Abstract

The propagation of any general astigmatic laser beam through any paraxial optical system can be characterized to second order by a 4 × 4 matrix of second moments in the transverse spatial and angular coordinates, with this matrix having at most ten independent elements. We show how to interpret the two existing beam invariants that contain these elements in terms of an overall beam quality factor Meff4 and an intrinsic antigmatism factor a. We describe a general procedure for determining all ten parameters for any arbitrary beam by measuring only second-order moments of the beam irradiance in a single transversal plane after the beam has passed through at most four simple astigmatic optical systems. A number of such systems containing, in particular, only one or two rotating cylindrical lenses and a CCD camera are proposed and analyzed in more detail.

© 1994 Optical Society of America

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  1. H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
  2. J. A. Arnaud, H. Kogelnik, “Light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
    [CrossRef] [PubMed]
  3. J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
  4. S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
    [CrossRef]
  5. M. W. Sasnett, “Propagation of multimode laser beams—the M2factor,” in Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, Bristol, U.K., 1989), pp. 132–142.
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    [CrossRef]
  7. P. A. Belanger, “Beam propagation and the ABCD ray matrices,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef] [PubMed]
  8. H. Weber, “Propagation of the higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
    [CrossRef]
  9. F. Tappert, “Diffractive ray tracing of laser beams,” J. Opt. Soc. Am. 66, 1368–1373 (1976).
    [CrossRef]
  10. M. J. Bastiaans, “Wigner distribution function and its application to first order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
    [CrossRef]
  11. M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
    [CrossRef]
  12. R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  14. J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  15. M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).
  16. R. Martinez-Herero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
    [CrossRef]
  17. R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
    [CrossRef]
  18. M. R. Teague, “Image analysis via general theory of moments,” J. Opt. Soc. Am. 70, 920–930 (1980).
    [CrossRef]
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    [CrossRef]
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  23. V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984), Chap. I, pp. 1–150.
  24. J. Serna, G. Nemes, “Decoupling coherent Gaussian beams with general astigmatism,” Opt. Lett. 18, 1774–1776 (1993).
    [CrossRef] [PubMed]
  25. R. Simon, N. Mukunda, “Twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 10, 95–109 (1993).
    [CrossRef]
  26. V. Bagini, F. Gori, M. Santasiero, F. Frezza, G. Schettini, M. Richetti, G. Scirripa Spagnolo, “On a class of twisting beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 31–40.
  27. A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
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    [CrossRef]
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  31. G. Nemes, A. G. Kostenbauder, “Optical systems for rotating a beam,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 99–109.
  32. H. H. Arsenault, “A matrix representation for nonsymmetrical optical systems,” J. Opt. (Paris) 11, 87–91 (1980).
    [CrossRef]
  33. B. Macukow, H. H. Arsenault, “Matrix decompositions for nonsymmetrical optical systems,” J. Opt. Soc. Am. 73, 1360–1366 (1983).
    [CrossRef]
  34. J. Shamir, “Cylindrical lens systems described by operator algebra,” Appl. Opt. 18, 4195–4202 (1979).
    [CrossRef] [PubMed]
  35. M. Nemes, G. Nemes, “Optical quadrupole,” Romanian Patent No. 72789 (1977).
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    [CrossRef]
  37. H. Braunecker, O. Bryngdahl, B. Schnell, “Optical system for image rotation and magnification,” J. Opt. Soc. Am. 70, 137–141 (1980).
    [CrossRef]
  38. H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
    [CrossRef]

1994 (1)

1993 (5)

1992 (3)

R. Martinez-Herero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

H. Weber, “Propagation of the higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

J. A. Ruff, A. E. Siegman, “Single-pulse laser beam quality measurements using a CCD camera system,” Appl. Opt. 31, 4907–4909 (1992).
[CrossRef] [PubMed]

1991 (3)

1989 (1)

1988 (1)

1986 (1)

1985 (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

1983 (1)

1980 (4)

H. Braunecker, O. Bryngdahl, B. Schnell, “Optical system for image rotation and magnification,” J. Opt. Soc. Am. 70, 137–141 (1980).
[CrossRef]

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

H. H. Arsenault, “A matrix representation for nonsymmetrical optical systems,” J. Opt. (Paris) 11, 87–91 (1980).
[CrossRef]

M. R. Teague, “Image analysis via general theory of moments,” J. Opt. Soc. Am. 70, 920–930 (1980).
[CrossRef]

1979 (2)

1977 (1)

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

1976 (2)

1971 (1)

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

1970 (1)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).

