Abstract

We analyze the coherence properties of a partially coherent field emerging from two pinholes in an opaque screen and show that the spectral degree of coherence possesses phase singularities on certain surfaces in the region of superposition. To our knowledge, this is the first illustration of the singular behavior of the spectral degree of coherence, and the results extend the field of singular optics to the study of phase singularities of correlation functions.

© 2003 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, 1999).
    [CrossRef]
  2. D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
    [CrossRef]
  3. D. F. V. James and E. Wolf, Opt. Commun. 145, 1 (1998).
    [CrossRef]
  4. H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, D. S. Metha, and K. C. Joshi, Phys. Lett. A 167, 114 (1992).
    [CrossRef]
  5. H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, and K. C. Joshi, Phys. Lett. A 167, 120 (1992).
    [CrossRef]
  6. V. N. Kumar and D. N. Rao, J. Mod. Opt. 48, 1455 (2001).
  7. L. Basan, P. Ottonelloa, G. Rottigni, and M. Vicari, Opt. Commun. 207, 77 (2002).
    [CrossRef]
  8. M. Santarsiero and F. Gori, Phys. Lett. A 167, 123 (1992).
    [CrossRef]
  9. H. F. Schouten, T. D. Visser, and E. Wolf, “New effects in Young’s interference experiment with light of any state of coherence,” Opt. Lett. (to be published).
  10. S. A. Ponomarenko and E. Wolf, Opt. Commun. 170, 1 (1999).
    [CrossRef]
  11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995).
    [CrossRef]

2002

L. Basan, P. Ottonelloa, G. Rottigni, and M. Vicari, Opt. Commun. 207, 77 (2002).
[CrossRef]

2001

V. N. Kumar and D. N. Rao, J. Mod. Opt. 48, 1455 (2001).

1999

S. A. Ponomarenko and E. Wolf, Opt. Commun. 170, 1 (1999).
[CrossRef]

1998

D. F. V. James and E. Wolf, Opt. Commun. 145, 1 (1998).
[CrossRef]

1992

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, D. S. Metha, and K. C. Joshi, Phys. Lett. A 167, 114 (1992).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, and K. C. Joshi, Phys. Lett. A 167, 120 (1992).
[CrossRef]

M. Santarsiero and F. Gori, Phys. Lett. A 167, 123 (1992).
[CrossRef]

1991

D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[CrossRef]

Basan, L.

L. Basan, P. Ottonelloa, G. Rottigni, and M. Vicari, Opt. Commun. 207, 77 (2002).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

Chander, M.

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, D. S. Metha, and K. C. Joshi, Phys. Lett. A 167, 114 (1992).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, and K. C. Joshi, Phys. Lett. A 167, 120 (1992).
[CrossRef]

Gori, F.

M. Santarsiero and F. Gori, Phys. Lett. A 167, 123 (1992).
[CrossRef]

James, D. F. V.

D. F. V. James and E. Wolf, Opt. Commun. 145, 1 (1998).
[CrossRef]

D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[CrossRef]

Joshi, K. C.

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, and K. C. Joshi, Phys. Lett. A 167, 120 (1992).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, D. S. Metha, and K. C. Joshi, Phys. Lett. A 167, 114 (1992).
[CrossRef]

Kandpal, H. C.

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, D. S. Metha, and K. C. Joshi, Phys. Lett. A 167, 114 (1992).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, and K. C. Joshi, Phys. Lett. A 167, 120 (1992).
[CrossRef]

Kumar, V. N.

V. N. Kumar and D. N. Rao, J. Mod. Opt. 48, 1455 (2001).

Mandel, L.

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

Metha, D. S.

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, D. S. Metha, and K. C. Joshi, Phys. Lett. A 167, 114 (1992).
[CrossRef]

Ottonelloa, P.

L. Basan, P. Ottonelloa, G. Rottigni, and M. Vicari, Opt. Commun. 207, 77 (2002).
[CrossRef]

Ponomarenko, S. A.

S. A. Ponomarenko and E. Wolf, Opt. Commun. 170, 1 (1999).
[CrossRef]

Rao, D. N.

V. N. Kumar and D. N. Rao, J. Mod. Opt. 48, 1455 (2001).

Rottigni, G.

L. Basan, P. Ottonelloa, G. Rottigni, and M. Vicari, Opt. Commun. 207, 77 (2002).
[CrossRef]

Santarsiero, M.

M. Santarsiero and F. Gori, Phys. Lett. A 167, 123 (1992).
[CrossRef]

Saxena, K.

