## Abstract

A simple and inexpensive optical setup to phase-shifting interferometry is proposed. This optical setup is based on the Twyman-Green Interferometer where the phase shift is induced by the lateral displacement of the point laser source. A theoretical explanation of the induced phase by this alternative method is given. The experimental results are consistent with the theoretical expectations. Both, the phase shift and the wrapped phase are recovered by a generalized phase-shifting algorithm from two or more interferograms with arbitrary and unknown phase shift. The experimental and theoretical results show the feasibility of this unused phase-shifting technique.

© 2013 OSA

## 1. Introduction

Phase-Shifting Interferometry (PSI) has become a versatile powerful tool used widely in a variety of precise optical measurement applications [1]. In contrast with the spatial carrier based methods, PSI is especially attractive because it can ensure major sensitivity, higher accuracy, and maximum spatial resolution of the calculated phase [2].

The principle of the PSI method is very simple, it consists of the following two steps: First, an interferogram is recorded for each phase step induced to the reference beam. Second, the acquired phase shifted interferograms are processed to obtain the wrapped phase distribution. Note that, currently, it is not necessary to induce calibrated and regular phase-shifting because the so called Generalized Phase-Shifting Interferometry (GPSI) algorithms can process phase shifted interferograms with arbitrary and unknown phase steps [3, 4, 5]. This allows focusing only on simplified critical optical setup issues; in particular, on the phase shifter technology.

The piezoelectric transducer (PZT) is the most common element applied as the phase shifter in PSI techniques [6, 7, 8]. However, the piezoelectric devices are expensive elements and, additionally, they require amplifiers and control systems. Another alternative phase shifter uses liquid-crystal phase modulators [9], diffraction gratings [10], Bragg cells [11], polarizing elements [12], multiple wavelengths [13] or wavelength-shifting [14, 15], amplitude modulators [16], among others. Unfortunately, these methods demand complex and delicate equipment. A different approach exploits the environmental vibrations [17]; however, the computer processing is time consuming.

In this work, a simple and inexpensive phase-shifting technique is proposed. This technique is based on a phase shift induced by changing the angle of illumination into a plane parallel interferometer [18]. This effect is exploited in the Twyman-Green Interferometer where a point laser source is moved perpendicularly to the optical axis. It is worth mentioning that the configuration proposed is different from inducing a tilt in the reference mirror (spatial carrier) or shearing interference.

This paper is organized as follows: First, we describe the principle behind our technique. Next, we present some experimental results that demonstrate the validity of the proposal. Finally, a conclusion is presented.

## 2. Principles of the phase-shifting by lateral source displacement

In this section the principles of the phase-shifting induction by a lateral displacement of the light source is given. We explain and show the application of it using the Twyman-Green setup for PSI.

A centered point laser source, in the configuration shown in Fig. 1(a), is usually employed as the illumination system in the Twyman-Green Interferometer. Here, we consider the case where the point source is laterally displaced an amount given by *d⃗* = *dd̂*, where *d̂* is a unit vector perpendicular to the optical axis, as it is shown in Fig. 1(b). The spherical wavefront which reaches the input plane of the lens *L* is given by

*p⃗*= (

*x*,

*y*) is a spatial variable with Cartesian components

*x*and

*y*,

*r*= ||

*r⃗*|| = ||

*f⃗*+

*p⃗*−

*d⃗*|| (where ||·|| denoting the Euclidean norm) is the distance between the point source at the point

*p⃗*in the input plane of the lens

*L*,

*A*

_{0}is a real value amplitude,

*k*= 2

*π*/

*λ*is the wavenumber,

*λ*is the wavelength, and

*i*is the imaginary unit (

*i*

^{2}= −1). We consider the paraxial approximation of

*A*(

*p⃗*,

*d⃗*) as:

