Abstract

The anomalous behavior of a polychromatic beam reflected from the interface of two homogeneous dielectric media at the neighborhood of Brewster’s angle is investigated theoretically and examined experimentally. An explicit expression is derived for the spectrum at an arbitrary observation point for a given incident spectrum. By introducing a modifier function it is shown that the spectrum at each point sensitively depends on the observation angle just in the immediate neighborhood of Brewster’s angle and wavelength. Finally, the obtained results are applied to account for the horizontal parts of the supernumerary rainbows.

© 2008 Optical Society of America

Full Article  |  PDF Article

Corrections

Mohammad Amiri and Mohammad Taghi Tavassoly, "Spectral anomalies near phase singularities in reflection at Brewster’s angle and colored catastrophes: publisher’s note," Opt. Lett. 33, 2571-2571 (2008)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-33-21-2571

References

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  1. J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
    [CrossRef]
  2. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.8.4.
  3. M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.
    [CrossRef]
  4. G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
    [CrossRef] [PubMed]
  5. G. Popescu and A. Dogariu, Phys. Rev. Lett. 8, 183902 (2002).
    [CrossRef]
  6. S. A. Ponomarenko and E. Wolf, Opt. Lett. 27, 211 (2002).
  7. G. Gbur, T. D. Visser, and E. Wolf, J. Opt. Soc. Am. A 19, 1694 (2002).
    [CrossRef]
  8. M. V. Berry, New J. Phys. 4, 66.1 (2002).
  9. T. D. Visser and E. Wolf, J. Opt. A 5, 371 (2003).
    [CrossRef]
  10. M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, Opt. Commun. 255, 23 (2005).
    [CrossRef]
  11. M. M. Brundavanam, N. K. Viswanathan, and N. Rao Desai, Opt. Lett. 32, 2279 (2007).
    [CrossRef] [PubMed]
  12. M. Amiri and M. T. Tavassoly, Opt. Commun. 272, 349 (2007).
    [CrossRef]
  13. R. L. Lee, Appl. Opt. 30, 3401 (1991).
    [CrossRef] [PubMed]
  14. M. V. Berry and A. N. Wilson, Appl. Opt. 33, 4714 (1994).
    [CrossRef] [PubMed]
  15. T. Poston and I. Stewart, Catastrophe Theory and Its Applications (Dover, 1996).
  16. M. V. Berry and S. Klein, Proc. Natl. Acad. Sci. USA 93, 2614 (1996).
    [CrossRef] [PubMed]
  17. R. L. Lee and A. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science (Pennsylvania State U. and SPIE, 2001).
  18. D. K. Lynch and W. Livingstone, Color and Light in Nature, 2nd ed. (Cambridge U. Press, 2001), Chap. 4.

2007 (2)

2005 (1)

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, Opt. Commun. 255, 23 (2005).
[CrossRef]

2003 (1)

T. D. Visser and E. Wolf, J. Opt. A 5, 371 (2003).
[CrossRef]

2002 (5)

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

G. Popescu and A. Dogariu, Phys. Rev. Lett. 8, 183902 (2002).
[CrossRef]

S. A. Ponomarenko and E. Wolf, Opt. Lett. 27, 211 (2002).

G. Gbur, T. D. Visser, and E. Wolf, J. Opt. Soc. Am. A 19, 1694 (2002).
[CrossRef]

M. V. Berry, New J. Phys. 4, 66.1 (2002).

1996 (1)

M. V. Berry and S. Klein, Proc. Natl. Acad. Sci. USA 93, 2614 (1996).
[CrossRef] [PubMed]

1994 (1)

1991 (1)

1974 (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Amiri, M.

M. Amiri and M. T. Tavassoly, Opt. Commun. 272, 349 (2007).
[CrossRef]

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, Opt. Commun. 255, 23 (2005).
[CrossRef]

Berry, M. V.

M. V. Berry, New J. Phys. 4, 66.1 (2002).

M. V. Berry and S. Klein, Proc. Natl. Acad. Sci. USA 93, 2614 (1996).
[CrossRef] [PubMed]

M. V. Berry and A. N. Wilson, Appl. Opt. 33, 4714 (1994).
[CrossRef] [PubMed]

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.8.4.

Brundavanam, M. M.

Dogariu, A.

G. Popescu and A. Dogariu, Phys. Rev. Lett. 8, 183902 (2002).
[CrossRef]

Fraser, A.

R. L. Lee and A. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science (Pennsylvania State U. and SPIE, 2001).

Gbur, G.

G. Gbur, T. D. Visser, and E. Wolf, J. Opt. Soc. Am. A 19, 1694 (2002).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Karimi, E.

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, Opt. Commun. 255, 23 (2005).
[CrossRef]

Khalesifard, H. R.

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, Opt. Commun. 255, 23 (2005).
[CrossRef]

Klein, S.

M. V. Berry and S. Klein, Proc. Natl. Acad. Sci. USA 93, 2614 (1996).
[CrossRef] [PubMed]

Lee, R. L.

R. L. Lee, Appl. Opt. 30, 3401 (1991).
[CrossRef] [PubMed]

R. L. Lee and A. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science (Pennsylvania State U. and SPIE, 2001).

Livingstone, W.

D. K. Lynch and W. Livingstone, Color and Light in Nature, 2nd ed. (Cambridge U. Press, 2001), Chap. 4.

Lynch, D. K.

D. K. Lynch and W. Livingstone, Color and Light in Nature, 2nd ed. (Cambridge U. Press, 2001), Chap. 4.

Nye, J. F.

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Ponomarenko, S. A.

