4. Phase-shifting interferometry

Interference fringe patterns obtained by means of interferometric techniques can be evaluated by using digital image processing techniques for the estimation of phase map distributions. The intensity of an image in an interference fringe pattern, according with Eq. (7), can be represented by Eq. (13)

$$I(\mathbf{x}, y) = a(\mathbf{x}, y) + b(\mathbf{x}, y) \cos[\phi(\mathbf{x}, y)] \tag{13}$$

where φ(x, y) is the phase difference between the reference and testing wave fronts that interfere, a(x, y) = I1(x, y) + I2(x, y) is the mean intensity and b(x, y) = 2(I1I2) 1/2 is the intensity modulation registered by each pixel (x, y) of a CCD camera. When analyzing a single image for the estimation of the relative phase difference in applied metrology, the sign of the phase cannot be assessed because of the argument in the sinusoidal function of the modulation term in Eq. (13) results in an identical interferogram for ±φ(x, y) values. In order to overcome this difficulty, multiple interferograms can be sequentially registered introducing a known small phase change amount which is linear in time; this method is well known as phase shifting [4]. In practical phase shifting, a piezoelectric device PZT is included in the interferometric system to produce the phase-shifted interferograms. In general, when a voltage signal is used to polarize a PZT actuator, this electrical signal is converted directly into linear displacement motion. Then, phase-displaced intensity images can be represented by

$$I(\mathbf{x}, y) = a(\mathbf{x}, y) + b(\mathbf{x}, y)\cos[\phi(\mathbf{x}, y) + \theta]. \tag{14}$$

Since there are three unknown terms in the representation of the interference intensity equation, then the measurement of at least three interferograms at known phase shifts is needed to determine the relative phase difference. However, one of the simplest modes to determinate the phase considers the use of four interferograms equally spaced by θ = π/2, obtaining:

$$I\_1 = a + b\cos(\phi),\tag{15}$$

$$I\_2 = a \text{--} b \text{sim}(\phi),\tag{16}$$

$$I\_3 = a \text{--} b \text{cos}(\phi),\tag{17}$$

and

attribute obtained from the Fourier transform is that it gives the frequencies content of the image. Due to this property, frequency filter design is a very straight forward task for image processing. Low frequencies are located into the Fourier domain around the central coordinates; as frequencies gradually increase, they spread out from the center in a radial form. This characteristic is ideal for frequency filtering (low-pass, high-pass, band-pass and band-stop) [3]. The frequency filtering development consists in the multiplication between the Fourier transform with some kind convolution function. The kind of convolution mask will determine the class of filtering to be performed. The following is a summary of the most usual filtering masks: a white centered circle for low-pass; a black centered circle for high-pass; a white

In Figure 5, it is showed the masks described above along with the results of the filtering process. Low-pass filtering masks and results are presented in Figure 5b, band-pass filters and filtered images are seen in Figure 5c and finally, band-stop filters and filtering results can be appreciated in Figure 5d. The kind of filter or the size of geometrical mask depends directly in the image and in its noise content. There is no ideal filter; the kind of applied filter to process an image is dependent of the characteristics that are pretended to enhance or eliminate.

Interference fringe patterns obtained by means of interferometric techniques can be evaluated by using digital image processing techniques for the estimation of phase map distributions. The intensity of an image in an interference fringe pattern, according with Eq. (7), can be

where φ(x, y) is the phase difference between the reference and testing wave fronts that

modulation registered by each pixel (x, y) of a CCD camera. When analyzing a single image for the estimation of the relative phase difference in applied metrology, the sign of the phase cannot be assessed because of the argument in the sinusoidal function of the modulation term in Eq. (13) results in an identical interferogram for ±φ(x, y) values. In order to overcome this difficulty, multiple interferograms can be sequentially registered introducing a known small phase change amount which is linear in time; this method is well known as phase shifting [4]. In practical phase shifting, a piezoelectric device PZT is included in the interferometric system to produce the phase-shifted interferograms. In general, when a voltage signal is used to polarize a PZT actuator, this electrical signal is converted directly into linear displacement

Since there are three unknown terms in the representation of the interference intensity equation, then the measurement of at least three interferograms at known phase shifts is needed to

interfere, a(x, y) = I1(x, y) + I2(x, y) is the mean intensity and b(x, y) = 2(I1I2)

motion. Then, phase-displaced intensity images can be represented by

Iðx, yÞ ¼ aðx, yÞ þ bðx, yÞcos½φðx, yÞ� (13)

Iðx, yÞ ¼ aðx, yÞ þ bðx, yÞcos½φðx, yÞ þ θ�: (14)

1/2 is the intensity

centered ring for band-pass and a black centered ring for band-stop.

4. Phase-shifting interferometry

represented by Eq. (13)

8 Optical Interferometry

$$I\_4 = a + b \sin(\phi). \tag{18}$$

Rearranging the simultaneous equation system from the above formulas, in which the spatial dependence (x, y) was not included, the phase distribution from intensity images can be calculated with:

$$
tan(\phi) = \frac{I\_4 - I\_2}{I\_1 - I\_3}.\tag{19}
$$

The phase difference estimated from the four equations is determined in the range between −π and π when using the arctangent function, hence producing a wrapped phase distribution. In order to analyze an interference pattern by the phase-shifting techniques, a Twyman-Green interferometer is considered for the testing of optical elements; the basic arrangement is shown in Figure 6. A laser diode is used in the interferometer as illumination source, and then the emerging laser beam is collimated by a lens to obtain a plane wave front that propagates throughout a 50:50 beam splitter to produce two wave fronts of same amplitude. One beam is deviated to the reference mirror M1, and the second beam, to the mirror M2 under test. Next, the beams are reflected back toward the beam splitter, and part of the intensity overlaps on the

Figure 6. Twyman-Green interferometer for the testing of optical elements.

observation plane, where a CCD camera sensor is placed to register an image of the resulting interferogram that is then stored in a PC for subsequent processing.

In Figure 7, a set of four phase-shifted interferograms with phase shifts of π/2 is shown. A first interferogram with θ = 0 is seen in Figure 7a, and subsequent interferograms with θ = π/2, θ = π and θ = 3π/2 are observed in Figure 7b–d, respectively. The wrapped phase is showed in Figure 7e, and the unwrapped phase related with the shape of the optical element being tested is shown in Figure 7f. A three-dimensional representation of the unwrapped phase seen in Figure 7f is presented in Figure 8.

In applied phase-shifting interferometry, there are concerns about the presence of errors that may affect the accurate phase extraction from phase-shifted interferograms. A typical systematical source of error introduced PZT arises when there is miscalibration of the phaseshifting actuator, causing detuning in the phase extraction process. The inclusion of phaseshifting algorithms with more than three or four interferograms can be implemented in order to reduce this systematic error [5].

Figure 7. Four-step phase-shifted interferograms from (a) to (d), the wrapped phase (e) and the unwrapped phase (f).

Figure 8. The unwrapped continuous phase obtained from a set of four phase-shifted fringe patterns and an unwrapping method.
