (1989).
[Crossref] [PubMed]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[Crossref]

L. A. Thompson, C. S. Gardner, “Excimer laser guide star techniques for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 184–190 (1989).
[Crossref]

C. S. Gardner, L. A. Thompson, B. M. Welsh, “Sodium laser guide star technique for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 191–202 (1989).
[Crossref]

B. M. Welsh, C. S. Gardner, L. A. Thomspon, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[Crossref]

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Chap. 3, p. 482.

Herrmann, J.

Humphreys, R. A.

Ishimaru, A.

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 293–305 (1969).
[Crossref]

Marichev, O. I.

O. I. Marichev, Integral Transforms of Higher Transcendental Functions (Horwood, Chichester, UK, 1983).

Murphy, D. V.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London)141–143 (1991).
[Crossref]

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 15, 1797–1799 (1991).
[Crossref]

Noll, R. J.

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London)141–143 (1991).
[Crossref]

Parenti, R.

R. Parenti, R. J. Sasiela, SWAT System Performance Predictions, SWP-1 (MIT Lincoln Laboratory, Lexington, Mass., 1993).

Primmerman, C. A.

R. A. Humphreys, C. A. Primmerman, L. C. Bradley, J. Herrmann, “Atmospheric-turbulence measurements using a synthetic beacon in the mesospheric sodium layer,” Opt. Lett. 16, 1367–1369 (1991).
[Crossref] [PubMed]

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 15, 1797–1799 (1991).
[Crossref]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London)141–143 (1991).
[Crossref]

Roberts, P. H.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors; expected values for the SOR-2 experiment,” (Optical Sciences Company, Placentia, Calif., 1984).

Ruane, R. E.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Chap. 3, p. 482.

Sasiela, R. J.

R. J. Sasiela, J. D. Shelton, “Mellin transform techniques applied to integral evaluation: Taylor series and asymptotic approximations,”J. Math. Phys. 34, 2572–2617 (1993).
[Crossref]

R. J. Sasiela, “Strehl ratios with various types of anisoplanatism,” J. Opt. Soc. Am. A 9, 1398–1405 (1992).
[Crossref]

R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1992).
[Crossref]

R. Parenti, R. J. Sasiela, SWAT System Performance Predictions, SWP-1 (MIT Lincoln Laboratory, Lexington, Mass., 1993).

Shelton, J. D.

R. J. Sasiela, J. D. Shelton, “Mellin transform techniques applied to integral evaluation: Taylor series and asymptotic approximations,”J. Math. Phys. 34, 2572–2617 (1993).
[Crossref]

R. J. Sasiela, J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1992).
[Crossref]

Strohbehn, J. W.

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, Berlin, 1978).
[Crossref]

Tallon, M.

R. Foy, M. Tallon, “ATLAS experiment to test the laser probe technique for wavefront measurements,” in Active Telescope Systems, J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 174–183 (1989).
[Crossref]

Tatarski, V. I.

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961), Chap. 7, pp. 122–163.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Washington, D.C., 1971), Chap. 3, pp. 218–258.

Thompson, L. A.

B. M. Welsh, L. A. Thompson, “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]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[Crossref]

L. A. Thompson, C. S. Gardner, “Excimer laser guide star techniques for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 184–190 (1989).
[Crossref]

C. S. Gardner, L. A. Thompson, B. M. Welsh, “Sodium laser guide star technique for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 191–202 (1989).
[Crossref]

Thomspon, L. A.

B. M. Welsh, C. S. Gardner, L. A. Thomspon, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[Crossref]

Tyler, G. A.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Welsh, B. M.

B. M. Welsh, L. A. Thompson, “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]

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

B. M. Welsh, C. S. Gardner, “Nonlinear resonant absorption effects on the design of resonance fluorescence lidars and laser guide stars,” Appl. Opt. 28, 4141–4153 (1989).
[Crossref] [PubMed]

B. M. Welsh, C. S. Gardner, “Performance analysis of adaptive-optics systems using laser guide stars and slope sensors,” J. Opt. Soc. Am. A 6, 1913–1923 (1989).
[Crossref]

B. M. Welsh, C. S. Gardner, L. A. Thomspon, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[Crossref]

C. S. Gardner, L. A. Thompson, B. M. Welsh, “Sodium laser guide star technique for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 191–202 (1989).
[Crossref]

Wopat, L. M.

