Abstract

The evolution of a complex beam width parameter g describing focusing or defocusing of a paraxial Gaussian beam is considered as the cause of an additional topological (Berry) phase of the electromagnetic field associated with the beam. It is pointed out that the well-known Gouy phase is a special case of such a phase that should arise in general in all symplectic systems.

© 1995 Optical Society of America

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References

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  1. M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
    [CrossRef]
  2. B. Simon, Phys. Rev. Lett. 51, 2167 (1983).
    [CrossRef]
  3. R. Y. Chiao, C. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Chiao, Phys. Rev. Lett. 57, 193 (1986); M. V. Berry, Nature (London) 326, 277 (1987).
    [CrossRef] [PubMed]
  4. A. Shapere, F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989).
  5. S. Pancharatnam, Proc. Indian Acad. Sci. 44, 51 (1956) (reprinted in Ref. 4).
  6. S. Ramaseshan, R. Nityananda, Commun. Sci. (India) 55, 1225 (1986).
  7. J. Samuel, S. Bhandari, Phys. Rev. Lett. 66, 2339 (1988); S. Bhandari, J. Samuel, Phys. Rev. Lett. 60, 1211 (1988).
    [CrossRef]
  8. M. V. Berry, J. Mod. Opt. 34, 1401 (1987) (reprinted in Ref. 4).
    [CrossRef]
  9. R. M. A. Azzam, N. M. Bashare, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1977).
  10. H. Poincaré, Theorie Mathematique de la Lumière, Vol. 2 (Georges Cavé, Paris, 1892).
  11. L. G. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890); E. H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69, 827 (1956).
  12. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1989).
  13. H. Kogelnik, Appl. Opt. 4, 1562 (1965).
    [CrossRef]
  14. D. Marcuse, Light Transmission Optics(Van Nostrand Reinhold, New York, 1972); A. Yariv, Optical Electronics, 4th ed. (Harcourt Brace Jovanovich, Fort Worth, Tex., 1991), Chap. 2, p. 35.
  15. H. Schwerdtfeger, Geometry of Complex Numbers (Dover, New York, 1979), Chap. 12.
  16. D. Subbarao, R. Uma, H. Singh, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 97 (1993); R. Uma, H. Singh, D. Subbarao, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 109 (1993).
  17. D. Subbarao, M. S. Sodha, Self-Focusing Theory—A Perspective in Contemporary Plasma Physics (McMillan India, New Delhi, 1984), p. 249; J. Appl. Phys. 50, 4604 (1979); D. Subbarao, R. Uma, A. K. Ghatak, Laser Part. Beams 1, 383 (1983).
  18. C. L. Siegal, Topics in Complex Function Theory (Wiley Interscience, New York, 1973).
  19. R. Simon, N. Mukunda, Phys. Lett. A 138, 474 (1989); R. Bhandari, Phys. Lett. A 138, 469 (1986).
    [CrossRef]
  20. M. Martinelli, P. Vavassori, Opt. Commun. 80, 166 (1990).
    [CrossRef]

1993 (1)

D. Subbarao, R. Uma, H. Singh, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 97 (1993); R. Uma, H. Singh, D. Subbarao, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 109 (1993).

1990 (1)

M. Martinelli, P. Vavassori, Opt. Commun. 80, 166 (1990).
[CrossRef]

1989 (1)

R. Simon, N. Mukunda, Phys. Lett. A 138, 474 (1989); R. Bhandari, Phys. Lett. A 138, 469 (1986).
[CrossRef]

1988 (1)

J. Samuel, S. Bhandari, Phys. Rev. Lett. 66, 2339 (1988); S. Bhandari, J. Samuel, Phys. Rev. Lett. 60, 1211 (1988).
[CrossRef]

1987 (1)

M. V. Berry, J. Mod. Opt. 34, 1401 (1987) (reprinted in Ref. 4).
[CrossRef]

1986 (2)

R. Y. Chiao, C. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Chiao, Phys. Rev. Lett. 57, 193 (1986); M. V. Berry, Nature (London) 326, 277 (1987).
[CrossRef] [PubMed]

S. Ramaseshan, R. Nityananda, Commun. Sci. (India) 55, 1225 (1986).

1984 (1)

M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
[CrossRef]

1983 (1)

B. Simon, Phys. Rev. Lett. 51, 2167 (1983).
[CrossRef]

1965 (1)

1956 (1)

S. Pancharatnam, Proc. Indian Acad. Sci. 44, 51 (1956) (reprinted in Ref. 4).

1890 (1)

L. G. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890); E. H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69, 827 (1956).

