The image structure produced by a periodic hexagonal pattern of mirror
surface undulations has been analyzed. Such undulations form a two-di
mensional phase grating that can result from the polishing of honeycom
b mirrors or, for example, meniscus mirrors with a hexagonal pattern o
f axial supports. For monochromatic light of wavelength lambda, undula
tions having uniform peak-to-valley amplitude H << lambda and period L
cause a decrease in the central intensity of the point spread functio
n (PSF), and a fraction, similar to 13(H/lambda)(2), of the total powe
r is diffracted into an infinite hexagonal array of satellite images.
These have angular separations of 2 lambda/root 3L and intensity profi
les in the form of perfect diffraction Limited PSF's, but with intensi
ties decreasing with increasing diffraction order. The six innermost (
first-order) satellites each have central intensities approximately 2(
H/lambda)(2) times that of the central image. If the am, plitudes of t
he surface bumps are of random size with a normal frequency distributi
on, then the intensity of the diffracted orders decreases, and an addi
tional weak structure appears over the image plane; the positions and
heights of the peaks in this grasslike structure depend on the particu
lar two-dimensional distribution of the random bumps. When the input i
s polychromatic, the diffracted orders other than zero give images tha
t are elongated radially and decrease outward in intensity with a 1/la
mbda(4) dependence. (C) 1997 Optical Society of America.