1. Introduction

Interferometry is a well-known technique for determining the phase profile of phase objects. In particular phase-shifting (PS) [1] has proved to be very efficient technique for phase retrieval.

However, most classical PS interferometers present some drawbacks: Firstly, they present some inherent sensitivity to mechanical vibrations associated to the proliferation of mirrors and beam splitters necessary for bringing the reference and test waves to interfere. And secondly, the phase-shifting mechanisms described in the literature are often associated to mechanical movements of optical elements inside the interferometer, e.g. translation of mirrors with PZT, which affects the stability of the interference between reference and object waves.

Thus, the paradigm of a stable interferometer is a device with the minimal number of optical elements, without moving parts, and with phase-shift control placed outside the interferometer itself. In the past decades, several devices satisfying some of these requirements have been published in the literature (see e.g [2–4]. and the references therein).

The purpose of this work is to present an interferometer architecture somewhat similar to the Jamin-Lebedev configuration [5] with the advantage that only requires a calcite plate as interference element and with a novel external phase-shift control without moving parts. In the proposed device the reference and test waves propagate together in one collimated beam that is incident on a calcite plate, which produces a polarization selective lateral translation and superposition of the waves with a controllable phase difference (as required for applying PS-algorithms with arbitrary number of phase-steps). The proposed interferometer is robust and insensitive to environment vibrations and temperature fluctuations, and does not require carefully alignment to achieve interference. The device is described in detail in Sect. 2. Experimental results are presented in Sect. 3 and conclusions in Sect. 4.

2. Proposed setup

The proposed setup is shown in Fig. 1. The light source is a polarized laser with its polarization direction at ± 45° with respect to the principal polarization directions ($\widehat{x}$ and$\widehat{y}$) of a liquid crystal retarder (LCR), whose birefringence is electrically controlled by a PC. Thus, after the LCR we have two orthogonal linearly polarized waves in directions $\widehat{x}$ and$\widehat{y}$ of the same amplitude $\left(E_0\right)$and with a voltage induced phase difference $\phi \left(\nu \right)$in the interval $\left[0,\pi \right]$, where $\nu$ is the applied voltage.

 

Fig. 1 Proposed interferometer. L₂ is an imaging lens and C is a camera. The inset shows the calcite plate (BD): OA is the optical axis, and d is the lateral displacement of the extraordinary beam.

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The light beam is filtered by a spatial filter (SF) and collimated by the lens L₁. Then, the light beam travels through a test sample (T), e.g. a pure phase object, so that the electric field after T can be written as

Then the light beam passes through a polarizer (P) with its transmission direction at 45° with respect to the$\widehat{x}$and$\widehat{y}$directions. Disregarding a factor $1/\sqrt{2}$, after the polarizer the field amplitude will be

The lens L₂ images the plane of the test object across the photodetector array of the camera (C), so that the intensity distribution acquired by the camera will be $I\left(x,y\right)={\left|E_3\left(x,y\right)\right|}^2$, i.e.

Let D be the size of the phase object along the $x$-direction (see Fig. 1). If $d<<D$, the two replicas of the phase object, $\varphi \left(x,y\right)$and$\varphi \left(x-d,y\right)$, partially superpose, so that in the limit it results $\varphi \left(x,y\right)-\varphi \left(x-d,y\right)\approx \frac{\partial \varphi }{\partial x}d$ . And thus, the described interferometer becomes a shear (Nomarski’s) interferometer [6].

In our case, on the contrary, we assume that $d\ge D$ so that $\varphi \left(x,y\right)$ and $-\varphi \left(x-d,y\right)$ does not superpose. Thus, the proposed device produces simultaneously two laterally separated complementary interferograms of the same test object, as schematically illustrated in the inset of Fig. 1. Specifically, if the phase object verifies $\left|\varphi \left(x,y\right)\right|\ne 0$ for $d<x<2d$ and zero elsewhere, one can consider two independent interferograms of the same phase object,

3 Experimental results

The electro-optical phase-shift mechanism was a liquid crystal retarder (LCR, Thorlabs Inc.) connected to a power supply controlled by a computer. The phase sample (T) was fabricated onto an ITO-layer (150nm width) deposited on a glass substrate; the selected design –a capital letter “F” with a size $D\approx$2.8mm – was imprinted by etching in the ITO-layer (a detailed description is in [7]). The calcite plate (BD40, Thorlabs Inc.) has a width of 41mm and generates a lateral displacement $d=4$ mm of the extraordinary with respect to the ordinary light beam. The images were acquired using a CCD camera of 1024 × 768 pixels. The lenses L_1,2 have focal lengths 50cm and 15cm, respectively.

In a first series of experiments we measured the phase-shift (between the waves polarized in directions $\widehat{x}$ and$\widehat{y}$) produced by the liquid crystal retarder (LCR) as function of the applied voltage ($v$) in the range 0-20 volts. In order to obtain the calibration curve, we generated a high density of interference fringes at the system output by suppressing momentarily the collimating lens (L₁), and measuring the fringe-shift as function of the applied voltage. The measured phase-shift versus the applied voltage is shown in Fig. 2.

