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Abstract
In this manuscript it is presented what we believe to be a novel and simple phase-shifting technique based on polarizer rotations, using a right angle prism as the interferometric device. It is possible to generate interference when the right angle edge of the prism is placed parallel to a linearly polarized collimated beam. In this configuration optical phase changes are presented due to the internal reflections in the prism; therefore, a linearly polarized beam turns into elliptical polarization. If the output elliptically polarized interference beam passed through a lineal polarizer, phase-shifting is generated if it rotates by an angle theta. This proposal will be sustained with simulated and experimental shifted interference patterns and with the calculation of wrapped and unwrapped phases.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Phase shifting interferometry (PSI) [1] is a widely used technique for precise optical testing. In PSI, the phase difference between the interfering beams is either changed in discrete steps or it is changed at a constant rate. The irradiance due to the interference of two waves for each point $(x,y)$ can be expressed by
where $a(x,y)$ is a bias intensity, $b(x,y)$ is a modulation intensity, $\phi (x,y)$ is the phase difference between the two interfering waves and ${\psi _j}$ with $j = 0,1,\ldots ,N - 1$ is the phase step generated to obtain N equations. In this way a set of N equations (interferograms) changed in phase are created; then a $\textrm{Nx3}$ system is formed that can be solved when $\textrm{N} \ge \textrm{3}$ [2,3]. There are many well-known methods used to generate ${\psi _j}$ for example: changing the optical path difference by displacing a mirror with a piezoelectric transducer [4,5] or changing the refraction index by means of tilting a glass plate [6], by displacing a diffraction grating, by modulating the amplitude of the reference beams in a three beam interferometer [7] and others [8].An alternative method to generate phase steps is by using the tunability of a laser diode (LD) [9–11], in which the wavelength of a single-mode laser can be changed by the variation of the injection current and/or the temperature of active region. It is possible to modulate the laser current by applying analog voltages to its input, producing a phase shift in the interferograms [12]. This characteristic of the LD has been used for example in speckle shearing interferometer [13], distance measurements [14], surface contouring [15], digital phase measuring [16], among others.
Some other methods are based on changing the direction of polarization under the scheme of Michelson [8], Mach-Zehnder [17] or common path interferometers, just to name a few [18]. Ghosh and K. Bhattacharya [19] proposed a cube beam-splitter interferometric setup with input circular polarized light generated with a polarizer and a quarter wave-plate, with the aim of generating phase-shifting by rotations of a polarizer at its output plane.
In this manuscript it is presented a novel and simple setup under the scheme of a right-angle prism interferometer that can be used for PSI by rotating a linear polarizer at the output plane. This is due to the internal combination of two elliptically polarized light beams, which are generated due to total internal reflection when a linearly polarized beam passes through the prism [20].
2. Principle
In a single right-angle interferometric setup, if its right angle edge is placed parallel to a collimated input beam and perpendicular to its propagation direction, as shown in Fig. 1, internal reflections will generate interference. Unlike a cube-prism interferometric setup [21], this proposal doesn’t perform as a two-window common-path because the interfering beams travel through separate optical paths inside the prism, so this configuration can by affected by unwanted vibrations.
Let’s consider two elliptical polarized light beams formed inside the right-angle prism
With a vectorial treatment of Eq. (1), the irradiance due to the interference of these two beams is given by
If the polarization of the waves turns in opposite directions ${\delta _1} ={-} {\delta _2} = {\delta _0}$, then $\delta = 2{\delta _0}$, so
According to this equation, it is known for the particular case of two circular polarized light with opposite rotation ${\delta _0}$=${\pi / 2}$ and ${b_x} = {b_y}$, then $I = a$. Under these conditions the will be no interference term due to the orthogonality of the components [22].
Equation (3) can be seen as two interference pattern, one of them with x-components and the other with y-components
As a particular case and considering ${a_x} = {a_y} = {b_x} = {b_y} = {a_0}$ and $\delta = \pi $
Again, under these conditions the interference term is not presented.
If a beam defined by the components of Eq. (2) (with ${E_{1x}},{E_{1y}} = {E_1}$ and ${E_{2x}},{E_{2y}} = {E_2}$) traverses a linear polarizer with a transmission axis at an angle $\theta$
Therefore, both elliptically polarized beams can be described by
As a particular case, we will calculate the output irradiance due to the interference of two circularly polarized beams with opposite polarization direction
It can be observed that there will be phase-shifting added by rotating a polarizer at an angle $\theta $. However, in the general case of elliptical polarization and according to Eq. (11), the ${E_x}$ and ${E_y}$ components will be also modulated by $\theta $, affecting the modulation light term in the interferogram.
