Abstract

A new, to our knowledge, technique for determining the modal content of partially coherent beams that are made up of an incoherent superposition of Hermite–Gaussian modes is studied. The algorithm makes use of the intensity profile of the beam at an arbitrarily chosen transverse plane. Analytical derivations are presented for a Gaussian Schell-model source and flat-topped beams, as well as an analysis of their performances in the presence of experimental errors and noise. Numerical simulations are performed to test the accuracy and the stability of the recovery algorithm.

© 1999 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
    [CrossRef]
  3. When two or more modes oscillate at the same frequency (i.e., in the case of degeneracy), the hypothesis of total noncorrelation has to be removed, and the treatment becomes more complex. See, for example, Ref. 2.
  4. 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]
  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. 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]
  7. 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]
  8. 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]
  9. R. Borghi, M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313–315 (1998).
    [CrossRef]
  10. F. Gori, M. Santarsiero, G. Guattari, “Coherence and space distribution of intensity,” J. Opt. Soc. Am. A 10, 673–679 (1993).
    [CrossRef]
  11. 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]
  12. 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]
  13. R. Gase, T. Gase, K. Blüthner, “Complex wave-field reconstruction by means of the Page distribution function,” Opt. Lett. 20, 2045–2047 (1995).
    [CrossRef] [PubMed]
  14. G. Iaconis, I. A. Walmsley, “Direct measurement of the two-point field correlation function,” Opt. Lett. 21, 1783–1785 (1996).
    [CrossRef] [PubMed]
  15. 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]
  16. F. Gori, “Shape-invariant propagation of the cross-spectral density,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 363.
  17. 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]
  18. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  19. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
    [CrossRef]
  20. E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
    [CrossRef]
  21. P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductors lasers,” Opt. Commun. 33, 265–270 (1980).
    [CrossRef]
  22. E. G. Johnson, “Direct measurements of the spatial mode of a laser pulse: theory,” Appl. Opt. 25, 2967–2975 (1986).
    [CrossRef]
  23. A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
    [CrossRef]
  24. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  25. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  26. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
    [CrossRef]
  27. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
    [CrossRef]
  28. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  29. C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
    [CrossRef]
  30. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  31. M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
    [CrossRef]
  32. R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
    [CrossRef]
  33. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).
  34. 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]
  35. R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed., revised (McGraw-Hill, New York, 1986).
  36. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.
  37. A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 2.

1999 (1)

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]

1998 (4)

1997 (1)

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

1996 (3)

1995 (3)

1994 (3)

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]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

A. T. Friberg, E. Tervonen, J. Turunen, “Interpretation and experimental demonstration of twisted Gaussian Schell-model beams,” J. Opt. Soc. Am. A 11, 1818–1826 (1994).
[CrossRef]

1993 (3)

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

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]

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)

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

1986 (1)

1984 (1)

1982 (1)

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
[CrossRef]

1980 (2)

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

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

1978 (1)

Abramowitz, M.

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

Agarwal, G. S.

Aiello, D.

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

Ambrosini, D.

Bagini, V.

Beck, M.

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]

Blüthner, K.

Borghi, R.

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]

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (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, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed., revised (McGraw-Hill, New York, 1986).

Brychkov, Yu. A.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.

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]

Collett, E.

Cutolo, A.

De Silvestri, S.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Friberg, A. T.

Gase, R.

Gase, T.

Gori, F.

Gradshtein, I. S.

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

Guattari, G.

Gureyev, T. E.

Iaconis, G.

Isernia, T.

Izzo, I.

Johnson, E. G.

Laporta, P.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

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]

Magni, V.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Mandel, L.

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

Marichev, O. I.

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.

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

McAlister, D. F.

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]

Nugent, K. A.

Pacileo, A. M.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

Pierri, R.

Prudnikov, A. P.

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

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.

Raymer, M. G.

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]

Roberts, A.

Ryzhik, I. M.

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

Saghafi, S.

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

Santarsiero, M.

Schirripa Spagnolo, G.

