Abstract

Phase-space representations of optical beams such as the ambiguity function or the Wigner distribution function have recently gained considerable importance for the characterization of coherent and partially coherent beams. There is growing interest in beam properties such as the beam propagation factor and the coherence and phase information that can be extracted from these phase-space representations. A method is proposed to decompose a partially coherent beam into Hermite–Gaussian modes by using the ambiguity function. The modal weights and the possible phase relations of the Hermite–Gaussian modes are retrieved. The method can also be applied for the decomposition of the Wigner distribution function. Some examples are discussed in the scope of beam characterization.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  41. H. Laabs, B. Ozygus, “Excitation of Hermite–Gaussian-modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
    [CrossRef]
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    [CrossRef]
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2000 (4)

1999 (4)

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

Z. Y. Wang, T. Chen, P. He, T. C. Zou, “Calculation of mode contents of high power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (1999).
[CrossRef]

H. Laabs, C. Gao, H. Weber, “Twisting of three-dimensional Hermite–Gaussian-beams,” J. Mod. Opt. 46, 709–719 (1999).

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

1998 (4)

1997 (1)

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

1996 (2)

H. Laabs, B. Ozygus, “Excitation of Hermite–Gaussian-modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

C. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
[CrossRef] [PubMed]

1995 (5)

1994 (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

1993 (1)

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from multimode stable-cavity laser,” IEEE J. Quantum Electron. 19, 1212–1217 (1993).
[CrossRef]

1992 (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1050 (1992).
[CrossRef]

1989 (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]

1984 (1)

K. H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

1981 (1)

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

1980 (1)

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[CrossRef]

1979 (1)

1946 (1)

H. J. Groenewold, “On the principles of elementary quantum mechanics,” Physica 12, 405ff (1946).
[CrossRef]

Agarwal, G. S.

Anderson, B. L.

C. M. Warnky, B. L. Anderson, C. A. Klein, “Determining spatial modes of lasers with spatial coherence measurements,” Appl. Opt. 39, 6109–6117 (2000).
[CrossRef]

L. J. Pelz, B. L. Anderson, “Practical use of the spatial coherence function for determining laser transverse mode structure,” Opt. Eng. 34, 3323–3328 (1995).
[CrossRef]

Bagini, V.

V. Bagini, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, M. Richetta, G. S. Spagmola, “On a class of twisting beams,” in Proceedings of the First Workshop on Laser Beam Characterization, Madrid, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonazales-Urena-Sociedad, eds. (Espanola de Optica, Madrid, 1993), pp. 31ff.

Bastiaans, M. J.

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

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

Beck, M.

Borghi, R.

Brenner, K. H.

K. H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Chen, T.

Z. Y. Wang, T. Chen, P. He, T. C. Zou, “Calculation of mode contents of high power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (1999).
[CrossRef]

Cheng, C. C.

C. C. Cheng, M. G. Raymer, H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[CrossRef]

Clarke, L.

Cutolo, A.

Eppich, B.

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity,” Opt. Laser Technol. 30, 337–240 (1998).
[CrossRef]

H. Laabs, B. Eppich, S. Johansson, H. Weber, “Determination of phase and coherence parameters from simple caustic measurements,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 120–128.

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Simultaneous determination of spatial phase and coherence properties by the measurement of the Wigner distribution,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 56–70.

B. Eppich, A. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

B. Eppich, N. Reng, “Measurement of the Wigner distribution function based in the inverse Radon transformation,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 261–268 (1995).
[CrossRef]

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Measuring laser beam parameters: phase and spatial coherence using the Wigner function,” in Laser Resonators, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE3930, 76–86 (2000).
[CrossRef]

Erhard, J.

J. Erhard, H. Laabs, B. Ozygus, H. Weber, “Diode-pumped multipath laser oscillators,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 2–10 (1999).
[CrossRef]

Frezza, F.