 

Fig. 2 Experimental calibration curve.

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In order to demonstrate the mechanical stability of the proposed interferometric optical scheme in a long term experiment, we mounted the interferometer over a table without special mechanism to preclude vibration noise. We found that in a time interval of two hours the interference pattern fluctuates less than 1/25 of fringe.

In order to compare the proposed setup with other optical architectures, we performed experiments involving simultaneously two interferometers. Next to the calcite interferometer, we built a standard Mach–Zehnder interferometer using similar optical and mechanical elements. On the same table we mounted a mechanical oscillator –a loudspeaker – working at 125Hz. The fluctuations of the interference patterns due to the strong mechanical vibrations induced by the loudspeaker were measured with the help of point-like photodetectors connected to a computer. Figure 3 shows simultaneously the fringe displacements of both interferometers.

 

Fig. 3 Fringe displacement due to mechanical vibrations of the Mach-Zehnder (blue curve) and the proposed interferometer (red curve).

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Thus, in principle the proposed optical architecture is inherently more stable than the standard ones, e.g. Mach–Zehnder. [We realize that the experimental results shown in Fig. 3 have only an illustrative value, because the mechanical stability of a particular optical architecture depends largely on the specific mechanical mounting of the optical elements.]

The Mach–Zehnder interferometer shows excursions of the order of 2-3 fringes per period of mechanical oscillation, while the calcite interferometer shows excursions of a fraction of fringe (1/10 of fringe or less).

In a third series of experiments we measured the 3D phase profile of the phase sample etched on the ITO-layer. We performed the acquisition of fifty interferograms applying a voltage to the LCR in the range 0-20 volts. Figures 4(a)-4(c) are excerpts from a video sequence (see Visualization 1) showing the fifty interferograms. The interferograms ($I_{1,2,3}$) shown in Figs. 4(a)-4(c) were obtained for the (arbitrary) voltage values ${\nu }_1=0$volt, ${\nu }_2=2.0$volt and ${\nu }_3=7.3$ volt. Without loss of generality one can assume $\phi \left({\nu }_1\right)=0$(rad), and thus, from the calibration curve one gets $\phi \left({\nu }_2\right)=1.49$ (rad) and $\phi \left({\nu }_3\right)=3.17$ (rad). As expected, the interferograms show two replicas of the same phase object. [The inverted replica (Eq. (6) is generated by the extraordinary wave and is inherently blurred.] Figs. 4(d)-4(f) show three additional interferograms with the same phase-shifts but without test sample. These interferograms show a low-contrast spurious interference pattern, which is probably due to multiple reflections in the collimating lens L₁.

 

Fig. 4 (a)-(c) Interferograms showing the two replicas of the phase sample, acquired for$\phi \left({\nu }_1\right)=0$(rad), $\phi \left({\nu }_2\right)=1.49$ (rad) and $\phi \left({\nu }_3\right)=3.17$ (rad), respectively; (d)-(f) Interferograms acquired without the test sample showing spurious low-contrast interference.

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In order to retrieve the true phase profile of the sample, this spurious phase pattern must be subtracted from the phase obtained from $I_{1,2,3}$.

For reconstructing the phase profile of the test object we used the generalized 3-frame, 10-frame, 30-frame and 50-frame algorithms for unevenly spaced phase-shifts developed by Ayubi et al. For the case of three frames we used the algorithm described in [8] Eq. (16), and for the other cases we used the algorithms described in [9] Sect. 4 (see the erratum in [10]). The results obtained with the different algorithms do not present significant differences:

The RMS difference between the 3-, 10-, 30-frame algorithm and the 50-frame one is shown in Table 1. The reconstructed 3D phase profile of the sample using the 50-frame algorithm is depicted in Fig. 5, which is consistent with an ITO refractive index of the order of 1.7 and a depth $\le 150$ nm.

Table 1. RMS versus No. of frames.

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Fig. 5 Reconstructed phase profile of the test object (in false color). The figure on the right side show a horizontal cut of the phase profile along the dashed line.

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4 Conclusions

We presented a robust interferometer with an electro-optical phase-shift control (without moving parts). In principle the phase-shift mechanism could reach high phase modulation rates which is potentially useful for the study of dynamical processes. The main advantage of the proposed system is that the phase retarder is placed before the spatial filter, i.e. it lies outside the interferometer, so it does not need to have neither high optical quality nor high numerical aperture, which reduces the cost of the system.

Unlike a conventional interferometric scheme, the proposed setup involves few optical elements and does not involve mirrors and/or beamsplitters which are prone to induce mechanical fluctuations. Also, both orthogonally polarized waves (reference and sample wave) propagate together as parts of the same beam, so that the proposed device is more stable against thermal and mechanical fluctuations of the optical path lengths than other interferometers reported in the literature. The drawbacks of the proposed architecture are the small working area (depending on the size of the calcite plate), and the inherently low numerical aperture and image resolution. Despite these drawbacks, the proposed interferometer is conceptually simple, mechanically stable and easy to adjust.