1969 (1)

1965 (1)

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Arnaud, J. A.

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).

J. A. Arnaud, H. Kogelnik, “Light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
[CrossRef] [PubMed]

Arsenault, H. H.

B. Macukow, H. H. Arsenault, “Matrix decompositions for nonsymmetrical optical systems,” J. Opt. Soc. Am. 73, 1360–1366 (1983).
[CrossRef]

H. H. Arsenault, “A matrix representation for nonsymmetrical optical systems,” J. Opt. (Paris) 11, 87–91 (1980).
[CrossRef]

Bagini, V.

V. Bagini, F. Gori, M. Santasiero, F. Frezza, G. Schettini, M. Richetti, G. Scirripa Spagnolo, “On a class of twisting beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 31–40.

Bartelt, H. O.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bastiaans, M. J.

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

M. J. Bastiaans, “Application of the Wigner distribution function to partially coherent light,” J. Opt. Soc. Am. A 3, 1227–1238 (1986).
[CrossRef]

M. J. Bastiaans, “Wigner distribution function and its application to first order optics,” J. Opt. Soc. Am. 69, 1710–1716 (1979).
[CrossRef]

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

Belanger, P. A.

Braunecker, H.

Brenner, K.-H.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Bryngdahl, O.

Frezza, F.

V. Bagini, F. Gori, M. Santasiero, F. Frezza, G. Schettini, M. Richetti, G. Scirripa Spagnolo, “On a class of twisting beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 31–40.

Friberg, A. T.

Gori, F.

V. Bagini, F. Gori, M. Santasiero, F. Frezza, G. Schettini, M. Richetti, G. Scirripa Spagnolo, “On a class of twisting beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 31–40.

Guillemin, V.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984), Chap. I, pp. 1–150.

Kauderer, M. H.

Keren, E.

Kogelnik, H.

J. A. Arnaud, H. Kogelnik, “Light beams with general astigmatism,” Appl. Opt. 8, 1687–1693 (1969).
[CrossRef] [PubMed]

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

Kostenbauder, A. G.

G. Nemes, A. G. Kostenbauder, “Optical systems for rotating a beam,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 99–109.

Lavi, S.

Lohmann, A. W.

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Lysenko, W. P.

W. P. Lysenko, M. S. Overley, “Moments invariants for particle beams,” in “Linear Accelerators and Beam Optics Codes,” American Institute of Physics Conference Proceedings 177, Ch. R. Eminhizer, ed. (American Institute of Physics, New York, 1988), pp. 323–335.

Macukow, B.

Mandel, L.

Martinez-Herero, R.

R. Martinez-Herero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

Martinez-Herrero, R.

R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Mejias, P. M.

R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

R. Martinez-Herero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

J. Serna, R. Martinez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
[CrossRef]

Mukunda, N.

Neira, J. L. H.

R. Martinez-Herero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

Nemes, G.

J. Serna, G. Nemes, “Decoupling coherent Gaussian beams with general astigmatism,” Opt. Lett. 18, 1774–1776 (1993).
[CrossRef] [PubMed]

M. Nemes, G. Nemes, “Optical quadrupole,” Romanian Patent No. 72789 (1977).

G. Nemes, “Measuring and handling general astigmatic beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 325–358.

G. Nemes, A. G. Kostenbauder, “Optical systems for rotating a beam,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 99–109.

Nemes, M.

M. Nemes, G. Nemes, “Optical quadrupole,” Romanian Patent No. 72789 (1977).

Overley, M. S.

W. P. Lysenko, M. S. Overley, “Moments invariants for particle beams,” in “Linear Accelerators and Beam Optics Codes,” American Institute of Physics Conference Proceedings 177, Ch. R. Eminhizer, ed. (American Institute of Physics, New York, 1988), pp. 323–335.

Petrishchev, V. A.

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Prochaska, R.

Richetti, M.

V. Bagini, F. Gori, M. Santasiero, F. Frezza, G. Schettini, M. Richetti, G. Scirripa Spagnolo, “On a class of twisting beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 31–40.

Ruff, J. A.

Sanchez, M.

R. Martinez-Herero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

Santasiero, M.

V. Bagini, F. Gori, M. Santasiero, F. Frezza, G. Schettini, M. Richetti, G. Scirripa Spagnolo, “On a class of twisting beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 31–40.

Sasnett, M. W.

M. W. Sasnett, “Propagation of multimode laser beams—the M2factor,” in Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, Bristol, U.K., 1989), pp. 132–142.

Schettini, G.