3. Simulations
In this section numerical simulations (Figs. 1–5) are carried out in order to show which combination of polarization states generates more adequate phase-shifting by rotating a liner polarizer in order to be applied in Phase-shifting interferometry (considering for simplicity that the interfering fields have the same amplitude and angular steps of $\theta = 60^\circ$). Four simulated patterns were evaluated on $x \in ( - 1,1)$ and $y \in ( - 1,1)$ in a rectangular grid of 401 by 401 points for particular cases with different field components phases … In these figures, phase-shifting is indicated as the number of pixels in which a crest of a cross section is translated between consecutive interference patterns.
It can be observed that when the phase difference ${\delta _2} - {\delta _1} = \pi$, a wide range of phase-shifting is obtained. Figure 2 shows the ideal case when two orthogonal circular polarization states interfere and by rotating a polarizer in front of the resulting field, constant phase-shifting with constant amplitude is obtained. However if the phase difference of $\pi$ is still preserved but ${\delta _2},{\delta _1} \ne |{{\pi / 2}} |$ and in accordance with Fig. 3, the resulting phase-shifting can be suitable to use in 3-Step PSI algorithm. This can be verified also in Fig. 6 in which by decreasing the angle of rotation $\theta = 30^\circ$, ${I_3},{I_4},{I_5}$ can be used for phase retrieval purposes.
Fig. 2. Phase-shifting with a phase difference of $\pi $(${\delta _1} = {{ - \pi } / 2}$, ${\delta _2} = {\pi / 2}$).
Fig. 3. Phase-shifting with a phase difference of $\pi $(${\delta _1} = {{ - 5\pi } / 6}$, ${\delta _2} = {\pi / 6}$).
Fig. 4. Phase-shifting with same polarization ellipticity and opposite rotation ${\delta _1} = {{ - \pi } / 3}$, ${\delta _2} = {\pi / 3}$.
Fig. 5. No phase-shifting with equal elliptical polarization and sense of rotation (${\delta _1} = {\pi / 3}$, ${\delta _2} = {\pi / 3}$).
Fig. 6. Simulation of constant and consecutive phase-shifting with ${\delta _1} = {{ - 5\pi } / 6}$, ${\delta _2} = {\pi / 6}$, showing the viability for applying 3-step PSI algorithm.
4. Experimental setup
The setup of the proposal is depicted in Fig. 7 in which internal reflections in the right-angle prism will generate the interference of two elliptically polarized light beams (avoiding the use of extra optical components in order to modify the polarization state of the input beam [23]); therefore phase-shifting is obtained with rotations of the output linear polarizer (P) at an angle $\theta $ (Visualization 1).
Fig. 7. Schematic of the experimental proposal in which phase-shifts are added by rotating a polarizer at the output plane.
According to the simulations shown in Figs. 3 and 6 and in order to get different elliptical polarizations with counter rotations (verified also with a Thorlabs PAX1000 polarimeter), the angle of the linearly polarized input beam is adjusted till ${\delta _2} - {\delta _1} = \pi$ (visually detected when the interference fringes disappear, corresponding to an input polarization angle of 45°). Once this condition is achieved, phase-shifted interference patterns are obtained by rotating the linear polarizer from 0 to 180° at a constant speed and recorded by a CCD camera at a constant interval time. In this simple configuration, the portion of the incoming beam that passes through each side of the prism can be either used as reference or probe beam.
Experimental phase-shifted interference patterns related to the prism without probe object are shown in Fig. 8. Individual visibilities and the phase-shift value ${\psi _j}$ [24] between two consecutive interferograms are also indicated.
Fig. 8. Experimental phase-shifted interferograms by rotating a polarizer in a right-angle prism interferometer.
In order to know the phase of a probe object, it should be placed in either side of the prism as shown in Fig. 7. Considering that the total output phase ${\phi _T}$ contains the object phase ${\phi _o}$ and the phase of the prism itself ${\phi _P}$, then ${\phi _o} = {\phi _T} - {\phi _P}$.
