## Abstract

We presented the interference setup which can produce interesting two-dimensional patterns in polarization state of the resulting light wave emerging from the setup. The main element of our setup is the Wollaston prism which gives two plane, linearly polarized waves (eigenwaves of both Wollaston’s wedges) with linearly changed phase difference between them (along the *x*-axis). The third wave coming from the second arm of proposed polarization interferometer is linearly or circularly polarized with linearly changed phase difference along the *y*-axis. The interference of three plane waves with different polarization states (LLL – linear-linear-linear or LLC – linear-linear-circular) and variable change difference produce two-dimensional light polarization and phase distributions with some characteristic points and lines which can be claimed to constitute singularities of different types. The aim of this article is to find all kind of these phase and polarization singularities as well as their classification. We postulated in our theoretical simulations and verified in our experiments different kinds of polarization singularities, depending on which polarization parameter was considered (the azimuth and ellipticity angles or the diagonal and phase angles). We also observed the phase singularities as well as the isolated zero intensity points which resulted from the polarization singularities when the proper analyzer was used at the end of the setup. The classification of all these singularities as well as their relationships were analyzed and described.

©2012 Optical Society of America

## 1. Introduction

Optical point singularities, also called optical vortices (OV) [1], have recently become a subject of numerous investigations. They were used to develop some devices based on existing single optical vortex points or on the creation of entire lattices formed by numerous optical vortex points [2]. The second approach allowed developing a new kind of interferometry and the development of a new device – Optical Vortex Interferometer (OVI) – which uses the optical vortices as phase markers. Interferometry with phase singularities allows obtaining a new type of measurement standard with a stable and regular markers lattice. This lattice can be generated in many different ways, for example, by way of nonlinear optical phenomena [3] and three homogenous plane-waves interference [4]. A new approach to this generation techniques, using polarization elements and formation of a new kind of singularities of polarization type, was also presented [5–8].

The phase singularities can be investigated using the scalar description of a light field. Much more complicated and less investigated is the vector approach, which has to be used for the description of polarization type singularities. The morphology and relationships between topological characteristics of polarization singularities have been studied by a narrow circle of scientists [9–13]. Most of these studies concern the cases of computer generated polarization fields, however, the authors of this article attempted to present two setups for producing a regular lattice of polarization singularities [14]. These setups include Wollaston prisms as well as special circular compensators. They generated L- or C-type singularity points, which form the lattice of the same shape and behavior. What is more, we have shown the equivalence of L- and C-type singularity points in the measurement setup using two different polarization descriptions: phase and diagonal angles set ($\delta ,\beta $) for L-points equivalent to azimuth and ellipticity angles set ($\alpha ,\vartheta $) for C-points. Both of these polarization markers can be detected as phase point singularities using a proper analyzer (circular for C-points and linear for L-points). What is more, the existence and evaluation of L-points in the polarized light field could be described in the same manner as C-points by indefiniteness of one of the polarization parameters using different polarization descriptions. Our first attempt made us take a closer look at our other setup presented earlier as an alternative version of three homogenous plane-waves interference [7]. Application of Wollaston prism allowed using the simple Mach-Zehnder interferometer to obtain the stable lattice of optical singularities identical to that derived with three plane-waves interference. However, the use of the linear analyzer at the output of the proposed system revealed that the observed singularities are of phase character – all information about the polarization state of the interfering waves were lost. The results obtained in [14] prompted us to look more closely to the polarization state distribution of the light field just prior to the analyzer in the setup presented in [7]. We were hoping that we would find more interesting structures (such as points, lines, etc.) in a polarization field than in a simple phase distribution.