V. Bagini, F. Gori, M. Santasiero, F. Frezza, G. Schettini, M. Richetti, G. Scirripa Spagnolo, “On a class of twisting beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 31–40.

Schnell, B.

Scirripa Spagnolo, G.

V. Bagini, F. Gori, M. Santasiero, F. Frezza, G. Schettini, M. Richetti, G. Scirripa Spagnolo, “On a class of twisting beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 31–40.

Serna, J.

Shamir, J.

Siegman, A. E.

J. A. Ruff, A. E. Siegman, “Single-pulse laser beam quality measurements using a CCD camera system,” Appl. Opt. 31, 4907–4909 (1992).
[CrossRef] [PubMed]

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).
[CrossRef]

Simon, R.

Sternberg, S.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984), Chap. I, pp. 1–150.

Sudarshan, E. C. G.

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Sundar, K.

Talanov, V. I.

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Tappert, F.

Teague, M. R.

Tervonen, E.

Turunen, J.

Vlasov, S. N.

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Weber, H.

R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

H. Weber, “Propagation of the higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

Wolf, E.

Appl. Opt. (4)

Bell Syst. Tech. J. (2)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).

H. Kogelnik, “Imaging of optical modes-resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).

J. Opt. (Paris) (1)

H. H. Arsenault, “A matrix representation for nonsymmetrical optical systems,” J. Opt. (Paris) 11, 87–91 (1980).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (7)

Opt. Acta (1)

M. J. Bastiaans, “A frequency-domain treatment of partial coherence,” Opt. Acta 24, 261–274 (1977).
[CrossRef]

Opt. Commun. (1)

H. O. Bartelt, K.-H. Brenner, A. W. Lohmann, “The Wigner distribution function and its optical production,” Opt. Commun. 32, 32–38 (1980).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (3)

H. Weber, “Propagation of the higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, S1027–S1049 (1992).
[CrossRef]

R. Martinez-Herero, P. M. Mejias, M. Sanchez, J. L. H. Neira, “Third- and fourth-order parametric characterization of partially coherent beams propagating through ABCD optical systems,” Opt. Quantum Electron. 24, S1021–S1026 (1992).
[CrossRef]

R. Martinez-Herrero, P. M. Mejias, H. Weber, “On the different definitions of laser beam moments,” Opt. Quantum Electron. 25, 423–428 (1993).
[CrossRef]

Optik (Stuttgart) (1)

M. J. Bastiaans, “Second-order moments of the Wigner distribution function in first-order optical systems,” Optik (Stuttgart) 88, 163–168 (1991).

Phys. Rev. A (1)

R. Simon, E. C. G. Sudarshan, N. Mukunda, “Anisotropic Gaussian Schell-model beams: passage through optical systems and associated invariants,” Phys. Rev. A 31, 2419–2434 (1985).
[CrossRef] [PubMed]

Radiophys. Quantum Electron. (1)

S. N. Vlasov, V. A. Petrishchev, V. I. Talanov, “Averaged description of wave beams in linear and nonlinear media (the method of moments),” Radiophys. Quantum Electron. 14, 1062–1070 (1971).
[CrossRef]

Other (8)

M. W. Sasnett, “Propagation of multimode laser beams—the M2factor,” in Physics and Technology of Laser Resonators, D. R. Hall, P. E. Jackson, eds. (Hilger, Bristol, U.K., 1989), pp. 132–142.

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1224, 2–14 (1990).
[CrossRef]

G. Nemes, “Measuring and handling general astigmatic beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 325–358.

V. Guillemin, S. Sternberg, Symplectic Techniques in Physics (Cambridge U. Press, Cambridge, 1984), Chap. I, pp. 1–150.

V. Bagini, F. Gori, M. Santasiero, F. Frezza, G. Schettini, M. Richetti, G. Scirripa Spagnolo, “On a class of twisting beams,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 31–40.

W. P. Lysenko, M. S. Overley, “Moments invariants for particle beams,” in “Linear Accelerators and Beam Optics Codes,” American Institute of Physics Conference Proceedings 177, Ch. R. Eminhizer, ed. (American Institute of Physics, New York, 1988), pp. 323–335.

G. Nemes, A. G. Kostenbauder, “Optical systems for rotating a beam,” in Proceedings of the Workshop on Laser Beam Characterization, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonzales-Urena, eds. (Sociedad Española de Optica, Madrid, 1993), pp. 99–109.