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Zollars, B. G.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London)141–143 (1991).
[Crossref]

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 15, 1797–1799 (1991).
[Crossref]

Appl. Opt. (1)

J. Math. Phys. (1)

R. J. Sasiela, J. D. Shelton, “Mellin transform techniques applied to integral evaluation: Taylor series and asymptotic approximations,”J. Math. Phys. 34, 2572–2617 (1993).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

Nature (London) (3)

L. A. Thompson, C. S. Gardner, “Experiments on laser guide stars at Mauna Kea Observatory for adaptive imaging in astronomy,” Nature (London) 328, 229–231 (1987).
[Crossref]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London)141–143 (1991).
[Crossref]

R. Q. Fugate, D. L. Fried, G. A. Ameer, B. R. Boeke, S. L. Browne, P. H. Roberts, R. E. Ruane, G. A. Tyler, L. M. Wopat, “Measurement of atmospheric distortion using scattered light from a laser guide star,” Nature (London) 353, 144–146 (1991).
[Crossref]

Opt. Lett. (2)

R. A. Humphreys, C. A. Primmerman, L. C. Bradley, J. Herrmann, “Atmospheric-turbulence measurements using a synthetic beacon in the mesospheric sodium layer,” Opt. Lett. 16, 1367–1369 (1991).
[Crossref] [PubMed]

D. V. Murphy, C. A. Primmerman, B. G. Zollars, H. T. Barclay, “Experimental demonstration of atmospheric compensation using multiple synthetic beacons,” Opt. Lett. 15, 1797–1799 (1991).
[Crossref]

Proc. IEEE (1)

C. S. Gardner, B. M. Welsh, L. A. Thompson, “Design and performance analysis of adaptive optical telescopes using laser guide stars,” Proc. IEEE 78, 1721–1743 (1990).
[Crossref]

Radio Sci. (1)

A. Ishimaru, “Fluctuations of a beam wave propagating through a locally homogeneous medium,” Radio Sci. 4, 293–305 (1969).
[Crossref]

Other (11)

R. Parenti, R. J. Sasiela, SWAT System Performance Predictions, SWP-1 (MIT Lincoln Laboratory, Lexington, Mass., 1993).

B. L. Ellerbroek, P. H. Roberts, “Turbulence induced angular separation errors; expected values for the SOR-2 experiment,” (Optical Sciences Company, Placentia, Calif., 1984).

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1980), Chap. 3, p. 482.

B. M. Welsh, C. S. Gardner, L. A. Thomspon, “Effects of nonlinear resonant absorption on sodium laser guide stars,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 203–214 (1989).
[Crossref]

J. W. Strohbehn, Laser Beam Propagation in the Atmosphere (Springer-Verlag, Berlin, 1978).
[Crossref]

V. I. Tatarski, Wave Propagation in a Turbulent Medium (Dover, New York, 1961), Chap. 7, pp. 122–163.

V. I. Tatarski, The Effects of the Turbulent Atmosphere on Wave Propagation (U.S. Department of Commerce, Washington, D.C., 1971), Chap. 3, pp. 218–258.

R. Foy, M. Tallon, “ATLAS experiment to test the laser probe technique for wavefront measurements,” in Active Telescope Systems, J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 174–183 (1989).
[Crossref]

L. A. Thompson, C. S. Gardner, “Excimer laser guide star techniques for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 184–190 (1989).
[Crossref]

C. S. Gardner, L. A. Thompson, B. M. Welsh, “Sodium laser guide star technique for adaptive imaging in astronomy,” in Active Telescope Systems, F. J. Roddier, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1114, 191–202 (1989).
[Crossref]

O. I. Marichev, Integral Transforms of Higher Transcendental Functions (Horwood, Chichester, UK, 1983).

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Figures (15)

Fig. 1
Fig. 1

Geometry of focused and collimated beams of different diameters propagating in different directions with a separation that depends on the axial coordinate.

Fig. 2
Fig. 2

Geometry of a distributed source and a finite-size aperture.

Fig. 3
Fig. 3

Phase variance caused by turbulence below the beacon for a 60-cm aperture with HV-21 turbulence at 0.5-μm wavelength. The total, piston-removed, and piston- and tilt-removed variances are shown.

Fig. 4
Fig. 4

Piston-removed and piston- and tilt-removed phase variances caused by turbulence of the entire atmosphere for a 60-cm aperture with HV-21 turbulence at 0.5-μm wavelength.

Fig. 5
Fig. 5

Piston- and tilt-removed phase variances caused by turbulence of the entire atmosphere for a 4-m aperture with HV-21 turbulence at 0.5-μm wavelength.

