Abstract

We use a generalized refractive index that includes diffraction effects to show that the Gouy phase shift can be seen as an intensity averaged optical path difference between the generalized eikonal and the geometrical eikonal. This approach generalizes previous treatments to include the effects of phase distortion and confirms the role of transverse spatial confinement in the Gouy shift.

© 2006 Optical Society of America

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  1. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890).
  2. Gouy, Ann. Chim. Phys. Ser. 6, XXIV, 145 (1891).
  3. A. E. Siegman, Lasers (University Science, 1986).
  4. F. Gittes and C. F. Schmidt, Opt. Lett. 23, 7 (1998).
    [CrossRef]
  5. R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), p. 111.
  6. S. Feng, H. G. Winful, and R. W. Hellwarth, Opt. Lett. 23, 385 (1998).
    [CrossRef]
  7. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
    [CrossRef]
  8. R. W. McGowan, R. A. Cheville, and D. Grischkowsky, Appl. Phys. Lett. 76, 670 (2000).
    [CrossRef]
  9. F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
    [CrossRef] [PubMed]
  10. N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, Appl. Phys. Lett. 83, 392 (2003).
    [CrossRef]
  11. R. W. Boyd, J. Opt. Soc. Am. 70, 877 (1980).
    [CrossRef]
  12. P. Hariharan and P. A. Robinson, J. Mod. Opt. 43, 219 (1996).
  13. S. Feng and H. G. Winful, Opt. Lett. 26, 485 (2001).
    [CrossRef]
  14. D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982), Chap. 1.
  15. X. Deng and H. Fang, Chin. J. Lasers 7, 14 (1980).
  16. H. Guo and X. Deng, J. Opt. Soc. Am. A 12, 600 (1995).
    [CrossRef]
  17. S. C. Yap, B. C. Quek, and K. S. Low, J. Opt. Soc. Am. A 15, 2725 (1998).
    [CrossRef]
  18. P. A. Belanger, Opt. Lett. 16, 196 (1991).
    [CrossRef] [PubMed]
  19. J. Yang and D. Fan, J. Opt. Soc. Am. A 16, 2488 (1999).
    [CrossRef]

2004 (1)

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef] [PubMed]

2003 (1)

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, Appl. Phys. Lett. 83, 392 (2003).
[CrossRef]

2001 (1)

2000 (1)

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

1999 (2)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

J. Yang and D. Fan, J. Opt. Soc. Am. A 16, 2488 (1999).
[CrossRef]

1998 (3)

1996 (1)

P. Hariharan and P. A. Robinson, J. Mod. Opt. 43, 219 (1996).

1995 (1)

1991 (1)

1980 (2)

X. Deng and H. Fang, Chin. J. Lasers 7, 14 (1980).

R. W. Boyd, J. Opt. Soc. Am. 70, 877 (1980).
[CrossRef]

1891 (1)

Gouy, Ann. Chim. Phys. Ser. 6, XXIV, 145 (1891).

Baltuska, A.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef] [PubMed]

Belanger, P. A.

Boyd, R. W.

R. W. Boyd, J. Opt. Soc. Am. 70, 877 (1980).
[CrossRef]

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), p. 111.

Cheville, R. A.

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

Daly, B. C.

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, Appl. Phys. Lett. 83, 392 (2003).
[CrossRef]

Deng, X.

H. Guo and X. Deng, J. Opt. Soc. Am. A 12, 600 (1995).
[CrossRef]

X. Deng and H. Fang, Chin. J. Lasers 7, 14 (1980).

Fan, D.

Fang, H.

X. Deng and H. Fang, Chin. J. Lasers 7, 14 (1980).

Feng, S.

Gittes, F.

Goulielmakis, E.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef] [PubMed]

Gouy,

Gouy, Ann. Chim. Phys. Ser. 6, XXIV, 145 (1891).

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

Grischkowsky, D.

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

Guo, H.

Hariharan, P.

P. Hariharan and P. A. Robinson, J. Mod. Opt. 43, 219 (1996).

Hellwarth, R. W.

Holme, N. C.

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, Appl. Phys. Lett. 83, 392 (2003).
[CrossRef]

Krausz, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef] [PubMed]

Lezius, M.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef] [PubMed]

Lindner, F.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef] [PubMed]

Low, K. S.

