Abstract

Mellin transform techniques are applied to evaluate the covariance of the integrated turbulence-induced phase distortions along a pair of ray paths through the atmosphere from two points in a telescope aperture to a pair of sources at finite or infinite range. The derivation is for the case of a finite outer scale and a von Karman turbulence spectrum. The Taylor hypothesis is assumed if the two phase distortions are evaluated at two different times and amplitude scintillation effects are neglected. The resulting formula for the covariance is a power series in one variable for the case of a fixed atmospheric wind velocity profile and a power series in two variables for a fixed wind-speed profile with a random and uniformly distributed wind direction. These formulas are computationally efficient and can be easily integrated into computer codes for the numerical evaluation of adaptive optics system performance. Sample numerical results are presented to illustrate the effect of a finite outer scale on the performance of natural and laser guide star adaptive optics systems for an 8-m astronomical telescope. A hypothetical outer scale of 10 m significantly reduces the magnitude of tilt anisoplanatism, thereby improving the performance of a laser guide star adaptive optics system if the auxiliary natural star used for full-aperture tip/tilt sensing is offset from the science field. The reduction in higher-order anisoplanatism that is due to a 10-m outer scale is smaller, and the off-axis performance of a natural guide star adaptive optics system is not significantly improved.

© 1997 Optical Society of America

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1996 (3)

1994 (5)

1993 (1)

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, and D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time delay and isoplanicity errors in partial adaptive correction,” J. Opt. Soc. Am A 10, 957–965 (1993).
[CrossRef]

1991 (4)

B. M. Welsh and 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]

D. M. Winker, “Effect of a finite outer scale on the Zernike decomposition of atmospheric optical turbulence,” J. Opt. Soc. Am. A 8, 1568–1573 (1991).
[CrossRef]

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

F. Roddier, M. Northcott, and J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

1990 (1)

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

1984 (1)

1983 (1)

1982 (1)

1977 (1)

1976 (1)

1966 (1)

Angel, J. R. P.

D. G. Sandler, M. Lloyd-Hart, T. Martinez, P. M. Gray, J. R. P. Angel, T. K. Barrett, D. G. Bruns, and S. M. Stahl, “6.5-meter MMT infrared adaptive optics system: detailed design and progress report,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 372–377 (1995).
[CrossRef]

Barclay, H. T.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Barrett, T. K.

D. G. Sandler, M. Lloyd-Hart, T. Martinez, P. M. Gray, J. R. P. Angel, T. K. Barrett, D. G. Bruns, and S. M. Stahl, “6.5-meter MMT infrared adaptive optics system: detailed design and progress report,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 372–377 (1995).
[CrossRef]

Belsher, J. F.

Boeke, B. R.

Boreman, G. D.

Boyer, C.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Bruns, D. G.

D. G. Sandler, M. Lloyd-Hart, T. Martinez, P. M. Gray, J. R. P. Angel, T. K. Barrett, D. G. Bruns, and S. M. Stahl, “6.5-meter MMT infrared adaptive optics system: detailed design and progress report,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 372–377 (1995).
[CrossRef]

Cleis, R. A.

Crochet, M.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of “sheets” in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Dainty, C.

Dalaudier, F.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of “sheets” in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Ellerbroek, B. L.

B. L. Ellerbroek, “First-order performance evaluation of adaptive optics systems for atmospheric turbulence compensation in extended field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

R. Q. Fugate, B. L. Ellerbroek, C. H. Higgins, M. P. Jelonek, W. J. Lange, A. C. Slavin, W. J. Wild, D. M. Winker, J. M. Wynia, J. M. Spinhirne, B. R. Boeke, R. E. Ruane, J. F. Moroney, M. D. Oliker, D. W. Swindle, and R. A. Cleis, “Two generations of laser guide star adaptive optics experiments at the Starfire Optical Range,” J. Opt. Soc. Am. A 11, 310–324 (1994).
[CrossRef]

B. L. Ellerbroek, S. M. Pompea, D. J. Robertson, and C. M. Mountain, “Adaptive optics performance analysis for the Gemini 8-meter Telescopes Project,” in Adaptive Optics in Astronomy, M. A. Ealey and F. Merkle, eds., Proc. SPIE 2201, 421–436 (1994).
[CrossRef]

R. Racine and B. L. Ellerbroek, “Profiles of night-time turbulence above Mauna Kea and isoplanatism extension in adaptive optics,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 248–257 (1995).
[CrossRef]

Fontanella, J. C.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Fried, D. L.