Fig. 6
Fig. 6

Contributions to the phase variance of a multibeacon system include focal anisoplanatism below the beacon, unmeasured turbulence above the beacon, and errors in determining the relative positions of the beacons.

Fig. 7
Fig. 7

Jitter variance caused by turbulence below the beacon resulting from focal anisoplanatism for a 60-cm aperture with 30-cm beacon spacing at 0.5-μm wavelength for HV-21 turbulence.

Fig. 8
Fig. 8

Jitter variance caused by turbulence below the beacon resulting from focal anisoplanatism for a 4-m aperture with 2-m beacon spacing at 0.5-μm wavelength for HV-21 turbulence.

Fig. 9
Fig. 9

Stitching difference between correlated and uncorrelated tilts.

Fig. 10
Fig. 10

Correlation function for parallel components of tilt in which the displacement is parallel, perpendicular, and at 45° to the tilt.

Fig. 11
Fig. 11

Correlation function for perpendicular components of tilt in which the displacement is 45° to the tilt.

Fig. 12
Fig. 12

Aperture model for stitching.

Fig. 13
Fig. 13

Figure variance of a 60-cm aperture with four beacons at various altitudes for the HV-21 turbulence model at 0.5-μm wavelength.

Fig. 14
Fig. 14

Figure variance of a 4-m aperture with four beacons at various altitudes for the HV-21 turbulence model at 0.5-μm wavelength.

Fig. 15
Fig. 15

Focal anisoplanatic geometry.

Equations (111)