M. Nemes, G. Nemes, “Optical quadrupole,” Romanian Patent No. 72789 (1977).

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Figures (4)

Fig. 1
Fig. 1

Proposed measuring system. Plane (1) is the input plane at which the beam parameters are to be determined, and plane (2) is the output plane at which the CCD camera actually measures the transformed beam.

Fig. 2
Fig. 2

Example of a SA generalized thin lens (configuration in Subsection 4.A): (a) aligned SA type and (b) rotated SA type.

Fig. 3
Fig. 3

Example of a SA generalized thin lens (configuration in Subsection 4.B): (a) aligned SA type and (b) rotated SA type.

Fig. 4
Fig. 4

Another SA generalized thin lens (configuration in Subsection 4.C).

Equations (37)

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W ( r , Θ ; z ) k 2 + Γ ( r + r / 2 , r r / 2 ; z ) × exp ( ± j k r Θ ) d r .
I ( r ; z ) Γ ( r , r ; z ) = 1 ( 2 π ) 2 + W ( r , Θ ; z ) d Θ ,
p 0 = + I ( r ; z ) d r = 1 ( 2 π ) 2 + W ( r , Θ ; z ) d r d Θ .
f ¯ f = 1 p 0 + f ( x , y , u , υ ) W ( r , Θ ; z ) d r d Θ ,
P = RR T = [ x 2 ¯ x y ¯ x u ¯ x υ ¯ x y ¯ y 2 ¯ y u ¯ y υ ¯ x u ¯ y u ¯ u 2 ¯ u υ ¯ x υ ¯ y υ ¯ u υ ¯ υ 2 ¯ ] .
P = [ W 11 W 12 M 11 M 12 W 12 W 22 M 21 M 22 M 11 M 21 U 11 U 12 M 12 M 22 U 12 U 22 ] = [ W M M T U ] .
S = [ A B C D ] ,
R 2 = S R 1 ,
SJ S T = J ,
J = [ 0 I I 0 ] ,
P 2 = S P 1 S T ,
W 11 = W 22 , M 11 = M 22 , U 11 = U 22 , W 12 = M 12 = M 21 = U 12 = 0 .
W 12 = M 12 = M 21 = U 12 = 0 ,
M 12 = M 21 ,
W 12 W 11 W 22 = W 12 M 11 M 22 = U 12 U 11 U 22 = ( 1 2 ) tan ( 2 ϕ ) .
M 2 = 2 k ( det P ) 1 / 4 = 2 k ( W 11 U 11 M 11 2 ) 1 / 2 1 .
M x 2 = 2 k ( W 11 U 11 M 11 2 ) 1 / 2 1 ,
M y 2 = 2 k ( W 22 U 22 M 22 2 ) 1 / 2 1 .
M x 2 M y 2 = 4 k 2 ( det P ) 1 / 2 1 ,
det P det P 00 = 1 / ( 4 k 2 ) 2 .
T = ( W 11 U 11 M 11 2 ) + ( W 22 U 22 M 22 2 ) + 2 ( W 12 U 12 M 12 M 21 ) T 00 = 1 / 2 k 2 .
T ST = ( 1 / 2 k 2 ) ( M 2 ) 2 ,
T ASA = ( 1 / 4 k 2 ) [ ( M x 2 ) 2 + ( M y 2 ) 2 ] .
M eff 4 [ ( det P / ( det P 00 ) ] 1 / 2 = 4 k 2 ( det P ) 1 / 2 1 ,
t T / T 00 = 2 k 2 T 1 .
( M eff 4 ) ST = ( M 2 ) 2 ,
t ST = ( M 2 ) 2 = ( M eff 4 ) ST ,
( M eff 4 ) ASA = M x 2 M y 2 ,
t ASA = ( 1 / 2 ) [ ( M x 2 ) 2 + ( M y 2 ) 2 ] .
a t M eff 4 0 .
