Abstract

Laguerre–Gaussian laser modes carry orbital angular momentum as a consequence of their helical-phase front screw dislocation. This torsional beam structure interacts with rotating targets, changing the orbital angular momentum (azimuthal Doppler) of the scattered beam because angular momentum is a conserved quantity. I show how to measure this change independently from the usual longitudinal momentum (normal Doppler shift) and derive the apropos coherent mixing efficiencies for monostatic, truncated Laguerre and Gaussian-mode ladar antenna patterns.

© 2002 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
    [CrossRef] [PubMed]
  2. V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
    [CrossRef]
  3. W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
  4. M. S. Soskin, V. N. Vasnetsov, J. T. Malow, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
    [CrossRef]
  5. M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
    [CrossRef]
  6. G. Indebetauw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
    [CrossRef]
  7. S. A. Ponomarenko, “A class of partially coherent beams carrying optical vortices,” J. Opt. Soc. Am. A 18, 150–156 (2001).
    [CrossRef]
  8. S. Ramee, R. Simon, “Effect of holes and vortices on beam quality,” J. Opt. Soc. Am. A 17, 84–93 (2000).
    [CrossRef]
  9. M. S. Soskin, ed., International Conference on Singular Optics, Proc. SPIE3487 (1998).
  10. J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
    [CrossRef]
  11. L. Allen, M. Babiker, W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
    [CrossRef]
  12. J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
    [CrossRef]
  13. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  14. R. L. Phillips, L. C. Andrew, “Spot size and divergence for Laguerre Gaussian beams of any order,” Appl. Opt. 22, 643–644 (1983).
    [CrossRef] [PubMed]
  15. The actual circulation of energy is determined by the Gouy phase term (2p + m + 1)ψ(z), as well as the mθ term, hence there is a z dependence on the rate of rotation of the field energy. However, in the far field, ψ(z) → π/2, a constant, where the phase advance is then determined by the mθ term alone.
  16. M. W. Beijersbergen, M. Kristensen, J. P. Woerdman, “Spiral phase-plate used to produce helical wavefront laser beams,” in Conference on Lasers and Electro-Optics, 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CFA5.
  17. G. F. Brand, “Phase singularities in beams,” Am. J. Phys. 67 (1), 55–60 (1999).
  18. R. J. Hull, D. G. Biron, S. Marcus, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-83-238/ESD-TR-83-72 (Lincoln Laboratory, Lexington, Mass., 1983).
  19. D. U. Fluckiger, S. Marcus, R. J. Hull, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-84-306/ESE-TR-85-03 (Lincoln Laboratory, Lexington, Mass., 1984).
  20. J. H. Shapiro, B. A. Capron, R. C. Harney, “Imaging and target detection with a heterodyne-reception optical radar,” Appl. Opt. 20, 3292–3313 (1981).
    [CrossRef] [PubMed]
  21. J. F. Corum, “Relativistic rotation and the anholonomic object,” J. Math. Phys. 18, 770–776 (1977).
    [CrossRef]
  22. J. F. Corum, “Relativistic covariance and rotational electrodynamics,” J. Math. Phys. 21, 2360–2364 (1980).
    [CrossRef]
  23. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 3 and 4.
  24. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), formula entry 9.1.80.
  25. N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
    [CrossRef] [PubMed]

2001 (1)

2000 (1)

1999 (1)

G. F. Brand, “Phase singularities in beams,” Am. J. Phys. 67 (1), 55–60 (1999).

1998 (2)

M. S. Soskin, V. N. Vasnetsov, J. T. Malow, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

1997 (1)

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

1994 (2)

L. Allen, M. Babiker, W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

1993 (2)

G. Indebetauw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992 (3)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

1983 (1)

1981 (1)

1980 (1)

J. F. Corum, “Relativistic covariance and rotational electrodynamics,” J. Math. Phys. 21, 2360–2364 (1980).
[CrossRef]

1977 (1)

J. F. Corum, “Relativistic rotation and the anholonomic object,” J. Math. Phys. 18, 770–776 (1977).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), formula entry 9.1.80.