Azzam, R. M. A.

R. M. A. Azzam, N. M. Bashare, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1977).

Bashare, N. M.

R. M. A. Azzam, N. M. Bashare, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1977).

Berry, M. V.

M. V. Berry, J. Mod. Opt. 34, 1401 (1987) (reprinted in Ref. 4).
[CrossRef]

M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
[CrossRef]

Bhandari, S.

J. Samuel, S. Bhandari, Phys. Rev. Lett. 66, 2339 (1988); S. Bhandari, J. Samuel, Phys. Rev. Lett. 60, 1211 (1988).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1989).

Chiao, R. Y.

R. Y. Chiao, C. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Chiao, Phys. Rev. Lett. 57, 193 (1986); M. V. Berry, Nature (London) 326, 277 (1987).
[CrossRef] [PubMed]

Gouy, L. G.

L. G. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890); E. H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69, 827 (1956).

Kogelnik, H.

Marcuse, D.

D. Marcuse, Light Transmission Optics(Van Nostrand Reinhold, New York, 1972); A. Yariv, Optical Electronics, 4th ed. (Harcourt Brace Jovanovich, Fort Worth, Tex., 1991), Chap. 2, p. 35.

Martinelli, M.

M. Martinelli, P. Vavassori, Opt. Commun. 80, 166 (1990).
[CrossRef]

Mukunda, N.

R. Simon, N. Mukunda, Phys. Lett. A 138, 474 (1989); R. Bhandari, Phys. Lett. A 138, 469 (1986).
[CrossRef]

Nityananda, R.

S. Ramaseshan, R. Nityananda, Commun. Sci. (India) 55, 1225 (1986).

Pancharatnam, S.

S. Pancharatnam, Proc. Indian Acad. Sci. 44, 51 (1956) (reprinted in Ref. 4).

Poincaré, H.

H. Poincaré, Theorie Mathematique de la Lumière, Vol. 2 (Georges Cavé, Paris, 1892).

Ramaseshan, S.

S. Ramaseshan, R. Nityananda, Commun. Sci. (India) 55, 1225 (1986).

Samuel, J.

J. Samuel, S. Bhandari, Phys. Rev. Lett. 66, 2339 (1988); S. Bhandari, J. Samuel, Phys. Rev. Lett. 60, 1211 (1988).
[CrossRef]

Schwerdtfeger, H.

H. Schwerdtfeger, Geometry of Complex Numbers (Dover, New York, 1979), Chap. 12.

Shapere, A.

A. Shapere, F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989).

Siegal, C. L.

C. L. Siegal, Topics in Complex Function Theory (Wiley Interscience, New York, 1973).

Simon, B.

B. Simon, Phys. Rev. Lett. 51, 2167 (1983).
[CrossRef]

Simon, R.

R. Simon, N. Mukunda, Phys. Lett. A 138, 474 (1989); R. Bhandari, Phys. Lett. A 138, 469 (1986).
[CrossRef]

Singh, H.

D. Subbarao, R. Uma, H. Singh, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 97 (1993); R. Uma, H. Singh, D. Subbarao, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 109 (1993).

Sodha, M. S.

D. Subbarao, M. S. Sodha, Self-Focusing Theory—A Perspective in Contemporary Plasma Physics (McMillan India, New Delhi, 1984), p. 249; J. Appl. Phys. 50, 4604 (1979); D. Subbarao, R. Uma, A. K. Ghatak, Laser Part. Beams 1, 383 (1983).

Subbarao, D.

D. Subbarao, R. Uma, H. Singh, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 97 (1993); R. Uma, H. Singh, D. Subbarao, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 109 (1993).

D. Subbarao, M. S. Sodha, Self-Focusing Theory—A Perspective in Contemporary Plasma Physics (McMillan India, New Delhi, 1984), p. 249; J. Appl. Phys. 50, 4604 (1979); D. Subbarao, R. Uma, A. K. Ghatak, Laser Part. Beams 1, 383 (1983).

Uma, R.

D. Subbarao, R. Uma, H. Singh, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 97 (1993); R. Uma, H. Singh, D. Subbarao, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 109 (1993).

Vavassori, P.

M. Martinelli, P. Vavassori, Opt. Commun. 80, 166 (1990).
[CrossRef]

Wilczek, F.

A. Shapere, F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1989).

Wu, C. S.