Fugate, R. Q.

Gaffard, J. P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Gardner, C. S.

Gigan, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Gleckler, A. D.

A. D. Gleckler and P. L. Wizinowich, “W. M. Keck Observatory adaptive optics program,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 386–400 (1995).
[CrossRef]

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, Orlando, Fla., 1980), pp. 686–687.

Graves, J. E.

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, and D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time delay and isoplanicity errors in partial adaptive correction,” J. Opt. Soc. Am A 10, 957–965 (1993).
[CrossRef]

F. Roddier, M. Northcott, and J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Gray, P. M.

D. G. Sandler, M. Lloyd-Hart, T. Martinez, P. M. Gray, J. R. P. Angel, T. K. Barrett, D. G. Bruns, and S. M. Stahl, “6.5-meter MMT infrared adaptive optics system: detailed design and progress report,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 372–377 (1995).
[CrossRef]

Higgins, C. H.

Jagourel, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Jelonek, M. P.

Jenkins, C. R.

R. W. Wilson and C. R. Jenkins, “Adaptive optics for astronomy: theoretical performance and limitations,” Mon. Not. R. Astron. Soc. 278, 39–61 (1996).
[CrossRef]

Kern, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Lange, W. J.

Lena, P.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Link, D.

D. Link, “Comparison of the effects of near field and distributed atmospheric turbulence on the performance of an adaptive optics system,” in Laser Beam Propagation and Control, H. Weichel and L. DeSandere, eds., Proc. SPIE 2120, 87–94 (1994).
[CrossRef]

Lloyd-Hart, M.

D. G. Sandler, M. Lloyd-Hart, T. Martinez, P. M. Gray, J. R. P. Angel, T. K. Barrett, D. G. Bruns, and S. M. Stahl, “6.5-meter MMT infrared adaptive optics system: detailed design and progress report,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 372–377 (1995).
[CrossRef]

Martinez, T.

D. G. Sandler, M. Lloyd-Hart, T. Martinez, P. M. Gray, J. R. P. Angel, T. K. Barrett, D. G. Bruns, and S. M. Stahl, “6.5-meter MMT infrared adaptive optics system: detailed design and progress report,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 372–377 (1995).
[CrossRef]

McKenna, D. L.

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, and D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time delay and isoplanicity errors in partial adaptive correction,” J. Opt. Soc. Am A 10, 957–965 (1993).
[CrossRef]

Merkle, F.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Moroney, J. F.

Mountain, C. M.

B. L. Ellerbroek, S. M. Pompea, D. J. Robertson, and C. M. Mountain, “Adaptive optics performance analysis for the Gemini 8-meter Telescopes Project,” in Adaptive Optics in Astronomy, M. A. Ealey and F. Merkle, eds., Proc. SPIE 2201, 421–436 (1994).
[CrossRef]

Murphy, D. V.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Noll, R. J.

Northcott, M.

F. Roddier, M. Northcott, and J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

Northcott, M. J.

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, and D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time delay and isoplanicity errors in partial adaptive correction,” J. Opt. Soc. Am A 10, 957–965 (1993).
[CrossRef]

Oliker, M. D.

Page, D. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Parenti, R. R.

Pompea, S. M.

B. L. Ellerbroek, S. M. Pompea, D. J. Robertson, and C. M. Mountain, “Adaptive optics performance analysis for the Gemini 8-meter Telescopes Project,” in Adaptive Optics in Astronomy, M. A. Ealey and F. Merkle, eds., Proc. SPIE 2201, 421–436 (1994).
[CrossRef]

Primmerman, C. A.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Racine, R.

R. Racine and B. L. Ellerbroek, “Profiles of night-time turbulence above Mauna Kea and isoplanatism extension in adaptive optics,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 248–257 (1995).
[CrossRef]

Rigaut, F.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Robertson, D. J.

