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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References

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  1. K. A. Nugent, “Wave field determination using three-dimensional intensity information,” Phys. Rev. Lett. 68, 2261–2264 (1993).
    [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]
  18. 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]
  19. R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
    [CrossRef]
  20. 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]
  21. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  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).
    [CrossRef]
  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]
  27. 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]
  28. A. Papoulis, “Ambiguity function in Fourier optics,” J. Opt. Soc. Am. 64, 779 (1974).
    [CrossRef]
  29. B. Eppich, “Definition, meaning and measurements of coherence parameters,” in Laser Resonators IV, A. V. Kudryashov, A. H. Paxton, eds., Proc. SPIE4270, 71 (2001).
    [CrossRef]
  30. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  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 (2)

2001 (2)

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

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]

2000 (1)

1999 (3)

1998 (5)

1995 (2)

1993 (4)

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]

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]

1992 (1)

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

1989 (2)

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 (1)

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

1974 (1)

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).
[CrossRef]

Friberg, A. T.

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]

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).
[CrossRef]

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.

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]

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]

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, 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.

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]

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.

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]

Weber, H.

Wei, H.

Wolf, E.

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

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

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

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).
[CrossRef]

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]

IEEE J. Quantum Electron. (1)

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. (1)

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

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

J. Quantum Electron. (1)

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. (2)

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

Opt. Lett. (5)

Phys. Rev. Lett. (3)

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 (10)

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).

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).
[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.

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

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.

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Figures (3)

Fig. 1
Fig. 1

Plot of the degree of coherence ψ0(x1, x2), given by Eq. (32) (solid curve) and the approximated coherence ψ0(N,L)(x1, x2) given by Eq. (31), with L=0 (squares), L=1 (triangles), and L=2 (circles), with N=10. The value of x1 has been set to zero.

Fig. 2
Fig. 2

Behaviors of ψn(2l)(x), as functions of x, for some values of n and l.

Fig. 3
Fig. 3

Behaviors of ψn(2l+1)(x), as functions of x, for some values of n and l.

Equations (51)