V. Bagini, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, M. Richetta, G. S. Spagmola, “On a class of twisting beams,” in Proceedings of the First Workshop on Laser Beam Characterization, Madrid, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonazales-Urena-Sociedad, eds. (Espanola de Optica, Madrid, 1993), pp. 31ff.

Friberg, A.

B. Eppich, A. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Friberg, A. T.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

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]

Gao, C.

H. Laabs, C. Gao, H. Weber, “Twisting of three-dimensional Hermite–Gaussian-beams,” J. Mod. Opt. 46, 709–719 (1999).

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity,” Opt. Laser Technol. 30, 337–240 (1998).
[CrossRef]

B. Eppich, A. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Gori, F.

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

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

V. Bagini, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, M. Richetta, G. S. Spagmola, “On a class of twisting beams,” in Proceedings of the First Workshop on Laser Beam Characterization, Madrid, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonazales-Urena-Sociedad, eds. (Espanola de Optica, Madrid, 1993), pp. 31ff.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Series and Products (Academic, San Diego, Calif., 1980).

Groenewold, H. J.

H. J. Groenewold, “On the principles of elementary quantum mechanics,” Physica 12, 405ff (1946).
[CrossRef]

Guattari, G.

He, P.

Z. Y. Wang, T. Chen, P. He, T. C. Zou, “Calculation of mode contents of high power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (1999).
[CrossRef]

Heier, H.

C. C. Cheng, M. G. Raymer, H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[CrossRef]

Iaconis, C.

Isernia, T.

Izzo, I.

Johansson, S.

H. Laabs, B. Eppich, S. Johansson, H. Weber, “Determination of phase and coherence parameters from simple caustic measurements,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 120–128.

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Measuring laser beam parameters: phase and spatial coherence using the Wigner function,” in Laser Resonators, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE3930, 76–86 (2000).
[CrossRef]

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Simultaneous determination of spatial phase and coherence properties by the measurement of the Wigner distribution,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 56–70.

Kebian, P. L.

Kirk, A. G.

Klein, C. A.

Laabs, H.

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

H. Laabs, C. Gao, H. Weber, “Twisting of three-dimensional Hermite–Gaussian-beams,” J. Mod. Opt. 46, 709–719 (1999).

H. Laabs, B. Ozygus, “Excitation of Hermite–Gaussian-modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Simultaneous determination of spatial phase and coherence properties by the measurement of the Wigner distribution,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 56–70.

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Measuring laser beam parameters: phase and spatial coherence using the Wigner function,” in Laser Resonators, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE3930, 76–86 (2000).
[CrossRef]

J. Erhard, H. Laabs, B. Ozygus, H. Weber, “Diode-pumped multipath laser oscillators,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 2–10 (1999).
[CrossRef]

H. Laabs, B. Eppich, S. Johansson, H. Weber, “Determination of phase and coherence parameters from simple caustic measurements,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 120–128.

Liesenhoff, A.

A. Liesenhoff, F. Ruehl, “An interferometric method of laser beam analysis,” Rev. Sci. Instrum. 38, 4059–4065 (1995).
[CrossRef]

Mayer, A.

McAlister, D. F.

McManus, J. B.

Ojeda-Castañeda, J.

K. H. Brenner, J. Ojeda-Castañeda, “Ambiguity function and Wigner distribution function applied to partially coherent imagery,” Opt. Acta 31, 213–223 (1984).
[CrossRef]

Ozygus, B.

H. Laabs, B. Ozygus, “Excitation of Hermite–Gaussian-modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

J. Erhard, H. Laabs, B. Ozygus, H. Weber, “Diode-pumped multipath laser oscillators,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 2–10 (1999).
[CrossRef]

Pelz, L. J.

L. J. Pelz, B. L. Anderson, “Practical use of the spatial coherence function for determining laser transverse mode structure,” Opt. Eng. 34, 3323–3328 (1995).
[CrossRef]

Pierri, R.

Raymer, M. G.

