Abstract

It was recently shown that planar sources of entirely different states of coherence may generate fields that have the same intensity distribution throughout the far zone. With the help of a certain new correlation coefficient (which we call the coefficient of directionality), a new formulation of the underlying equivalence theorem is obtained which elucidates, in a very clear manner, the underlying physical principles involved. It is also shown that a quasihomogeneous source may always be specified that will generate the same far-zone intensity distribution as any given finite steady-state planar source. This result is illustrated by several examples. In particular a quasihomogeneous source is described that will generate the well-known Airy pattern intensity distribution.

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  1. E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?" Opt. Lett. 2, 27–29 (1978).
  2. 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).
  3. F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional beams," Opt. Commun. 27, 185–188 (1978).
  4. Footnote added in proofs: Since this paper was written the first experimental realization of such a source appears to have been achieved by P. De Santis, F. Gori, G. Guattari, and C. Palma, "An Example of a Collett-Wolf Source," Opt. Commun. (in press). Some of the theoretical predictions of Refs. 1 and 2 were also recently confirmed by experiments of J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from globally incoherent sources," submitted to Opt. Lett.
  5. E. Wolf, "The radiant intensity from planar sources of any state of coherence," J. Opt. Soc. Am. 68, 1597–1605 (1978).
  6. W. H. Carter, "Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density," Opt. Commun. 26, 1–4 (1978).
  7. See, for example, L. Mandel and E. Wolf, "Spectral Coherence and the Concept of Cross-Spectral Purity," J. Opt. Soc. Am. 66, 529–535 (1976).
  8. Superscript zero is used throughout this paper to denote some of the quantities defined on the source plane z = 0. The dependence of ν(r), W(0)(r1,r2), etc. on the frequency ω is not shown explicitly.
  9. Certain refinements that are needed in order to define the crossspectral density function in a mathematically rigorous manner are ignored here.
  10. The radiant intensity J(s) represents the rate at which energy, at the temporal frequency ω, is radiated per unit solid angle around the s direction. It is related to the optical intensity I(Rs)= 〈ν(Rs)-ν*(Rs)〉 at the point in the far zone, specified by the position vector Rs, by the formula (to be understood in the asymptotic sense as kR → ∞) [equation].
  11. Instead of requiring that the two sources have the same coefficient of directionality η(0).(r′), it is sufficient to demand, because |s⊥| ≤ 1, that the low-frequency (|f| < k) elements of the two-dimensional spatial Fourier transform η¯(0)(f) of η(0)(r′) are the same. However, when the sources are finite, as we assume, the two requirements are equivalent because of analytic properties of η˜(0)(f), that will be discussed in Sec. V.
  12. That the two versions of the theorem are equivalent can be seen at once with the help of the results given in footnote 4 of Ref. 4. [See also Ref. 5].
  13. To keep the calculations as simple as possible we ignore the fact that the laser source is finite. This procedure does not lead to any difficulties even though it formally contradicts an earlier assumption. In fact many of the results obtained in this paper hold also for fields generated by other planar sources of infinite extent, provided appropriate convergence conditions are satisfied.
  14. W. H. Carter and E. Wolf, "Coherence and radiometry with quasihomogeneous planar sources," J. Opt. Soc. Am. 67, 785–796 (1977).
  15. For the special case when the source is spatially fully coherent this result was also recently established by P. De Santis, F. Gori and C. Palma in a paper entitled "Generalized Collett-Wolf sources," Opt. Commun. 28, 151–155 (1979).
  16. This fact is an immediate consequence of the well-known result that the function 2J1(ξ)/ξ is an entire analytic function of the complex variable ξ [See, for example, E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford, Clarendon, 1935), p. 315].
  17. J. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 120.
  18. See, for example, R. R. Goldberg, Fourier Transforms (Cambridge University, Cambridge, England, 1965), Chap. 5.
  19. J. Peřina, Coherence of Light (Van Nostrand, London, 1972), Sec. 4.2.
  20. E. W. Marchand and E. Wolf, "Radiometry with sources of any state of coherence," J. Opt. Soc. Am. 64, 1219–1226 (1974).
  21. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

1979

For the special case when the source is spatially fully coherent this result was also recently established by P. De Santis, F. Gori and C. Palma in a paper entitled "Generalized Collett-Wolf sources," Opt. Commun. 28, 151–155 (1979).

1978

E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?" Opt. Lett. 2, 27–29 (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).

F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional beams," Opt. Commun. 27, 185–188 (1978).

E. Wolf, "The radiant intensity from planar sources of any state of coherence," J. Opt. Soc. Am. 68, 1597–1605 (1978).

W. H. Carter, "Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density," Opt. Commun. 26, 1–4 (1978).

1977

1976

1974

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

Carter, W. H.

W. H. Carter, "Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density," Opt. Commun. 26, 1–4 (1978).

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

Collett, E.

E. Collett and E. Wolf, "Is complete spatial coherence necessary for the generation of highly directional light beams?" Opt. Lett. 2, 27–29 (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).

De Santis, P.

For the special case when the source is spatially fully coherent this result was also recently established by P. De Santis, F. Gori and C. Palma in a paper entitled "Generalized Collett-Wolf sources," Opt. Commun. 28, 151–155 (1979).

Goldberg, R. R.

See, for example, R. R. Goldberg, Fourier Transforms (Cambridge University, Cambridge, England, 1965), Chap. 5.

Goodman, J. W.

J. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 120.

Gori, F.

