Florin Munteanu

Iterative least square phase-measuring method that tolerates extended finite bandwidth illumination

Iterative least square phase-measuring techniques address the phase-shifting interferometry issue of sensitivity to vibrations and scanner nonlinearity. In these techniques the wavefront phase and phase steps are determined simultaneously from a single set of phase-shifted fringe frames where the phase shift does not need to have a nominal value or be a priori precisely known. This method is commonly used in laser interferometers in which the contrast of fringes is constant between frames and across the field. We present step-by-step modifications to the basic iterative least square method. These modifications allow for vibration insensitive measurements in an interferometric system in which fringe contrast varies across a single frame, as well as from frame to frame, due to the limited bandwidth light source and the nonzero numerical aperture of the objective. We demonstrate the efficiency of the new algorithm with experimental data, and we analyze theoretically the degree of contrast variation that this new algorithm can tolerate.

Iterative algorithm for phase shifting interferometry with finite bandwidth illumination

While phase shifting interferometry (PSI) has clearly established itself as a powerful tool for surface profiling, two main experimental drawbacks still exist, namely phase-shift errors and non-sinusoidal interferometric signals. These problems cause a fringe print-through pattern in measured phase having a frequency typically double the original fringe frequency. The main tool used to compensate for non-linear phase-shift errors is the least-square phase-stepping method, which is capable of accurately determining the phase separation between consecutive frames as well as phase distribution for each frame. Our work here extends the capabilities of this algorithm so as to account for the effects of a wide bandwidth light source and a higher numerical aperture, which make the interference signal deviate from the ideal sinusoidal signal. We present a simple variable change mechanism that overcomes the coupling that occurs between the planar and depth coordinates and allows for a seamless integration with the standard least-square procedure.

Limitations of iterative least squares methods in phase shifting interferometry in presence of vibrations

Least square (LS) methods are an alternate approach to phase shifting interferometry algorithms for object shape measurement in the presence of unequal phase steps. One of the sources of unequal steps is vibrations. We study the influence of vibrations on intensity data and compare the obtained results with a LS and classic PSI algorithms. We find that in the case when no vibrations are present, both methods give similar results, while when vibrations are present LS methods generated slightly better results than the PSI algorithm. The results for both methods worsen significantly as the amplitude of the vibrations increases. We also show that a different set of phase steps than those found by LS method, while not generating minimal error function, can yield a better phase result with no residual phase error.

Self-calibrating lateral scanning white-light interferometer

The concept of lateral scanning white-light interferometer (LSWLI) has been introduced nearly a decade ago [1] as an alternative to the conventional white-light (WL) interferometers [2-14], capable of improved speed and image stitching. The general principle of this type of measurement is shown in Figure 1. A conventional white light interferometer is equipped with an XYZ stage which can perform an accurate lateral (XY) translation. The interferometer objective is tilted with respect to this stage such that the zero optical path difference (OPD) makes an angle α with respect to the direction of the translation. By convention, the tilt angle will be measured from the direction of the translation. For the case when this angle is different than zero, an object placed on the stage will present a specific fringe pattern whose density is dictated by the magnitude of the angle. In Figure 2, a linear fringe pattern obtained from a flat surface is shown. As the profiled object is translated at a constant speed, the CCD will record interference frames at a constant rate. Figure 3 shows how different pixels of the object (marked by up or down pointing arrows) will be recorded in consecutive frames during the object translation. In the case when the CCD frame rate and the stage speed are properly correlated, a given point of the object will be translated by exactly one pixel from one CCD frame to the other. The correlogram of each object point can thus be recovered by taking a diagonal section through the stack of recorded frames (Figure 4). Because during the scan the optical path difference of each point of the sample changes continuously, the LSWLI correlogram looks similar with its counterpart obtained by using WL interferometers. As mentioned before, the LSWLI measurements allow for a continuous data acquisition process, eliminating thus the need for a cumbersome stitching procedure that must be done for large samples when measured by using a standard WL interferometer. It also allows for a faster data acquisition and, in principle, it is possible for very large samples to be measured during a single pass.