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Abstract
Lateral shear interferometer, being a self-referenced interferometer, has proven to be an important tool in scalar optics. Here we employ a vectorial counterpart - polarization lateral shear interferometer, in which the two interfering beams apart from being derived from the test wavefront, are in orthogonal states of polarization. Therefore when the test wavefront has spatially varying phase gradient across the beam cross-section, the resulting shearogram produces polarization fringes instead of intensity fringes. Further, the shearogram becomes inhomogeneously polarized. This polarization lateral shear interferometer may have potential uses in metrology, but in this article we demonstrate the ability of the interferometer in the generation of all Stokes singularities in the single beam by launching a phase singular beam into it. It is found that a vortex dipole is formed along with other generic Stokes singularities. Experimental observations support the results and they are discussed in the article.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Topological defects, namely phase, and polarization singularities have been studied widely by the scientific community for many years. Due to the orbital angular momentum (OAM) associated with phase singularities, these beams have found various applications in optical science. Polarization singularities are extensions of phase singularities in the vector regime in which the spin and orbital angular momentum of light are coupled together [1–3]. These singularities naturally occur in skylight [4], speckle fields [5], and scattered optical field by sub-wavelength particles [6]. Polarization singularities are a subset of Stokes singularities that are of a more general type [7,8]. This letter reports the generation of an optical beam containing all Stokes singularities using a polarization lateral shear interferometer. In polarization lateral shear interferometry, the two sheared wavefronts are in the orthogonal states of polarization (SOP).
In scalar fields, lateral shear interferometry [9,10] is a traditional optical metrology [11,12] technique that has been extensively used. In modern optics, it is useful in the creation [13,14] and detection, [15–17] of phase singularities, coherence length measurement [18], incoherent holography [19] and so on. In vector optics regime, by shearing using Wollaston prism, the properties of birefringent media was studied [20]. There is a vectorial shear interferometer [21], with control on the direction of the shear. It is vectorial not in the sense of polarization but the word vectorial refers to the direction in which shear can be introduced. There are studies where vectorial shear interferometers were used to introduce shear but the objective of these articles were to observe intensity fringes by inducting a polarizer in the setup rather than polarization fringes as such [22–25]. There are two reports worth mentioning. In one of them, non-coaxial superposition of orthogonally linearly polarized vortex beams was studied to produce polarization singularities, which is a subset of Stokes singularities [26]. In the other study, the non-coaxial superposition was achieved by using uniaxial birefringent crystal, but here the aim was to study the instability of phase singularities [27,28]. None of these studies deals with the experimental generation of all Stokes singularities within the single beam, using polarization shear interferometer.
In a lateral shear interferometer, the shearogram represents the gradient of the phase distribution of the test wavefront.
where $\Delta W = W(x,y)-W(x+\Delta x,y)$. Here $W(x,y)$ is the path function describing the test wavefront, $k$ is the propagation constant, $\Phi$ is phase distribution and $\Delta x$ is the magnitude of shear in the $x$ direction. In our experiments, we use phase singular beams as test beams. Phase singularity (vortex) is characterized by beams having helical wavefront structure with a topological charge $m$ given by $\oint \nabla \Phi \cdot dl = 2 \pi m$ where $dl$ is the line element surrounding the singularity. At the phase singularity, the phase is undefined and amplitude is zero. Shearograms of vortices were first reported in 2008 [15,29,30]. In a polarization shear interferometer, the fringe formation is governed by phase difference at two different points like in a conventional shear interferometer but the phases at two different points correspond to orthogonal polarization states. Therefore, these fringes are polarization fringes. Further, if the beam has amplitude variation, that also will contribute to the polarization fringe formation.In this article, we study the generation of all Stokes singularities in a single beam using a polarization lateral shear interferometer. This interferometer as such can be recommended as a new type of instrument. Being new, the metrological advantages it may offer is unknown. It may have potential uses in metrology, but in this article we show the ability of the interferometer in the generation of Stokes singularities. A phase vortex beam is launched into the interferometer and it is observed that the resultant beam comprises of all Stokes singularities. It is found that a vortex dipole is formed along with other generic Stokes singularities. Experimental observations support the results discussed in the article. The article is organized as follows. The requisite background/theory of Stokes singularities is provided in section 2. Polarization lateral shear interferometer is introduced in section 3. The types of shearograms when a scalar vortex beam is launched into the interferometer is discussed in section 4 with concluding remarks made in section 5.