Allen, L.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

L. Allen, M. Babiker, W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Andrew, L. C.

Babiker, M.

L. Allen, M. Babiker, W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

Bazhenov, V. Yu.

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

M. W. Beijersbergen, M. Kristensen, J. P. Woerdman, “Spiral phase-plate used to produce helical wavefront laser beams,” in Conference on Lasers and Electro-Optics, 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CFA5.

Beijersbergen, W.

W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Biron, D. G.

R. J. Hull, D. G. Biron, S. Marcus, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-83-238/ESD-TR-83-72 (Lincoln Laboratory, Lexington, Mass., 1983).

Brand, G. F.

G. F. Brand, “Phase singularities in beams,” Am. J. Phys. 67 (1), 55–60 (1999).

Capron, B. A.

Corum, J. F.

J. F. Corum, “Relativistic covariance and rotational electrodynamics,” J. Math. Phys. 21, 2360–2364 (1980).
[CrossRef]

J. F. Corum, “Relativistic rotation and the anholonomic object,” J. Math. Phys. 18, 770–776 (1977).
[CrossRef]

Courtial, J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

Dholakia, K.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

Fluckiger, D. U.

D. U. Fluckiger, S. Marcus, R. J. Hull, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-84-306/ESE-TR-85-03 (Lincoln Laboratory, Lexington, Mass., 1984).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 3 and 4.

Harney, R. C.

Harris, M.

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Heckenberg, N. R.

M. S. Soskin, V. N. Vasnetsov, J. T. Malow, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

N. R. Heckenberg, R. McDuff, C. P. Smith, A. G. White, “Generation of optical phase singularities by computer-generated holograms,” Opt. Lett. 17, 221–223 (1992).
[CrossRef] [PubMed]

Hill, C. A.

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

Hull, R. J.

R. J. Hull, D. G. Biron, S. Marcus, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-83-238/ESD-TR-83-72 (Lincoln Laboratory, Lexington, Mass., 1983).

D. U. Fluckiger, S. Marcus, R. J. Hull, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-84-306/ESE-TR-85-03 (Lincoln Laboratory, Lexington, Mass., 1984).

Indebetauw, G.

G. Indebetauw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

Kristensen, M.

M. W. Beijersbergen, M. Kristensen, J. P. Woerdman, “Spiral phase-plate used to produce helical wavefront laser beams,” in Conference on Lasers and Electro-Optics, 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CFA5.

Malow, J. T.

M. S. Soskin, V. N. Vasnetsov, J. T. Malow, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

Marcus, S.

D. U. Fluckiger, S. Marcus, R. J. Hull, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-84-306/ESE-TR-85-03 (Lincoln Laboratory, Lexington, Mass., 1984).

R. J. Hull, D. G. Biron, S. Marcus, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-83-238/ESD-TR-83-72 (Lincoln Laboratory, Lexington, Mass., 1983).

McDuff, R.

Padgett, M. J.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

Phillips, R. L.

Ponomarenko, S. A.

Power, W. L.

L. Allen, M. Babiker, W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

Ramee, S.

Robertson, D. A.

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

Shapiro, J. H.

J. H. Shapiro, B. A. Capron, R. C. Harney, “Imaging and target detection with a heterodyne-reception optical radar,” Appl. Opt. 20, 3292–3313 (1981).
[CrossRef] [PubMed]

D. U. Fluckiger, S. Marcus, R. J. Hull, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-84-306/ESE-TR-85-03 (Lincoln Laboratory, Lexington, Mass., 1984).

R. J. Hull, D. G. Biron, S. Marcus, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-83-238/ESD-TR-83-72 (Lincoln Laboratory, Lexington, Mass., 1983).

Siegman, A. E.

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

Simon, R.

Smith, C. P.