Sheppard, C. J. R.

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[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]

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

Spano, P.

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

Starikov, A.

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
[CrossRef]

Stegun, I.

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

Svelto, O.

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Tamura, S.

Tervonen, E.

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.

Vicalvi, S.

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

Walmsley, I. A.

Wolf, E.

E. Wolf, G. S. Agarwal, “Coherence theory of laser resonator modes,” J. Opt. Soc. Am. A 1, 541–546 (1984).
[CrossRef]

A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and of their radiation fields,” J. Opt. Soc. Am. A 72, 923–928 (1982).
[CrossRef]

E. Collett, E. Wolf, “Is complete coherence necessary for the generation of highly directional light beams?” Opt. Lett. 2, 27–29 (1978).
[CrossRef]

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
[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. (3)

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]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[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]

J. Mod. Opt. (1)

M. Santarsiero, D. Aiello, R. Borghi, S. Vicalvi, “Focusing of axially symmetric flattened Gaussian beams,” J. Mod. Opt. 44, 633–650 (1997).
[CrossRef]

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

Opt. Commun. (6)

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]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

C. J. R. Sheppard, S. Saghafi, “Flattened light beams,” Opt. Commun. 132, 144–152 (1996).
[CrossRef]

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

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

R. Borghi, M. Santarsiero, S. Vicalvi, “Focal shift of focused flat-topped beams,” Opt. Commun. 154, 243–248 (1998).
[CrossRef]

Opt. Lett. (6)

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).
[CrossRef] [PubMed]

Other (10)

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

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

F. Gori, “Shape-invariant propagation of the cross-spectral density,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds. (Plenum, New York, 1984), p. 363.

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

When two or more modes oscillate at the same frequency (i.e., in the case of degeneracy), the hypothesis of total noncorrelation has to be removed, and the treatment becomes more complex. See, for example, Ref. 2.

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

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

R. N. Bracewell, The Fourier Transform and its Applications, 2nd ed., revised (McGraw-Hill, New York, 1986).

A. P. Prudnikov, Yu. A. Brychkov, O. I. Marichev, Integrals and Series (Gordon, New York, 1986), Vol. 1.

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

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Figures (9)

Fig. 1
Fig. 1

Intensity distribution I(x) for a Gaussian Schell-model source with I 0 = 1 and σ I = 1.

Fig. 2
Fig. 2

FG intensity profiles for w 0 = 1, I 0 = 1, and several values of N.

Fig. 3
Fig. 3

Normalized coefficients c n for FG intensity profiles with several values of the order n.

Fig. 4
Fig. 4

Values of E n , as defined in Eq. (25), plotted versus orders of n.

Fig. 5
Fig. 5

(a) Intensity distribution for the case of additive noise with ∊ = 0.05 (solid curve) and for the noiseless case (dashed curve). (b) Average values of the real part of the expansion coefficients (filled circles) and the standard deviation (error bars). The parameters of the Gaussian Schell-model source are the same as for Fig. 1, with I 0 = 1 and x 0 = 0.05.

Fig. 6
Fig. 6

(a) Intensity distribution for the case of additive noise with ∊ = 0.05 (solid curve) and for the noiseless case (dashed curve). (b) Average values of the real part of the expansion coefficients (filled circles) and the standard deviation (error bars). The parameters of the FG partially coherent source are the same as for Fig. 2, with N = 10, w 0 = 1, I 0 = 1, and x 0 = 0.05.

Fig. 7
Fig. 7

Behavior of the mean-squared errors 〈|c n - n |2〉 plotted versus n. The values of the mean-squared errors were suitably normalized to make them comparable with those shown in Fig. 4 [See Eq. (24)]. The open circles represent the Gaussian Schell-model source of Fig. 5, whereas the filled triangles represent the partially coherent FG source of Fig. 6. The errors were evaluated for more than 1000 realizations of the random-noise process.

