However, they typically present aThe Lissajous figures. Piezoelectric translators (PZTs) are very often used as phase shifters in interferometry. The Lissajous figures are mainly used in analogue electronics to analyse the intersection of two or more. The Lissajous figure is of high importance in physics in order to study the sinusoidal waves. The Lissajous figures use mainly consist of measurement of the frequency and measurement of the phase difference.

The equation of the contour of the figure is quadratic and one gets atmost. Speed of Sound Using Lissajous FiguresAppendix I Lissajous figures in some pathological cases Let us represent the. We have used this technique in our Physics of Music class for nonscience stu-dents, both as a way to measure the speed of sound and to introduce discussion of some basic concepts such as phase differences between waves. If B:E 1:1, It makes the circle.sound using Lissajous figures that can be readi-ly shown as a lecture demonstration.

A less expensive approach uses a non-linear calibration. This approach works well, but makes the device more complexAnd expensive. The use of an additional resistive or capacitive sensor make possible toLinearize the response of the PZT by feedback control.

By fitting an ellipsis equation to the resulting 2D ellipsis it is possible toAccurately compute the resulting phase value for each interferogram. It can be verified that the resulting NdimensionalEllipsis lays in a 2D plane. N-dimensional Lissajous figures are formed when N sinusoidal signals areCombined in an N-dimensional space, where one signal is assigned to each axis.

Despite the positive aspects, PZTs also have drawbacks. They are frequently used to translate mirrors by a very small amount, usually in the sub-wavelength range, to produce phase shifts in interferometry. The results in speckle interferometry show that, after two or three interactions, the calibration error isPiezoelectric translators (PZT) are quite compact and stiff elements that present small expansions of their dimensions in response to an applied charge to it.

As result, a set of calibrated voltage values, to produce previously defined phase shifts, are determined. No a priori knowledge of the PZT behavior is needed. Hildebrand , Okada and Cundy have developed different approaches to retrieve phase values of interferograms in the absence of a rigorous PZT calibration.This paper presents a calibration approach to handle the nonlinear and hysteresis behaviors of PZTs using multidimensional Lissajous figures. However, the resulting device, and the control electronics, become more expensive and complex.Previous works have reported different strategies for calibration of piezoelectric actuators to reduce measurement errors. That solution is commercially available, and produces very good results. Both nonlinear response and hysteresis can be minimized by adding resistive or capacitive sensors to a PZT device by a closed loop feedback control.

For example, if the light intensity of a third pixel in the interferogram is now associated with the z axis, a three dimensional (3D) Lissajous parametric figure is obtained. Instead of using the xy plane only, additional dimensions can be added to form N-dimensional Lissajous figures. To use more data, the authors introduce the concept of generalized N-dimensional Lissajous figures. A better idea is using several of them instead of only two. In the real world, there are imperfections, like noise, non-linear sensor responses, quantization errors, that make phase determination more difficult.The image of an interferogram has typically from few to several hundreds of thousand pixels.

When badly-behaved pixel positions are chosen, the corresponding dimensions will not participate actively in the figure formation. Among those 250 pixels, most likely some will have a bad behavior. For example, if 250 pixel positions are selected from a interferogram, for every value of the phase Оё an ellipsis point in a 250-dimensional space will be obtained. The resulting N-dimensional Lissajous figures, formed by N signals with same frequency, will be also an ellipsis in an N-dimensional space.From the statistical point of view, this approach is quite robust. Before continuing the discussion, letвЂ™s change the axis notation do X 1, X 2, X 3, вЂ¦ instead of x, y, z, вЂ¦ A fourth, fifth, sixth, and, finally, a N th additional signal and axis can be added to form a N-dimensional Lissajous figure.

To do that, a orthonormal base of two N-dimensional vectors is formed from a set of at least three non-aligned N-dimensional points. The N-dimensional points of the ellipsis are first projected on the 2D plane. The point location in the N-dimensional space will follow the 2D ellipsis path as Оё varies from 0В° to 360В°.In order to compute the ellipsis parameters, the coordinates of a minimum of 5 N-dimensional points are acquired for different unknown values of Оё.

Both can be minimized by using an additional sensor and a feedback loop control. Quite expressive nonlinear behavior and hysteresis are frequently observed. Its response is not ideal. The mirror displacement is a function of the applied electric signal. Therefore, the first orthonormal vector n 1 is computed by equation (10), where the symbol вЂ– вЂ– means the norm (length) of a vector.Piezoelectric translators (PZT) are frequently used to move mirrors for a very small amount in order to apply a relative phase shift between two interfering signals. Let T g and T h be those points.

The nonlinear calibration algorithm is performed in the seven steps S1 to S7 described next:It is not an easy task to compensate the PZT hysteresis effect. An approximate value is sufficient. This is a quite reasonable and feasible assumption in most optical systems.A linear calibration constant, that converts voltage values applied to the PZT into approximate phase values, is required as a starting point. No reference displacement standard is required.A basic assumption of this approach is that the phase shift resulting of the PZT motion is uniform over the entire image, i.e, the same value of phase shifting is simultaneously applied to all pixels of an interferogram. The idea is to determine a correction function that turns the response of the PZT linear.

Other phase increments also can be adapted. However, if only the values for 0В°, 90В°, 180В° and 270В° are required, they can extracted from the set of 12 values. Indeed, F( Оё) is a nonlinear PZT calibration function determined in the same way the PZT will be used.This algorithm is very convenient for a 12 equally spaced phase steps algorithm, what is the case of interest of the authors. However, since the four degrees polynomial function F( Оё) is able to represent this nonlinear effect, the hysteresis handling is done automatically by the algorithm. That is a basic assumption to handle the hysteresis effect.The strongest nonlinear PZT behavior due to hysteresis is expected when the PZT comes from V 11 to V 0 and moves to V 1. However, if a same sequence of voltages is repeated, the displaced value is nicely repeatable.

The reference phase values are in column two. The first column has the resulting voltage values due to nonlinear calibration algorithm. This approach must be repeated in the same way during calibration time as well as during usage time.вЂ“ Annular region of interest for the DSPI experiment.Table 1 shows the main results. Therefore, the PZT reaches the voltage V 0 coming from a lower voltage, what reduces substantially the hysteresis effect.

Figure 7 plot both errors as functions of the reference phase value. The errors range from -0.46В° to 0.57В°. Note that the performance is a lot better. Finally, the last two columns show the results for a nonlinear calibration using the proposed algorithm. The errors are ranging from -4.6В° to 2.7В°.

Lissajous Figures Equation