2. Stokes singularities
The state of polarization of light can be described using the four Stokes parameters $S_0$, $S_1$, $S_2$, and $S_3$ [31]
Here, $E_{x,y}$ represent electric field components. Using these Stokes parameters, a complex Stokes field namely $S_{jk}=S_j+iS_k$ can be constructed. The argument of this complex field is called as Stokes phase which is defined as $\phi _{jk}=arg(S_{jk})=tan^{-1}(S_k/S_j)$. Each of these Stokes phases represent the phase difference between the orthogonal polarization components i.e., $\phi _{12}= \phi _L-\phi _R$, $\phi _{23}= \phi _Y-\phi _X$, $\phi _{31}= \phi _A-\phi _D$. Here, the suffix $R$ stands for right circular, $L$ (left circular), $X$ (horizontal), $Y$ (vertical), $D$ (diagonal), and $A$ (anti-diagonal) polarization states. Stokes singularities [8,32] are points of phase singularities in Stokes phases. At the Stokes singular point, the curl of gradient of Stokes phase is non-zero. This means $\oint \nabla \phi _{jk} \cdot dl = 2 \pi \sigma _{jk}$ where $dl$ is the line element surrounding the singularity and $\sigma _{jk}$ is the Stokes index. At the $\phi _{12}$ Stokes singular point, the SOP is circular whereas at $\phi _{23}$ and $\phi _{31}$ Stokes singularity, the SOP is linear. Here, $\phi _{12}$ singularities are known as polarization singularities whereas $\phi _{23}$ and $\phi _{31}$ singularities are known as Poincaré singularities. The Poincaré singularities were initially introduced to detect the position of phase singularities in a scalar field without limitation to fringe contrast by using an orthogonally polarized field [7]. These singularities result from the choice of coordinate basis of the Stokes vector which leads to points of linear polarization along the coordinate axes and diagonals, but omitting any other orientation of linear polarization.
For every polarization distribution, there are three possible Stokes phase distributions namely, $\phi _{12}$, $\phi _{23}$ and $\phi _{31}$. A cyclic switch on superposition from one basis to another is possible and there are Stokes’ singularity relations available [8,32,33]. However, there is a difference between a cyclic switch between one basis to another basis of observation and a cyclic switch from one superposition basis to another. In other words, observation of a same polarization shearogram distribution in three different basis is different from having superpositons in three different basis that would have produced three different polarization shearogram distributions. It cannot be construed as the polarization shearogram obtained by superposition in one polarization basis is same as that produced by superposition in another basis. The polarization distributions, the (relative) positions of the Stokes singularities, their types and therefore the properties of these beams are totally different.
At the polarization singular point, some parameter that defines the SOP of light is undefined. Various types of polarization singularities are discussed in literature including point (C-points, V-points) and line (L-line) singularities [2]. At the point polarization singularity (C-point, V-point), azimuth is undefined and at the Poincaré singularity, the handedness is undefined. Polarization singularities present in 3D optical fields (Möbius strips, links) are also studied by many authors [34–36]. Generation of all Stokes singularities in the single beam by coaxial superposition of vector beams [33,37,38] and in lattices by multiple beam interference [32,39] are reported. The polarization modulation of resultant beam in the interference of orthogonally polarized waves was first studied in [40,41]. Chains of C point polarization singularities were reported in the interference of orthogonally polarized waves where the small intensity gradient was introduced in one of the superposing beams [42]. The superposition of phase vortex beams in different polarization states produces inhomogeneous polarization distribution, containing one type of Stokes singularity at the center of the beam, which strongly depends on the state of polarization (SOP) of the superposing beams. Focusing of each of these Stokes singularities will produce different focal plane intensity distributions [43,44]. The exotic polarization patterns of beams containing Stokes singularities may offer great advantage in optical trapping and manipulation of the particles. Further, the energy flow (orbital flow and spin flow) in each of the Stokes singularities is different [45]. This implies that the Poynting vector flow and hence the optical angular momentum associated with different Stokes singularities will be different [46]. Spatial modulation of the Poynting vector can be achieved in this way which is useful in the micromanipulators and tweezers [47]. The structured beams containing Stokes singularities have found various applications in optical beam shaping [43], polarimetry [48], in robust beam engineering [49] and so on.