B. L. Ellerbroek, S. M. Pompea, D. J. Robertson, and C. M. Mountain, “Adaptive optics performance analysis for the Gemini 8-meter Telescopes Project,” in Adaptive Optics in Astronomy, M. A. Ealey and F. Merkle, eds., Proc. SPIE 2201, 421–436 (1994).
[CrossRef]

Roddier, D.

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, and D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time delay and isoplanicity errors in partial adaptive correction,” J. Opt. Soc. Am A 10, 957–965 (1993).
[CrossRef]

Roddier, F.

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, and D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time delay and isoplanicity errors in partial adaptive correction,” J. Opt. Soc. Am A 10, 957–965 (1993).
[CrossRef]

F. Roddier, M. Northcott, and J. E. Graves, “A simple low-order adaptive optics system for near-infrared applications,” Publ. Astron. Soc. Pac. 103, 131–149 (1991).
[CrossRef]

F. Roddier, Report on the Seeing on Mauna Kea (University of Hawaii, Honolulu, 1992).

Rousset, G.

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

Ruane, R. E.

Rudin, W.

W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1994).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, Orlando, Fla., 1980), pp. 686–687.

Sandler, D. G.

D. G. Sandler, M. Lloyd-Hart, T. Martinez, P. M. Gray, J. R. P. Angel, T. K. Barrett, D. G. Bruns, and S. M. Stahl, “6.5-meter MMT infrared adaptive optics system: detailed design and progress report,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 372–377 (1995).
[CrossRef]

Sasiela, R. J.

Sidi, C.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of “sheets” in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Slavin, A. C.

Spinhirne, J. M.

Stahl, S. M.

D. G. Sandler, M. Lloyd-Hart, T. Martinez, P. M. Gray, J. R. P. Angel, T. K. Barrett, D. G. Bruns, and S. M. Stahl, “6.5-meter MMT infrared adaptive optics system: detailed design and progress report,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 372–377 (1995).
[CrossRef]

Swindle, D. W.

Tatarskii, V. I.

V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation (National Technical Information Service, Springfield, Va., 1971).

Tyler, G. A.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic, San Diego, Calif., 1991).

Vernin, J.

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of “sheets” in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

Wallner, E. P.

Welsh, B. M.

Wild, W. J.

Wilson, R. W.

R. W. Wilson and C. R. Jenkins, “Adaptive optics for astronomy: theoretical performance and limitations,” Mon. Not. R. Astron. Soc. 278, 39–61 (1996).
[CrossRef]

Winker, D. M.

Wizinowich, P. L.

A. D. Gleckler and P. L. Wizinowich, “W. M. Keck Observatory adaptive optics program,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 386–400 (1995).
[CrossRef]

Wynia, J. M.

Zollars, B. G.

C. A. Primmerman, D. V. Murphy, D. A. Page, B. G. Zollars, and H. T. Barclay, “Compensation of atmospheric optical distortion using a synthetic beacon,” Nature (London) 353, 141–143 (1991).
[CrossRef]

Astron. Astrophys. (1)

G. Rousset, J. C. Fontanella, P. Kern, P. Gigan, F. Rigaut, P. Lena, C. Boyer, P. Jagourel, J. P. Gaffard, and F. Merkle, “First diffraction limited astronomical images with adaptive optics,” Astron. Astrophys. 230, L29–L32 (1990).

J. Atmos. Sci. (1)

F. Dalaudier, C. Sidi, M. Crochet, and J. Vernin, “Direct evidence of “sheets” in the atmospheric temperature field,” J. Atmos. Sci. 51, 237–248 (1994).
[CrossRef]

J. Opt. Soc. Am A (1)

F. Roddier, M. J. Northcott, J. E. Graves, D. L. McKenna, and D. Roddier, “One-dimensional spectra of turbulence-induced Zernike aberrations: time delay and isoplanicity errors in partial adaptive correction,” J. Opt. Soc. Am A 10, 957–965 (1993).
[CrossRef]

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. A (8)

B. M. Welsh and 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]

B. L. Ellerbroek, “First-order performance evaluation of adaptive optics systems for atmospheric turbulence compensation in extended field-of-view astronomical telescopes,” J. Opt. Soc. Am. A 11, 783–805 (1994).
[CrossRef]

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Nature (London) (1)