C. C. Cheng, M. G. Raymer, H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[CrossRef]

D. F. McAlister, M. Beck, L. Clarke, A. Mayer, M. G. Raymer, “Optical phase retrieval by phase-space tomography and fractional-order Fourier transforms,” Opt. Lett. 20, 1181–1183 (1995).
[CrossRef] [PubMed]

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
[CrossRef] [PubMed]

Reng, N.

B. Eppich, N. Reng, “Measurement of the Wigner distribution function based in the inverse Radon transformation,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 261–268 (1995).
[CrossRef]

Richetta, M.

V. Bagini, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, M. Richetta, G. S. Spagmola, “On a class of twisting beams,” in Proceedings of the First Workshop on Laser Beam Characterization, Madrid, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonazales-Urena-Sociedad, eds. (Espanola de Optica, Madrid, 1993), pp. 31ff.

Ruehl, F.

A. Liesenhoff, F. Ruehl, “An interferometric method of laser beam analysis,” Rev. Sci. Instrum. 38, 4059–4065 (1995).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Series and Products (Academic, San Diego, Calif., 1980).

Santarsiero, M.

Schettini, G.

V. Bagini, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, M. Richetta, G. S. Spagmola, “On a class of twisting beams,” in Proceedings of the First Workshop on Laser Beam Characterization, Madrid, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonazales-Urena-Sociedad, eds. (Espanola de Optica, Madrid, 1993), pp. 31ff.

Siegman, A. E.

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from multimode stable-cavity laser,” IEEE J. Quantum Electron. 19, 1212–1217 (1993).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Simon, R.

Spagmola, G. S.

V. Bagini, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, M. Richetta, G. S. Spagmola, “On a class of twisting beams,” in Proceedings of the First Workshop on Laser Beam Characterization, Madrid, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonazales-Urena-Sociedad, eds. (Espanola de Optica, Madrid, 1993), pp. 31ff.

Spano, P.

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[CrossRef]

Tamura, S.

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]

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

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]

Toft, P.

P. Toft, “The Radon transform—theory and implementation,” Ph.D. dissertation (Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark, 1996).

Townsend, S. W.

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from multimode stable-cavity laser,” IEEE J. Quantum Electron. 19, 1212–1217 (1993).
[CrossRef]

Tu, J.

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]

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

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]

Walmsley, I. A.

Wang, Z. Y.

Z. Y. Wang, T. Chen, P. He, T. C. Zou, “Calculation of mode contents of high power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (1999).
[CrossRef]

Warnky, C. M.

Weber, H.

H. Laabs, C. Gao, H. Weber, “Twisting of three-dimensional Hermite–Gaussian-beams,” J. Mod. Opt. 46, 709–719 (1999).

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity,” Opt. Laser Technol. 30, 337–240 (1998).
[CrossRef]

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1050 (1992).
[CrossRef]

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Simultaneous determination of spatial phase and coherence properties by the measurement of the Wigner distribution,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 56–70.

B. Eppich, A. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

J. Erhard, H. Laabs, B. Ozygus, H. Weber, “Diode-pumped multipath laser oscillators,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 2–10 (1999).
[CrossRef]

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Measuring laser beam parameters: phase and spatial coherence using the Wigner function,” in Laser Resonators, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE3930, 76–86 (2000).
[CrossRef]

H. Laabs, B. Eppich, S. Johansson, H. Weber, “Determination of phase and coherence parameters from simple caustic measurements,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 120–128.

Wei, H.

Xue, X.

Zahniser, M. S.

Zeni, L.

Zou, T. C.

Z. Y. Wang, T. Chen, P. He, T. C. Zou, “Calculation of mode contents of high power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (1999).
[CrossRef]

Appl. Opt. (4)

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]

IEEE J. Quantum Electron. (2)

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[CrossRef]

H. Laabs, A. T. Friberg, “Nonparaxial eigenmodes of stable resonators,” IEEE J. Quantum Electron. 35, 198–207 (1999).
[CrossRef]

