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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References

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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)

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]

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]

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)

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]

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)

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]

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]

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)

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

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

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).

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

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.

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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Figures (9)

Fig. 1
Fig. 1

Normalized far-field truncated Gaussian beam transmitter antenna pattern as a function of normalized transverse distance x for μ T = 0.5, 1.7, and 3.0 (solid curves). The two dashed curves are for untruncated Gaussians for μ T = 1.0 and 2.0.

Fig. 2
Fig. 2

Normalized far-field truncated Gaussian beam receiver antenna pattern as a function of normalized transverse distance x for μ L = 1.5 and γ = 0.5, 1.5, and 2.0.

Fig. 3
Fig. 3

Normalized far-field truncated LG beam transmitter antenna pattern as a function of normalized transverse distance x for μ T = 0.5, 1.7, and 3.0 (solid curves). The two dashed curves are for untruncated LG for μ T = 1.0 and 2.0.

Fig. 4
Fig. 4

Normalized far-field truncated Gaussian beam receiver antenna pattern as a function of normalized transverse distance x for μ L = 1.5 and γ = 0.5, 1.5, and 3.0.

Fig. 5
Fig. 5

All-Gaussian beam heterodyne speckle target mixing efficiency μ T = 1.7 as a function of normalized detector size γ for various values of LO truncation μ L . The mixing efficiency is close to 0.4 for modest values of truncation.

Fig. 6
Fig. 6

All-LG beam heterodyne speckle target mixing efficiency μ T = 1.2 as a function of normalized detector size γ for various values of LO truncation μ L . The mixing efficiency is close to 0.15 for modest values of truncation.

Fig. 7
Fig. 7

Gaussian LO and LG transmitter beam heterodyne speckle target mixing efficiency μ T = 1.4 as a function of normalized detector size γ for various values of LO truncation μ L . The detector is assumed to be (electrically) cut in half along a diameter. The difference signal from the two halves will contain the beat signal between the transmitter and the LO offset frequency.

Fig. 8
Fig. 8

Monstatic, shared-optics dual (LH and RH) LG beam ladar. The offset angle in the far field is given by δ. The cross talk is defined as the coherent speckle target mixing efficiency of beam 2 as can be seen in LO channel 1.

Fig. 9
Fig. 9

Loss for an all-LG beam and a Gaussian LO with a LG transmitter speckle target coherent mixing efficiency for μ T = μ L = 1.2, γ = 1.8 (all LG), μ T = μ L = 1.4, γ = 1.5 (Gaussian–LG) for the transmitter offset by x = D ρL from the boresight determined by the backpropagated LO beam normalized to the on-axis (x = 0) mixing efficiency.

Tables (1)

Tables Icon

Table 1 Detector Integration Regions (in θ) for Combinations of Gaussian and (LH or RH) LG beamsa

Equations (36)

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

ũpmr, θ, z=2p!1+δ0mπm+p!1/21ωz×expi2p+m+1ψz-ψ0×2rωzmLpm2r2ωz2×exp-ik r2ωz+imθ,
L0mx=1.
r=r,
θ=γθ+Ωt,
z=z,
t=γt+r2Ωθc2,
γ=11-r2Ω2c21/2,
phase=kz-ωt+mθ.
Δphaset=mΩ.
ξTρ=2πωT21/2 exp-|ρ|2ωT-2+ik/2R,
ξLρ=2πωDL21/2 exp-|ρ|2ωDL-2-ik/2l,
ωL=λlπωDL.
LξTρ=2πωT21/2|ρ|exp-|ρ|2ωT-2+ik/2R+iθ,
LξLρ=2πωDL21/2|ρ|exp-|ρ|2ωDL-2-ik/2l+iθ.
μT=4ωTD,
μL=4ωLD,
γ=Dd2λf,
hLρ, ρ=1iλLexpikL1+|ρ-ρ|2/2L2.
ξTρ=2ππωTλL002π ydydθ exp-πμTy/42J1πz2πz,
ξLρ=2ππωLλL0γ02π ydydθ exp-πμLy/42J1πz2πz1-exp-2πμLy/421/2,
LξTρ=2ππωTλLexpiθ002π ydydθ 2πμTy4exp-πμTy/42exp-iθJ1πz2πz,
LξLρ=2ππωLλLexpiθ0γ02π ydydθ 2πμTy4exp-πμLy/42expiθJ1πz2πz1-1+πμLγ28exp-2πμLγ/421/2.
ζTx=ξTρξTρmax2
02πdθ J1πz2πz=2πJ1πxπxJ1πyπy+3J3πxπxJ3πyπy+Λ,
02πdθ expiθJ1πz2πz=2π2 J2πxπxJ2πyπy+4J4πxπxJ4πyπy+Λ,
εhetS=4ARSπD2,
ARS=λL2  dρ|ξTρ|2|ξLρ|2
εhetS=2π2πμT22πμL220 xdx|ηTx|2|ηγLx|21-exp-πμLγ2/8,
ηsix=2 n=1,3,KnJnπxπx0SdyJnπy×exp-πμiy/42,
Lηsix=2 n=2,4,KnJnπxπx0SdyJnπy2πμiy/4×exp-πμiy/42,
εhetSx=2π2πμT22πμL22×02πdθ 0 xdx|η˜Tz|2|ηγLx|21-1+πμLγ28exp-πμLγ2/8,
η˜Tz=ηTx2+x2-2xx cosθ1/22π.
sigt  expi-θ+ω-ωAO+ωD-ωT+expi+θ+ω+ωAO+ωD+ωT,
LOt  expiθ+ωt
Q1=Q3  π2cosωAO+ωD+ωT-sinωAO-ωD+ωT, Q2=Q4  π2cosωAO+ωD+ωT+sinωAO-ωD+ωT, D1=Q1+Q2+Q3+Q4 2π cosωAO+ωD+ωT,
D1Q2+Q4-Q1-Q3 4π-sin2ωDt+sin2ωAO+ωT,

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