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ϕ ( r , L ) = k 0 0 L d z d ν ( κ , z ) cos [ P ( γ , κ , z ) ] exp ( i γ κ · r ) ,
P ( γ , κ , z ) = γ κ 2 ( L - z ) 2 k 0 ,
P ( γ , κ , z ) = γ κ 2 z 2 k 0 .
G ( γ κ ) = d r g ( r ) exp ( i γ κ · r ) .
4 π D 2 d r g 2 ( r ) = 1.
ϕ ( r , L ) = k 0 0 L d z d ν ( κ , z ) { G 1 ( γ 1 κ ) cos [ P 1 ( γ 1 , κ , z ) ] - A ( κ , z ) G 2 ( γ 2 κ ) cos [ P 2 ( γ 2 , κ , z ) ] } .
A ( κ , z ) = exp [ i κ · d ( z ) ] .
σ ϕ R 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × { G 1 ( γ 1 κ ) cos [ P 1 ( γ 1 , κ , z ) ] - A ( κ , z ) G 2 ( γ 2 κ ) × cos [ P 2 ( γ 2 , κ , z ) ] } × { G 1 * ( γ 1 κ ) cos [ P 1 ( γ 1 , κ , z ) ] - A * ( κ , z ) G 2 * ( γ 2 κ ) × cos [ P 2 ( γ 2 , κ , z ) ] } ,
f ( κ ) = κ - 11 / 3 .
σ ϕ R 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × cos 2 [ P ( γ , κ , z ) ] F ( γ κ ) ,
F ( γ κ ) = G ( γ κ ) G * ( γ κ ) .
σ ϕ R 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) G 1 ( γ 1 κ ) - G 2 ( γ 2 κ ) 2 .
σ ϕ R 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ f ( κ ) × cos 2 [ P ( γ , κ , z ) ] F ( γ κ ) 2 [ 1 - cos ( κ · d ) ] .
F ( κ ) = [ 2 J 1 ( κ D / 2 ) κ D / 2 ] 2 .
F x ( κ ) F y ( κ ) F ( κ ) } = [ 4 J 2 ( κ D / 2 ) κ D / 2 ] 2 { cos 2 ( φ ) sin 2 ( φ ) 1 .
F ( κ ) = 1 - [ 2 J 1 ( κ D / 2 ) κ D / 2 ] 2 - [ 4 J 2 ( κ D / 2 ) κ D / 2 ] 2 .
exp ( i κ · ρ ) - exp { i κ · [ ρ ( 1 - z / L ) + ( ρ + b ) z / L ] } .
ϕ R ( L ) = k 0 0 L d z d ν ( κ , z ) cos [ P ( γ , κ , z ) ] ( exp ( i κ · ρ ) - d ρ S ( ρ ) exp { i κ · [ ρ ( 1 - z / L ) + ( ρ + b ) z / L ] } d ρ S ( ρ ) ) .
F ( κ ) = | 1 - d ρ S ( ρ ) exp { i κ · [ - ρ z / L + ( ρ + b ) z / L ] } d ρ S ( ρ ) | 2 .
F ( κ ) = 1 - 2 2 J 1 ( D s h ) D s h cos [ κ · z ( b - ρ ) L ] + [ 2 J 1 ( D s h ) D s h ] 2 ,
h = κ z / 2 L .
F ( κ ) = 1 - 2 2 J 1 ( D s h ) D s h 2 J 1 ( D h ) D h cos ( κ · z b L ) + [ 2 J 1 ( D s h ) D s h ] 2 .
F ( κ ) = 1 - 2 2 J 1 ( D s h ) D s h 2 J 1 ( D h ) D h J 0 ( κ b z L ) + [ 2 J 1 ( D s h ) D s h ] 2 .
F ( κ ) = 1 - 2 2 J 1 ( D s h ) D s h 2 J 1 ( D h ) D h + [ 2 J 1 ( D s h ) D s h ] 2 .
F ( κ ) = 2 [ 1 - 2 J 1 ( D h ) D h ] .
ϕ R ( L ) = k 0 0 L d z d ν ( κ , z ) cos [ P ( γ , κ , z ) ] × d ρ d ρ g ( ρ ) [ exp ( i κ · ρ ) - S ( ρ ) exp ( i κ · { ρ ( 1 - z / L ) + [ ( ρ + b ) z / L ] } ) d ρ S ( ρ ) ] .
F ( κ ) = | 2 ν J ν ( κ D / 2 ) κ D / 2 - exp ( i κ · b z L ) 2 ν J ν [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 d ρ S ( ρ ) exp ( i κ · ρ z / L ) d ρ S ( ρ ) | 2 ,