3. Polarization lateral shear interferometer
In a polarization lateral shear interferometer, the incident beam is polarization split and are made to interfere with each other with a lateral shift between them [22–25]. The intensity fringe formation in a conventional shear interferometer is governed by the phase gradient of the test wavefront, whereas here the phase gradient leads to the polarization fringe formation. The SOP distribution in the shearogram is inhomogeneous.
By launching a phase vortex beam into the polarization lateral shear interferometer, all three types of Stokes singularities can be realized in the shearogram. The number and location of Stokes singularities depend on the amount of shear, the topological charge, and the phase difference between the two superposing vortex beams. The location of Stokes singularities interchanges on changing the basis of superposition. The field corresponding to the test wavefront carrying an optical vortex of topological charge $m$ can be written as:
In the polarization shear interferometer, this vortex beam is split into orthogonal polarization components and shear is given between them. Here we have considered a phase vortex beam with a particular amplitude distribution, but vortex beams with other amplitude profiles can also be used in general. The resultant field $E_R$ is therefore due to the superposition of the test beam and its sheared version in an orthogonal polarization state.
Here, $r=\sqrt {(x^2+y^2)}$, $r_s=\sqrt {((x+\Delta x)^2+(y)^2)}$, $\phi =tan^{-1}(y/x)$, and $\phi _s=tan^{-1}(y/x+\Delta x)$. In Eq. (3) "$A$" is a real constant and $\Delta x$ is the amount of shear introduced in the x direction. $\hat {n}_1$ and $\hat {n}_2$ represent the orthogonal polarization states. $a$ and $b$ define the scaling factors and $\phi _0$ represents the constant phase shift between the two beams in the superposition. The phase distribution that governs the SOP distribution in the shearogram for small values of shear can be defined as:
Here, $W_i$ defines the $i^{th}$ component of path function where $i$ denotes the component of SOP. Unlike the superposition in circular basis, in the linear bases superposition, the phase differences $(k \Delta W_{XY})$ and $(k \Delta W_{DA})$ decide both ellipticity and the azimuth of SOPs in the shearogram. For smaller shear ($\Delta x$) values, the phase gradient as given in Eqs. (5–7) and the amplitude difference (between the sheared beams in the region of overlap) in the direction of shear decide the resultant polarization distribution [9].
