## Abstract

We propose a novel birefringent interferometer setup for the study of unfolding points, and obtain for the first time to our knowledge the spatial polarization structure very near the unfolding point of a uniformly polarized optical vortex beam propagating in a birefringent crystal. The unfolding point is reconstructed by folding back the two separated eigen-beams at the output of the birefringent crystal into a single beam using another identical birefringent crystal, resulting in a birefringent interferometer of Mach-Zehnder type. We also demonstrate that the separation near the unfolding point can be varied by a small rotation of the second crystal.

© 2012 OSA

## 1. Introduction

In the last two decades, there has been a great interest in studying the natural singularities of scalar optics, namely, optical vortices. The vortices are rotational phase structures, with a singularity at center where the phase is undefined. Interest in such structures has become a separate topic in modern optics called Singular Optics [1]. The concept has been extended to vector optics to study the polarization singularities, where the geometric polarization parameters of coherent fields cannot be defined [2]. The two types of polarization singularities are the C-points where the major axis of the polarization ellipse is not defined and the L-lines where the polarization handedness is not defined. These structures have been observed in the transverse plane of different fields [3–5]. The polarization singularities have significant importance in understanding the fine details of beams with complex polarization structure.

Recently, there has been a tremendous interest in studying the connection between the singularities of scalar optics and vector optics using birefringent crystals [6–8]. A detailed study of the spatial variation of polarization due to a circularly polarized input beam with Gaussian and Laguerre-Gaussian envelopes propagating through a birefringent crystal along and at an angle to its optic axis is presented in Ref [6]. The formation of a complicated network of polarization singularities at the output of a birefringent crystal due to unfolding of a diagonally polarized optical vortex (OV) beam have been studied [7,8] to show the topological interplay between scalar and vector optics. However, in these previous works, the polarization structures were studied at the output of a finite-length crystal where centers of the two beams are well separated. The spatial polarization structure near the unfolding point where the beam initially enters the crystal and splits into two eigen-beams has not been studied so far. In order to investigate the spatial polarization structure near the unfolding point one needs to attain the unfolding point at the output of the crystal.

In the present paper, we propose a novel setup to reconstruct the unfolding point at the crystal output and study the polarization structure near the unfolding point of a uniformly polarized OV beam. Using this setup, we obtain for the first time to our knowledge the polarization structure very near the unfolding/refolding point in a birefringent crystal. We also demonstrate that the separation between the two eigen-beams can be controlled, allowing observation of the beam structure further from the unfolding point.

At the input surface of a birefringent crystal, an initially diagonally polarized OV beam is split into two orthogonally polarized eigen-beams, namely, ordinary ray (O-ray) and extra-ordinary ray (E-ray). The point where this occurs is called the unfolding point. The resulting two eigen-beams are separated in space at the output of the crystal. In our setup a half wave plate (HWP) interchanges the polarization of the two beams, and propagation through a second identical crystal reconstructs the unfolding point at the output, folding back the two eigen-beams into a single beam. Such an arrangement will result in an interferometer of Mach-Zehnder type. The beam at the output of this optical vortex birefringent interferometer can be unfolded by rotating the second crystal with respect to the first crystal about its vertical axis. Small rotation of the second crystal produces a varying separation of the two eigen-beams at the output of the interferometer. The polarization of the beam at the output of the interferometer is characterized by measuring the Stokes parameters in the conventional way.

In previous works, the usefulness of the birefringent interferometer with Gaussian input beam is demonstrated to measure the surface deformations due to photo-thermal effects [9] and for signal processing techniques [10]. The birefringent interferometer has also been used as a stable interferometer for quantum information processing [11]. In the present paper the birefringent Mach-Zehnder interferometer using OV beam is demonstrated and characterized for the first time to show its applicability in reconstruction of unfolding points and to study the spatially varying polarization structure near the unfolding point. Using the proposed configuration, the polarization structure of the OV beam very near and far from the unfolding point is created. It can also be observed from the results that it is easy to generate and control spatially varying polarization using the designed setup. The present study can also find potential applications in spectroscopy and microscopy dealing with spatially varying polarization structures.

The paper is organized as follows. Basic theoretical model regarding the present work is given in section 2. Section 3 describes the experimental setup and the results obtained using the stable optical vortex birefringent Mach-Zehnder interferometer (BR-MZI). The conclusions of the present work are given in section 4.

