Abstract

A recently formulated theory of partial coherence in the space-frequency domain is used to determine the mode structure of an important class of partially coherent sources and of the radiation fields generated by them. The effective number of modes is found to depend in a fundamental way on the ratio of the coherence length to the effective size of the source. The contribution of the effective modes to the far-field intensity is also analyzed.

© 1982 Optical Society of America

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  1. E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38, 3–6 (1981).
  2. E. Wolf, "New theory of partial coherence in the space-frequency domain. Part I: spectra and cross-spectra of steady-state sources," J. Opt. Soc. Am. 72, 343–351 (1982).
  3. H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and correlation properties of bounded planar sources," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441; W. H. Carter and M. Bertollotti, "An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals," J. Opt. Soc. Am. 68, 329–333 (1978); E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293–296 (1978); J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).
  4. F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301–305 (1980).
  5. L. Mandel and E. Wolf, "Spectral coherence and the concept of cross-spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).
  6. Actually a third case is possible: Eq. (2.1) may represent the cross-spectral density of the field across a primary source.
  7. To keep the notation as simple as possible, we have suppressed in Eqs. (2.7) and (2.8), and also in some subsequent equations, the explicit dependence of some of the quantities on the frequency ω.
  8. E. D. Rainville, Special Functions (Macmillan, New York, 1960), p. 198, Eq. (2).
  9. W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).
  10. A. Sommerfeld, Optics (Academic, New York, 1954), p. 275, Eq. (6) (corrected here for an error).
  11. Our argument leading from Eq. (4.1) to Eq. (4.5) is obviously appropriate when the source is finite. When the source is infinite, a more careful analysis is required, but it seems intuitively obvious that, in view of the exponential decrease of the mode functions øn (x′) with increasing |x′|, formula (4.5) will hold in the present case also.
  12. A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), pp. 98–101.
  13. H. Hochstadt, Integral Equations (Wiley, New York, 1973), p. 146, Eq. (4).
  14. W. H. Carter and E. Wolf, "Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources: II. Radiation from isotropic model sources," Opt. Acta 28, 245–259 (1981).
  15. H. Gamo, "Matrix treatment of partial coherence" in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–336, especially Sec. 4.4.
  16. R. Martínez-Herrero, "Expansion of the complex degree of coherence," Nuovo Cimento 54B, 205–210 (1979).
  17. E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6–17 (1978), Eqs. (53).

1982 (1)

1981 (2)

E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38, 3–6 (1981).

W. H. Carter and E. Wolf, "Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources: II. Radiation from isotropic model sources," Opt. Acta 28, 245–259 (1981).

1980 (1)

F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301–305 (1980).

1979 (1)

R. Martínez-Herrero, "Expansion of the complex degree of coherence," Nuovo Cimento 54B, 205–210 (1979).

1978 (1)

1977 (1)

1976 (1)

Antes, G.

H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and correlation properties of bounded planar sources," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441; W. H. Carter and M. Bertollotti, "An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals," J. Opt. Soc. Am. 68, 329–333 (1978); E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293–296 (1978); J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).

Baltes, H. P.

H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and correlation properties of bounded planar sources," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441; W. H. Carter and M. Bertollotti, "An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals," J. Opt. Soc. Am. 68, 329–333 (1978); E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293–296 (1978); J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).

Carter, W. H.

W. H. Carter and E. Wolf, "Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources: II. Radiation from isotropic model sources," Opt. Acta 28, 245–259 (1981).

W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).

Gamo, H.

H. Gamo, "Matrix treatment of partial coherence" in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–336, especially Sec. 4.4.

Gori, F.

F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301–305 (1980).

Hochstadt, H.

H. Hochstadt, Integral Equations (Wiley, New York, 1973), p. 146, Eq. (4).

Mandel, L.

Martínez-Herrero, R.

R. Martínez-Herrero, "Expansion of the complex degree of coherence," Nuovo Cimento 54B, 205–210 (1979).

Rainville, E. D.

E. D. Rainville, Special Functions (Macmillan, New York, 1960), p. 198, Eq. (2).

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, New York, 1954), p. 275, Eq. (6) (corrected here for an error).

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), pp. 98–101.

Steinle, B.

H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and correlation properties of bounded planar sources," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441; W. H. Carter and M. Bertollotti, "An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals," J. Opt. Soc. Am. 68, 329–333 (1978); E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293–296 (1978); J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).

Wolf, E.

E. Wolf, "New theory of partial coherence in the space-frequency domain. Part I: spectra and cross-spectra of steady-state sources," J. Opt. Soc. Am. 72, 343–351 (1982).

E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38, 3–6 (1981).

W. H. Carter and E. Wolf, "Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources: II. Radiation from isotropic model sources," Opt. Acta 28, 245–259 (1981).

E. Wolf, "Coherence and radiometry," J. Opt. Soc. Am. 68, 6–17 (1978), Eqs. (53).

W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).

L. Mandel and E. Wolf, "Spectral coherence and the concept of cross-spectral purity," J. Opt. Soc. Am. 66, 529–535 (1976).

J. Opt. Soc. Am. (4)

Nuovo Cimento (1)

R. Martínez-Herrero, "Expansion of the complex degree of coherence," Nuovo Cimento 54B, 205–210 (1979).

Opt. Acta (1)

W. H. Carter and E. Wolf, "Correlation theory of wavefields generated by fluctuating, three-dimensional, scalar sources: II. Radiation from isotropic model sources," Opt. Acta 28, 245–259 (1981).

Opt. Commun. (2)

E. Wolf, "New spectral representation of random sources and of the partially coherent fields that they generate," Opt. Commun. 38, 3–6 (1981).

F. Gori, "Collett-Wolf sources and multimode lasers," Opt. Commun. 34, 301–305 (1980).

Other (9)

H. P. Baltes, B. Steinle, and G. Antes, "Radiometric and correlation properties of bounded planar sources," in Coherence and Quantum Optics IV, L. Mandel and E. Wolf, eds. (Plenum, New York, 1978), pp. 431–441; W. H. Carter and M. Bertollotti, "An analysis of the far-field coherence and radiant intensity of light scattered from liquid crystals," J. Opt. Soc. Am. 68, 329–333 (1978); E. Wolf and E. Collett, "Partially coherent sources which produce the same far-field intensity distribution as a laser," Opt. Commun. 25, 293–296 (1978); J. T. Foley and M. S. Zubairy, "The directionality of Gaussian Schell-model beams," Opt. Commun. 26, 297–300 (1978).

Actually a third case is possible: Eq. (2.1) may represent the cross-spectral density of the field across a primary source.

To keep the notation as simple as possible, we have suppressed in Eqs. (2.7) and (2.8), and also in some subsequent equations, the explicit dependence of some of the quantities on the frequency ω.

E. D. Rainville, Special Functions (Macmillan, New York, 1960), p. 198, Eq. (2).

H. Gamo, "Matrix treatment of partial coherence" in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1964), Vol. 3, pp. 187–336, especially Sec. 4.4.

A. Sommerfeld, Optics (Academic, New York, 1954), p. 275, Eq. (6) (corrected here for an error).

Our argument leading from Eq. (4.1) to Eq. (4.5) is obviously appropriate when the source is finite. When the source is infinite, a more careful analysis is required, but it seems intuitively obvious that, in view of the exponential decrease of the mode functions øn (x′) with increasing |x′|, formula (4.5) will hold in the present case also.

A. Sommerfeld, Partial Differential Equations in Physics (Academic, New York, 1949), pp. 98–101.

H. Hochstadt, Integral Equations (Wiley, New York, 1973), p. 146, Eq. (4).

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