Marcuse, D.

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982), Chap. 1.

McGowan, R. W.

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

Myaing, M. T.

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, Appl. Phys. Lett. 83, 392 (2003).
[CrossRef]

Norris, T. B.

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, Appl. Phys. Lett. 83, 392 (2003).
[CrossRef]

Paulus, G. G.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef] [PubMed]

Quek, B. C.

Robinson, P. A.

P. Hariharan and P. A. Robinson, J. Mod. Opt. 43, 219 (1996).

Rudd, J. V.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Ruffin, A. B.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Schmidt, C. F.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Walther, H.

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef] [PubMed]

Whitaker, J. F.

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

Winful, H. G.

Yang, J.

Yap, S. C.

Ann. Chim. Phys. (1)

Gouy, Ann. Chim. Phys. Ser. 6, XXIV, 145 (1891).

Appl. Phys. Lett. (2)

R. W. McGowan, R. A. Cheville, and D. Grischkowsky, Appl. Phys. Lett. 76, 670 (2000).
[CrossRef]

N. C. R. Holme, B. C. Daly, M. T. Myaing, and T. B. Norris, Appl. Phys. Lett. 83, 392 (2003).
[CrossRef]

Chin. J. Lasers (1)

X. Deng and H. Fang, Chin. J. Lasers 7, 14 (1980).

J. Mod. Opt. (1)

P. Hariharan and P. A. Robinson, J. Mod. Opt. 43, 219 (1996).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Lett. (4)

Phys. Rev. Lett. (2)

A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
[CrossRef]

F. Lindner, G. G. Paulus, H. Walther, A. Baltuska, E. Goulielmakis, M. Lezius, and F. Krausz, Phys. Rev. Lett. 92, 113001 (2004).
[CrossRef] [PubMed]

Other (4)

R. W. Boyd, Nonlinear Optics, 2nd ed. (Academic, 2003), p. 111.

A. E. Siegman, Lasers (University Science, 1986).

D. Marcuse, Light Transmission Optics, 2nd ed. (Van Nostrand Reinhold, 1982), Chap. 1.

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

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Figures (1)

Fig. 1
Fig. 1

Focused beam with wavefront and path at point P, where d S is the real ray path with the generalized refractive index n G and d s is the imaginary geometrical path with the classical refractive index n.

Equations (17)

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2 E + ( E ϵ ϵ ) + n 2 k 2 E = 0 .
2 E + n 2 k 2 E = 0 .
( L ̃ ) 2 = n 2 + 1 k 2 2 E ̃ E ̃ n G 2 ,
( E ̃ 2 L ̃ ) = 0 .
2 E 0 + 2 i k n 0 E 0 z + k 2 ( n 2 n 0 2 ) E 0 = 0 .
( L ) 2 + 2 n 0 L z + n 0 2 = 1 k 2 2 ψ ψ + n 2 n G 2 ,
( ψ 2 L ) + n 0 ψ 2 z = 0 .
( L ) 2 1 k 2 2 ψ ψ = r = x , y ( M r 4 k 2 W r 2 + W r 2 4 R r 2 ) r = x , y θ r 2 .
2 n 0 L z = r = x , y ( M r 4 k 2 W r 2 + W r 2 4 R r 2 + n 0 n 2 W r 2 4 ) .
k L = r = x , y k r 2 2 R r + Φ a ( x , y , z ) + Φ G ( z ) .
2 n 0 z ( r 2 2 R r ) = M r 4 k 2 W r 2 W r 2 4 R r 2 n 0 n 2 W r 2 4 .
n 0 ( Φ a + Φ G ) z = r = x , y M r 4 k W r 2 .
Φ G z + Φ a z = 1 n 0 k 2 ψ ψ k n 0 [ ( L ) 2 W r 2 4 R r 2 ] .
Φ G z + Φ a z = 1 n 0 k 2 ψ ψ .
Φ G ( z ) + Φ a ( z ) = 0 z 1 n 0 k 2 ψ ψ d z = 0 z ( k x 2 + k y 2 ) n 0 k d z ,
Φ G ( z ) + Φ a ( z ) = k 0 z n G d S n d s .
Φ G = r = x , y M r 2 2 tan 1 [ B ( A + B R 1 r ) π W 1 r 2 λ M r 2 ] ,

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