Abstract

It is shown that any partially polarized, partially coherent source can be expressed in terms of a suitable superposition of transverse coherent modes with orthogonal polarization states. Such modes are determined through the solution of a system of two coupled integral equations. An example, for which the modal decomposition is obtained in closed form in terms of fully linearly polarized Hermite Gaussian modes, is given.

© 2003 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  26. F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt., 7, 941–951 (1998).
    [CrossRef]
  27. F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 3, 1–9 (2001).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  35. F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
    [CrossRef]
  36. C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).
  37. F. Gori, “Collett–Wolf sources and multimode laser,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  38. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]
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    [CrossRef]

2002 (1)

2001 (2)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

J. Pu, B. Lu, “Focal shifts in focused nonuniformly polarized beams,” J. Opt. Soc. Am. A 18, 2760–2766 (2001).
[CrossRef]

2000 (2)

X. Xue, H. Wei, A. G. Kirk, “Intensity-based modal decomposition of optical beams in terms of Hermite–Gaussian functions,” J. Opt. Soc. Am. A 17, 1086–1091 (2000).
[CrossRef]

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 18–30 (2000).

1999 (3)

M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure for light beams with Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

1998 (8)

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

A. A. Tovar, “Production and propagation of cylindrically polarized Laguerre Gaussian laser beams,” J. Opt. Soc. Am. A 15, 2705–2711 (1998).
[CrossRef]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt., 7, 941–951 (1998).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “Intensity-based modal analysis for partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989–991 (1998).
[CrossRef]

F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
[CrossRef]

J. Tu, S. Tamura, “Analytic relation for recovering the mutual intensity by means of intensity information,” J. Opt. Soc. Am. A 15, 202–206 (1998).
[CrossRef]

1997 (2)

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spatial characterization of partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef]

1996 (1)

1995 (2)

1994 (2)

D. F. V. James, “Change of polarization of light beam on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

1993 (2)

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

1989 (2)

J. Turunen, E. Tervonen, A. T. Friberg, “Coherence theoretic algorithm to determine the transverse-mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
[CrossRef] [PubMed]

E. Tervonen, J. Turunen, A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

1988 (1)

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

1983 (1)

F. Gori, “Mode propagation of the field generated by Collet–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

1982 (2)

1980 (1)

F. Gori, “Collett–Wolf sources and multimode laser,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Abramowitz, M.

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

Berberian, S. K.

S. K. Berberian, Introduction to Hilbert Space (Oxford U. Press, Oxford, UK, 1961).

Blüthner, K.

Borghi, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure for light beams with Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “Intensity-based modal analysis for partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989–991 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt., 7, 941–951 (1998).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, “Modal-weight determination for a class of multimode beams,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Optisches Institut, Technische Universität Berlin, Berlin, 2000), pp. 161–170.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

Cai, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, Paris, 1977).

Cutolo, A.

Diu, B.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, Paris, 1977).

Encinas-Sanz, F.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 18–30 (2000).

F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
[CrossRef]

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

Eppich, B.

Friberg, A. T.

J. Turunen, E. Tervonen, A. T. Friberg, “Coherence theoretic algorithm to determine the transverse-mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
[CrossRef] [PubMed]

E. Tervonen, J. Turunen, A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

Gase, R.

Gase, T.

Gori, F.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure for light beams with Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “Intensity-based modal analysis for partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989–991 (1998).
[CrossRef]

F. Gori, “Matrix treatment for partially polarized, partially coherent beams,” Opt. Lett. 23, 241 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt., 7, 941–951 (1998).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

F. Gori, “Mode propagation of the field generated by Collet–Wolf Schell-model sources,” Opt. Commun. 46, 149–154 (1983).
[CrossRef]

F. Gori, “Collett–Wolf sources and multimode laser,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, “Modal-weight determination for a class of multimode beams,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Optisches Institut, Technische Universität Berlin, Berlin, 2000), pp. 161–170.

Greene, P. L.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Guattari, G.

M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure for light beams with Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt., 7, 941–951 (1998).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “Intensity-based modal analysis for partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989–991 (1998).
[CrossRef]

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

Hall, D. G.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Iaconis, G.

Isernia, T.

Izzo, I.

James, D. F. V.

D. F. V. James, “Change of polarization of light beam on propagation in free space,” J. Opt. Soc. Am. A 11, 1641–1643 (1994).
[CrossRef]

D. F. V. James, “Polarization of light radiated by black-body sources,” Opt. Commun. 109, 209–214 (1994).
[CrossRef]

Kirk, A. G.

Kreyszig, E.

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).

Laabs, H.

Laloe, F.

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, Paris, 1977).

Lu, B.

Lü, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Marti´nez, C.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 18–30 (2000).

F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
[CrossRef]

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

Marti´nez-Herrero, R.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 18–30 (2000).

F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
[CrossRef]

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spatial characterization of partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef]

Meji´as, P. M.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 18–30 (2000).

F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
[CrossRef]

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

R. Martı́nez-Herrero, P. M. Mejı́as, J. M. Movilla, “Spatial characterization of partially polarized beams,” Opt. Lett. 22, 206–208 (1997).
[CrossRef]

Mondello, A.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Movilla, J. M.