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

Vz(x)=αexp(ikz)expik2R x2×n=0cnΦn(xα)exp(-inϕ).
α=1[1+(z/L¯)2]1/2=cos ϕ,
ϕ=arctan(z/L¯),
Φn(x)=2πv021/412nn! Hnx2v0exp-x2v02,
I(x, z)=Vz*(x)Vz(x)=αn,mcn*cmexp[i(n-m)ϕ]Φn(xα)Φm(xα).
Iˆ(x, ϕ)=n,mcn*cmexp[i(n-m)ϕ]Φn(x)Φm(x),
Iˆ(x, ϕ)=1cos ϕ Ixcos ϕ, L ¯tan ϕ.
I^even(x, ϕ)=Iˆ(x, ϕ)+Iˆ(-x, ϕ)2=2n=0l=011+δl,0Re{cn*cn+2l×exp(-2ilϕ)}Φn(x)Φn+2l(x),
I^odd(x, ϕ)=Iˆ(x, ϕ)-Iˆ(-x, ϕ)2=2n=0l=0Re{cn*cn+2l+1×exp[-i(2l+1)ϕ]}Φn(x)Φn+2l+1(x),
Re{cn*cmexp(-ikϕ)}=Mn,mcos(kϕ-αn,m),
Jk(x)=-π/2+π/2I^even(x, ϕ)exp(ikϕ)dϕ,keven-π/2+π/2I^odd(x, ϕ)exp(ikϕ)dϕ,kodd,
J2l(x)=2n=0l=011+δl,0 Mn,n+2lΦn(x)Φn+2l(x)×-π/2+π/2cos(2lϕ-αn,n+2l)exp(2ilϕ)dϕ.
-π/2+π/2cos(2lϕ-α)exp(2ilϕ)dϕ
=π2 [exp(iα)δl,l+exp(-iα)δl,lδl,0].
J2l(x)=πn=0cn*cn+2lΦn(x)Φn+2l(x)
Jk(x)=πn=0cn*cn+kΦn(x)Φn+k(x),
J˜k(p)=-+Jk(x)exp(-i2πxp)dx.
-+Φn(x)Φn+k(x)exp(-i2πxp)dx
=(-i)kΨn(k)(π2v02p2),
Ψn(k)(t)=n!(n+k)!1/2tk/2Ln(k)(t)exp-t2,
0Ψn(k)(π2v02p2)Ψn(k)(π2v02p2)d(π2v02p2)=δn,nδk,k.
cn*cn+k=(-1)lπ0J˜k(p)Ψn(k)(π2v02p2)d(π2v02p2),
W0(x1, x2)=2πw21/2exp-x12+x22+2a2w2×cosh2a x1+x2w.
W0(x1, x2)=n=0m=0cn*cmΦn*(x1)Φm(x2),
cn*cm=W0(x1, x2)Φn(x2)Φm*(x2)dx1dx2,
cn*cm=1n!m!1/2aw(n+m)×exp-a2w2[1+(-1)n+m].
I(x, z)=12(πw2/2)1/21[1+(z/L)2]1/2×exp-2(x-a)2w211+(z/L)2+exp-2(x+a)2w211+(z/L)2,
Iˆ(x, ϕ)=1cos ϕ Ixcos ϕ, L tan ϕ=12(πw2/2)1/2exp-2(x-a cos ϕ)2w2+exp-2(x+a cos ϕ)2w2.
F{Iˆ(x, ϕ)}=exp-π2w2p22cos(2πpa cos ϕ),
J˜2l(p)=20π/2F{Iˆ}cos(2lϕ)dϕ=(-1)lπ exp-π2w2p22J2l(2πpa),
cn*cn+2l=n!(n+2l)!1/20J2l2awxxlLn(2l)(x)×exp(-x)dx=1n!(n+2l)!1/2a2w2(n+l)exp-a2w2,
W0(x1, x2)=n=0l=0cn*cn+2l1+δl,0 [Φn(x1)Φn+2l(x2)+Φn(x2)Φn+2l(x1)].
μ0(N,L)(x1, x2)=W0(N,L)(x1, x2)[W0(N,L)(x1, x1)W0(N,L)(x2, x2)]1/2.
μ0(x1, x2)=W0(x1, x2)[W0(x1, x1)W0(x2, x2)]1/2=cosh2aw (x1+x2)cosh4ax1wcosh4ax2w1/2.
-+Φn(x)Φn+k(x)ψn(k)(x; v0)dx=δnn,
cn*cn+2l=1π-+J2l(x)ψn(2l)(x; v0)dx=1π-+-π/2+π/2I^even(x, ϕ)exp(2ilϕ)ψn(2l)×(x; v0)dxdϕ,
cn*cn+2l+1=1π-+J2l+1(x)ψn(2l+1)(x; v0)dx=1π-+-π/2+π/2I^odd(x, ϕ)×exp[i(2l+1)ϕ]×ψn(2l+1)(x; v0)dxdϕ,
0Ψn(2l)(π2v02p2)Ψn(2l)(π2v02p2)d(π2v02p2)=δn,n.
Ψn(2l)(π2v02p2)=2(-1)l0Φn(x)Φn+2l(x)cos(2πpx)dx,
ψn(2l)(x; v0)=(-1)l0cos(2πpx)×Ψn(2l)(π2v02p2)d(π2v02p2),
ψn(2l)(x; v0)=(-1)ln!(n+2l)!1/20cos2 xv0 ξξ2l×Ln(2l)(ξ2)exp(-ξ2/2)d(ξ2).
ψn(0)(x; v0)=4(-1)n+1n!2π×ReD-(n+1)2ixv0D-(n+1)2ixv0,
ψn(2l)(x; v0)=(-1)lm=0n+lαm(n, l)ψm(0)(x; v0),
αm(n, l)=0tlLn(2l)(t)Lm(t)exp(-t)dt.
Ψn(2l+1)(π2v02p2)=2(-1)l0Φn(x)Φn+2l+1(x)×sin(2πpx)dx,
ψn(2l+1)(x; v0)=(-1)ln!(n+2l+1)!1/2×0sin2 xv0 ξξ2l+1Ln(2l+1)(ξ2)×exp(-ξ2/2)d(ξ2).
ψn(1)(x; v0)=2n+10sin2 xv0 ξξ2×Ln(1)(ξ2)exp(-ξ2/2)d(ξ).
tLn(1)(t)=(n+1)[Ln(t)-Ln+1(t)];
ψn(1)(x; v0)=2(n+1)π1/2(-1)n+1×n! ImD-(n+1)22ixv0+(n+1)! ImD-(n+2)22ixv0,
ψn(2l+1)(x; v0)=(-1)ln!(n+2l+1)!1/2×m=0n+lβm(n, l)ψm(1)(x; v0),
βm(n, l)=1m+10tl+1Ln(2l+1)(t)Lm(1)(t)exp(-t)dt.

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