Abstract

It is shown that a highly spatially incoherent light distribution may be generated from a highly coherent one on propagation in free space. This result essentially demonstrates that there exists an inverse of a classic result of optical coherence theory, namely, the van Cittert–Zernike theorem. The analysis also indicates that the technique of phase conjugation may be used to reverse changes in the coherence properties of light, at least those which are generated on propagation in free space.

© 1997 Optical Society of America

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References

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  1. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
    [CrossRef]
  2. J. R. Shewell and E. Wolf, J. Opt. Soc. Am. 58, 1596 (1968), Eqs.  (4.11) and (4.12). See also R. Mittra and P. L. Ransom, in Modern Optics, K. Fox, ed. (Polytechnic, Brooklyn, and Wiley, New York, 1967), p. 619.
    [CrossRef]
  3. By a free field we mean a field which can be represented by homogeneous plane-wave modes only. General fields in a half-space include also inhomogeneous evanescent waves, whose amplitudes decay exponentially with the distance of propagation. Free fields are usually excellent approximations to actual fields, except in the immediate neighborhood of scattering bodies (cf.  Ref.  1, Secs.  3.2.2 and 3.2.3).
  4. D. M. Pepper, Opt. Eng. 21, 156 (1982).
    [CrossRef]
  5. B. Ya Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).
    [CrossRef]
  6. R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), Sec.  6.1.
  7. G. S. Agarwal, A. T. Friberg, and E. Wolf, J. Opt. Soc. Am. 73, 529 (1983).
    [CrossRef]
  8. M. W. Bowers, R. W. Boyd, and A. K. Hankla, Opt. Lett. 22, 360 (1997).
    [CrossRef] [PubMed]

1997 (1)

1983 (1)

1982 (1)

D. M. Pepper, Opt. Eng. 21, 156 (1982).
[CrossRef]

1968 (1)

Agarwal, G. S.

Bowers, M. W.

Boyd, R. W.

M. W. Bowers, R. W. Boyd, and A. K. Hankla, Opt. Lett. 22, 360 (1997).
[CrossRef] [PubMed]

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), Sec.  6.1.

Friberg, A. T.

Hankla, A. K.

Mandel, L.

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

Pepper, D. M.

D. M. Pepper, Opt. Eng. 21, 156 (1982).
[CrossRef]

Pilipetsky, N. F.

B. Ya Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).
[CrossRef]

Shewell, J. R.

Shkunov, V. V.

B. Ya Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).
[CrossRef]

Wolf, E.

Ya Zel'dovich, B.

B. Ya Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

D. M. Pepper, Opt. Eng. 21, 156 (1982).
[CrossRef]

Opt. Lett. (1)

Other (4)

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

B. Ya Zel'dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, Berlin, 1985).
[CrossRef]

R. W. Boyd, Nonlinear Optics (Academic, New York, 1992), Sec.  6.1.

By a free field we mean a field which can be represented by homogeneous plane-wave modes only. General fields in a half-space include also inhomogeneous evanescent waves, whose amplitudes decay exponentially with the distance of propagation. Free fields are usually excellent approximations to actual fields, except in the immediate neighborhood of scattering bodies (cf.  Ref.  1, Secs.  3.2.2 and 3.2.3).

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Figures (1)

Fig. 1
Fig. 1

Schematic illustration of generation of a spatially highly incoherent distribution from a highly coherent one on propagation in free space. The highly coherent distribution may be generated, for example, in the plane 1, by propagating light from a highly spatially incoherent source in the plane 0 over a large distance d in free space (First stage). The field in the plane 1 is then phase conjugated (indicated in the figure by replacing 1 by 1+) and propagates another distance d to a plane 2. The field in the plane 2 will be spatially highly incoherent, being essentially identical with the original field in the field 0, except possibly for very small details (high-spatial-frequency components), which give rise to evanescent waves (whose amplitudes decay exponentially on propagation).

Equations (16)

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U2ρ, z0=U1ρ, z0*,
U2ρ, z0+d=U1ρ, z0-d*
U2*ρ1, z0U2ρ2, z0=U1*ρ1, z0U1ρ2, z0*,
U2*ρ1, z0+dU2ρ2, z0+d=U1*ρ1, z0-dU1ρ2, z0-d*,
W2ρ1, ρ2, z0=W1ρ1, ρ2, z0*,
W2ρ1, ρ2, z0+d=W1ρ1, ρ2, z0-d*
Iρ, z=Wρ, ρ, z,
μρ1, ρ2, z=Wρ1, ρ2, zW(ρ1, ρ1, z)Wρ2, ρ2, z,
I2ρ, z0=I1ρ, z0
μ2ρ1, ρ2, z0=μ1ρ1, ρ2, z0*,
I2ρ, z0+d=I1ρ, z0-d,
μ2ρ1, ρ2, z0+d=μ1ρ1, ρ2, z0-d*.
I2ρ, d=I1ρ, d,
μ2ρ1, ρ2, d=μ1ρ1, ρ2, d*,
I2ρ, 2d=I1ρ, 0,
μ2ρ1, ρ2, 2d=μ1ρ1, ρ2, 0*,

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