Olson, C.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

Palma, C.

F. Gori, G. Guattari, C. Palma, C. Padovani, “Specular cross-spectral density functions,” Opt. Commun. 68, 239–243 (1988).
[CrossRef]

Pierri, R.

Piquero, G.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Pu, J.

Rishton, S.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Santarsiero, M.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, G. Guattari, “Evaluation of the modal structure for light beams with Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “Intensity-based modal analysis for partially coherent beams with Hermite–Gaussian modes,” Opt. Lett. 23, 989–991 (1998).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
[CrossRef]

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt., 7, 941–951 (1998).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, “Modal-weight determination for a class of multimode beams,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Optisches Institut, Technische Universität Berlin, Berlin, 2000), pp. 161–170.

Serna, J.

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 18–30 (2000).

F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
[CrossRef]

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

Siegman, A. E.

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

Simon, R.

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

Starikov, A.

Stegun, I.

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

Tamura, S.

Tervonen, E.

E. Tervonen, J. Turunen, A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Coherence theoretic algorithm to determine the transverse-mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
[CrossRef] [PubMed]

Tovar, A. A.

Townsend, S. W.

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

Tu, J.

Turunen, J.

E. Tervonen, J. Turunen, A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

J. Turunen, E. Tervonen, A. T. Friberg, “Coherence theoretic algorithm to determine the transverse-mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
[CrossRef] [PubMed]

Vicalvi, S.

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt., 7, 941–951 (1998).
[CrossRef]

Walmsley, I. A.

Weber, H.

Wei, H.

Wicks, G. W.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Wolf, E.

Xue, X.

Yang, C.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Zeni, L.

Zhang, B.

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. B (1)

E. Tervonen, J. Turunen, A. T. Friberg, “Transverse laser-mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
[CrossRef]

Appl. Phys. Lett. (1)

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

IEEE J. Quantum Electron. (3)

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” IEEE J. Quantum Electron. 35, 745–750 (1999).
[CrossRef]

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

F. Encinas-Sanz, J. Serna, C. Martı́nez, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-varying beam quality factor and mode evolution in TEA CO2 laser pulses,” IEEE J. Quantum Electron. 34, 1835–1838 (1998).
[CrossRef]

J. Eur. Opt. Soc. A Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, S. Vicalvi, R. Borghi, G. Guattari, “Beam coherence-polarization matrix,” J. Eur. Opt. Soc. A Pure Appl. Opt., 7, 941–951 (1998).
[CrossRef]

J. Opt. Pure Appl. Opt. (1)

F. Gori, M. Santarsiero, G. Piquero, R. Borghi, A. Mondello, R. Simon, “Partially polarized Gaussian Schell-model beams,” J. Opt. Pure Appl. Opt. 3, 1–9 (2001).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Opt. Commun. (7)

B. Lü, B. Zhang, B. Cai, C. Yang, “A simple method for estimating the number of effectively oscillating modes and weighting factors of mixed mode laser beams behaving like Gaussian Schell-model beams,” Opt. Commun. 101, 49–52 (1993).
[CrossRef]

C. Martı́nez, F. Encinas-Sanz, J. Serna, P. M. Mejı́as, R. Martı́nez-Herrero, “On the parametric characterization of the transversal spatial structure of laser pulses,” Opt. Commun. 139, 299–305 (1997).
[CrossRef]

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[CrossRef]

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F. Gori, M. Santarsiero, R. Borghi, G. Guattari, “The irradiance of partially polarized beams in a scalar treatment,” Opt. Commun. 163, 159–163 (1999).
[CrossRef]

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[CrossRef]

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[CrossRef]

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Opt. Quantum Electron. (1)

C. Martı́nez, J. Serna, F. Encinas-Sanz, R. Martı́nez-Herrero, P. M. Mejı́as, “Time-resolved spatial structure of TEA CO2 laser pulses,” Opt. Quantum Electron. 32, 18–30 (2000).

Other (8)

M. Santarsiero, F. Gori, R. Borghi, “Modal-weight determination for a class of multimode beams,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Optisches Institut, Technische Universität Berlin, Berlin, 2000), pp. 161–170.

S. K. Berberian, Introduction to Hilbert Space (Oxford U. Press, Oxford, UK, 1961).

E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978).

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics (Wiley, Paris, 1977).

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

C. Brosseau, Fundamentals of Polarized Light (Wiley, New York, 1998).

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

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Equations (56)

