When shadow moiré is practiced in industry for the warpage of microelectronic devices, the required high basic measurement sensitivity limits a dynamic range due to the diffraction effect of the reference grating. An extensive understanding of the contrast and intensity of shadow moiré fringes is required to achieve optical configurations for the measurements. In Part I, an exact mathematical description for the contrast and intensity of shadow moiré fringe is developed using a diffraction theory for a monochromatic light source first. The analysis is extended to study the effect of a broad spectrum light source on the contrast and intensity of shadow moiré fringes. The effect of an aperture on the fringe contrast is defined to propose a complete expression for the contrast of shadow moiré fringe. The mathematical analysis is exploited to define the systematic error from the non-sinusoidal intensity distribution of shadow moiré fringe when the displacement resolution is enhanced using the phase-shifting technique. The results of the mathematical analysis provide a guideline for optimum optical configurations for the required basic measurement sensitivity, which results in a novel technique, called high sensitivity shadow moiré using non-zero Talbot distance (SM-NT). The SM-NT increases the dynamic range substantially and allows the warpage measurements of high-end microelectronics devices, which is not possible with the conventional shadow moiré using the zero Talbot distance.

In an achromatic moiré interferometry system, a compensator grating is translated to achieve phase-shifting. The phase-shifting in the achromatic system cannot be explained by the existing theories of moiré interferometry based on the concept of optical path length. In Part II, a diffraction theory is used to explain the phase shifting in the achromatic system. The results reveal that the amount of translation of the compensator grating is proportional to the diffraction order and the frequency of the compensator grating. The diffraction theory based the mathematical description is extended further to define the mini-order diffractions associated with a general deformations. The discrete Fourier transform is employed to characterize the mini-order from a generally deformed grating. The results explain that the magnitude of strain is only parameter to control the angle of mini-order.