S L ( g , ϕ ) = [ I 0 G g , ϕ I ] ,
G g , ϕ = [ g cos 2 ϕ ( 1 / 2 ) g sin ( 2 ϕ ) ( 1 / 2 ) g sin ( 2 ϕ ) g sin 2 ϕ ] .
S F = [ I d I 0 I ] .
S = [ I + d G 2 d I + d 2 G G I + d G ] .
( W 11 ) I = x 2 ¯ , ( W 12 ) I = x y ¯ 2 f x υ ¯ , ( W 22 ) I = y 2 ¯ + 4 f y υ ¯ + 4 f 2 υ 2 ¯ , ( W 11 ) II = f 2 u 2 ¯ , ( W 12 ) II = f 2 u υ ¯ , ( W 22 ) II = f 2 u 2 ¯ , ( W 11 ) III = x 2 ¯ + 4 f x u ¯ + 4 f 2 υ 2 ¯ , ( W 12 ) III = x y ¯ 2 f y u ¯ , ( W 22 ) III = y 2 ¯ , 4 ( W 11 ) IV = x 2 ¯ 2 f x u ¯ + f 2 u 2 ¯ + y 2 ¯ + 2 f y υ ¯ + f 2 υ 2 ¯ + 2 x y ¯ 2 f 2 u υ ¯ + 2 f x υ ¯ 2 f y u ¯ , 4 ( W 12 ) IV = x 2 ¯ f 2 u 2 ¯ y 2 ¯ 4 f y υ ¯ 3 f 2 υ 2 ¯ + 4 f 2 u υ ¯ 2 f x υ ¯ + 2 f y u ¯ , 4 ( W 22 ) IV = x 2 ¯ + 2 f x u ¯ + f 2 u 2 ¯ + y 2 ¯ + 6 f y υ ¯ + 9 f 2 υ 2 ¯ 2 x y ¯ 6 f 2 u υ ¯ 6 f x υ ¯ 2 f y u ¯ .
( W II ) I = x 2 ¯ , ( W 12 ) I = 3 x y ¯ 8 f x υ ¯ , ( W 22 ) I = 9 y 2 ¯ + 48 f y υ ¯ + 64 f 2 υ 2 ¯ , ( W 11 ) II = 4 f 2 u 2 ¯ , ( W 12 ) II = 12 f 2 u υ ¯ + 4 f y u ¯ , ( W 22 ) II = 4 y 2 ¯ + 24 f y υ ¯ + 36 f 2 υ 2 ¯ , ( W 11 ) III = 9 x 2 ¯ + 48 f x u ¯ + 64 f 2 u 2 ¯ , ( W 12 ) III = 3 x y ¯ 8 f y u ¯ , ( W 22 ) III = y 2 ¯ , ( W 11 ) IV = x 2 ¯ + 8 f x u ¯ + 16 f 2 u 2 ¯ + 4 y 2 ¯ + 16 f y υ ¯ + 16 f 2 υ 2 ¯ 4 x y ¯ 32 f 2 u υ ¯ 8 f x υ ¯ 16 f y u , ¯ ( W 12 ) IV = 2 x 2 ¯ 12 f x u ¯ 16 f 2 u 2 ¯ 2 y 2 ¯ 12 f y υ ¯ 16 f 2 υ 2 ¯ + 5 x y ¯ + 32 f 2 u υ ¯ + 12 f x υ ¯ + 12 f y u ¯ , ( W 22 ) IV = 4 x 2 ¯ + 16 f x u ¯ + 16 f 2 u 2 ¯ + y 2 ¯ + 8 f y υ ¯ + 16 f 2 υ 2 ¯ 4 x y ¯ 32 f 2 u υ ¯ 16 f x υ ¯ 8 f y u ¯ .
( W 11 ) I = x 2 ¯ , ( W 12 ) I = f x υ ¯ , ( W 22 ) I = f 2 υ 2 ¯ , 4 ( W 11 ) II = x 2 ¯ + 2 x y ¯ + y 2 ¯ + f 2 u 2 ¯ 2 f 2 u υ ¯ + f 2 υ 2 ¯ 2 f x u ¯ + 2 f y υ ¯ + 2 f x υ ¯ 2 f y u ¯ , 4 ( W 12 ) II = x 2 ¯ + 2 x y ¯ + y 2 ¯ f 2 u 2 ¯ + 2 f 2 u υ ¯ f 2 υ 2 ¯ , 4 ( W 22 ) II = x 2 ¯ + 2 x y ¯ + y 2 ¯ + f 2 u 2 ¯ 2 f 2 u υ ¯ + f 2 υ 2 ¯ + 2 f x u ¯ 2 f y υ ¯ 2 f x υ ¯ + 2 f y u ¯ , ( W 11 ) III = f 2 u 2 ¯ , ( W 12 ) III = f y u ¯ , ( W 22 ) III = y 2 ¯ , ( W 11 ) IV = x 2 ¯ + 4 f x u ¯ + 4 f 2 u 2 ¯ ( W 12 ) IV = x y ¯ + 2 f x υ ¯ + 2 f y u ¯ + 4 f 2 u υ ¯ , ( W 22 ) IV = y 2 ¯ + 4 f y υ ¯ + 4 f 2 υ 2 ¯ .

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