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D. Link, “Comparison of the effects of near field and distributed atmospheric turbulence on the performance of an adaptive optics system,” in Laser Beam Propagation and Control, H. Weichel and L. DeSandere, eds., Proc. SPIE 2120, 87–94 (1994).
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A. D. Gleckler and P. L. Wizinowich, “W. M. Keck Observatory adaptive optics program,” in Adaptive Optical Systems and Applications, R. K. Tyson and R. Q. Fugate, eds., Proc. SPIE 2534, 386–400 (1995).
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[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Coordinate system for ray paths and atmospheric profiles. Here r is a point in the telescope aperture plane, ψ is the zenith angle, and the z axis of the coordinate system is the optical axis of the telescope. The ray path from a wave-front sensor (WFS) beacon to the point r is denoted by the curve [ p (r, z), z], where z is the range from the aperture and p (r, z) is the offset of the ray path from the optical axis at this range. The same coordinate system is also used for the atmospheric turbulence and wind velocity profiles.

Fig. 2
Fig. 2

Representative C n 2 (h) profile for Mauna Kea. This profile was obtained by Roddier during 1987 with the scidar measurement technique. The corresponding values of r 0 and θ0 are 0.235 m and 9.90 µrad, respectively, at a wavelength of 0.5 µm.

Fig. 3
Fig. 3

Impact of a 10-m outer scale on the off-axis performance of LGS and NGS AO systems. These results assume the atmospheric C n 2 (h) profile plotted in Fig. 2, an observing wavelength of 1.25 µm, and the Gemini telescope AO system parameters described in the text. For the LGS AO case, only the auxiliary NGS used for full-aperture tip/tilt sensing is offset from the science field, with the LGS projected directly at the object of interest. (a) Tilt-included Strehl ratio, (b) tilt-removed Strehl ratio, (c) tilt-included phase variance, and (d) tilt-removed phase variance.

Fig. 4
Fig. 4

Impact of a 10-m outer scale on the off-axis performance of LGS and NGS AO systems with the DM and WFS optically conjugate to an altitude of 6.5 km. Similar to Fig. 3 except that the DM and WFS used by the AO system are optically conjugate to an altitude of 6.5 km. This is approximately the mean height of the turbulence profile plotted in Fig. 2 (a) Tilt-included Strehl ratio, (b) tilt-removed Strehl ratio, (c) tilt-included phase variance, and (d) tilt-removed phase variance.

Fig. 5
Fig. 5

Aperture stop configuration for AO systems with the DM and WFS optically conjugate to a nonzero altitude. For the NGS AO case, the pupil of the science instrument is vignetted at the DM by a stop matching the NGS beam print. For the LGS AO system, the pupil of the tip/tilt sensor is similarly vignetted by a stop matching the LGS beam print. The purpose of these stops is to limit the potential impact of unsensed DM actuators on AO system performance and is described further in the text.

Equations (47)