## 2. Theoretical model

The OV beam passing through the first crystal will split into two eigen-beams which are orthogonally polarized. In our setup the O-ray is vertically polarized, and the E-ray is horizontally polarized. The polarizations of these eigen-beams are interchanged using a HWP and the beams are passed through a second crystal identical to the first one to obtain the unfolding point. This arrangement will result in an interferometer in the Mach-Zehnder configuration with orthogonal polarizations. The output of the interferometer is given by

where*ê*and

_{H}*ê*are unit vectors in the horizontal and vertical directions respectively. The complex amplitudes

_{V}*E*and

_{e}*E*of the E-ray and O-ray respectively are given as

_{o}*w*is the beam radius and

_{0}*d*is the spatial separation between the two eigen-beams at the output of the BR-MZI. The small rotation of the second crystal will result in a spatial separation

*d*between the two eigen-beams due to the angle dependent refractive index of the extraordinary beam [12] inside the crystal. The beam propagation is considered in z direction.

The polarization structure of the beam at the output of the interferometer due to the small rotation of the second crystal is characterized by Stokes parameters S_{0}, S_{1}, S_{2} and S_{3} defined as [13]

The normalized Stokes parameters with respect to the total intensity S_{0} are defined as *s _{1}* = S

_{1}/S

_{0},

*s*= S

_{2}_{2}/S

_{0}and

*s*= S

_{3}_{3}/S

_{0}. Using the normalized Stokes parameters we construct the abstract fields

*s*and

_{1}+ is_{2}*s*. The argument of the abstract field

_{2}+ is_{3}*s*gives the information on the orientation of the major axis of polarization ellipse and the argument of

_{1}+ is_{2}*s*gives the phase difference between the two eigen-beams.

_{2}+ is_{3}## 3. Experimental details and results

The experimental setup to demonstrate the proposed optical vortex birefringent interferometer is shown in Fig. 1. The polarization of the laser beam from a He-Ne laser is set to 45 degrees using a polarizer (P_{1}). The diagonally polarized laser beam then enters a transmission type fork structured phase hologram (HG) made by electron beam lithography (EBL) [14]. The Laguerre-Gaussian (LG) beam with a topological charge of + 1 generated in the first order diffraction from the hologram is selected using an aperture (A). The size of the beam (2*w _{0}*) at the detector plane is estimated to be 1.37 mm. The singly charged and diagonally polarized LG beam is then allowed pass through a birefringent crystal (Cr

_{1}) made of yttrium vanadate (YVO

_{4}) with dimensions 30 mm (length) X 12 mm (width) X 10 mm (height) (from CASTECH, China) and its optic axis in the horizontal plane making an angle of 45 degrees to the propagation direction.

The ordinary and extraordinary refractive indices *n _{o}* and

*n*of the crystals are given by 1.9929 and 2.2154 respectively. The birefringence (Δ

_{e}*n*) of the crystals and the walk-off angle (

*ρ*) between the two eigen-beams are calculated to be 0.2225 and 6.04 degrees respectively at 630 nm [15]. The two eigen-beams with orthogonal polarizations separated at the output of the first crystal due to birefringence are passed through a half wave plate (HWP) to interchange the polarizations of the two beams. The two beams after the HWP plate then enter a second crystal (Cr

_{2}) identical to the first one with its optic axis parallel to the first crystal and mounted on a rotational stage. The two eigen-beams are folded back together to form a single beam at the output of Cr

_{2}. Such an arrangement will result in a stable interferometer of Mach-Zehnder type.

The unfolding point occurring at the input surface of the first crystal is reconstructed at the output of the second crystal. Small rotation of the second crystal will result in a small separation of the two eigen-beams at the output of the interferometer. Similar rotation techniques have been previously used to control the relative phase between the two eigen-beams with a very small rotation angle [7,8], and also for a different crystal orientation [16]. The polarization structure of the resultant beam at the output of the interferometer is characterized by measuring the Stokes parameters using a quarter wave plate (QWP) and polarizer (P_{2}) as shown in Fig. 1. The intensity distribution for different combinations of QWP and P_{2} are recorded using a CCD (Model No. C5948, Hamamatsu) connected to a personal computer. As the BR-MZI is mounted on a translation stage it can be moved in and out of the input beam. Initially, the BR-MZI is moved out from the setup and we characterized the input beam. The polarization of the input OV beam is characterized by measuring the Stokes parameters using [17]

*I*(

*β,γ*) is the output intensity when the axes of the QWP and P

_{2}are at

*β*and

*γ*respectively as measured from horizontal direction.

