## Abstract

A general solution of undersampling frequency conversion and its optimization for parallel photodisplacement imaging is presented. Phase-modulated heterodyne interference light generated by a linear region of periodic displacement is captured by a charge-coupled device image sensor, in which the interference light is sampled at a sampling rate lower than the Nyquist frequency. The frequencies of the components of the light, such as the sideband and carrier (which include photodisplacement and topography information, respectively), are downconverted and sampled simultaneously based on the integration and sampling effects of the sensor. A general solution of frequency and amplitude in this downconversion is derived by Fourier analysis of the sampling procedure. The optimal frequency condition for the heterodyne beat signal, modulation signal, and sensor gate pulse is derived such that undesirable components are eliminated and each information component is converted into an orthogonal function, allowing each to be discretely reproduced from the Fourier coefficients. The optimal frequency parameters that maximize the sideband-to-carrier amplitude ratio are determined, theoretically demonstrating its high selectivity over $80\text{\hspace{0.17em} dB}$. Preliminary experiments demonstrate that this technique is capable of simultaneous imaging of reflectivity, topography, and photodisplacement for the detection of subsurface lattice defects at a speed corresponding to an acquisition time of only $\text{0 .26}\mathrm{s}$ per $256\times 256$ pixel area.

© 2006 Optical Society of America

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