Abstract

It is shown that under very general conditions monochromatic scalar wave fields that have a focus in the sense of geometrical optics possess some simple symmetry properties with respect to the focus. Certain well-known but poorly understood symmetry properties of the three-dimensional amplitude distribution and phase distribution in the focal region of a uniform converging spherical wave diffracted at a circular aperture are found to be immediate consequences of our results.

© 1980 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, England, 1975), Sec. 8.8.
  2. A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Sec. 8.2.
  3. That Eq. (2.2) obeys the Helmholtz equation (▽2 + k2)U = 0 (as appropriate to a monochromatic field of frequency ω in free space) everywhere on the right of the aperture plane z > −R follows at once from the fact that each plane wave, exp[ik(sxx + syy + szz)], under the integral sign in Eq. (2.2) satisfies this equation. That Eq. (2.2) satisfies the asymptotic boundary condition (2.1), as kR → ∞, can be readily verified by the use of the principle of stationary phase [cf. Appendix in K. Miyamoto, E. Wolf, “Generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave—Part I,” J. Opt. Soc. Am. 52, 615–625 (1962)].
    [CrossRef]
  4. P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30 (4), 755–776 (1909).
    [CrossRef]
  5. J. Picht, Optische Abbildung (Vieweg, Braunschweig, Germany, 1931), Chap. 8.
  6. R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Secs. 46–48.
  7. E. Wolf, “Electromagnetic diffraction in optical systems. I. an integral representation of the image field,” Proc. Roy. Soc. A, 253, 349–357 (1959).
    [CrossRef]
  8. Indeterminacy in the phase of an amount 2nπ (with n being any integer) is ignored in Eq. (2.6b) because it has no physical significance.
  9. E. Wolf, “Phase conjugacy and symmetries in spatially bandlimited wavefields,” submitted to J. Opt. Soc. Am.

1962 (1)

1959 (1)

E. Wolf, “Electromagnetic diffraction in optical systems. I. an integral representation of the image field,” Proc. Roy. Soc. A, 253, 349–357 (1959).
[CrossRef]

1909 (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30 (4), 755–776 (1909).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, England, 1975), Sec. 8.8.

Debye, P.

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30 (4), 755–776 (1909).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Secs. 46–48.

Miyamoto, K.

Picht, J.

J. Picht, Optische Abbildung (Vieweg, Braunschweig, Germany, 1931), Chap. 8.

Siegman, A. E.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Sec. 8.2.

Wolf, E.

Ann. Phys. (1)

P. Debye, “Das Verhalten von Lichtwellen in der Nähe eines Brennpunktes oder einer Brennlinie,” Ann. Phys. 30 (4), 755–776 (1909).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. Roy. Soc. A (1)

E. Wolf, “Electromagnetic diffraction in optical systems. I. an integral representation of the image field,” Proc. Roy. Soc. A, 253, 349–357 (1959).
[CrossRef]

Other (6)

Indeterminacy in the phase of an amount 2nπ (with n being any integer) is ignored in Eq. (2.6b) because it has no physical significance.

E. Wolf, “Phase conjugacy and symmetries in spatially bandlimited wavefields,” submitted to J. Opt. Soc. Am.

J. Picht, Optische Abbildung (Vieweg, Braunschweig, Germany, 1931), Chap. 8.

R. K. Luneburg, Mathematical Theory of Optics (U. California Press, Berkeley, Calif., 1964), Secs. 46–48.

M. Born, E. Wolf, Principles of Optics, 5th ed. (Pergamon, Oxford, England, 1975), Sec. 8.8.

A. E. Siegman, An Introduction to Lasers and Masers (McGraw-Hill, New York, 1971), Sec. 8.2.

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 (5)

Fig. 1
Fig. 1

Isophotes (contours of the intensity) in a meridional plane in the neighborhood of the focus of a uniform, converging, monochromatic, spherical wave diffracted at a circular aperture. The intensity is normalized to unity at the focus. The dotted lines represent the boundary of the geometrical shadow, making an angle θ with the z axis (normal to the aperture plane). u = kz sin2 θ, v = k(x2 + y2)1/2 sin θ (k = wave number). [Adapted from E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956)].

Fig. 2
Fig. 2

Profiles of the surfaces of constant phase in a meridional plane in the neighborhood of the geometrical focal plane of a uniform, converging, monochromatic spherical wave diffracted at a circular aperture. The angle of convergence is that appropriate to an f/3.5 pencil of rays. [Adapted from E. H. Linfoot and E. Wolf, “Phase distribution near focus in an aberration-free diffraction image,” Proc. Phys. Soc. B 69, 823–832 (1956)].

Fig. 3
Fig. 3

Notation relating to Eqs. (2.1) and (2.2).

Fig. 4
Fig. 4

Illustrating the symmetry relations (2.6) in focused fields, viz., |U(Q)| = |U(Q+)|, ϕ(Q) = −ϕ(Q+) − π, where Q+ and Q are any two points related by inversion symmetry with respect to the geometrical focus, i.e., such that Q+ ≡ (x,y,z), Q ≡ (−x,−y,−z).

Fig. 5
Fig. 5

Illustrating the symmetry relations (2.11) in focused fields that possess rotational symmetry about the z axis, viz., |U(S)| = |U(S+)|, ϕ(S) = −ϕ(S+) − π, where S+ and S are any two points related by reflection symmetry about the focal plane z = 0, i.e., such that S+ ≡ (r,z), S ≡ (r,−z).

Equations (14)

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

U ( - R ˙ s ) = a ( s ) exp ( - i k R ) R             when s Ω ,
U ( - R s ) = 0             when s Ω .
U ( x , y , z ) = - i k 2 π Ω a ( s x , s y ) s z × exp [ i k ( s x x + s y y + s z z ) ] d s x d s y .
U ( - x , - y , - z ) = - i k 2 π Ω a ( s x , s y ) s z × exp [ - i k ( s x x + s y y + s z z ) ] d s x d s y = { i k 2 π Ω a ( s x , s y ) s z × exp [ i k ( s x x + s y y + s z z ) ] d s x d s y } * ,
U ( - x , - y , - z ) = - [ U ( x , y , z ) ] * .
U ( x , y , z ) = U ( x , y , z ) exp [ i ϕ ( x , y , z ) ] .
U ( - x , - y , - z ) = U ( x , y , z ) ,
ϕ ( - x , - y , - z ) = - ϕ ( x , y , z ) - π .
I ( x , y , z ) = U ( x , y , z ) 2
U ( x , y , z ) = U ( r ; z ) ,
r = x 2 + y 2
U ( r , - z ) = - [ U ( r , z ) ] * ,
U ( r ; - z ) = U ( r ; z ) ,
ϕ ( r , - z ) = - ϕ ( r , z ) - π .

Metrics