Abstract

We demonstrate in theory and experiment superluminal properties of optical field propagation in the shadow area behind an opaque disk (Poisson’s spot). The wave front of the field in the shadow zone is initially delayed with respect to the unperturbed field. This phase shift gradually diminishes along the optical path and therefore ensures variation of the phase velocity of the field at the axis. A resulting excess of about c×105 was detected.

© 2007 Optical Society of America

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References

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  1. M. Gouy, Acad. Sci., Paris, C. R. 110, 1251 (1890).
  2. P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
    [CrossRef]
  3. D. Mignai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000).
    [CrossRef]
  4. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
    [CrossRef]
  5. A. N. Khoroshun, M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, Opt. Commun. 271, 316 (2007).
    [CrossRef]
  6. A. Sommerfeld, Optics (Academic, 1954).
  7. R. L. Lucke, Eur. J. Phys. 27, 193 (2006).
    [CrossRef]
  8. A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity, Collected Papers (Dover, 1952).
  9. B. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge U. Press, 1997).

2007

A. N. Khoroshun, M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, Opt. Commun. 271, 316 (2007).
[CrossRef]

2006

R. L. Lucke, Eur. J. Phys. 27, 193 (2006).
[CrossRef]

2000

D. Mignai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000).
[CrossRef]

1997

P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

B. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge U. Press, 1997).

1987

1954

A. Sommerfeld, Optics (Academic, 1954).

1952

A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity, Collected Papers (Dover, 1952).

1890

M. Gouy, Acad. Sci., Paris, C. R. 110, 1251 (1890).

Born, B.

B. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge U. Press, 1997).

Durnin, J.

Einstein, A.

A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity, Collected Papers (Dover, 1952).

Gouy, M.

M. Gouy, Acad. Sci., Paris, C. R. 110, 1251 (1890).

Khoroshun, A. N.

A. N. Khoroshun, M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, Opt. Commun. 271, 316 (2007).
[CrossRef]

Lorentz, H. A.

A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity, Collected Papers (Dover, 1952).

Lucke, R. L.

R. L. Lucke, Eur. J. Phys. 27, 193 (2006).
[CrossRef]

Mignai, D.

D. Mignai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000).
[CrossRef]

Minkowski, H.

A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity, Collected Papers (Dover, 1952).

Pas'ko, V. A.

A. N. Khoroshun, M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, Opt. Commun. 271, 316 (2007).
[CrossRef]

Ranfagni, A.

D. Mignai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000).
[CrossRef]

Reivelt, K.

P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

Ruggeri, R.

D. Mignai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000).
[CrossRef]

Saari, P.

P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, Optics (Academic, 1954).

Soskin, M. S.

A. N. Khoroshun, M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, Opt. Commun. 271, 316 (2007).
[CrossRef]

Vasnetsov, M. V.

A. N. Khoroshun, M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, Opt. Commun. 271, 316 (2007).
[CrossRef]

Weyl, H.

A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity, Collected Papers (Dover, 1952).

Wolf, E.

B. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge U. Press, 1997).

Acad. Sci., Paris, C. R.

M. Gouy, Acad. Sci., Paris, C. R. 110, 1251 (1890).

Eur. J. Phys.

R. L. Lucke, Eur. J. Phys. 27, 193 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

A. N. Khoroshun, M. V. Vasnetsov, V. A. Pas'ko, and M. S. Soskin, Opt. Commun. 271, 316 (2007).
[CrossRef]

Phys. Rev. Lett.

P. Saari and K. Reivelt, Phys. Rev. Lett. 79, 4135 (1997).
[CrossRef]

D. Mignai, A. Ranfagni, and R. Ruggeri, Phys. Rev. Lett. 84, 4830 (2000).
[CrossRef]

Other

A. Sommerfeld, Optics (Academic, 1954).

A. Einstein, H. A. Lorentz, H. Minkowski, and H. Weyl, The Principle of Relativity, Collected Papers (Dover, 1952).

B. Born and E. Wolf, Principle of Optics, 7th ed. (Cambridge U. Press, 1997).

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Figures (4)

Fig. 1
Fig. 1

(a) Dependence of the phase shift Δ Φ ( N ) between input plane wave in free-space propagation and Poisson’s spot wave field on Fresnel number N. Dashed line shows linear dependence Δ Φ ( N ) = π N . (b) Relative phase velocity ν c as a function of Fresnel number. The geometric parameter α was chosen 10 3 .

Fig. 2
Fig. 2

(a) Calculated transversal intensity distribution at the distance z 0 = 30 cm behind an opaque disk with radius 700 μ m , illuminated with a monochromatic plane wave ( λ = 0.63 μ m ) . (b) Calculated profile of the wave front Φ ( ρ , z ) = const for the field at the same distance ( z = z z 0 ) . (c) Calculated interferogram showing the fringes bending at the center of the shadow area (scales are in millimeters).

Fig. 3
Fig. 3

(a) Experimental scheme. Laser beam is expanded and collimated with lenses ( L 1 , L 2 ). The reference plane wave is produced with a beam splitter (BS). An opaque dielectric disk (D) is placed on the rear face of the beam splitter. The mirrors ( M 1 , M 2 ) and glass wedge (W) compose an interferometer. The interference patterns on different distances are observed with a CCD camera and processed with the computer (C). (b) Microscopic view of the disk, with a scale ( 2 mm over the ruler). (c) Experimental interferogram at the distance 32 cm from the disk.

Fig. 4
Fig. 4

(a) Experimental dependence of the phase shift Δ Φ ( z ) , for wavelengths 0.63 μ m . The solid curve is the theoretical calculation. (b) Comparison of the experimentally obtained dependences [Eq. (8)], for wavelengths λ = 0.44 and 0.63 μ m with theoretically calculated phase velocity (thin curve).

Equations (8)

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E ( z ) = E 0 z z 2 + a 2 exp ( i k z 2 + a 2 ) ,
k z 2 + a 2 ω t = const .
ν = d z d t = c 1 + a 2 z 2 .
Δ Φ ( z ) = k z 2 + a 2 k z ,
Δ Φ ( N ) = 2 π ( 1 + α 2 N 2 1 ) α 2 N ,
ν ( N ) = c ( 1 + α 2 N 2 ) 1 2 ,
F d ( r , z ) + F o ( r , z ) = P ( z ) ,
ν exp ( z ) = c ( 1 B k z 2 ) 1 .

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