Soskin, M. S.

M. S. Soskin, V. N. Vasnetsov, J. T. Malow, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), formula entry 9.1.80.

van der Ween, H. E. L. O.

W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Vasnetsov, M. V.

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

Vasnetsov, V. N.

M. S. Soskin, V. N. Vasnetsov, J. T. Malow, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

Vaughan, J. M. R.

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

White, A. G.

Woerdman, J. P.

W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

M. W. Beijersbergen, M. Kristensen, J. P. Woerdman, “Spiral phase-plate used to produce helical wavefront laser beams,” in Conference on Lasers and Electro-Optics, 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CFA5.

Am. J. Phys. (1)

G. F. Brand, “Phase singularities in beams,” Am. J. Phys. 67 (1), 55–60 (1999).

Appl. Opt. (2)

J. Math. Phys. (2)

J. F. Corum, “Relativistic rotation and the anholonomic object,” J. Math. Phys. 18, 770–776 (1977).
[CrossRef]

J. F. Corum, “Relativistic covariance and rotational electrodynamics,” J. Math. Phys. 21, 2360–2364 (1980).
[CrossRef]

J. Mod. Opt. (2)

G. Indebetauw, “Optical vortices and their propagation,” J. Mod. Opt. 40, 73–87 (1993).
[CrossRef]

V. Yu. Bazhenov, M. S. Soskin, M. V. Vasnetsov, “Screw dislocations in light wavefronts,” J. Mod. Opt. 39, 985–990 (1992).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (4)

W. Beijersbergen, L. Allen, H. E. L. O. van der Ween, J. P. Woerdman, “Astigmatic laser mode converter and transfer of angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

M. Harris, C. A. Hill, J. M. R. Vaughan, “Optical helices and spiral interference fringes,” Opt. Commun. 106, 161–166 (1994).
[CrossRef]

L. Allen, M. Babiker, W. L. Power, “Azimuthal Doppler shift in light beams with orbital angular momentum,” Opt. Commun. 112, 141–144 (1994).
[CrossRef]

J. Courtial, K. Dholakia, L. Allen, M. J. Padgett, “Gaussian beams with very high orbital angular momentum,” Opt. Commun. 144, 210–213 (1997).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (2)

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992).
[CrossRef] [PubMed]

M. S. Soskin, V. N. Vasnetsov, J. T. Malow, N. R. Heckenberg, “Topological charge and angular momentum of light beams carrying optical vortices,” Phys. Rev. A 56, 4064–4075 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

J. Courtial, K. Dholakia, D. A. Robertson, L. Allen, M. J. Padgett, “Measurement of the rotational frequency shift imparted to a rotating light beam possessing orbital angular momentum,” Phys. Rev. Lett. 80, 3217–3219 (1998).
[CrossRef]

Other (8)

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

The actual circulation of energy is determined by the Gouy phase term (2p + m + 1)ψ(z), as well as the mθ term, hence there is a z dependence on the rate of rotation of the field energy. However, in the far field, ψ(z) → π/2, a constant, where the phase advance is then determined by the mθ term alone.

M. W. Beijersbergen, M. Kristensen, J. P. Woerdman, “Spiral phase-plate used to produce helical wavefront laser beams,” in Conference on Lasers and Electro-Optics, 1994 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1994), paper CFA5.

M. S. Soskin, ed., International Conference on Singular Optics, Proc. SPIE3487 (1998).

R. J. Hull, D. G. Biron, S. Marcus, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-83-238/ESD-TR-83-72 (Lincoln Laboratory, Lexington, Mass., 1983).

D. U. Fluckiger, S. Marcus, R. J. Hull, J. H. Shapiro, “Coherent laser radar remote sensing,” Final Rep. MIT Lincoln Laboratory ESD-TR-84-306/ESE-TR-85-03 (Lincoln Laboratory, Lexington, Mass., 1984).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chaps. 3 and 4.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972), formula entry 9.1.80.

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