5. Phase retrieval
According to the simulation presented and the experimental implementation (the internal reflections in a right-angle prism generate two elliptical polarized light beams. However their ellipticity changes between sizes, models and manufacturers). Hence, phase-shifts are limited and dependent on each prism. Due to this restriction, the more appropriate PSA in the proposal is a three-step algorithm [25] to calculate the output phase $\phi (x,y)$, for which three irradiance measurements are needed ${I_{1\ldots 3}}(x,y)$
So, for the calculation of the wrapped ${\phi _w}$ and unwrapped phase $\phi$ [26] shown in Fig. 9, three phase-shifted interferograms (190px x 410px) from the above figure where used, obtained with three continuous rotations of the polarizer by $\theta = 30^\circ$.
Fig. 9. Three phase shifted interferograms and the corresponding wrapped ${\phi _w}$ and unwrapped phase $\phi $.
The results are also sustained by also calculating the resulting phase from a phase object with the phase-shifting by using a cube beam splitter and rotating a polarizer at the interferometer output [27]. Figure 10 depicts the schematic o both interferometric setups, in which a phase object (test object) has been place in front of one face of each prism.
Fig. 10. Schematic of the experimental proposal and PSI with CBS used to calculate the retrieved phase of the same object.
The unwrapped phase of an object (Fig. 11c)) was calculated by subtracting the unwrapped phase of the right-angle prism without object (Fig. 11b)) and the unwrapped phase of the prism with the phase object (Fig. 11a)). That is: ${\phi _o} = {\phi _{p\_o}} - {\phi _p}$
Fig. 11. a) The unwrapped phase of the prism + phase object. b) The unwrapped phase of the prism and c) the phase of the object.
Figure 12 depicts the calculated unwrapped phase of the same object (a piece of an acetate sheet), by using a cube beam splitter as the interference device, as shown in Fig. 10. In order to accomplish this task, the phase of the cube prism (${\phi _{cp}}$) was subtracted from the phase of the object + the prism (${\phi _{cp\_o}}$); that is: ${\phi _o} = {\phi _{cp\_o}} - {\phi _{cp}}$; validating with this, the feasibility of the proposal.
It worth mentioning that the cube-beam splitter interferometer was chosen as a validation method due to its simplicity both in the experimental setup and in the way of generating phase-shifts. Besides, these methods (considering the one proposed in this manuscript) use the inherent change in polarization due to the particular characteristics of both prisms. However, another simple method could be used in order to retrieve the phase of an object. For example, by adding carrier fringes or phase-shifting by means of displacing a lens [28] or a Ronchi ruling in a 4f interferometric setup [29].
6. Conclusions
This proposal is based on how elliptically polarized light is formed upon total internal reflection when a linearly polarized beam enters a right-angle prism. In the presented implementation it was concluded that PSI can be achieved when the input beam is linearly polarized at 45°, originating two different elliptical polarizations with counter rotations and with a phase difference between them of 180°. The multiple experimental phase-shifted interferograms obtained with this technique allow increasing the value of ${\psi _j}$ by taking non-consecutive but equal spaced interferograms, by changing the rotation angle $\theta$ or with a combination of both.
The principal feature of the implementation relies on its low-cost which is due to the fact that unlike other proposed experimental arrangements, it uses the minimum of optical components to generate interference and phase-shifts such as a right-angle prism and a linear polarizer, respectively. Therefore, it also presents a simple experimental implementation that allows the proposal to be used for demonstrative purposes in topics such as interference and polarization in graduate or undergraduate levels.
It is important to emphasize and point out that this technique presents constant phase-shifting only when the linear polarizer covers some regions of the output elliptical polarization, so if this implementation is used for PSI, it was found that the three-step algorithm is suitable to compute phase of the output wavefront. However, it could still be possible to obtain an accurate wavefront because techniques and algorithms have been developed for non-constant phase displacements [30], or to allow the estimation of the relative differences in the unknown phase-step [31].
One of the potential applications of the proposed right-angle prism interferometer is that it can test the quality of a prism. That is, a good-quality prism should present straight line interference fringes due to the superposition of plane waves. Also, it can be quantitative tested how elliptical the resulting polarization is at the exit of the prism due to its inner reflections, according to the amount of phase-shifting obtained. Therefore, the present proposal can be used to characterize prisms in order to choose then according to the user needs.
Due to the characteristics, simplicity and cost of the setup, it can be suitable for measuring, research, demonstrative or even educational purposes.
Funding
Universidad Tecnológica de Puebla.
Acknowledgments
We thank the funding given by Universidad Tecnológica de Puebla.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this Letter are not publicly available at this time but may be obtained from the authors upon reasonable request.
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