F ( κ ) = 4 ν 2 | J ν ( κ D / 2 ) κ D / 2 - exp ( i κ · b z L ) × J ν [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 2 J 1 ( D s h ) D s h | 2 .
F ( κ ) = 4 ν 2 [ J ν ( κ D / 2 ) κ D / 2 - J ν [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 2 J 1 ( D s h ) D s h ] 2 .
σ - 2 = 1.303 k 0 2 0 L d z C n 2 ( z ) 0 d κ κ - 8 / 3 × { 1 - 2 J 1 ( κ D s z / 2 L ) κ D s z / 2 L 2 J 1 ( κ D z / 2 L ) κ D z / 2 L J 0 ( κ b z L ) + [ 2 J 1 ( κ D s z / 2 L ) κ D s z / 2 L ] 2 } .
σ - 2 = 2.606 k 0 2 0 L d z C n 2 ( z ) 0 d κ κ - 8 / 3 [ 1 - 2 J 1 ( κ D z / 2 L ) κ D z / 2 L ] .
σ - 2 = 0.5 k 0 2 ( D L ) 5 / 3 μ 5 / 3 - ( H ) = ( 0.348 D L θ 0 - ) 5 / 3 .
( θ 0 - ) - 5 / 3 = 2.91 k 0 2 μ 5 / 3 - ( H ) .
σ - Z 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) 4 ν 2 d κ κ - 11 / 3 × { J ν ( κ D / 2 ) κ D / 2 - J ν [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 } 2 ,
σ - Z 2 = 1.642 k 0 2 ν 2 D 5 / 3 0 L d z C n 2 ( z ) I ,
I = 0 d x x - 14 / 3 { [ J ν 2 ( x ) - a x 2 4 ] + [ y 2 J ν 2 ( x / y ) - a x 2 4 ] - 2 [ J ν ( x ) y J ν ( x / y ) - a x 2 4 ] } .
σ - P 2 = 0.5 k 0 2 D 5 / 3 0 L d z C n 2 ( z ) × { F 2 1 [ - 5 6 , - 11 6 ; 2 ; ( 1 - z / L ) 2 ] - 1 + ( 1 - z / L ) 5 / 3 π 2 - 8 / 3 Γ [ / ²³ / ] } ,
σ - T 2 = 0.8345 k 0 2 D 5 / 3 0 L d z C n 2 ( z ) { [ 1 + ( 1 - z / L ) 5 / 3 ] π 2 - 11 / 3 Γ [ / ²⁹ / ] - ( 1 - z / L ) F 2 1 [ 1 6 , - 11 6 ; 3 ; ( 1 - z / L ) 2 ] } .
σ - P 2 0.0833 k 0 2 D 5 / 3 μ 2 - ( H ) L 2 .
σ - T 2 0.368 k 0 2 D 5 / 3 μ 2 - ( H ) L 2 .
σ - PR 2 = ( 0.348 D H θ 0 - ) 5 / 3 - σ - P 2 k 0 2 D 5 / 3 2 [ μ 5 / 3 - ( H ) L 5 / 3 - 0.167 μ 2 - ( H ) L 2 ] .
σ - PTR 2 = ( 0.348 D H θ 0 - ) 5 / 3 - σ - P 2 - σ - T 2 k 0 2 D 5 / 3 2 [ μ 5 / 3 - ( H ) L 5 / 3 - 0.903 μ 2 - ( H ) L 2 ] .
σ + PR 2 = 1.033 ( D / r 0 + ) 5 / 3 ,
( r 0 + ) - 5 / 3 = 0.423 k 0 2 μ 0 + ( H ) .
σ + PTR 2 = 0.134 ( D / r 0 + ) 5 / 3 .
σ PR 2 = σ - PR 2 + σ + PR 2
σ PTR 2 = σ - PTR 2 + σ + PTR 2 .
T Z 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ κ - 11 / 3 ( 16 k 0 D ) 2 × { J 2 [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 } 2 2 [ 1 - cos ( κ · b z / L ) ] .
T Z 2 = 10.68 D 1 / 3 0 z t d z C n 2 ( z ) [ ( b z D L ) 2 ( 1 - z / L ) - 1 / 3 - 2.067 ( b z D L ) 4 ( 1 - z / L ) - 7 / 3 - 1.472 ( b z D L ) 14 / 3 × ( 1 - z / L ) - 3 + 0.339 ( b z D L ) 6 ( 1 - z / L ) - 13 / 3 + ] + 12.16 D 1 / 3 z t L d z C n 2 ( z ) [ ( 1 - z / L ) 5 / 3 - 0.6657 ( D L b z ) 1 / 3 × ( 1 - z / L ) 2 - 0.00308 ( D L b z ) 7 / 3 ( 1 - z / L ) 4 - 3.06 × 10 - 6 ( D L b z ) 13 / 3 ( 1 - z / L ) 6 - 6.15 × 10 - 6 ( D L b z ) 19 / 3 ( 1 - z / L ) 8 + ] .
z t = L b / D + 1 .
m = A ϕ + n g .