4. Experiment and discussion
After developing the theoretical background in the previous sections, various shearograms are simulated and presented in Fig. 1 and Fig. 2 and the shearograms are analyzed. Experimentally these shearograms are realized using the setup shown in Fig. 3. The shearograms shown in Fig. 1 consist of all the three types of Stokes singularities when a scalar vortex beam of topological charge +1 is launched into the interferometer. Figure 1 ($a_1 - a_2$) show the shearograms, in which sheared beams are in orthogonal circular polarization states, whereas Fig. 1 ($b_1, b_2, c_1$ and $c_2$) show the shearograms, in which the sheared beams are in linear orthogonal polarization states. Irrespective of the basis in which the superposition happens, each of all the polarization distributions shown in Fig. 1 contains all the Stokes vortices. For each SOP distribution there are three Stokes phases ($\phi _{12}, \phi _{23},$ and $\phi _{31}$), possible and they are depicted on the right side of each SOP distribution from top to bottom. On the left top corner in each case, the intensity distribution is shown. All the SOP distributions contain Stokes singularities of all the three types and the polarization states occupy the whole of the Poincaré sphere. Such beams are known as full Poincaré beams [50]. If a Poincaré beam contains a Stokes vortex ring [38] within the beam size, then it will contain all the Stokes singularities independent of the extent of occupancy on the Poincare sphere. The amplitudes and phases in the orthogonal component states satisfy Stokes vortex formation condition at different locations of the polarization shearogram. Undefined azimuth occurs at isolated points where the SOP is circular and undefined handedness with fixed azimuths ($\gamma =n \pi /4$ where $n=0,1,2,3$) occurs at isolated points where the SOP is linear. Along with these conditions, around the Stokes singularity, $\nabla \times \nabla \phi _{jk} \ne 0$. For the beam with complex amplitude as given in Eqn.4, due to shearing, a unique combination of all these conditions is satisfied at discrete points in the same shearogram. The polarization gradient in such beams may be useful in optical trapping and manipulation of particles. Due to shear, the asymmetry in amplitude/phase introduced in the region of superposition may lead to non-coincidence of C-point and Poynting singularity [51] in this polarization lateral shear interferometer. For non-zero values of $\phi _0$ in Eq. (4), the resulting polarization fringes will be due to the phase difference between the orthogonal polarization components at two different spatial points as well as at two different times. This is because the two interfering beams travel different optical path lengths. For example, in a plane parallel plate lateral shear interferometer the front and back reflected beams have path difference in addition to shear between them. Figure 2 presents the resultant polarization distributions, intensity distributions, and respective Stokes phases for different phase shifts ($\phi _0$) between the superposing beams and for different topological charges $m$. The results are presented for a particular value of shear ($=0.4d$) and in two different polarization bases. The respective basis and topological charge ($m$) for each row is mentioned in the left side of the Figure. The index of C-points ($\phi _{12}$ Stokes vortices) that forms the dipole in the polarization distribution increases on increasing the topological charge ($m$) of vortex beam which can be explicitly seen in Fig. 2 ($a_1$ and $b_1$). The ellipse fields around these singularities are seen to rotate on varying the phase shift $\phi _0$ between the two superposing beams. Similarly, in linear basis superposition, the increase in index of $\phi _{23}$ singularities is observed on increasing the value of topological charge $m$ (refer $\phi _{23}$ in Fig. 2 ($c_1$ and $d_1$)). The variation in number and location of other Stokes singularities with $m$ and $\phi _0$ is also observed. Notably, in circular basis superposition, the resultant beam has zero net helicity independent of the amount of shear, phase shift $\phi _0$, and topological charge of vortex beams. This is in contrast with the previous observation in the case of coaxial superposition of vortex beams where the resultant beam in linear basis superposition has zero net helicity [52].
Fig. 1. Simulation, resultant SOP distributions due to superposition of two laterally sheared vortex beams in orthogonal states of polarization. Insets of SOP distributions show resultant intensity distribution. The respective Stokes phases $\phi _{12}$, $\phi _{23}$, and $\phi _{31}$ are shown on the right side of SOP distributions. Shear increases from left to right (1-2). The values of lateral shear in terms of beam diameter (d $=2w_0$) are (1) 0.24d, and (2) 0.48d. In subsequent shearograms, only the overlap region shown by a dotted black circle as indicated in $c_2$ are shown.
Fig. 2. Simulation: resultant SOP distributions due to superposition of laterally sheared orthogonally polarized vortex beams for different charge and phase difference $\phi _0$ between them. The results are shown in two different superposition bases. Respective intensities are shown as insets. The corresponding Stokes phases are shown on the right side of the SOP distribution.
Fig. 3. Experimental setup, SF: spatial filter assembly; L1,2: lenses, P: polarizer, H(Q)WP: half (quarter) wave plate, PBS: polarizing beam splitter, BS: beam splitter, M1,2: mirrors, SPP: spiral phase plate, SC: Stokes camera.
