Abstract

An intuitive argument is presented for the phase anomaly, that is, the 180° phase shift of a light wave in passing through a focus. The treatment is based on the geometrical properties of Gaussian light beams, and suggests a new viewpoint for understanding the origin of the phase shift. Generalizing the argument by including higher-order modes of the light field allows the case of a spherical wave to be treated.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Gouy, C. R. Acad. Sci. Paris 110, 1251–1253 (1890).
  2. Gouy, Ann. Chim. Phys. 6, XXIV, 145–213 (1891).
  3. P. Debye, Ann. Phys. 30, 755 (1909).A discussion of Debye’s work can be found in J. Picht, Optische Abbildung (Vieweg & Sohn, Braunschweig, 1931).References to much of the early work in this field, including that of F. Reiche and K. Schwarzschild, can also be found in this book.
    [Crossref]
  4. A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931–936 (1938).
    [Crossref]
  5. E. H. Linfoot and E. Wolf, “Phase Distribution near Focus in an Aberration-free Diffraction Image,” Proc. Phys. Soc. London 69, 823–832 (1956).
    [Crossref]
  6. Due to an algebraic error, the original paper states that the wavelength is decreased by this factor (E. Wolf, private communication).
  7. H. Kogelnik and T. Li, “Laser Beams and Resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [Crossref] [PubMed]
  8. P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954), p. 3.
  9. M. Abramowitz and I. A. Segen, Handbook of Mathematical Functions (Dover, New York, 1965), Eqs. (17.3.11) and (17.3.12).
  10. N. N. Lebedev, Special Functions and their Applications (Dover, New York, 1972), p. 88.

1966 (1)

1956 (1)

E. H. Linfoot and E. Wolf, “Phase Distribution near Focus in an Aberration-free Diffraction Image,” Proc. Phys. Soc. London 69, 823–832 (1956).
[Crossref]

1938 (1)

A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931–936 (1938).
[Crossref]

1909 (1)

P. Debye, Ann. Phys. 30, 755 (1909).A discussion of Debye’s work can be found in J. Picht, Optische Abbildung (Vieweg & Sohn, Braunschweig, 1931).References to much of the early work in this field, including that of F. Reiche and K. Schwarzschild, can also be found in this book.
[Crossref]

1891 (1)

Gouy, Ann. Chim. Phys. 6, XXIV, 145–213 (1891).

1890 (1)

Gouy, C. R. Acad. Sci. Paris 110, 1251–1253 (1890).

Abramowitz, M.

M. Abramowitz and I. A. Segen, Handbook of Mathematical Functions (Dover, New York, 1965), Eqs. (17.3.11) and (17.3.12).

Byrd, P. F.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954), p. 3.

Debye, P.

P. Debye, Ann. Phys. 30, 755 (1909).A discussion of Debye’s work can be found in J. Picht, Optische Abbildung (Vieweg & Sohn, Braunschweig, 1931).References to much of the early work in this field, including that of F. Reiche and K. Schwarzschild, can also be found in this book.
[Crossref]

Friedman, M. D.

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954), p. 3.

Gouy,

Gouy, Ann. Chim. Phys. 6, XXIV, 145–213 (1891).

Gouy, C. R. Acad. Sci. Paris 110, 1251–1253 (1890).

Kogelnik, H.

Lebedev, N. N.

N. N. Lebedev, Special Functions and their Applications (Dover, New York, 1972), p. 88.

Li, T.

Linfoot, E. H.

E. H. Linfoot and E. Wolf, “Phase Distribution near Focus in an Aberration-free Diffraction Image,” Proc. Phys. Soc. London 69, 823–832 (1956).
[Crossref]

Rubinowicz, A.

A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931–936 (1938).
[Crossref]

Segen, I. A.

M. Abramowitz and I. A. Segen, Handbook of Mathematical Functions (Dover, New York, 1965), Eqs. (17.3.11) and (17.3.12).

Wolf, E.

E. H. Linfoot and E. Wolf, “Phase Distribution near Focus in an Aberration-free Diffraction Image,” Proc. Phys. Soc. London 69, 823–832 (1956).
[Crossref]

Due to an algebraic error, the original paper states that the wavelength is decreased by this factor (E. Wolf, private communication).

Ann. Chim. Phys. (1)

Gouy, Ann. Chim. Phys. 6, XXIV, 145–213 (1891).

