Asymptotic analysis of imaging in the presence of a sinusoidal phase modulation

When a mirror is produced by numerically controlled diamond turning, its su rface may be optically smooth except for a sinusoidal height modulation in the radial direction. If the amplitude of this modulation is small compared with its wavelength and the mirror has no aberrations, the reflected light will be a converging spherical wavefront with a radial, sinusoidal modulat ion superimposed on it. To determine how such a modulation affects the imag e produced by the mirror we study the reflected light in the focal region. We show that if the aperture radius is much larger than the modulation wave length, the focal-plane intensity distribution of the zeroth-order image re mains unchanged except for a drop in the peak intensity by a factor J(0)(2) (2 pi delta(1)), where delta(1) is the modulation amplitude measured in uni ts of the radiation wavelength. Also we show that higher-order diffracted i mages appear at equally spaced positions along the optical axis, and that t hese high-order images give concentric circles of light in the geometrical focal plane.