Abstract

The problem of recovering the coherence features of a partially coherent quasi-monochromatic scalar optical source, starting solely from intensity measurements on the emitted beam, is addressed in the most general way, under the paraxial approximation. In particular, it is shown that on expanding the beam emitted by the source as a bundle of partially correlated Hermite–Gaussian beams, the correlation coefficients can be recovered, in principle, simply by performing scalar products between transverse intensity distributions and suitably defined functions.

© 2003 Optical Society of America

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    [CrossRef]
  2. G. Hazak, “Comments on ‘Wave field determination using three-dimensional intensity information’,” Phys. Rev. Lett. 69, 2874 (1992).
    [CrossRef]
  3. F. Gori, M. Santarsiero, G. Guattari, “Coherence and space distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]
  4. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  5. 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]
  6. 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]
  7. A. Cutolo, T. Isernia, I. Izzo, R. Pierri, L. Zeni, “Transverse mode analysis of a laser beam by near- and far-field intensity measurements,” Appl. Opt. 34, 7974–7978 (1995).
    [CrossRef] [PubMed]
  8. J.-P. Hermier, A. Bramati, A. Z. Khoury, E. Giacobino, J.-Ph. Poizat, T. J. Chang, Ph. Grangier, “Spatial quantum noise of semiconductor lasers,” J. Opt. Soc. Am. B 16, 2140–2145 (1999).
    [CrossRef]
  9. C. L. Garrido Alzar, S. M. de Paula, M. Martinelli, R. J. Horowicz, A. Z. Khoury, G. A. Barbosa, “Transverse Fourier analysis of squeezed light in diode lasers,” J. Opt. Soc. Am. B 18, 1189–1195 (2001).
    [CrossRef]
  10. A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quan-titative optical phase microscopy,” Opt. Lett. 223, 817–819 (1998).
    [CrossRef]
  11. D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
    [CrossRef]
  12. G. Gbur, E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27, 1890–1892 (2002).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  14. M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73, 1434–1441 (1983).
    [CrossRef]
  15. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  16. 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]
  17. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  22. 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]
  23. R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” J. Quantum Electron. 35, 745–750 (1999).
    [CrossRef]
  24. X. Xue, A. G. Kirk, “Transverse modal characterization of VCSELs based on intensity measurement,” in Optoelectronic Interconnects VII; Photonics Packaging and Integration II, M. R. Feldman, R. L. Q. Li, W. B. Matkin, S. Tang, eds., Proc. SPIE3952, 144–153 (2000).
  25. M. Santarsiero, F. Gori, R. Borghi, “Modal weight determination for a class of multimode beams,” Proceedings of the 5th International Workshop on Laser Beams and Optics Characterization (Technische Universität Berlin, Berlin, Germany, 2000), pp. 161–170.
  26. 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]
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    [CrossRef]
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    [CrossRef]
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  31. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  32. H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 233–243.
  33. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.
  34. R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
    [CrossRef]

2002

2001

C. L. Garrido Alzar, S. M. de Paula, M. Martinelli, R. J. Horowicz, A. Z. Khoury, G. A. Barbosa, “Transverse Fourier analysis of squeezed light in diode lasers,” J. Opt. Soc. Am. B 18, 1189–1195 (2001).
[CrossRef]

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

2000

1999

1998

1995

1993

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]

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1993).
[CrossRef]

F. Gori, M. Santarsiero, G. Guattari, “Coherence and space distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

1992

G. Hazak, “Comments on ‘Wave field determination using three-dimensional intensity information’,” Phys. Rev. Lett. 69, 2874 (1992).
[CrossRef]

1989

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]

1983

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73, 1434–1441 (1983).
[CrossRef]

1974

Abramowitz, M.

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

Barbosa, G. A.

Barty, A.

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quan-titative optical phase microscopy,” Opt. Lett. 223, 817–819 (1998).
[CrossRef]

Borghi, R.

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

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

M. Santarsiero, F. Gori, R. Borghi, “Modal weight determination for a class of multimode beams,” Proceedings of the 5th International Workshop on Laser Beams and Optics Characterization (Technische Universität Berlin, Berlin, Germany, 2000), pp. 161–170.

Born, M.

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

Bramati, A.

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

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]

Chang, T. J.

Cutolo, A.

de Paula, S. M.

Eppich, B.