Ann. Phys. (1)

P. Debye, Ann. Phys. 30, 755 (1909).A discussion of Debye’s work can be found in J. Picht, Optische Abbildung (Vieweg & Sohn, Braunschweig, 1931).References to much of the early work in this field, including that of F. Reiche and K. Schwarzschild, can also be found in this book.
[Crossref]

Appl. Opt. (1)

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

Gouy, C. R. Acad. Sci. Paris 110, 1251–1253 (1890).

Phys. Rev. (1)

A. Rubinowicz, “On the anomalous propagation of phase in the focus,” Phys. Rev. 54, 931–936 (1938).
[Crossref]

Proc. Phys. Soc. London (1)

E. H. Linfoot and E. Wolf, “Phase Distribution near Focus in an Aberration-free Diffraction Image,” Proc. Phys. Soc. London 69, 823–832 (1956).
[Crossref]

Other (4)

Due to an algebraic error, the original paper states that the wavelength is decreased by this factor (E. Wolf, private communication).

P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Physicists (Springer-Verlag, Berlin, 1954), p. 3.

M. Abramowitz and I. A. Segen, Handbook of Mathematical Functions (Dover, New York, 1965), Eqs. (17.3.11) and (17.3.12).

N. N. Lebedev, Special Functions and their Applications (Dover, New York, 1972), p. 88.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

FIG. 1
FIG. 1

Amplitude distribution of a Gaussian beam.

FIG. 2
FIG. 2

Profile of a Gaussian beam, showing the change of w with z, the far-field divergence angle θff, and the wave-front radius of curvature R.

FIG. 3
FIG. 3

The wave fronts AB and DE of a Gaussian beam are separated by the optical path length BCD. The difference between the path length BCD and the geometrical separation BE gives rise to the phase anomaly.

FIG. 4
FIG. 4

The phase anomaly ΔΦ predicted by the heuristic argument described here is shown as a function of (2θff)−1f#. For large values of f#, the predicted phase anomaly agrees with the actual value of π radians. The broken portion of the curve corresponds to f# ≲ 1, where the approximations used in the derivation are invalid.

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

2 u + k 2 u = 0
u ( r , z ) = w 0 w ( z ) exp [ i [ k z Φ ( z ) ] r 2 ( 1 w 2 ( z ) i k 2 R ( z ) ) ] ,
w 2 ( z ) = w 0 2 [ 1 + ( λ z / π w 0 2 ) 2 ] .
θ ff = λ / π w 0 .
R ( z ) = z [ 1 + ( π w 0 2 / λ z ) 2 ] .
Φ ( z ) = arctan ( λ z / π w 0 2 ) .
e ikz 2 z 2 [ u ( r , z ) e ikz ]
L = 2 π w 0 2 λ 0 z 1 / b d x ( 1 + ( 1 + θ ff 2 ) x 2 1 + x 2 ) 1 / 2 ,
L = 2 w 0 1 + θ ff 2 ( 1 1 + θ ff 2 F ( ϕ , κ ) E ( ϕ , κ ) + z 1 θ ff w 0 ( 1 + θ ff 2 ) ( w 0 2 θ ff 2 + z 1 2 ) w 0 2 θ ff 4 + z 1 2 ( 1 + θ ff 2 ) ) ,
ϕ = sin 1 z 1 2 / [ z 1 2 + w 0 2 ( θ ff 2 + θ ff 4 ) 1 ]
κ = θ ff 2 / ( 1 + θ ff 2 ) .
L = 2 z 1 1 + θ ff 2 + w 0 2 z 1 2 .
Δ Φ = lim z 1 2 π λ ( L L ) ,
Δ Φ = 4 θ ff 2 1 + θ ff 2 ( 1 1 + θ ff 2 F ( π / 2 , κ ) E ( π / 2 , κ ) ) .
u p ( r , z ) = L p ( 2 r 2 w 2 ( z ) ) w 0 w ( z ) exp [ i [ k z Φ p ( z ) ] r 2 ( 1 w 2 ( z ) i k 2 R ( z ) ) ] ,
Φ p ( z ) = ( 2 p + 1 ) arctan ( λ z / π w 0 2 ) .
f ( x ) = p = 0 C p L p ( x ) .
u s ( r , z ) = { A r 2 + z 2 i k r 2 + z 2 r | z | θ 0 r | z | > θ for z 0 ,
u s ( r , z ) = p C p u p ( r , z ) ,
u x ( r , z ) = { A r 2 + z 2 e i k r 2 + z 2 r | z | θ 0 r | z | > θ for z 0 ,