## 2. Theoretical analysis

Let us present the setup used in [7] with some modifications that will allow us to set up more complex distributions of the light polarization state. The scheme of the proposed setup is presented in Fig. 1
. Four beam splitters BS form a classical Mach-Zehnder interferometer. The linear polarizer P with the azimuth angle equal to 45° at the input of the setup allow generating both Wollaston prism’s eigenwaves with the same amplitude (note that the azimuth angle of the Wollaston prism’s wedge is equal to 0° so we assume that the waves coming out of the prism are inclined relative to the *x*-axis). The Wollaston prism WP is placed in one of the interferometer’s arm (we called it “a reference arm” in [7]) resulting in two plane, linearly polarized waves (eigenwaves of both Wollaston’s wedges) with linearly changed phase difference between them (along the *x*-axis). The second arm was previously used as a place where the measured object could be placed – now we decided to either leave it empty or put there a quarter wave plate Q to change the polarization state of the third wave which interfere with two Wollaston prism’s eigenwaves. The analyzing module AM at the end of the setup allows us to observe and measure the polarization state distribution of the emerging light as well as its intensity distribution. This configuration should give us some interesting two-dimensional patterns in the polarization state of the resulting light wave emerging from the setup. The azimuth angle of the resultant wave emerging from the Wollaston prism is equal to 0° or 90° and its ellipticity angle varies from −45° to 45°. The third wave (from the “measurement arm”) will interfere with this resultant reference wave and, depending on its polarization state and the phase difference, can give as a consequence a polarization state 2D distribution with some characteristic points and lines which can be said to be singularities of different types. The aim of this article is to find all kinds of these singularities (phase and polarization singularities as well as isolated zero intensity points) and their classification.

To describe the behavior of the setup let us use the Jones formalism [15]. Let us denote the Jones vectors of the waves emerging from the Wollaston prism (with linearly changed phase difference along the wedge *x*-axis) as ${E}_{A}$ and ${E}_{B}$, respectively:

*y*-axis to obtain the phase changes in the direction vertical to that in which the phase changes are introduced by the Wollaston prism (the

*x*-axis). The final result of the interference of the three coherent waves is represented by the Jones vector $E$ and is equal to:

The parameters which we want to calculate are: the intensity $I$ distribution and polarization parameters distributions. We can use the standard set of polarization parameters related to Jones formalism: the diagonal angle $\beta $ and the phase angle $\delta $ (the phase difference between Jones vector components) as well as the set of polarization parameters related to Stokes formalism: the azimuth angle $\alpha $and the ellipticity angle $\vartheta $. The first three distributions ($I$,$\beta $,$\delta $) can be calculated directly from the Jones vector components:

while the last two ($\alpha $and $\vartheta $) can be obtained from the well-known formula for the components of the reduced Stokes vector $V$:## 3. Numerical simulations

To study the occurrence of different singularities we first made some numerical simulations using the equations from the previous section. We studied two cases: 1) the setup without the quarter wave plate Q, when the interference of the three linearly polarized light beams was observed (two Wollaston prism’s eigenwaves and the third wave from the “measurement arm”) and 2) with the quarter wave plate Q placed with the azimuth angle equal to 0°, when the interference of the two linearly polarized (again, from Wollaston prism) and one circularly polarized (from the “measurement arm”) light beams occurred. We called this first case as “LLL interference” while the second one as “LLC interference”.

#### 3.1 LLL interference

To simulate the possible measurement’s results (using Stokes polarimeter) we firstly calculated the intensity $I$ distribution as well as all three components (${V}_{1}$,${V}_{2}$ and ${V}_{3}$) of the reduced Stokes vector distributions – see Fig. 2 .

The occurrence of the specific points in the distributions of ${V}_{1}$, ${V}_{2}$ and ${V}_{3}$ parameters and dark spots in the distribution of intensity $I$ leads to the conclusion that there exist some singularities in this light field. However, the detailed calculations of ($\alpha ,\vartheta $) and ($\delta ,\beta $) parameters distributions allowed us to recognize and classify the singularities of different types – see Fig. 3 .

At first sight the points in which the intensity is equal to zero (${P}_{0}$ – isolated zero intensity points in Fig. 3) seem the most interesting. In these points the polarization parameters distributions show different kinds of singularities: a) in the azimuth angle $\alpha $ distribution they appeared to be singularities with the topological charge equal to −2, b) in the ellipticity angle $\vartheta $ distribution – saddle points (topological charge equal to 0), c) in the diagonal angle $\beta $ distribution – singularities with topological charge equal to 0, and d) in the phase angle $\delta $ distribution – cross edge dislocations points. Moreover, the closer look at the polarization parameters distributions reveals the existence of other interesting points in which the intensity differs from zero – the points ${P}_{R}$ and ${P}_{L}$ with circular polarization states (right- and left-handed, respectively). Both these points are the azimuth angle $\alpha $ singularities with the same topological charge equal to + 1 while all the other polarization parameters are well defined. Note that the total topological charge for the three azimuth angles $\alpha $ singularities (${P}_{0}$,${P}_{R}$ and ${P}_{L}$) is equal to zero as it should be expected.