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φ|Jˆ|ψ=ψ|Jˆ|φ*
φ|Jˆ|φ0
Jˆ=nλn|nn|,
m|n=δm,n,
Jˆ|n=λn|n,n=0, 1, 2, ,
xˆ|r=x|r,yˆ|r=y|r,
r1|r2=δ(2)(r1-r2),
 d2r|rr|=1.
J(r1, r2)=r1|Jˆ|r2.
Jˆ= d2r1d2r2J(r1, r2)|r1r2|.
Tr{Jˆ2}= d2r1d2r2|J(r1, r2)|2<.
J(r1, r2)=nλnΦn(r1)Φn*(r2),λn>0,
 d2rΦm*(r)Φn(r)=δm,n.
 d2r2J(r1, r2)Φn(r2)=λnΦn(r1);
cu|u+cv|v=αcα|α,
α; r1|β; r2=δα,βδ(2)(r1-r2),
α  d2r|α; rα; r|=1,
Jαβ(r1, r2)=α; r1|Jˆ|β; r2,
Jˆ=α,βd2r1d2r2Jαβ(r1, r2)|α; r1β; r2|.
J(r1, r2)=Juu(r1, r2)Juv(r1, r2)Jvu(r1, r2)Jvv(r1, r2).
Tr{Jˆ2}=α,β  d2r1d2r2|Jαβ(r1, r2)|2<,
φ|Jˆ|φ0|φC2L2(R2),
α,β  d2r1d2r2Jαβ(r1, r2)φα*(r1)φβ(r2)0
Jαβ(r1, r2)=nΛnΦn;α(r1)Φn;β*(r2),
α  d2rΦn;α(r)Φn;β*(r)=δm,n,
Φn(r)=Φn;u(r)Φn;v(r),
J(r1, r2)=nΛnΦn(r1)Φn(r2),
β  d2r2Jαβ(r1, r2)Φn;β(r2)=ΛnΦn;α(r1)(α, β=u, v),
J0(r1, r2)=Juu(r1, r2)00Jvv(r1, r2).
Juu(r1, r2)=nλnϕn(r1)ϕn*(r2),
Jvv(r1, r2)=nμnψn(r1)ψn*(r2),
Λ2n=λn,Φ2n(r)=ϕn(r)0,Λ2n+1=μn,Φ2n+1(r)=0ψn(r),n=0, 1, 2, ,
J0(x1, x2)=I0 exp-β2 (x12+x22)exp[-γ(x1-x2)2]exp[-γ(x1+x2)2]exp[-γ(x1+x2)2]exp[-γ(x1-x2)2],
Jeq(x1, x2)=2I0 exp-β2 (x12+x22)×exp[-γ(x1-x2)2],
g(x1, x2)=exp[-γ(x1+x2)2],
J0(x, x)=I0 exp(-βx2)1exp(-4γx2)exp(-4γx2)1.
s0=Jxx+Jyy=2I0 exp(-βx2),
s1=Jxx-Jyy=0,
s2=2 Re(Jxy)=2I0 exp[-(β+4γ)x2],
s3=2 Im(Jxy)=0,
P=(Jxx-Jyy)2+4|Jxy|2(Jxx+Jyy)21/2=exp(-4γx2).
J0(x1, x2)=Jξξ(x1, x2)00Jηη(x1, x2),
Jξξ(x1, x2)=I0 exp-β2 (x12+x22)×{exp[-γ(x1-x2)2]+exp[-γ(x1+x2)2]},Jηη(x1, x2)=I0 exp-β2 (x12+x22)×{exp[-γ(x1-x2)2]-exp[-γ(x1+x2)2]}.
J±(x1, x2)=I0 exp-β x12+x222-γ(x1±x2)2.
Jξξ(x1, x2)=J-(x1, x2)+J+(x1, x2),Jηη(x1, x2)=J-(x1, x2)-J+(x1, x2).
J-(x1, x2)=cπ1/2n=0 λ0qn2nn! Hn(x1c)Hn(x2c)×exp-c x12+x222,
c=2(β2+2βγ)1/2,q=γ/(β+γ+c),λ0=I0π(β+γ+c)1/2.
J+(x1, x2)=J-(x1, -x2)=cπ1/2n=0 λ0(-q)n2nn! Hn(x1c)×Hn(x2c)exp-c x12+x222,
Hn(-s)=(-1)nHn(s).
Jξξ(x1, x2)=2λ0cπ1/2n=0 q2n22n2n!×H2n(x1c)H2n(x2c)×exp-c x12+x222,Jηη(x1, x2)=2λ0cπ1/2n=0 q2n+122n+1(2n+1)!×H2n+1(x1c)H2n+1(x2c)×exp-c x12+x222,
An=2λ0qn,ϕn(x)=cπ1/4 122n2n! H2n(xc)exp-cx22,ψn(x)=cπ1/4 1[22n+1(2n+1)!]2 H2n+1(xc)×exp-cx22,
ψ|Jˆ|ψ= dx1dx2{[g1*(x1)g1(x2)+g2*(x1)g2(x2)]exp[-α(x1-x2)2]+[g1*(x1)g2(x2)+g2*(x1)g1(x2)]×exp[-α(x1+x2)2]},
exp[-α(x1±x2)2]=πα  du exp-π2u2α×exp[i2π(x1±x2)u],
ψ|Jˆ|ψ=πα  du exp-π2u2α×[g˜1*(u)g˜1(u)+g˜2*(u)g˜2(u)+g˜1*(u)g˜2(-u)+g˜2*(u)g˜1(-u)],
g˜j(u)=E˜j(u)+O˜j(u),j=1, 2,
ψ|Jˆ|ψ=πα  du exp-π2u2α[|E˜1(u)+E˜2(u)|2+|D˜1(u)-D˜2(u)|2],

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