Fig. 8
Fig. 8

Average values of the real part of the expansion coefficients (filled circles) for the case of the Gaussian Schell-model source of Fig. 1 in the presence of uncertainty in the value of the spot size v 0. The open triangles (squares) denote the values obtained when an error of δv 0/v 0 = 0.05 (-0.05) is added to the spot size.

Fig. 9
Fig. 9

Average values of the real part of the expansion coefficients (filled circles) for the case of the FG partially coherent source of Fig. 2, with N = 10 and w 0 = 1, in the presence of uncertainty in the value of the spot size v 0. The open triangles (squares) denote the values obtained when an error of δv 0/v 0 = 0.05 (-0.05) is added to the spot size.

Equations (45)

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

Gnx; v0=2πv021/412nn!1/2 Hnx2v0exp-x2v02,
-+ Gn2x; v0dx=1,  n.
Ix=n=0 cnGn2x; v0,
Gn2x; v0p=Ψnπ2v02p2,
Ψnt=Lntexp-t2
0 ΨntΨmtdt=δn,m,
Ĩp=n=0 cnΨnπ2v02p2.
cn=2π2v020 ĨpΨnπ2v02p2pdp.
Ix=I0 exp-x22σI2,
Ĩp=I02πσI exp-2π2σI2p2.
cn=I02πσI0 Lntexp-t21+4σI2v02dt=I0π21/2v02σI1+v02σI21-v02σI21+v02σI2n, n=0, 1, 2, ,
c0=I0π21/2v02σI1+v02σI2,
q=1-v02σI21+v02σI2,
cn=c0qn,  n=0, 1, 2,.
Wx1, x2=I0 exp-x12+x224σI2exp-x1-x222σμ2,
1σμ2=14σI22σIv04-1.
Ix=I0 exp-N+1w02 x2n=0N1n!N+1w02 x2n,
v0=w02N+11/2.
Ĩp=I0-1Nπ22N+1N!p×H2N+1πw0pN+11/2exp-π2w02p2N+1.
cn=I0w0-1N22N-1N!2πN+11/2×0 H2N+1ξLn2ξ2exp-2ξ2dξ.
I¯x=Ix+Nx,
|cn-c¯n|2=4π4v0400 Ñ*pÑp×Ψnπ2v02p2Ψnπ2v02p2ppdpdp,
Ñ*pÑp=Aδp-p,
|cn-c¯n|2=4π4v04A 0 Ψn2π2v02p2p2dp=2πAv0En,
En=0tLn2texp-tdt
cn=2π2Fnv02,
Fnξ=ξ 0 ĨpLnπ2p2ξexp-π2p2ξ2pdp.
δcndcndv0δv0.
δcncnδv0v02ξFnξFnξξ=v02.
δcncnδv0v01-n cn-1cn+n+1cn+1cn.
δcncn2 δv0v0,  n.
δcncnδv0v01-nq+n+1q,
δcncnδv0v01-nq.
A=x0σN2,
Gn2x; v0p=-+ Gn2x; v0exp-i2πxpdx=2πv021/212nn! Hn2x2v0×exp-2 x2v02p,
Gn2x; v0p=12nn!π Hn2xexp-x2×pv0/2.
Hn2xexp-x2p=-+ Hn2xexp-x2×exp-i2πxpdx=2 0+ Hn2xexp-x2cos2πxpdx=2nn!πLn2π2p2exp-π2p2,
Gn2x; v0p=Lnπ2v02p2exp-π2v02p22,
Ix=I0UNN+11/2w0 x,
UNt=exp-t2n=0Nt2nn!.
Ĩp=I0w0N+11/2 ŨNw0N+11/2 p,
ŨNν=-+ UNtexp-i2πνtdt=n=0N1n!-+ t2n exp-t2exp-i2πνtdt.
ŨNν=π exp-π2ν2n=0N-1n22nn! H2nπν.
ŨNν=-1Nπ22N+1N!H2N+1πνν exp-π2ν2.
Ĩp=I0-1Nπ22N+1N!p×H2N+1πw0pN+11/2exp-π2w02p2N+1,

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