J. Mod. Opt. (2)

H. Laabs, C. Gao, H. Weber, “Twisting of three-dimensional Hermite–Gaussian-beams,” J. Mod. Opt. 46, 709–719 (1999).

C. C. Cheng, M. G. Raymer, H. Heier, “A variable lateral-shearing Sagnac interferometer with high numerical aperture for measuring the complex spatial coherence of light,” J. Mod. Opt. 47, 1237–1246 (2000).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (2)

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[CrossRef]

M. J. Bastiaans, “The Wigner distribution function of partially coherent light,” Opt. Acta 28, 1215–1224 (1981).
[CrossRef]

Opt. Commun. (2)

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[CrossRef]

Z. Y. Wang, T. Chen, P. He, T. C. Zou, “Calculation of mode contents of high power CO2 laser beam according to the changes of transverse intensity distribution,” Opt. Commun. 175, 215–220 (1999).
[CrossRef]

Opt. Eng. (1)

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[CrossRef]

Opt. Laser Technol. (2)

H. Laabs, B. Ozygus, “Excitation of Hermite–Gaussian-modes in end-pumped solid-state lasers via off-axis pumping,” Opt. Laser Technol. 28, 213–214 (1996).
[CrossRef]

B. Eppich, C. Gao, H. Weber, “Determination of the ten second order intensity,” Opt. Laser Technol. 30, 337–240 (1998).
[CrossRef]

Opt. Lett. (5)

Opt. Quantum Electron. (1)

H. Weber, “Propagation of higher-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1050 (1992).
[CrossRef]

Phys. Rev. E (1)

J. Tu, S. Tamura, “Wave field determination using tomography of the ambiguity function,” Phys. Rev. E 55, 1946–1949 (1997).
[CrossRef]

Phys. Rev. Lett. (1)

M. G. Raymer, M. Beck, D. F. McAlister, “Complex wave-field reconstruction using phase space tomography,” Phys. Rev. Lett. 72, 1137–1140 (1994).
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Other (15)

H. Laabs, B. Eppich, S. Johansson, H. Weber, “Determination of phase and coherence parameters from simple caustic measurements,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 120–128.

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Simultaneous determination of spatial phase and coherence properties by the measurement of the Wigner distribution,” in Laser Beam and Optics Characterization, H. Weber, H. Laabs, eds. (Technische Universität Berlin, Optisches Institut, Berlin, 2000), pp. 56–70.

B. Eppich, N. Reng, “Measurement of the Wigner distribution function based in the inverse Radon transformation,” in Beam Control, Diagnostics, Standards, and Propagation, L. W. Austin, A. Giesen, D. H. Leslie, H. Weichel, eds., Proc. SPIE2375, 261–268 (1995).
[CrossRef]

B. Eppich, S. Johansson, H. Laabs, H. Weber, “Measuring laser beam parameters: phase and spatial coherence using the Wigner function,” in Laser Resonators, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE3930, 76–86 (2000).
[CrossRef]

“Lasers and laser-related equipment—Test methods for laser beam parameters—Beam positional stability,” 9 (1999).

“Optics and optical instruments–Lasers and laser-related equipment–Test methods for laser beam parameters: phase distribution,” (1999).

“Lasers and laser-related equipment—Test methods for laser beam parameters—Beam widths, divergence angle and beam propagation factor,” 9 (1999).

“Optics and optical instruments—Lasers and laser-related equipment—Test methods for laser beam power (energy) density distribution,” 9 (2000).

“Lasers and laser-related equipment—Test methods for laser beam parameters—Polarization,” 9 (1999).

V. Bagini, F. Gori, M. Santarsiero, F. Frezza, G. Schettini, M. Richetta, G. S. Spagmola, “On a class of twisting beams,” in Proceedings of the First Workshop on Laser Beam Characterization, Madrid, P. M. Mejias, H. Weber, R. Martinez-Herrero, A. Gonazales-Urena-Sociedad, eds. (Espanola de Optica, Madrid, 1993), pp. 31ff.

