Abstract

We show explicitly that the well-known Gouy phase shift of any focused beam originates from transverse spatial confinement, which, through the uncertainty principle, introduces a spread in the transverse momenta and hence a shift in the expectation value of the axial propagation constant. A general expression is given for the Gouy phase shift in terms of expectation values of the squares of the transverse momenta. Our result also explains the phase shift in front of the Kirchhoff diffraction integral.

© 2001 Optical Society of America

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References

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  1. C. R. Gouy, C. R. Acad. Sci. Paris 110, 1251 (1890).
  2. C. R. Gouy, Ann. Chim. Phys. Ser. 6,  24, 145 (1891).
  3. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986). Note that Gouy’s name is misspelled as “Guoy” in this classic reference.
  4. S. Feng, H. G. Winful, and R. W. Hellwarth, Opt. Lett. 23, 385 (1998).
    [CrossRef]
  5. S. Feng, H. G. Winful, and R. W. Hellwarth, Opt. Lett. 23, 1141 (1998), erratum of Ref.  4.
    [CrossRef]
  6. A. B. Ruffin, J. V. Rudd, J. F. Whitaker, S. Feng, and H. G. Winful, Phys. Rev. Lett. 83, 3410 (1999).
    [CrossRef]
  7. R. W. McGowan, R. A. Cheville, and D. Grischkowsky, Appl. Phys. Lett. 76, 670 (2000).
    [CrossRef]
  8. R. W. Boyd, J. Opt. Soc. Am. 70, 877 (1980).
  9. P. Hariharan and P. A. Robinson, J. Mod. Opt. 43, 219 (1996).
  10. M. V. Berry, Proc. R. Soc. London Ser. A 392, 45 (1984).
    [CrossRef]
  11. D. Subbarao, Opt. Lett. 20, 2162 (1995).
    [CrossRef]
  12. R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993).
    [CrossRef] [PubMed]
  13. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N. J., 1984).

2000 (1)

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

1999 (1)

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

1998 (2)

1996 (1)

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

1995 (1)

1993 (1)

R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993).
[CrossRef] [PubMed]

1984 (1)

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

1980 (1)

1891 (1)

C. R. Gouy, Ann. Chim. Phys. Ser. 6,  24, 145 (1891).

1890 (1)

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

Berry, M. V.

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

Boyd, R. W.

Cheville, R. A.

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

Feng, S.

Gouy, C. R.

C. R. Gouy, Ann. Chim. Phys. Ser. 6,  24, 145 (1891).

C. R. 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]

Hariharan, P.

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

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N. J., 1984).

Hellwarth, R. W.

McGowan, R. W.

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

Mukunda, N.

R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993).
[CrossRef] [PubMed]

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]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986). Note that Gouy’s name is misspelled as “Guoy” in this classic reference.

Simon, R.

R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993).
[CrossRef] [PubMed]

Subbarao, D.

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.

Ann. Chim. Phys. Ser. 6 (1)

C. R. Gouy, Ann. Chim. Phys. Ser. 6,  24, 145 (1891).

Appl. Phys. Lett. (1)

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

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

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

J. Mod. Opt. (1)

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

J. Opt. Soc. Am. (1)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

R. Simon and N. Mukunda, Phys. Rev. Lett. 70, 880 (1993).
[CrossRef] [PubMed]

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

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

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

Other (2)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N. J., 1984).

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986). Note that Gouy’s name is misspelled as “Guoy” in this classic reference.

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Equations (24)

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k2=kx2+ky2+kz2,
ξ-+ξfξ2dξ-+fξ2dξ,
k¯zkz2k=k-kx2k-ky2k.
ϕG=-1kzkx2+ky2dz.
fx,y=2π1wzexp-x2+y2w2z,
w2z=w021+zzR2
F˜kx,ky=12π-+-+fx,y×exp-ikxx-ikyydxdy.
F˜kx,ky=wz2πexp-w2z4kx2+ky2,
-+-+fx,y2dxdy=-+-+F˜kx,ky2dkxdky=1.
kx2=-+-+kx2F˜kx,ky2dkxdky=1w2z=ky2.
ϕG=-1kzkx2+ky2dz=-2kz1w2zdz.
ϕG=-arctanz/zR.
ϕGcz=ϕGz+ilnz2+zR2.
fmnx,y=Cmn2wzΘm2xwzΘn2ywz,
Cmn=1π2m+nm!n!1/2
ΘmξHmξexp-ξ2/2.
F˜mnkx,ky=-im+nCmnwz2Θmwzkx2×Θnwzky2.
Hn+1-2ξHn+2nHn-1=0.
kx2mn=2w2zm+12,ky2mn=2w2zn+12.
ϕG,mnz=-m+n+1arctanz/zR,
-p+1/2arctanz/zR,p=m for x axis,p=n for y axis
ΔkxΔxconst.,
Δξξ-ξ21/2.
ΔkxmΔxm=m+1/2.

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