*B*

_{0}= (

*A*

_{0}/

*f*)exp[

*ikf*] is a complex amplitude and

*p*= ||

*p⃗*||. Due to the propagation through the lens

*L*, the amplitude

*A*(

*p⃗*,

*d⃗*) left its quadratic phase term exp[

*ikp*

^{2}/(2

*f*)] out. Thus, at the output plane of

*L*we have the tilted plane wavefront:

Now, the Twyman-Green Interferometer is illuminated with *B*(*p⃗*, *d⃗*) as shown in Fig. 1(c) or, equivalently, Fig. 1(d). It is not loss of generality to suppose that the collimated beam *B*(*p⃗*, *d⃗*) in the output plane of lens *L* is at the plane of the mirror *M _{r}* because the additional constant phase due to the propagation from

*L*to

*M*can be dropped. Thus, for the point

_{r}*p⃗*in the observation plane

*OP*, we have the intensity

*I*(

*p⃗*,

*d⃗*) due to the interference of the beams

*B*(

*p⃗*

_{2},

*d⃗*) and

*B̃*(

*p⃗*

_{2},

*d⃗*) as

*B̃*(

*p⃗*

_{2},

*d⃗*) is the beam

*B*(

*p⃗*

_{1},

*d⃗*) reflected from the test mirror, namely

*ρ*= 2

*D*[1+(

*d / f*)

^{2}]

^{1/2}and the test mirror’s aberrations

*ϕ*(

*p⃗*

_{0}). Since

*p⃗*

_{2}=

*p⃗*

_{1}+

*σ⃗*, where

*σ⃗*=

*σd̂*with

*σ*= −2

*Dd / f*is a translation, we can rewrite the beam

*B*(

*p⃗*

_{2},

*d⃗*) as

*p⃗*

_{0}=

*p⃗*−

*τ⃗*, and going through some algebraic operations, we can rewrite the Eq. (4) as

*δ*(

*d*) = −

*kρ*−

*kσd / f*is the phase shift, and

*τ⃗*is an linear image translation given by Substituting the variables

*ρ*and

*σ*in

*δ*(

*d*), and since

*d*

^{2}/

*f*

^{2}≪ 1, the phase shift

*δ*(

*d*) can be approximated to where the constant offset phase −2

*Dk*was omitted.

From Eq. (7) we can see that, when the magnitude of *d⃗* changes, two effects appear simultaneously in the interference intensity *I*(*p⃗*, *d⃗*). The first one is the quadratic phase shift *δ*(*d*), given by Eq. (9). The second one is the linear translation *τ⃗*, given by Eq. (8), of the fringe-pattern. Both effects are depicted in Fig. 2.

It is worth mentioning that the phase shifter sensibility can be tuned by the relative distance *D*. For example, for a lateral source displacement of 2 mm, a lens with *f* = 0.5 m, and a laser source with *λ* = 633 nm; we can obtain a phase shift of 2*π* rad if the distance *D* is set to *δf*^{2}/(*kd*^{2}) = 3.96 cm. If more phase shift gain is required, a greater distance *D* is need and vice versa.

The Eq. (9) and its application for phase-shifting interferometry is the main result of this paper. It predicts a phase shift by changing the location of the source which to the best of our knowledge has not been previously considered. In the next section we are going to give the experimental results which show the feasibility of this principle.

## 3. Optical experiment

The feasibility of this proposal is experimentally verified as follows. We consider the standard Twyman-Green Interferometer arrangement. The light source used was a He–Ne laser, with wavelength *λ* = 633 nm. The laser beam was expanded and filtered by a microscope objective and a pinhole, respectively. The pinhole was located at the focal point of a collimating lens with focal length of *f* = 0.5 m.

The collimated wavefront obtained was split by a non-polarizing cube beamsplitter. Two beams were produced and they were reflected by the reference *M _{r}* and test

*M*mirrors. The test mirror surface was deformed in order to obtain a distorted wavefront. The fringe pattern due to the interference of these two reflected beams was observed on a screen on the observation plane

_{t}*OP*. The fringe patterns was acquired by a gray-scale 8-bit CCD camera with a resolution of 768 × 1008 pixels.