For the special case when the source is spatially fully coherent this result was also recently established by P. De Santis, F. Gori and C. Palma in a paper entitled "Generalized Collett-Wolf sources," Opt. Commun. 28, 151–155 (1979).

F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional beams," Opt. Commun. 27, 185–188 (1978).

Footnote added in proofs: Since this paper was written the first experimental realization of such a source appears to have been achieved by P. De Santis, F. Gori, G. Guattari, and C. Palma, "An Example of a Collett-Wolf Source," Opt. Commun. (in press). Some of the theoretical predictions of Refs. 1 and 2 were also recently confirmed by experiments of J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from globally incoherent sources," submitted to Opt. Lett.

Guattari, G.

Footnote added in proofs: Since this paper was written the first experimental realization of such a source appears to have been achieved by P. De Santis, F. Gori, G. Guattari, and C. Palma, "An Example of a Collett-Wolf Source," Opt. Commun. (in press). Some of the theoretical predictions of Refs. 1 and 2 were also recently confirmed by experiments of J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from globally incoherent sources," submitted to Opt. Lett.

Mandel, L.

Marchand, E. W.

Palma, C.

For the special case when the source is spatially fully coherent this result was also recently established by P. De Santis, F. Gori and C. Palma in a paper entitled "Generalized Collett-Wolf sources," Opt. Commun. 28, 151–155 (1979).

F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional beams," Opt. Commun. 27, 185–188 (1978).

Footnote added in proofs: Since this paper was written the first experimental realization of such a source appears to have been achieved by P. De Santis, F. Gori, G. Guattari, and C. Palma, "An Example of a Collett-Wolf Source," Opt. Commun. (in press). Some of the theoretical predictions of Refs. 1 and 2 were also recently confirmed by experiments of J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from globally incoherent sources," submitted to Opt. Lett.

Perina, J.

J. Peřina, Coherence of Light (Van Nostrand, London, 1972), Sec. 4.2.

Santis, P. De

Footnote added in proofs: Since this paper was written the first experimental realization of such a source appears to have been achieved by P. De Santis, F. Gori, G. Guattari, and C. Palma, "An Example of a Collett-Wolf Source," Opt. Commun. (in press). Some of the theoretical predictions of Refs. 1 and 2 were also recently confirmed by experiments of J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from globally incoherent sources," submitted to Opt. Lett.

Wolf, E.

J. Opt. Soc. Am.

Opt. Commun.

For the special case when the source is spatially fully coherent this result was also recently established by P. De Santis, F. Gori and C. Palma in a paper entitled "Generalized Collett-Wolf sources," Opt. Commun. 28, 151–155 (1979).

W. H. Carter, "Radiant intensity from inhomogeneous sources and the concept of averaged cross-spectral density," Opt. Commun. 26, 1–4 (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).

F. Gori and C. Palma, "Partially coherent sources which give rise to highly directional beams," Opt. Commun. 27, 185–188 (1978).

Opt. Lett.

Other

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, 1975).

This fact is an immediate consequence of the well-known result that the function 2J1(ξ)/ξ is an entire analytic function of the complex variable ξ [See, for example, E. T. Copson, An Introduction to the Theory of Functions of a Complex Variable (Oxford, Clarendon, 1935), p. 315].

J. W. Goodman: Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 120.

See, for example, R. R. Goldberg, Fourier Transforms (Cambridge University, Cambridge, England, 1965), Chap. 5.

J. Peřina, Coherence of Light (Van Nostrand, London, 1972), Sec. 4.2.

Footnote added in proofs: Since this paper was written the first experimental realization of such a source appears to have been achieved by P. De Santis, F. Gori, G. Guattari, and C. Palma, "An Example of a Collett-Wolf Source," Opt. Commun. (in press). Some of the theoretical predictions of Refs. 1 and 2 were also recently confirmed by experiments of J. D. Farina, L. M. Narducci, and E. Collett, "Generation of highly directional beams from globally incoherent sources," submitted to Opt. Lett.

Superscript zero is used throughout this paper to denote some of the quantities defined on the source plane z = 0. The dependence of ν(r), W(0)(r1,r2), etc. on the frequency ω is not shown explicitly.

Certain refinements that are needed in order to define the crossspectral density function in a mathematically rigorous manner are ignored here.

The radiant intensity J(s) represents the rate at which energy, at the temporal frequency ω, is radiated per unit solid angle around the s direction. It is related to the optical intensity I(Rs)= 〈ν(Rs)-ν*(Rs)〉 at the point in the far zone, specified by the position vector Rs, by the formula (to be understood in the asymptotic sense as kR → ∞) [equation].

Instead of requiring that the two sources have the same coefficient of directionality η(0).(r′), it is sufficient to demand, because |s⊥| ≤ 1, that the low-frequency (|f| < k) elements of the two-dimensional spatial Fourier transform η¯(0)(f) of η(0)(r′) are the same. However, when the sources are finite, as we assume, the two requirements are equivalent because of analytic properties of η˜(0)(f), that will be discussed in Sec. V.

That the two versions of the theorem are equivalent can be seen at once with the help of the results given in footnote 4 of Ref. 4. [See also Ref. 5].

To keep the calculations as simple as possible we ignore the fact that the laser source is finite. This procedure does not lead to any difficulties even though it formally contradicts an earlier assumption. In fact many of the results obtained in this paper hold also for fields generated by other planar sources of infinite extent, provided appropriate convergence conditions are satisfied.

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