The experimental setup for polarization lateral shear interferometer is shown in Fig. 3. The beam from HeNe laser is spatially filtered and collimated using lens $L1$. The collimated beam diameter is 3.5 mm. The $45^{\circ }$ polarized beam is passed through a spiral phase plate of desired topological charge. The vortex beam thus generated enters a modified Mach-Zehnder type interferometer. After the polarizing beam splitter, the vortex beam splits into two equal amplitude beams in orthogonal SOP (linear basis). The two beam from the two arms of the interferometer combines after the beam splitter. One mirror is mounted on a translation stage to introduce the lateral shear between the two superposing beams. In addition to the lateral shear, such translation in one arm of the interferometer introduces a phase shift between the two superposing beams. By adopting to the common path configuration (Sagnac), one can easily tackle problem of phase instability in the interferometer. The quarter-wave plate at an appropriate angle is used after the beam splitter to switch from linear basis to circular basis superposition. The lens L2 is used to observe the far-field intensity and polarization distributions. The resultant intensities and Stokes parameters are recorded using a Stoke camera (Salsa full Stokes polarization imaging camera, 1040x1040 pixels, BOSSA Nova, USA). The SALSA Stokes camera executes live measurement of the full Stokes vector for each pixel of the image at a video frame rate. Stokes phases (say $\phi _{jk}$) are calculated using the Stokes parameters as: $\tan ^{-1}(S_k /S_j)$. Notably, the phase singular beam generated in the setup is due to the phase modulation by a spiral phase plate. Hence, just after the spiral phase plate, the beam has uniform amplitude with azimuthally varying phase. On propagation, the beam develops the dark core at the center. The transverse mode generated in this process is not a pure Laguerre Gaussian mode ($LG_{0l}$) but is the superposition of the required mode and other unwanted higher order modes. These modes are generally referred as hypergeometric modes [53]. To remove the effect of these unwanted higher order modes, the field is observed in the far field where the effect of higher order modes is negligible [54,55].
The experimental results obtained for the superposition of two laterally sheared vortex beams in orthogonal polarization basis are shown in Fig. 4. The respective bases and the input topological charges of the vortex beams are mentioned in the figure. All the Stokes singularities are observed in the beam. In the case of circular basis superposition, the polarization singularities are seen to depart from each other on increasing the amount of shear between the two vortex beams as per the theoretical predictions shown in Fig. 1 ($a_1, a_2$) (refer Fig. 4 $a_1-a_3$ and $c_1-c_3$ - experimental ). On the contrary, the increase in the number of polarization singularities with shear can be observed in the case of linear basis superposition as per Fig. 1 ($b_1, b_2$) (refer Fig. 4 $b_1-b_3$ - experimental). On increasing the topological charge of the vortex beam, the index of polarization singularities increases in the resultant beam in case of circular basis superposition as depicted in Fig. 4 ($c_1-c_3$) and discussed in Fig. 2 $b_1$. Note that the results shown in Fig. 4 are due to the combined effect of shear and phase shift between the two beams. The experimental results are in good agreement with theoretically predicted results.
Fig. 4. Experimental, resultant intensity distribution, Stokes phases, and polarization distributions due to superposition of two laterally sheared vortex beams: ($a_1$-$a_3$, $c_1$-$c_3$) in circular basis ($m=1$, $m=2$), and ($b_1$-$b_3$) in linear basis ($m=1$). The amount of lateral shear values in the three cases are (1) 0.15mm, (2) 0.45mm, and (3) 0.75mm.
5. Conclusions
To summarize, the experimental generation of Stokes singularities in polarization lateral shear interferometer is demonstrated for the first time. The resultant beam contains a pair of oppositely charged Stokes singularities of one type forming a dipole and other singularities which are of generic type depending on the polarization basis ($S_i=\pm 1$). The separation between the singularities that forms the dipole and their index can be varied by varying the shear and the topological charge of the vortex beam respectively. The number and location of other types of generic Stokes singularities depend on the shear, the topological charge, and the phase difference between the two superposing vortex beams. The location of Stokes singularities interchanges on changing the polarization basis of the superposing vortices.
Acknowledgments
Gauri Arora acknowledges UGC-SRF.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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