H. Laabs, B. Eppich, H. Weber, “Modal decomposition of partially coherent beams using the ambiguity function,” J. Opt. Soc. Am. A 19, 497–504 (2002).
[CrossRef]

B. Eppich, “Definition, meaning and measurements of coherence parameters,” in Laser Resonators IV, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE4270, 71 (2001).

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]

Gamo, H.

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 233–243.

Garrido Alzar, C. L.

Gbur, G.

Giacobino, E.

Gori, F.

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Grangier, Ph.

Guattari, G.

Gureyev, T. E.

Hazak, G.

G. Hazak, “Comments on ‘Wave field determination using three-dimensional intensity information’,” Phys. Rev. Lett. 69, 2874 (1992).
[CrossRef]

Hermier, J.-P.

Horowicz, R. J.

Isernia, T.

Izzo, I.

Khoury, A. Z.

Kirk, A. G.

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]

X. Xue, A. G. Kirk, “Transverse modal characterization of VCSELs based on intensity measurement,” in Optoelectronic Interconnects VII; Photonics Packaging and Integration II, M. R. Feldman, R. L. Q. Li, W. B. Matkin, S. Tang, eds., Proc. SPIE3952, 144–153 (2000).

Laabs, H.

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).

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

Martinelli, M.

Nugent, K. A.

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quan-titative optical phase microscopy,” Opt. Lett. 223, 817–819 (1998).
[CrossRef]

T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
[CrossRef]

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1993).
[CrossRef]

Paganin, D.

A. Barty, K. A. Nugent, D. Paganin, A. Roberts, “Quan-titative optical phase microscopy,” Opt. Lett. 223, 817–819 (1998).
[CrossRef]

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

Papoulis, A.

Pierri, R.

Piquero, G.

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

Poizat, J.-Ph.

Prudnikov, A. P.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

Roberts, A.

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

Santarsiero, M.

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

R. Borghi, M. Santarsiero, “Modal structure analysis for a class of axially symmetric flat-topped laser beams,” J. Quantum Electron. 35, 745–750 (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 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, G. Guattari, “Coherence and space distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
[CrossRef]

M. Santarsiero, F. Gori, R. Borghi, “Modal weight determination for a class of multimode beams,” Proceedings of the 5th International Workshop on Laser Beams and Optics Characterization (Technische Universität Berlin, Berlin, Germany, 2000), pp. 161–170.

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]

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

Stegun, I.

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

Tamura, S.

Teague, M. R.

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73, 1434–1441 (1983).
[CrossRef]

Tervonen, E.

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]

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]

Weber, H.

Wei, H.

Wolf, E.

G. Gbur, E. Wolf, “Diffraction tomography without phase information,” Opt. Lett. 27, 1890–1892 (2002).
[CrossRef]

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

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

Xue, X.

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]

X. Xue, A. G. Kirk, “Transverse modal characterization of VCSELs based on intensity measurement,” in Optoelectronic Interconnects VII; Photonics Packaging and Integration II, M. R. Feldman, R. L. Q. Li, W. B. Matkin, S. Tang, eds., Proc. SPIE3952, 144–153 (2000).

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.

Appl. Phys. B

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.

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]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Quantum Electron.

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

Opt. Commun.

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]

R. Borghi, G. Piquero, M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

D. Paganin, K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586–2589 (1998).
[CrossRef]

K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1993).
[CrossRef]

G. Hazak, “Comments on ‘Wave field determination using three-dimensional intensity information’,” Phys. Rev. Lett. 69, 2874 (1992).
[CrossRef]

Other

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

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

X. Xue, A. G. Kirk, “Transverse modal characterization of VCSELs based on intensity measurement,” in Optoelectronic Interconnects VII; Photonics Packaging and Integration II, M. R. Feldman, R. L. Q. Li, W. B. Matkin, S. Tang, eds., Proc. SPIE3952, 144–153 (2000).

M. Santarsiero, F. Gori, R. Borghi, “Modal weight determination for a class of multimode beams,” Proceedings of the 5th International Workshop on Laser Beams and Optics Characterization (Technische Universität Berlin, Berlin, Germany, 2000), pp. 161–170.

B. Eppich, “Definition, meaning and measurements of coherence parameters,” in Laser Resonators IV, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE4270, 71 (2001).

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

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).

H. Gamo, “Matrix treatment of partial coherence,” in Progress in Optics, Vol. III, E. Wolf, ed. (North-Holland, Amsterdam, 1964), pp. 233–243.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

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

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