The lateral source displacement was performed by mounting the laser source, the microscope objective and the pinhole on a manual linear translation stage. The displacement resolution reached with this mechanism is of 10 *μ*m. We considered a lateral source displacement of *d* = ±2 mm with steps of 100 *μ*m. We choose the relative distance *D* = 3.96 and 7.91 cm between the mirrors in order to obtain a phase shift *δ*(2 mm) = 2*π* and 4*π* rad, respectively. The observation plane is placed to the distance *g* = 11 cm from the reference mirror *M _{r}*. For each progressive displacement step, a phase shifted interferogram was recorded. Thus, for each value of

*D*, 41 interferograms were acquired. Of these interferograms, we show two adjacent interferograms in Fig. 3 as an example.

The interferograms were processed with the GPSI algorithm reported in [5]. The calculated phase shift and the image translation are shown in Figs. 4(a) and 4(b), respectively. The obtained experimental data values presents a mean error of 0.06 rad (with standard deviation of 0.10 rad) for the phase shift, and 1.31 *μ*m (with standard deviation of 11.42 *μ*m) for the image translation. These results are good considering that the displacement is induced manually.

Both the nonlinear phase shift and its deviation from the nominal values are not a problem because an appropriate GPSI algorithm can be implemented. In this work, we used the GPSI algorithm by parameter estimation [5]. Thereby, only two interferograms with an arbitrary and unknown phase shift are sufficient to phase demodulation. For example, the results obtained with this algorithm when it processes the two interferograms shown in Figs. 3(a) and 3(b) are: a phase step of 0.898 rad, and the wrapped phase distribution shown in Fig. 3(c).

With respect to the image translation issue, it is very small (in the described experiment, *τ* = 37.82 and 29.92 *μ*m for *D* = 7.91 and 3.96 cm, respectively). In addition, a scaling of this translation is performed by the camera’s imaging system. In our particular case, the size of the interferograms was of 3 × 3.9 cm and the target’s size of 2.7 × 3.5 mm. Thus, the image translation is reduced to 3.4 and 2.7 *μ*m for *D* = 7.91 and 3.96 cm, respectively. But, since the pixel size is 3.5 *μ*m, such translations are not observable. Moreover, for large translations, because the translation is a linear function of the displacement, the numerical correction is very simple and consists of a translation of all the pixels of the interferogram by a certain number of pixels.

## 4. Conclusion

A simple and inexpensive phase shifter by lateral source displacement to phase-shifting interferometry was proposed. Unlike the conventional PZT techniques, where fine nanometric translations are required, in this novel technique a coarse and miscalibrated translation stage obtained with a micrometric screw is sufficient. The phase shifter sensibility can be tuned by the optical path difference between the interferometer’s mirrors.

Some phase shift problems such as the quadratic phase shift, miscalibration, and other unknown possible error sources are overcome by an appropriate Generalized Phase-Shifting Interferometry (GPSI) algorithm to phase demodulation. In this work, the automatic real-time GPSI algorithm by parameter estimation was used.

The translation of the interferogram image can be easily numerically corrected by a simple pixels shift. Even, the translation effect can be negligible by either setting the relative distance between the mirrors or adjusting the camera’s image amplification.

A successful implementation of this technique in the Twyman-Green Interferometer was reported. The experimental results show that the proposed scheme is a simple and inexpensive alternative to interferometrical phase evaluation. We believe that other interferometric systems could incorporates this approach.

## Acknowledgments

This work was partially supported by VIEP-BUAP. R. Juarez-Salazar and V. Ixba-Santos appreciate the scholarship from Consejo Nacional de Ciencia y Tecnología, México and PROMEP. Authors thank N. Keranen for her advice on wording.

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