A = 0.5 [ - 1 1 0 - 1 1 0 0 0 0 0 - 1 1 0 - 1 1 0 0 0 0 0 0 - 1 1 0 - 1 1 0 0 0 0 0 - 1 1 0 - 1 1 - 1 - 1 0 1 1 0 0 0 0 0 - 1 - 1 0 1 1 0 0 0 0 0 0 - 1 - 1 0 1 1 0 0 0 0 0 - 1 - 1 0 1 1 ] .
C n = n g n g T
C n = [ 1 i p q 0 0 0 r i 1 q p 0 0 r 0 p q 1 i 0 r 0 0 q p i 1 r 0 0 0 0 0 0 r 1 i p q 0 0 r 0 i 1 q p 0 r 0 0 p q 1 i r 0 0 0 q p i 1 ] ,
ϕ ˜ = ( A T C n - 1 A ) + A T C n - 1 m = L m ,
ϕ ˜ TPR = Λ ( A T C n - 1 A ) + A T C n - 1 m = Λ L m .
Λ = 1 - t x T t x - t y T t y ,
t x = 0.5 [ - 1 0 1 - 1 0 1 - 1 0 1 ] ,
t y = 0.5 [ - 1 - 1 - 1 0 0 0 1 1 1 ] .
Error = L C n L T ,
E = 1 N Tr [ Λ ( A T C n - 1 A ) + Λ T ] .
σ 2 = σ PTR 2 + E σ - T 2 .
σ 2 = 0.2073 0 d z C n 2 ( z ) d κ κ - 11 / 3 ( 16 D ) 2 [ J 2 ( κ D / 2 ) κ D / 2 ] 2 × 2 { 1 - cos [ κ θ z cos ( φ ) ] } ,
σ 2 = 667 D 2 0 d z C n 2 ( z ) 0 d κ κ - 8 / 3 [ J 2 ( κ D / 2 ) κ D / 2 ] 2 [ 1 - J 0 ( κ θ z ) ] .
σ 2 = σ L 2 + σ H 2 .
μ m = 0 d z C n 2 ( z ) z m = sec m + 1 ( ξ ) 0 d h C n 2 ( h ) h m ,
μ m + ( L ) = L d z C n 2 ( z ) z m = sec m + 1 ( ξ ) H d h C n 2 ( h ) h m ,
μ m - ( L ) = 0 L d z C n 2 ( z ) z m = sec m + 1 ( ξ ) 0 H d h C n 2 ( h ) h m .
σ L 2 = 12.16 D 1 / 3 ( { 1 - 0 H c d z C n 2 ( z ) × F 3 2 [ 1 6 , - 23 6 , - 11 6 ; 1 , - 4 3 ; ( θ z D ) 2 ] } + 1.292 0 H c d z C n 2 ( z ) ( θ z / D ) 14 / 3 × F 3 2 [ 5 2 , 1 2 , - 3 2 ; 10 3 ; 10 3 ; ( θ z D ) 2 ] ) ,             z < H c .
σ H 2 = 12.16 D 1 / 3 { μ 0 + ( D θ ) - 0.6637 H c d z C n 2 ( z ) ( D θ z ) 1 / 3 × F 3 2 [ 1 6 , 5 2 , 1 6 ; 5 , 3 ; ( D θ z ) 2 ] } ,             z > H c .
σ x 2 σ y 2 } L = 6.08 D 1 / 3 0 H c d z C n 2 ( z ) × [ { 1 - F 4 3 [ 1 6 , - 23 6 , - 11 6 , 3 2 ; - 4 3 , 2 , 1 2 ; ( θ z D ) 2 ] 1 - F 3 2 [ 1 6 , - 23 6 , - 11 6 ; - 4 3 , 2 ; ( θ z D ) 2 ] } + { 2.19 ( θ z D ) 14 / 3 F 4 3 [ 5 2 , - 3 2 , 1 2 , 23 6 ; 17 6 , 13 3 , 10 3 ; ( θ z D ) 2 ] 0.388 ( θ z D ) 14 / 3 F 3 2 [ 5 2 , - 3 2 , 1 2 ; 13 3 , 10 3 ; ( θ z D ) 2 ] } ] .
σ x 2 σ y 2 } H = 6.08 D 1 / 3 { μ 0 + ( H c ) { 1 1 } - H c L d z C n 2 ( z ) × { 0.532 ( D θ z ) 1 / 3 F 4 3 [ - 5 6 , 5 2 , 1 6 , 2 3 ; 5 , 3 , - 1 3 ; ( D θ z ) 2 ] 0.798 ( D θ z ) 1 / 3 F 3 2 [ - 5 6 , 5 2 , 1 6 ; 5 , 3 ; ( D θ z ) 2 ] } } .
σ x 2 σ y 2 } = 2.67 μ 2 D 1 / 3 ( θ D ) 2 { 3 1 } - 3.68 μ 4 D 1 / 3 ( θ D ) 4 { 5 1 } + 2.35 μ 14 / 3 D 1 / 3 ( θ D ) 14 / 3 { ¹⁷ / 1 } + 0.304 μ 6 D 1 / 3 ( θ D ) 6 { 7 1 } - 0.306 μ 20 / 3 D 1 / 3 ( θ D ) 20 / 3 { ²³ / 1 } + .
θ T c = 0.184 λ D 1 / 6 μ 2 1 / 2 .
T Z 2 = 0.2073 k 0 2 0 L d z C n 2 ( z ) d κ κ - 11 / 3 ( 16 k 0 D ) 2 × { J 2 [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 } 2 2 { 1 - cos [ κ · d ( z ) ] } ,