Figure 2
shows the normalized stokes parameters of the input OV beam. S_{0} is normalized to its maximum value and S_{1}, S_{2}, S_{3} are normalized to S_{0} such that *s _{1}* = S

_{1}/S

_{0},

*s*= S

_{2}_{2}/S

_{0}and

*s*= S

_{3}_{3}/S

_{0}. The CCD images shown in the present paper are inverted in the vertical direction as indicated by the arrow showing the + x direction in the figures. It can be seen from Fig. 2(a), 2(b), 2(c) and 2(d) that the polarization of the input beam is 45 degrees and the contribution of the horizontal/vertical and circular polarizations are minimal. The two angles 2α = arg (

*s*) and δ = arg (

_{1}+ is_{2}*s*) calculated from the Stokes parameters are shown in Fig. 2(e) and 2(f) respectively. The angle α is the azimuthal angle of the major axis of the polarization ellipse and δ is the phase difference between the horizontal and vertical components. Figure 2(e) and 2(f) confirm that the major axis of the polarization ellipse is at 45 degrees and there is no phase difference between the two orthogonal components.

_{2}+ is_{3}Now the BR-MZI is introduced into the beam and the output beam is characterized. The crystal Cr_{2} is rotated about its vertical axis (x-axis) to a maximum angle on one side where we see the two eigen-beams separated well in horizontal direction (y-axis) and then it is rotated to the other side in steps of one degree. At each rotation angle the output beam is captured through the CCD and we estimated the position of the O-ray and E-ray separately by calculating the center of gravity of the intensity distribution of each beam. The positions of the O-ray and E-ray at each rotation angle are plotted in Fig. 3(a)
.

The relative separation *d _{r}* of the two eigen-beams with respect to the input beam size is defined as

*d*is the separation between the two eigen-beams and 2

*w*is 1/e

_{0}^{2}beam width of the input beam at the detector plane. Figure 3(b) shows the separation

*d*of the two eigen-beams as a function of rotation angle of the Cr

_{2}. As the length and birefringence of the crystal (YVO

_{4}) used in our experiment is large compared to that of reference [7], we get relative separation of about 220% between the two eigen-beams at the output of a single crystal. This larger separation per crystal and larger angles of rotation of the crystal Cr

_{2}compared to reference [7] allows us to control the separation over a wide range of values.

It can be seen from Fig. 3(b) that the separation decreases linearly with rotation angle of Cr_{2}. The zero value of the separation indicates the unfolding point and it is observed from Fig. 3(b) that this occurs between 346 and 347 degrees as read from meter of the rotational stage. The separation *d* between the O-ray and E-ray is found to be around 30.2 μm and −29 μm at 346 degrees and 347 degrees respectively. The relative separation, *d _{r}* corresponding to 346 degrees and 347 degrees is calculated using Eq. (5) to be 2.2% and −2.1% respectively at the output of the second crystal. The negative sign represents the interchange of the positions of the two-eigen beams after crossing the unfolding point. The dotted line in Fig. 3(b) represents the linear fit of the data. From the linear fit it is found that the separation changes −32.28$\pm $0.82 μm per unit angle rotation of Cr

_{2}. We note that the exact angle for the unfolding point depends on the initial separation at the input of the second crystal, which in turn depends on the orientation of the first crystal.

Now the Cr_{2} is adjusted to the angle near the crossing point where the relative separation is measured to be 1.7% in the y-direction, which is very close to the unfolding point. The polarization of the output beam is characterized with the same method as for the input beam. The normalized Stokes parameters measured at the output of the BR-MZI are shown in Fig. 4
. It can be seen from Fig. 4(c) that in the upper half of the beam as displayed the polarization is 45 degrees, which is identical to the input beam polarization. However, the lower half of the beam is closer to right circular polarization (Fig. 4(d)). This variation can also be seen in Fig. 4(f) as a slope in δ, indicating relative tilt in the wave front of the two eigen-beams. Figure 4(e) shows that 2α is more or less constant, indicating that the major axis of the polarization ellipse remains near 45 degrees throughout the beam. Figure 4(b) shows that *s*_{1} is relatively small and varies in the direction from top left to bottom right. Since *s*_{1} is the balance between the intensities of the two eigen-beams, this variation is expected to coincide with the direction of the separation between the two eigen-beams, inferring that the two eigen-beams are separated in the x (vertical) direction as well as the y (horizontal) direction.

In order to verify the effects of the tilt and the separation in x and y directions, we consider the following model which incorporate these details as well as the overall phase difference between the two eigen-beams. The tilt of the wave front will result in a phase variation in the direction of the relative wave vector of the two beams. To incorporate the wave vector changes due to the tilt we modify Eq. (2) to get the general form as follows:

*d*and

_{1}*d*represent the separation between the two eigen-beams in the x and y directions respectively.