#### 3.2 LLC interference

Our second attempt was to calculate the intensity $I$ distribution as well as all three components (${V}_{1}$,${V}_{2}$ and ${V}_{3}$) of Stokes vector distributions in the case of the presence of the quarter wave plate in the “measurement arm”, which realize the interference of two linearly polarized light wave emerging from Wollaston prism with the third circularly polarized wave from the “measurement arm”. The results are presented in Fig. 4 .

As in the LLL interference (see Sec. 3.1), the occurrence of the specific points in the distributions of ${V}_{1}$, ${V}_{2}$ and ${V}_{3}$ parameters and dark spots in the intensity $I$ distribution lead to the conclusion entailing the existence of the singularities in this light field. The detailed comparison of Fig. 2 and Fig. 4 lead to some immediate conclusions: as the ${V}_{1}$ parameter’s distribution is still the same while ${V}_{2}$ and ${V}_{3}$distributions turn to other places, the final $\beta $ distribution should not change while all other parameters distribution will. The detailed calculations of ($\alpha ,\vartheta $) and ($\delta ,\beta $) parameters are now presented in Fig. 5 .

And again, there are some interesting polarization singularities in the isolated zero intensity points ${P}_{0}$ with the different properties with regard to those described in Section 3.1. Now: a) in the azimuth angle $\alpha $ distribution they appeared to be singularities with the topological charge equal to 0, b) in the ellipticity angle $\vartheta $ distribution – also the singularities with the topological charge equal to 0, c) in the diagonal angle $\beta $ distribution – again the same type of singularities as in $\alpha $, $\vartheta $ distributions and only d) in the phase angle $\delta $ distribution – cross edge dislocations points. There are no more singularity points, however, there are lines of linear polarization states with different azimuth angles – we marked these lines as white in Fig. 5. We also marked some specific points with an azimuth angle equal to the azimuth angle of the light incident upon the setup ( + 45°, points marked as ${P}_{p}$) as well as points whose azimuth angle differs from the azimuth angle of the light incident upon the setup by 90° (−45°, points marked as ${P}_{m}$) – these points can be observed with the linear analyzer at the end of the setup.

#### 3.3 LLC interference with the linear analyzer at the end of the setup

Our analysis was focused on the distribution of the light polarization state at the output of the presented setup, however, we still remember that the original setup presented in [7] has a linear analyzer mounted just before CCD camera (we registered zero intensity points in intensity field only). It will be interesting to study the way in which polarization singularities change into phase one (called optical vortices) after passing the light through the analyzer. Our investigations allowed us to obtain another set of figures showing the intensity and phase distribution for LLC interference and the linear analyzer at the end of the setup (in AM module) for two different azimuth angles of this analyzer (Fig. 6 ). What is more, we simulated the continuous change of the analyzer’s azimuth angle which allows us to present the scheme of these singularities shift (Fig. 6(d)).

The points marked in Fig. 5 as ${P}_{0}$ (zero intensity points without the analyzer in the setup) remain in the same position as it was expected. The points marked in Fig. 5 as ${P}_{m}$ became the zero intensity points for the azimuth angle of the analyzer equal to + 45° and they move along the white lines shown in Fig. 5 while rotating the analyzer (reaching the points ${P}_{p}$ for the azimuth angle of the analyzer equal to + −45°). All ${P}_{m,p}$ and ${P}_{0}$ points are phase singularities (optical vortices) with the topological charge equal to ± 1 (which means that all “movable” points ${P}_{m,p}$ have the opposite topological charge sign to the points ${P}_{0}$).

Note that the same calculations can be made for the case of LLL interference and with the circular analyzer at the end of the setup. The properties of the outgoing optical field (the intensity and phase distributions) are the same as in the case of the LLC setup with the linear analyzer: optical vortices form lattices with the same hexagonal shape. Thus the figures presenting the intensity and phase distributions will be almost the same (note the vertical lines with the same azimuth angle and different ellipticity angles are not shown directly but one can see them carefully analyzing the alpha color scheme). Therefore we decided not to present them here – the more that the case of the analyzer with variable ellipticity (from −45° to + 45°) would be difficult for practical implementation (and experimental verification). Now the points marked as ${P}_{L}$ move to the points marked as ${P}_{R}$ while continuously changing the ellipticity of the analyzer.