S. A. Ponomarenko and E. Wolf, Opt. Lett. 27, 211 (2002).

Popescu, G.

G. Popescu and A. Dogariu, Phys. Rev. Lett. 8, 183902 (2002).
[CrossRef]

Poston, T.

T. Poston and I. Stewart, Catastrophe Theory and Its Applications (Dover, 1996).

Rao Desai, N.

Soskin, M. S.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.
[CrossRef]

Stewart, I.

T. Poston and I. Stewart, Catastrophe Theory and Its Applications (Dover, 1996).

Tavassoly, M. T.

M. Amiri and M. T. Tavassoly, Opt. Commun. 272, 349 (2007).
[CrossRef]

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, Opt. Commun. 255, 23 (2005).
[CrossRef]

Vasnetsov, M. V.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.
[CrossRef]

Visser, T. D.

T. D. Visser and E. Wolf, J. Opt. A 5, 371 (2003).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, J. Opt. Soc. Am. A 19, 1694 (2002).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

Viswanathan, N. K.

Wilson, A. N.

Wolf, E.

T. D. Visser and E. Wolf, J. Opt. A 5, 371 (2003).
[CrossRef]

G. Gbur, T. D. Visser, and E. Wolf, J. Opt. Soc. Am. A 19, 1694 (2002).
[CrossRef]

S. A. Ponomarenko and E. Wolf, Opt. Lett. 27, 211 (2002).

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.8.4.

Appl. Opt. (2)

J. Opt. A (1)

T. D. Visser and E. Wolf, J. Opt. A 5, 371 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

New J. Phys. (1)

M. V. Berry, New J. Phys. 4, 66.1 (2002).

Opt. Commun. (2)

M. T. Tavassoly, M. Amiri, E. Karimi, and H. R. Khalesifard, Opt. Commun. 255, 23 (2005).
[CrossRef]

M. Amiri and M. T. Tavassoly, Opt. Commun. 272, 349 (2007).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

G. Gbur, T. D. Visser, and E. Wolf, Phys. Rev. Lett. 88, 013901 (2002).
[CrossRef] [PubMed]

G. Popescu and A. Dogariu, Phys. Rev. Lett. 8, 183902 (2002).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

M. V. Berry and S. Klein, Proc. Natl. Acad. Sci. USA 93, 2614 (1996).
[CrossRef] [PubMed]

Proc. R. Soc. London Ser. A (1)

J. F. Nye and M. V. Berry, Proc. R. Soc. London Ser. A 336, 165 (1974).
[CrossRef]

Other (5)

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999), Sec. 8.8.4.

M. S. Soskin and M. V. Vasnetsov, in Progress in Optics, E.Wolf, ed. (Elsevier, 2001), Vol. 42, p. 219.
[CrossRef]

R. L. Lee and A. Fraser, The Rainbow Bridge: Rainbows in Art, Myth and Science (Pennsylvania State U. and SPIE, 2001).

D. K. Lynch and W. Livingstone, Color and Light in Nature, 2nd ed. (Cambridge U. Press, 2001), Chap. 4.

T. Poston and I. Stewart, Catastrophe Theory and Its Applications (Dover, 1996).

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Figures (3)

Fig. 1
Fig. 1

Schematic of the geometry and notation used for calculating the Fresnel–Kirchhoff integral and the spectral intensity at an arbitrary point on the screen ( n > 1 ) .

Fig. 2
Fig. 2

Plots of the normalized incident spectral intensity (dashed curves) for the central wavelength λ ¯ = 550 nm and the normalized spectral intensities S ( η , λ ) S 0 (solid curves) versus λ, which are obtained numerically in the immediate neighborhood of the Brewster’s angle for the five different relative angles.

Fig. 3
Fig. 3

CCD camera photographs of two adjacent different angles of a polarized (TM) white-light beam originated from a tungsten pencil lamp (OSRAM 500 W HALOLINE). The left picture corresponds to the angle near the Brewster’s angle, and the right picture corresponds to the uppermost angle ( η > 1 ) . These are diffracted from the air–water interface after being focused.

Equations (14)

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V ( r , λ , t ) = U ( r , λ ) exp ( i ω t ) ,
U ( r , λ ) = A ( λ ) r exp ( i k r ) ,
U P ( r , λ ) = K i λ A ( λ ) tan ( θ ϕ ) tan ( θ + ϕ ) exp [ i k ( r + r ) ] r r d x ,
r R + x 2 2 R x sin Θ ,
r R + x 2 2 R + x sin Θ ,
x = r sin θ R sin Θ ,
R ( sin θ sin Θ ) .
U P ( Θ , λ ) = U 0 ( λ ) exp [ i k ( R + R ) ] cos θ tan ( θ ϕ ) tan ( θ + ϕ ) × exp [ i π R ( R + R ) λ R ( sin θ sin Θ ) 2 ] d θ ,
= U 0 ( λ ) exp [ i k ( R + R ) ] [ C ( Θ , λ ) + i S ( Θ , λ ) ] ,
S ( P ) ( Θ , λ ) = U P ( Θ , λ ) 2 , = A ( λ ) 2 R K 2 R λ [ C 2 ( Θ , λ ) + S 2 ( Θ , λ ) ] .
S ( 0 ) ( λ ) = A ( λ ) 2 R ,
M ( Θ , λ ) = K 2 R λ [ C 2 ( Θ , λ ) + S 2 ( Θ , λ ) ] ,
S ( P ) ( Θ , λ ) = S ( 0 ) ( λ ) M ( Θ , λ ) .
S ( 0 ) ( λ ) = S 0 exp [ ( λ λ ¯ ) 2 2 σ 2 ] ,

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