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, D. S. Metha, and K. C. Joshi, Phys. Lett. A 167, 114 (1992).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, and K. C. Joshi, Phys. Lett. A 167, 120 (1992).
[CrossRef]

Schouten, H. F.

H. F. Schouten, T. D. Visser, and E. Wolf, “New effects in Young’s interference experiment with light of any state of coherence,” Opt. Lett. (to be published).

Vaishya, J. S.

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, and K. C. Joshi, Phys. Lett. A 167, 120 (1992).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, D. S. Metha, and K. C. Joshi, Phys. Lett. A 167, 114 (1992).
[CrossRef]

Vicari, M.

L. Basan, P. Ottonelloa, G. Rottigni, and M. Vicari, Opt. Commun. 207, 77 (2002).
[CrossRef]

Visser, T. D.

H. F. Schouten, T. D. Visser, and E. Wolf, “New effects in Young’s interference experiment with light of any state of coherence,” Opt. Lett. (to be published).

Wolf, E.

S. A. Ponomarenko and E. Wolf, Opt. Commun. 170, 1 (1999).
[CrossRef]

D. F. V. James and E. Wolf, Opt. Commun. 145, 1 (1998).
[CrossRef]

D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[CrossRef]

H. F. Schouten, T. D. Visser, and E. Wolf, “New effects in Young’s interference experiment with light of any state of coherence,” Opt. Lett. (to be published).

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

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

J. Mod. Opt.

V. N. Kumar and D. N. Rao, J. Mod. Opt. 48, 1455 (2001).

Opt. Commun.

L. Basan, P. Ottonelloa, G. Rottigni, and M. Vicari, Opt. Commun. 207, 77 (2002).
[CrossRef]

D. F. V. James and E. Wolf, Opt. Commun. 81, 150 (1991).
[CrossRef]

D. F. V. James and E. Wolf, Opt. Commun. 145, 1 (1998).
[CrossRef]

S. A. Ponomarenko and E. Wolf, Opt. Commun. 170, 1 (1999).
[CrossRef]

Phys. Lett. A

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, D. S. Metha, and K. C. Joshi, Phys. Lett. A 167, 114 (1992).
[CrossRef]

H. C. Kandpal, J. S. Vaishya, M. Chander, K. Saxena, and K. C. Joshi, Phys. Lett. A 167, 120 (1992).
[CrossRef]

M. Santarsiero and F. Gori, Phys. Lett. A 167, 123 (1992).
[CrossRef]

Other

H. F. Schouten, T. D. Visser, and E. Wolf, “New effects in Young’s interference experiment with light of any state of coherence,” Opt. Lett. (to be published).

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

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, 1999).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Illustration of the notation relating to Young’s interference experiment with partially coherent light.

Fig. 2
Fig. 2

Schematic illustration of surfaces on which points of observation P1 and P2 in the far zone are located for which μP1,P2,ω=0, i.e., at which the phase of μP1,P2,ω is singular. P1 and P2 lie on opposite cones.

Fig. 3
Fig. 3

Contours of equal phase of the spectral degree of coherence μP1,P2,ω near a singularity of its phase in a plane parallel to the screen. In this example k=0.333×107 m-1, d=0.1 cm, μ12=0.8+0.3i,r1=0,0,1.5m, and z2=1.5 m. (The significance of the points A–E will be discussed in connection with Fig. 5.)

Fig. 4
Fig. 4

Real and imaginary parts of the spectral degree of coherence μP1,P2,ω, with P1,y2, and z2 kept fixed while x2 is varied. y2=0.9 mm; all other parameters have the same value as in Fig. 3.

Fig. 5
Fig. 5

Spectral interference pattern formed along the x direction by combination of the light from pinholes P1 and P2 in a second Young’s interference experiment. The observation plane was taken to be at z=1.5 m, and the spacing of the pinholes was taken to be d=0.1 cm. The positions of points P2 (namely A–E) are illustrated in Fig. 3. S0 is a spectral intensity normalized by the value of the spectral intensity on the curve C. All other parameters are the same as in Fig. 3.

Equations (5)

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

WP1,P2,ω=kA2π2SωK11*K12+K21*K22+K11*K22μ12+K12K21*μ12*.
SPi,ω=kA2π22SωR1i21+μ12cosβ+kR2i-R1i,
μP1,P2,ω=WP1,P2,ωSP1,ωSP2,ω,
KijexpikRj-rˆj·diRj.
WP1,P2,ω=2kA2π2SωexpikR2-R1R1R2×coskd2cos θ1-cos θ2+μ12coskd2cos θ1+cos θ2+β,

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