R. Y. Chiao, C. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Chiao, Phys. Rev. Lett. 57, 193 (1986); M. V. Berry, Nature (London) 326, 277 (1987).
[CrossRef] [PubMed]

Appl. Opt. (1)

C. R. Acad. Sci. Paris (1)

L. G. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890); E. H. Linfoot, E. Wolf, Proc. Phys. Soc. B 69, 827 (1956).

Commun. Sci. (1)

S. Ramaseshan, R. Nityananda, Commun. Sci. (India) 55, 1225 (1986).

J. Mod. Opt. (1)

M. V. Berry, J. Mod. Opt. 34, 1401 (1987) (reprinted in Ref. 4).
[CrossRef]

Opt. Commun. (1)

M. Martinelli, P. Vavassori, Opt. Commun. 80, 166 (1990).
[CrossRef]

Phys. Lett. A (1)

R. Simon, N. Mukunda, Phys. Lett. A 138, 474 (1989); R. Bhandari, Phys. Lett. A 138, 469 (1986).
[CrossRef]

Phys. Rev. Lett. (3)

J. Samuel, S. Bhandari, Phys. Rev. Lett. 66, 2339 (1988); S. Bhandari, J. Samuel, Phys. Rev. Lett. 60, 1211 (1988).
[CrossRef]

B. Simon, Phys. Rev. Lett. 51, 2167 (1983).
[CrossRef]

R. Y. Chiao, C. S. Wu, Phys. Rev. Lett. 57, 933 (1986); A. Tomita, R. Chiao, Phys. Rev. Lett. 57, 193 (1986); M. V. Berry, Nature (London) 326, 277 (1987).
[CrossRef] [PubMed]

Proc. Indian Acad. Sci. (1)

S. Pancharatnam, Proc. Indian Acad. Sci. 44, 51 (1956) (reprinted in Ref. 4).

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

M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

D. Subbarao, R. Uma, H. Singh, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 97 (1993); R. Uma, H. Singh, D. Subbarao, Proc. Soc. Photo-Opt. Instrum. Eng. 2039, 109 (1993).

Other (8)

D. Subbarao, M. S. Sodha, Self-Focusing Theory—A Perspective in Contemporary Plasma Physics (McMillan India, New Delhi, 1984), p. 249; J. Appl. Phys. 50, 4604 (1979); D. Subbarao, R. Uma, A. K. Ghatak, Laser Part. Beams 1, 383 (1983).

C. L. Siegal, Topics in Complex Function Theory (Wiley Interscience, New York, 1973).

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1989).

D. Marcuse, Light Transmission Optics(Van Nostrand Reinhold, New York, 1972); A. Yariv, Optical Electronics, 4th ed. (Harcourt Brace Jovanovich, Fort Worth, Tex., 1991), Chap. 2, p. 35.

H. Schwerdtfeger, Geometry of Complex Numbers (Dover, New York, 1979), Chap. 12.

A. Shapere, F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989).

R. M. A. Azzam, N. M. Bashare, Ellipsometry and Polarised Light (North-Holland, Amsterdam, 1977).

H. Poincaré, Theorie Mathematique de la Lumière, Vol. 2 (Georges Cavé, Paris, 1892).

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Equations (9)

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g ˜ ( z ) = g ˜ 0 A 1 + B 1 g ˜ 0 C 1 + D 1 .
ϕ F = - 2 A c r d r d ν ( 1 + r 2 ) 2 = - 2 A c d g ˜ r d g ˜ i ( 1 + g ˜ r 2 + g ˜ i 2 ) = 1 i C g ˜ 2 d g ˜ g ˜ ( 1 + g ˜ 2 ) = 0 2 π d ν ( 1 + r 2 ) - 2 π ,
A c d F d g ˜ ¯ d g ˜ r d g ˜ i = 1 2 i C F d g ˜ .
ϕ F = - 2 π ρ 2 1 + ρ 2 = - 2 π sin 2 1 2 θ c = - π ( 1 - cos  θ c ) .
g ( z ) = g 0 A + B g 0 C + D ,
T = [ A B C D ] = [ cos  b z - sin  b z sin  b z cos  b z ] ,
S ( z ) = [ q ( z ) p ( z ) ] = [ A B C D ] [ q 0 p 0 ] = L S ( 0 ) ,
J = [ 0 I - I 0 ] ,             V = [ i I i I - I I ] , V - 1 L V = [ A 0 B 0 C 0 D 0 ] = L 0 ,
W * = ( A 0 W + B 0 ) ( C 0 W + D 0 ) - 1

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