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c= drwrϕr, t,
 drwr=0,
ϕr, t=2πλ0zdznpr, z,z, t.
nr, z, t=n0r-tvz, z,
ci= drwirϕir, ti, i=1, 2,
ϕir, ti=2πλ0Zidznpir, z, z, ti, i=1, 2.
c1c2=2πλ2 drdrw1rw2r0Z10Z2dzdz×n0p1r, z-t1vz, z×n0p2r, z-t2vz, z,
n0r, z=dκˆn0κ, zexp2πiκ·r.
c1c2=2πλ2 drdrw1rw2r0Z10Z2dzdz×dκdκnˆ0κ, znˆ0*κ, z×exp2πiκ·p1r, z-t1vz-κ·p2r,z-t2vz.
z+=z+z/2,
z-=z-z
c1c2=2πλ2 drdrw1rw2r0Z1+Z2/2×2 maxz+-Z2,-z+2 minZ1-z+, z+dz+dz-dκdκ×nˆ0κ, z++12 z-nˆ0*κ, z+-12z-×exp2πiκ·p1r, z++12 z--t1vz++12z-×exp-2πiκ·p2r, z+-12 z--t2vz+-12z-.
2 maxz+-Z2,-z+2 minZ1-z+,z+dz-nˆ0κ,z++12z-nˆ0*κ,z+-12z-=9.69×10-3Cn2hz+,ψδκ-κκ2+L0-2-11/6  if z+minZ1,Z20 otherwise.
c1c2=9.69×10-32πλ2drdrw1rw22r×0ZdzCn2z cos ψdκκ2+L0-2-11/6×exp2πiκ·p1r, z-p2r,z+t2-t1vz,
c1c2=9.69×10-32πλ2drdrw1r w2r×0ZdzCn2z cos ψdκκ2+L0-2-11/6×exp2πiκ·p1r, z-p2r,z+t2-t1vz-1.
c1c2=6.09×10-22πλ2drdrw1rw2r0ZdzCn2z cos ψ0dκκκ2+L0211/6×J02πκp1r, z-p2r, z+t2-t1vz-1 fixed vJ02πκp1r, z-p2r, zJ02πκt2-t1vz-1 fixed v.
x=L0κ
Δ=p1r,z-p2r,z+t2-t1vzfixed v,p1r,z-p2r,zfixed v,
δ=0fixed v,t2-t1vfixed v,
c1c2=6.09×10-22πλ2L05/3drdrw1rw2r×0ZdzCn2z cos ψ×0dxxx2+111/6J02πΔxL0J02πδxL0-1.
1=r0-5/32.916.882πλ20dzCn2z cos ψ-1,
c1c2=0.144 L0r05/30dhCn2h-1×drdrw1rw2r×0HdhCn2hdxxx2+111/6×J02πΔxL0 J02πδxL0-1.
c1c2=0.144 0dhCn2h-1drdrw1rw2r×0HdhCn2hL0hr05/3×0dxxx2+111/6J02πΔxL0J02πδxL0-1.
Ia, b=0dxxx2+111/6 J0axJ0bx,
I0,b=b5/6K5/6b25/6Γ11/6,
H*s=0dxHxxs-1.
Hx=12πidsH*sx-s,
Ha1,,an=0dxx H0xi=1n Hiai/x,
H*s1,,sn=H0*i=1nsii=1nHi*si.
Ha1,,an=12πin  ds1  dsnH0*i=1nsi×i=1n Hi*sii=1nai-si.
H0x=x2x2+111/6,
H1x=H2x=J0x-1.
Ha-1,b-1=Ia,b
Ia,b=12πi2dsdtH0*s+tH1*sH2*tasbt.
H0*s=12Γs2+1,56-s2116  -2s53 H1*s=H2*s=2-s-1 Γ-s21+s2  -320,
Γa1,,anb1,,bm=Γa1ΓanΓb1Γbm.
Ia, b=18Γ11/612πi2dsdt×Γ1+s2+t2,56-s2-t2,-s2,-t21+s2,1+t2a2sb2t,
u=a/22,
v=b/22,
Ia, b=12Γ11/612πi2dsdt×Γ1+s+t,56-s-t,-s,-t1+s, 1+tusvt.
Ia, b=12Γ11/6n=0-1nn!2un×12πidtΓn+1+t, 56-n-t, -t1+tvt+12Γ11/6n=0-1nn!u5/6+n×12πidtΓ116+n, t-56-n, -t116+n-t, 1+tv/ut.
Ia, b=12Γ11/6n=0m=0Γn+m+1, 56-n-m×-1nn!2-1mm!2unvm+12Γ11/6n=0m=nΓ116+m,n-m-56m-n+116×-1nn!2-1mm!unvm-n+5/6+12Γ11/6n=0m=1Γ116+n,m-561+n+m,116-m×-1nn!-1n+mn+m!u5/6+nv/u5/6-m,
Γx-k=-1kΓxΓ1-xΓk+1-x,
xk=Γx+kΓx=x+k-1x+k-2x+1x,
Ia, b=35n=0m=0n+m!1/6n+munn!2vmm!2+Γ-5/62Γ11/6v5/6n=0unn!2×m=0vmn+m!11/6m+n11/6m2+Γ-5/62Γ11/6v5/6n=0un11/6nn!3×m=1u/vm-5/6m2n+m!/n!2.
I0, 0=35,
I0, b=35m=0vmm!11/6m+Γ-5/62Γ11/6v5/6m=0vmm!111/6m.

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