d ( z ) = b z L .
x = κ D 1 - z / L 2 ,
t = D 2 b L z ( 1 - z / L )
T Z 2 = 210 D 1 / 3 0 L d z C n 2 ( z ) ( 1 - z / L ) 5 / 3 × 0 d x x x - 11 / 3 J 2 2 ( x ) [ 1 - J 0 ( x / t ) ] .
z t = L b / D + 1 .
T Z h 2 = 12.16 D 1 / 3 z t d z C n 2 ( z ) ( 1 - z / L ) 5 / 3 × ( 1 - 0.6637 t 1 / 3 F 3 2 [ 1 6 , 5 2 , 1 6 ; 5 , 3 ; t 2 ] ) ,             z > z t .
T Z l 2 = 12.16 D 1 / 3 { 0 z t d z C n 2 ( z ) ( 1 - z / L ) 5 / 3 × ( 1 - F 3 2 [ 1 6 , - 23 6 , - 11 6 ; 1 , - 4 3 ; t - 2 ] ) - 1.292 0 z t d z C n 2 ( z ) ( 1 - z / L ) 5 / 3 t - 14 / 3 × F 3 2 [ 5 2 , 1 2 , - 3 2 ; 10 3 , 10 3 ; t - 2 ] }             z < z t .
C T ( d ) = σ - T ( r ) σ - T ( r + d ) σ - T 2 ( r ) = 1 - [ σ - T ( r ) - σ - T ( r + d ) ] 2 2 σ - T 2 ( r ) = 1 - N T 2 σ - T 2 ( r ) .
N T a b = [ σ T a ( r ) - σ T b ( r + d ) ] 2 = 3.317 k 0 2 0 L d z C n 2 ( z ) d κ κ - 11 / 3 × G T a - G T b exp ( j κ · d ) 2 ,
N T x x = [ σ T x ( r ) - σ T x ( r + d ) ] 2 = 6.634 k 0 2 0 L d z C n 2 ( z ) d κ κ - 11 / 3 × { J 2 ( κ D / 2 ) κ D / 2 - J 2 [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 } 2 × cos 2 ( ϕ ) [ 1 - cos ( κ · d ) ] .
0 2 π d θ cos ( θ ) sin ( θ ) cos [ κ d cos ( θ ) ] = 2 0 π d θ cos ( θ ) sin ( θ ) cos [ κ d cos ( θ ) ] = 0.
cos ( a + b ) = cos ( a ) cos ( b ) - sin ( a ) sin ( b )
0 2 π d ϕ cos 2 ( ϕ ) { 1 - cos [ κ d cos ( ϕ + θ i ) ] } = 0 2 π d ϕ cos 2 ( ϕ - θ i ) { 1 - cos [ κ d cos ( ϕ ) ] } = 0 2 π d ϕ [ cos 2 ( θ i ) cos 2 ( ϕ ) + sin 2 ( θ i ) sin 2 ( ϕ ) - 2 cos ( θ i ) cos ( ϕ ) sin ( θ i ) sin ( ϕ ) ] { 1 - cos [ κ d cos ( ϕ ) ] } .
0 π d x exp [ i β cos ( x ) ] sin 2 ν ( x ) = π ( 2 β ) ν Γ [ ν + 1 2 ] J ν ( β ) ,             Re ν > - 1 2 .
N T x x = 41.68 k 0 2 0 L d z C n 2 ( z ) d κ κ - 8 / 3 × { J 2 ( κ D / 2 ) κ D / 2 - J 2 [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 } 2 × ( sin 2 ( θ i ) [ 1 / 2 - J 1 ( κ d ) / κ d ] + cos 2 ( θ i ) × { [ 1 - J 0 ( κ d ) ] - [ 1 / 2 - J 1 ( κ d ) / κ d ] } ) .
J ν ( λ x ) = λ ν k = 0 1 k ! J ν + k ( x ) ( 1 - λ 2 2 x ) k
σ T 2 0.190 k 0 2 D 5 / 3 μ 2 L 2 .
N T x x 13.13 k 0 2 D 5 / 3 μ 2 L 2 [ sin 2 ( θ i ) I 1 + cos 2 ( θ i ) ( I T - I 1 ) ] = 13.13 k 0 2 D 5 / 3 μ 2 L 2 [ I T cos 2 ( θ i ) - I 1 cos ( 2 θ i ) ] ,
I T I 1 } = ( 2 D ) 11 / 3 0 d κ κ - 14 / 3 J 2 2 ( κ D 2 ) { 1 - J 0 ( κ d ) 1 / 2 - J 1 ( κ d ) / κ d .
N T y y 41.7 k 0 2 ( D 2 ) 5 / 3 μ 2 L 2 [ sin 2 ( θ i - π / 2 ) I 1 + cos 2 ( θ i - π / 2 ) ( I T - I 1 ) ] .
