## Abstract

In digital phase-shifting interferometry the phase difference between the two interfering beams is varied in a known manner, and measurements are made of the intensity distribution across the pupil corresponding to at least three different phase shifts. If the values of these phase shifts are known, it is possible to calculate the original phase difference between the interfering beams.[1]–[4]

A simple way to introduce these phase shifts is to mount one of the mirrors of the interferometer on a piezoelectric transducer (PZT) and apply suitable voltages to the PZT, Accurate calibration of the PZT is then very important to obtain the desired phase shifts between data frames.[5] Two basic problems are encountered in this connection. The first is the unknown sensitivity of the PZT. The second is variations in the response of the PZT across a diameter which can introduce a tilt and result in a varying phase shift across the pupil.

These problems can be overcome by using an algorithm which implicitly evaluates the actual phase shifts at each point and uses them to calculate the values of the original phase difference between the interfering beams. The simplest form of such a self-calibrating algorithm requires four measurements of the intensity at each point corresponding to three equal phase steps.[6],[7] If *φ* is the original phase difference between the two beams in the interferometer and *I*_{1}, *I*_{2}, *I*_{3}, and *I*_{4} are the intensities corresponding to additional phase shifts of −3*α*, −*α*, +*α*, and +3*α*, respectively, we have

However, for measurements of the highest accuracy on flat and spherical surfaces, it is desirable to work with an almost uniform field. Difficulties can then arise with Eqs. (1) and (2) if the original phase difference between the two beams is close to *mπ*, where *m* is an integer. In these conditions, the numerator and denominator of Eqs. (1) and (2) tend to zero, increasing the uncertainty in the values of tan*α* and tan*φ*.

This problem is avoided with a phase-calculation algorithm which uses five measurements of the intensity *I*_{1}, *I*_{2}, *I*_{3}, *I*_{4}, and *I*_{5}, corresponding to additional phase steps of −2*α*, −*α*, 0, +*α*, and +2*α*, respectively. We can then write

where *A* and *B* are the intensities of the two beams. These equations yield the result

It can be shown that the variation of the right-hand side of Eq. (4) with *α* drops to zero when *α* = 90°. Equation (4) then reduces to

Since it is not possible for the numerator and denominator of Eq. (5) to go simultaneously to zero, it can be used for all values of *φ*.

The possibility of using this simple algorithm with five intensity measurements has been suggested by Schwider *et al*.,[5] but they appear to have evaluated the residual errors incorrectly and overlooked the fact that it gives very small errors for quite significant deviations of the phase step *α* from a nominal value of 90°. We shall assume that the PZT is adjusted initially so that the phase step *α* is nominally equal to 90°; that is, *α* = (*π*/2) + *∊*, where *∊* is a small quantity. It can then be shown from Eq. (4) that for an error *∊* in the phase step, the value of the original phase difference obtained from Eq. (5) is given to a first approximation by the relation

Accordingly, the error in the calculated value of *φ* is

It follows from Eq. (7) that a deviation in the phase step of 2° from its nominal value of 90° results in a maximum error in the calculated value of the phase difference of only ±0.02°. This value of the error is confirmed by a numerical calculation. On the other hand, with a conventional algorithm using three values of the intensity, the same deviation of the phase step from its nominal value gives an error of ±1°. The ability of this algorithm to compensate for quite large deviations in the phase step from its nominal value of 90° is shown by Fig. 1, which is derived from numerical calculations and shows the error as a function of *φ* for a phase step of 95°.

The use of this algorithm with five intensity readings also reduces substantially the effects of deviations from linearity of the PZT. We have evaluated the residual errors numerically in three typical cases characterized by the responses shown in Figs. 2(a)–(c). In the first case, shown in Fig. 2(a), the PZT has a nonlinear response with deviations from linearity of ±1° at nominal phase shifts of ±90°, respectively. Figure 3 shows that the maximum error in the value of *φ* in this case is less than +0.005°. In the second case, shown in Fig. 2(b), the PZT exhibits hysteresis with deviations from linearity of ±2° when the phase shift is nominally equal to 0 and ±1° when the phase shift is nominally equal to 90°. The prinicpal phase error then is an offset of ±1.00°, with a residual error due to nonlinearity which is less than ±0.005°. Since the offset is uniform over the whole field, it can be neglected for many purposes. However, since the sign of the offset depends on the direction in which the loop is traversed, it is possible to eliminate the offset where necessary by taking a second set of readings in the reverse order and averaging the two sets of values.

Another possible source of error, shown schematically in Fig. 2(c), is a linear drift of the PZT. The effects of such a drift can also be eliminated by taking a second set of readings in the reverse sequence and averaging the two sets of values. We have evaluated the residual errors numerically for a drift of 4° corresponding to actual values of phase shifts of −178, −88.5, 1, 90.5, and 180° in the first set of readings and 180, 89.5, −1, −91.5, and −182° in the second set. In this case the maximum error in *φ*, after averaging the two sets of readings, is only 0.001°.

The preliminary calibration of the PZT to ensure that the phase step is approximately equal to 90° can be carried out with four out of the five values of the intensity.[5] For this calibration, we use the equation

To avoid errors arising when both the numerator and denominator of Eq. (8) are close to zero, a few fringes are introduced across the field and data for those points for which (*I*_{4} − *I*_{2}) is less than a specified threshold are rejected. The mean value of the phase step is obtained by averaging the results over a number of data points. The voltages applied to the PZT are then adjusted so that cos*α* = 0.

Implementation of this algorithm with a microcomputer presents no problems, apart from the requirement for additional memory space to store one more frame. It can actually result in a decrease in computing time, since the formula used for calculating the phase difference is much simpler than Eq. (2). The range of movement required of the PZT (±180°) is also very similar to that required when Eq. (2) is used.

## Figures

## References

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**3. **L. M. Frantz, A. A. Sawchuk, and W. van der Ohe, “Optical Phase Measurement in Real Time,” Appl. Opt. **18**, 3301 (1979). [CrossRef] [PubMed]

**4. **P. Hariharan, B. F. Oreb, and N. Brown, “A Digital Phase-Measurement System for Real-Time Holographic Interferometry,” Opt. Commun. **41**, 393 (1982). [CrossRef]

**5. **J. Schwider, R. Burow, K-E. Elssner, J. Grzanna, R. Spolaczyk, and K. Merkel, “Digital Wave-Front Measuring Interferometry: Some Systematic Error Sources,” Appl. Opt. **22**, 3421 (1983). [CrossRef] [PubMed]

**6. **P. Carré, “Installation et Utilisation du Comparateur Photoelectrique et Interferentiel du Bureau International des Poids et Measures,” Metrologia **2**, 13 (1966). [CrossRef]

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