_{2}*θ*and

*ϕ*represent the spherical coordinates corresponding to the tilt angle and Δ is the overall phase difference between the two-eigen beams. Using Eq. (3) and Eq. (6) the normalized Stokes parameters are calculated and matched with experimental results. From the calculations,

*d*and

_{1}*d*are found to give a relative separation of 2.5% and 1.7% respectively, and

_{2}*θ*and

*ϕ*are estimated to be 35x10

^{−6}π and 0.1π respectively. The overall phase difference Δ between the two eigen-beams is found to be equal to 0.0012π. We note that the tilt can be a combined effect of a curved wave front and the separation between the two eigen-beams. The calculated results using Eq. (3) and Eq. (6) are shown in Fig. 5 . It can be observed that the calculated results coincide with the experimental results.

The spatial polarization structure for a large rotation angle of Cr_{2} which results in a large separation between the two eigen-beams is also investigated. The crystal Cr_{2} is rotated such that the rotation results in the relative separation of −42% in the y-direction between the eigen-beams at the output of the interferometer. The spatial distributions of the measured normalized Stokes parameters are shown in Fig. 6
. It can be observed that the large separation of the eigen-beams results in a non-uniform polarization across the beam cross section.

These polarization characteristics are identical to the results presented in the earlier research [18] for the large separation between the two eigen-beams using a single birefringent crystal. The experimental results are shown in Fig. 6. Figure 6(d) shows a horizontal line at center where S_{3} = 0, indicating linear polarization (L-line). In Fig. 6(e) two singular points can be observed which correspond to the positions where the major axis of the polarization ellipse is undefined. Such a situation will occur only for circular polarizations (C-points). The rotations of the major axis around the two C-points in Fig. 6(e) have opposite handedness.

The experimental results are compared with the computer simulations for the relative separation of −2.5% in the x-direction and −42% in the y-direction and using Eq. (3) and Eq. (6). The computer calculations are shown in Fig. 7
. It can be observed from Fig. 6 and Fig. 7 that the experimental results match well with the computer simulations, even though we have not included tilt and constant phase difference for these simulations. These results infer that for large relative separation the polarization structures are governed by the rotating phase of OVs and that the effect of the small wave front tilt and small overall phase difference is not as visible as for the case of small relative separation. The variation of *s _{1}* is slightly tilted from the horizontal direction, showing that the effect of separation in the y-direction is dominant but there is also a relatively small contribution from the separation in the x-direction. The separation in the x-direction is similar in magnitude to the one observed in Fig. 4 and Fig. 5, but opposite in sign. These vertical separations may be caused by small imperfections and misalignment in the experimental setup.

The present investigations show that the proposed birefringent interferometer in Mach-Zehnder configuration with an optical vortex input can be used to achieve the spatial separation of the eigen-beams very close to unfolding points and enable the study of the fine details of the spatial polarization structure very close to such points. The present setup can also be used to generate and control the spatial variation of polarization which results in polarization singularities in the output field. Further investigations on the sensitivity of the polarization structure to the relative separation between the eigen-beams and the evolution of the polarization singularities from uniform polarization are in progress.

## 4. Conclusions

In the present paper, we have demonstrated a novel experimental setup to reconstruct the unfolding point of an optical vortex beam inside the birefringent crystal where the input beam enters and splits into two eigen-beams. The experimental setup consists of two identical birefringent crystals with their optic axes parallel to each other and a half wave plate in between them. Such an arrangement will result in a birefringent interferometer in Mach-Zehnder configuration. The interferometer is characterized for the different rotation angles of the second crystal. It is found that the relative separation between the eigen-beams varies with the rotation angle. Beyond a particular rotation angle the relative separation changes its sign, inferring that the output has crossed the unfolding point, where the separation is zero.

Using this setup, we obtained for the first time to our knowledge the polarization structure very near the unfolding/refolding point (1.7% relative separation in y-direction) in a birefringent crystal. The deviation from the input spatial polarization structure observed near the unfolding point is explained based on a model using the relative separation in x and y-directions, tilt of the wave front of one of the two eigen-beams and the overall phase difference between them. We also demonstrated that the same setup can be used to observe the beam further from the unfolding point (−42% relative separation in y-direction). At large relative separation the spatial polarization structure contains the point singularities corresponding to circular polarization and the line singularities corresponding to linear polarization.

## Acknowledgments

The author M. M. Brundavanam thanks The University of Electro-Communications for the financial assistance.

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