B. Eppich, A. Friberg, C. Gao, H. Weber, “Twist of coherent fields and beam quality,” in Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

I. S. Gradshteyn, I. M. Ryzhik, Tables of Series and Products (Academic, San Diego, Calif., 1980).

P. Toft, “The Radon transform—theory and implementation,” Ph.D. dissertation (Department of Mathematical Modelling, Technical University of Denmark, Lyngby, Denmark, 1996).

J. Erhard, H. Laabs, B. Ozygus, H. Weber, “Diode-pumped multipath laser oscillators,” in Laser Resonators II, A. V. Kudryashov, ed., Proc. SPIE3611, 2–10 (1999).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Complex magnitude of the ambiguity function (AF) (left) and the modal weights (right) of a beam composed by the incoherent superposition of a Hermite–Gaussian beam of orders 3 and 4 with equal modal weights. For a better visibility of outer structures, the plot of the AF is scaled to values ranging from 0 to 0.4, although the magnitude of the AF is normalized to a maximum value of 1. The wavelength is λ=1064 nm, and the Gaussian spot size is w0=200 μm.

Fig. 2
Fig. 2

Complex magnitude of the AF (left) and the modal weights (right) of a beam composed by the partially coherent superposition of a Hermite–Gaussian beam of orders 1–6. The detail about the scaling of the AF from Fig. 1 also applies to this figure.

Fig. 3
Fig. 3

Setup of the mode generator. The Nd:YAG crystal has cross-section dimensions 9.5 mm×4 mm and contains 1.1% Nd. The resonator length is 9 cm, and the radius of curvature of the spherical mirror is 25 cm.

Fig. 4
Fig. 4

Experimental example of an AF (left) and modal weights (right). The AF is plotted as the complex magnitude.

Equations (17)

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HGm(x, z)=2π1/4σ(z)*2mm!w01/2σ(z)*σ(z)m/2×Hm[2|σ(z)|2]1/2xw0×exp-σ(z)*xw02,
I(x, z)=mcn|HGn(x, z)|2,
J(ξ)=-π/2π/2[I(ξ, z)+I(-ξ, z)]dθGouy=2πncn|HGn(ξ, 0)|2,
dθGouy=dz/zR1+(z/zR)2.
cn=πw020J˜(p)ψn(π2w02 p2)pdp,
cn=πw02-π/2π/20[I˜(p, z)+I˜(-p, z)]×ψn(π2w02 p2)pdpdθGouy=πw02-π/2π/2-I˜(p, z)ψn(π2w02p2)|p|dpdθGouy.
cn=πw0202π0AF(r, θ)ψn(π2w02r2)rdrdθ.
E(x, z, t)=m,nEmn(x, z, t)=m,ncmn(t)exp(ιwmnt)HGm(x, z).
I(x, z)=m,nEmn(x, z, t)2t.
I(x, z)=m,n1,n2cmn1(t)cmn2*(t)exp[ı(ωmn1-ωmn2)t]t|HGm(x, z)|2+m1,m2,n1,n2m1m2cm1n1(t)cm2n2*(t)exp[ι(ωm1n1-ωm2n2)t]t×HGm1(x, z)HGm2*(x, z).
I(x, z)=mWm|HGm(x, z)|2+m1,m2m1m2Cm1,m2HGm1(x, z)HGm2*(x, z),
F{HGm1(ξ, z)HGm2*(ξ, z)}(u)=-HGm1(ξ, z)HGm2*(ξ, z)exp(-2ιπuξ)dξ=π/2ιm1-m2LGFm1m2-m1(πu, θGouy(z))ifm2m1ιm2-m1LGFm2m1-m2(πu,-θGouy(z))otherwise,
LGFpq(X, θ)=2q+1p!π(p+q)!1/2XqLpq(2X2)×exp(-X2)exp(-ιqθ).
002πLGFpq(X, θ)LGFrs*(X, θ)XdθdX=δp,rδq,s,
Wm=2/π5 002πAF(u, θ)LGFm0*(πu, θ)udθdu,
Cm1,m2=2/π5ιm1-m2 002πAF(u, θ)LGFm1m2-m1*(πu, θ)udθdu.
M2=n|Wm|2(2m+1)

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