N T x y = [ σ T x ( r ) - σ T y ( r + d ) ] 2 = 3.318 k 0 2 0 L d z C n 2 ( z ) × d κ κ - 11 / 3 { J 2 ( κ D / 2 ) κ D / 2 - J 2 [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 } 2 × cos ( ϕ ) - sin ( ϕ ) exp ( j κ · d ) 2 .
N T x y = 3.318 k 0 2 0 L d z C n 2 ( z ) d κ κ - 11 / 3 × { J 2 ( κ D / 2 ) κ D / 2 - J 2 [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 } 2 × [ 1 - 2 sin ( ϕ ) cos ( ϕ ) cos ( κ · d ) ] .
I = 0 2 π d ϕ sin ( ϕ ) cos ( ϕ ) cos [ κ d cos ( ϕ - θ ) ] .
I = 0 2 π d θ sin ( θ + θ ) cos ( θ + θ ) cos [ κ d cos ( θ ) ] = 0 2 π d θ cos [ κ d cos ( θ ) ] { sin ( θ ) cos ( θ ) [ 1 - 2 sin 2 ( θ ) ] + sin ( θ ) cos ( θ ) [ cos 2 ( θ ) - sin 2 ( θ ) ] } .
I = 2 sin ( θ ) cos ( θ ) 0 π d θ [ 1 - 2 sin 2 ( θ ) ] cos [ κ d cos ( θ ) ] .
N T x y = 20.85 k 0 2 0 L d z C n 2 ( z ) 0 d κ κ - 8 / 3 × { J 2 ( κ D / 2 ) κ D / 2 - J 2 [ κ D ( 1 - z / L ) / 2 ] κ D ( 1 - z / L ) / 2 } 2 × { 1 - 2 sin ( θ ) cos ( θ ) [ J 0 ( κ d ) - 2 J 1 ( κ d ) / κ d ] } .
N T x y 6.57 k 0 2 μ 2 L 2 D 5 / 3 [ 0.0579 + sin ( 2 θ ) ( 2 I 1 - I T ) ] .
I T = 0.05789 - 0.03853 ( D d ) 1 / 3 F 3 2 [ 5 2 , 1 6 , 1 6 ; 5 , 3 ; ( D 2 d ) ] ,             d > D ,
I T = - 0.0748 ( d D ) 14 / 3 F 3 2 [ 5 2 , - 3 2 , 1 2 ; 10 3 , 10 3 ; ( d D ) 2 ] + 0.05789 { 1 - F 3 2 [ 1 6 , - 23 6 , - 11 6 ; 1 , - 4 3 ; ( d D ) 2 ] } ,             d < D .
I 1 = 0.02894 - 0.02312 ( D d ) 1 / 3 × F 3 2 [ 5 2 , 1 6 , - 5 6 ; 5 , 3 ; ( D d ) 2 ] ,             d > D ,
I 1 = 0.001122 ( d D ) 14 / 3 F 3 2 [ 5 2 , - 3 2 , 1 2 ; 13 3 , 10 3 ; ( d D ) 2 ] + 0.02894 { 1 - F 3 2 [ 1 6 , - 23 6 , - 11 6 ; 2 , - 4 3 ; ( d D ) 2 ] } ,             d > D .
C T x x ( d ) = - 0.7986 cos ( 2 θ i ) ( D d ) 1 / 3 × F 3 2 [ 5 2 , 1 6 , - 5 6 ; 5 , 3 ; ( D d ) 2 ] + 1.331 cos 2 ( θ i ) ( D d ) 1 / 3 × F 3 2 [ 5 2 , 1 6 , 1 6 ; 5 , 3 ; ( D d ) 2 ] ,             d > D ,
C T x x ( d ) = cos ( 2 θ i ) { - F 3 2 [ 1 6 , - 23 6 , - 11 6 ; 2 , - 4 3 ; ( d D ) 2 ] + 0.3877 ( d D ) 14 / 3 F 3 2 [ 5 2 , - 3 2 , 1 2 ; 13 3 , 10 3 ; ( d D ) 2 ] } - 2 cos 2 ( θ i ) { F 3 2 [ 1 6 , - 23 6 , - 11 6 ; 2 , - 4 3 ; ( d D ) 2 ] - 1.293 ( d D ) 14 / 3 F 3 2 [ 5 2 , - 3 2 , 1 2 ; 10 3 , 10 3 ; ( d D ) 2 ] } ,             d < D .
C T x y ( d ) = 17.28 sin ( 2 θ i ) ( I T - 2 I 1 ) .
C T x y ( d ) = 0.1331 sin ( 2 θ i ) ( D d ) 1 / 3 × F 3 2 [ 5 2 , 7 6 , - 5 6 ; 5 , 3 ; ( D d ) 2 ] ,             d > D ,
C T x y ( d ) = 0.9048 sin ( 2 θ i ) × { ( d D ) 14 / 3 F 3 2 [ 5 2 , - 3 2 , 1 2 ; 13 3 , 7 3 ; ( d D ) 2 ] + 0.4393 ( d D ) 2 × F 3 2 [ 7 6 , - 17 6 , - 5 6 ; 3 , - 1 3 ; ( d D ) 2 ] } ,             d < D .

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