## 4. Experimental results

To verify the results of our theoretical and numerical investigations we have made experiments in which both the LLC and LLL setups were investigated with and without the proper analyzer. We could not measure directly the ($\alpha ,\vartheta $) and ($\delta ,\beta $) parameters distributions so we decided to measure the intensity $I$ of the light as well as ${V}_{1}$, ${V}_{2}$ and ${V}_{3}$ Stokes parameter using the Stokes Polarimeter based on variable crystal retarders [16]. The following figures present the results of our experiments. Figure 7 shows the result for LLL setup without the analyzer at the end and should be compared with the Fig. 2 from Section 3.1. Analogically, Fig. 8 shows the result for LLC setup without the analyzer at the end and should be compared with the Fig. 3 from Section 3.2.

The last figure (Fig. 9 ) presents the intensity distributions for the LLC setup with the linear analyzer at the end of the setup – we have rotated this analyzer from the azimuth angle equal to −45° up to + 45° to validate the results presented in Fig. 6(d) but finally only two obtained intensity distributions for these two extreme analyzer’s position are presented. Let us note that the behavior of this setup (with the linear analyzer at the end) was described in detail in our previous work [7] and the results of the experiments with the fourth wave have been presented, in particular the characteristic “forks” confirming the presence of phase singularities in the points in which intensity is equal to zero have been shown (Fig. 5 in [7]).

We hope that the shapes of distributions presented in Figs. 7, 8 and 9 allow the reader to conclude that our theoretical considerations are correct. Some incorrectness of the measurement device (Stokes Polarimeter) deform the characteristic bow-tie shapes presented in Fig. 7(c) (in comparison to those shown in Fig. 2(c)) as well as presented in Fig. 8(d) (in comparison to those shown in Fig. 4(d)) but we believe that these deformations do not interfere with the positive reception of the results of our work.

## 4. Conclusions

We analyzed the behavior of the presented interference setup which can produce interesting two-dimensional patterns in the polarization state and light intensity of the resulting wave emerging from the setup. Various types of singularities were detected: isolated zero intensity points and polarization singularities of different types (depending on chosen set of polarization parameters). Isolated zero intensity points can be observed directly while polarization singularities can be detected using Stokes Polarimeter or proper analyzers at the end of the setup. Isolated zero intensity points form a specific rectangular lattice which turns to hexagonal one when using the analyzer; the additional isolated zero intensity points appear as a result of interaction of the polarization singularities with the analyzer. These vortices observed in phase distributions have the topological signs equal to ± 1. More interesting are polarization singularities present in the interference field at the end of the setup without the analyzer. We observed the polarization singularities with different topological signs: −2 and + 1 for azimuth angle distribution in the case of LLL interference; 0 for the diagonal angle distribution in LLL interference and for the azimuth, ellipticity and diagonal angles in the LLC case. We also noticed another characteristic points: saddles (for the ellipticity angle distribution in LLL interference) and cross edge dislocations (for the phase angle distributions in both cases). Moreover, the zero intensity points in the setup without the analyzer appeared to be double singularities where both polarization parameters (from the pair ($\alpha ,\vartheta $) or ($\delta ,\beta $)) are undetermined simultaneously. Note that in our previous work [14] we described the interference field characterized by the polarization distributions, only one of which has a singular nature.

Let us note that using the phase retarder instead of the quarter wave plate in the “measurement arm” one can analyze more complex cases of three plane waves interference. Thus the interference of two linearly polarized light waves from Wollaston prism and elliptically polarized light wave from the retarder can be observed (LLE interference). Our experiments with phase retarder (results are not presented here) were made only to confirm the proper identification of the topological signs of the singularities. The −2 and + 1 singularities for azimuth angle distribution in the case of LLL interference annihilate to 0 singularities in the case of LLC interference. The most complex case – the interference of three elliptically polarized plane light waves (EEE interference) – is also possible to obtain but we did not analyze this case in detail due to the complicated character of the resultant polarization distribution.

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