## 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

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### Equations (14)

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(1)
$$U(-\dot{R}\mathbf{s})=a(\mathbf{s})\frac{\text{exp}(-ikR)}{R}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{when}\hspace{0.17em}\mathbf{s}\in \mathrm{\Omega},$$
(2)
$$U(-R\mathbf{s})=0\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\mathrm{\hspace{0.17em}\u200a\u200a}\text{when}\hspace{0.17em}\mathbf{s}\notin \mathrm{\Omega}.$$
(3)
$$U(x,y,z)=-\frac{ik}{2\pi}{\iint}_{\mathrm{\Omega}}\frac{a({s}_{x},{s}_{y})}{{s}_{z}}\times \text{exp}[ik({s}_{x}x+{s}_{y}y+{s}_{z}z)]\text{d}{s}_{x}\text{d}{s}_{y}.$$
(4)
$$\begin{array}{l}U(-x,-y,-z)=-\frac{ik}{2\pi}{\iint}_{\mathrm{\Omega}}\frac{a({s}_{x},{s}_{y})}{{s}_{z}}\times \text{exp}[-ik({s}_{x}x+{s}_{y}y+{s}_{z}z)]\text{d}{s}_{x}\text{d}{s}_{y}\\ ={\left\{\frac{ik}{2\pi}{\iint}_{\mathrm{\Omega}}\frac{a({s}_{x},{s}_{y})}{{s}_{z}}\times \text{exp}[ik({s}_{x}x+{s}_{y}y+{s}_{z}z)]\text{d}{s}_{x}\text{d}{s}_{y}\right\}}^{*},\end{array}$$
(5)
$$U(-x,-y,-z)=-{[U(x,y,z)]}^{*}.$$
(6)
$$U(x,y,z)=\mid U(x,y,z)\mid \text{exp}[i\varphi (x,y,z)].$$
(7)
$$\mid U(-x,-y,-z)\mid =\mid U(x,y,z)\mid ,$$
(8)
$$\varphi (-x,-y,-z)=-\varphi (x,y,z)-\pi .$$
(9)
$$I(x,y,z)=\mid U(x,y,z){\mid}^{2}$$
(10)
$$U(x,y,z)=U(r;z),$$
(11)
$$r=\sqrt{{x}^{2}+{y}^{2}}$$
(12)
$$U(r,-z)=-{[U(r,z)]}^{*},$$
(13)
$$\mid U(r;-z)\mid =\mid U(r;z)\mid ,$$
(14)
$$\varphi (r,-z)=-\varphi (r,z)-\pi .$$