Singularity transformation through single-pass phase modulation of light

Abstract

The emergence of optical vortex phenomena has generated significant interest in optical technologies. By manipulating light parameters such as amplitude, phase, and polarization, new advances have been made in structured light techniques, which enable the creation of phase and polarization singularity. The present work demonstrates a novel and robust technique for converting phase singularity into polarization singularity through single-pass phase modulation of the input beam. The proposed approach simplifies the experimental setup and enables flexibility in the generation of different forms of polarization singularity, including topological conversion from C-point to V-point polarization singularity or otherwise. The method can be useful for diverse applications of singular beams, such as for classical and quantum information processing, and communications.

Introduction

Light is an electromagnetic wave having spatial and temporal parameters such as phase, amplitude, and wavelength¹. These parameters provide multi-dimensions for light field manipulation and thus are used in various optical systems such as optical communication, particle manipulation, image processing, and microscopy^2,3. Such tailored or shaped light is called Structured light⁴. A light beam with optical vortices is one example, which holds phase singularity along its propagation axis⁵.

In Optics, vortices are created at the point of singularity around the spiral structure of the wavefront^6,7. Optical singularities are points of unidentified parameters in the optical field surrounded by phase gradients. Specifically, these singularities illustrate the points at which the light’s phase becomes undetermined or undefined. ‘Singular Optics’ is one of the topics in modern optics where all these singularities are studied and analyzed^8–10.

Singularity arises because of the twist in the optical wavefront, at a point where the phase becomes undefined¹¹. Optical fields around singularity can have several fascinating characteristics. It has been shown that such an optical field holds well-defined orbital angular momentum (OAM) per photon^12–14. Mathematically, singularity in an optical field is expressed using the phase term where the term indicates the azimuthal angle distribution¹⁵. Light can also have spin angular momentum (SAM) associated with the circular states of polarization. Beyond phase singularity, optical fields can also have polarization singularities that arise when the parameters that describe the polarization ellipse become indeterminate^16,17. Such optical fields have spatially in-homogeneous polarization states around the singularity and are referred to as vector fields¹⁸. Depending on the geometric orientation of optical fields in their surroundings of singular points, polarization singularity can be of two types, such as C-points and V-points¹⁹.

C-points are observed in an optical field with spatially in-homogeneous polarization distribution across the transverse plane. The polarization state at the C-point is circular where the azimuth of the polarization ellipse is indeterminate. As a result, the line integral of the polarization azimuth gradient enclosing a C-point is non-zero^20,21. Around this point, the circulating gradient of the polarization azimuth angle can be observed. Polarization distributions around singular points can have different geometric patterns. Some commonly used patterns have been named lemon and star. Another type of polarization singularity is the V-point singularity which occurs at the vector fields at a point of zero intensity surrounded by the field with the circulating orientation of linear polarization states²². Different names have been assigned to some commonly used geometric patterns of polarization distribution around V-points, such as Type I, II, III, and Type IV. Their polarization distributions of C- and V-points²³ have been shown in Fig. 1.

Fig. 1.

The distribution of polarization in the optical field with polarization singularities. The polarization ellipse in star and lemon types is indicated by row (a) through columns (i) & (ii), whereas the linear state of polarization is indicated by row (b) through columns (iii-vi).

Optical singularities are crucial in various fields of optics, including optical tweezers for manipulating microscopic particles, high-dimensional quantum entanglement, and creating novel light beams with unique properties like OAM and SAM^24,25. They have also found applications in various fields of science and technology, including material processing optical communication, and quantum information processing^26,27.

Vortex beams with phase singularity can be generated by introducing a phase singularity into a Gaussian light beam, resulting in a helical wavefront caused by the spiral phase structure. One common method to generate vortex beams is using an electro-optic spatial light modulator (SLM) to make a desired phase pattern onto a laser beam²⁸. Another technique uses a spiral phase plate which introduces a phase delay that varies azimuthally around the beam axis^29,30. Besides this, a vortex beam can be generated using holographic techniques, where a hologram is designed to produce the desired phase distribution³¹.

In phase singularity, there are circulating phase gradients, while in polarization singularity, circulating stokes phase gradients. At these singularity points, some specific values become undefined which are discussed in the above paragraph. For direct transformation, a circulating phase gradient into a stokes phase gradient establishes a connection between phase and polarization singularities. We demonstrate this coordination is illustrated through theoretical as well as experimental methods where polarization singularities are directly generated through phase singularity³².

Generation of complex polarization singularity typically requires an elaborate system involving multiple phase modulations using SLM. One common approach to generating polarization singularity is by superposition of scalar vortex beams with orthogonal polarization³³. In this technique, an experimental arrangement is used to combine two beams with an orthogonal state of polarization (SOP) and with different phase distributions³⁴. Another method uses dual-phase modulation^35,36, where at least two light components are modulated with SLM to generate polarization singularity fields. Most of the reported articles are primarily focused on the theoretical and experimental generation of polarization singularities. For specific applications, direct conversion of singularity from phase to polarization³² would be useful. However, techniques for direct conversion from phase to polarization singularity is a relatively less explored area, and to the best of our knowledge, very few methods have been investigated and reported.

The present work reports a novel technique to directly convert phase singularity in a light beam to polarization singularity through a single-pass phase modulation with a single hologram encoded on SLM. The input vortex beam having phase singularity has been created with a spiral phase plate, and it is converted to C-point and V-point polarization singularity through a single reflection from a phase-only SLM³⁷. This streamlined design not only reduces system complexity but also improves alignment ease and reproducibility. It does not require setups with interferometric³⁸ configurations. Polarization singularities, typically characterized by their unique polarization textures in an optical field hold potential for various applications. In quantum information processing³⁹, their entanglement can be utilized for encoding and transmitting information security. Beyond quantum technologies, such beams are utilized in advanced imaging, optical trapping, and communication systems, making our method a valuable tool for both fundamental and applied research.

Many articles have been reported in the literature describing methods for generation, characterization and applications of phase and polarization singularity. However, techniques to transform a beam with isolated phase singularity into a desirable polarization singularity is a less explored area. To the best of our knowledge, explicit investigation on the direct conversion of a phase singularity embedded in a laser beam into desirable polarization singularity describing the scientific and technical details has not been reported in prior works. In this background, the present work demonstrates a straightforward and novel approach for the direct transformation of a phase singularity into a tunable C or V-type polarization singularity by utilizing an SLM and without any mechanical change in the setup. Topological conversion of C-point to V-point polarization singularity has been also presented.

Principle

Formation of phase singularity

One of the common modes for light beams is the Gaussian mode, which can be derived from the wave equation. Mathematically, the Gaussian beam propagates along direction can be represented using the following Eqs⁴⁰,

1

In the above equation, is the amplitude term, represents the 1/e radius of the Gaussian term given by with being the beam waist radius where z is the axial distance from the beam focus (or waist) being the beam waist and the Rayleigh range. denotes the Gouy phase given , and with x and y being the coordinates in the transverse plane. The terms and denotes the coordinates along the transverse plane of the beam.

To create a phase singularity, the Gaussian beam can be converted to a vortex beam through phase modulation. Mathematically, the modulated beam can be expressed as follows,

2

where represents the Electric field amplitude of the Gaussian beam and the term indicate the azimuthal phase term^41,42. The phase value distributions, can be expressed as,

3

The term denotes the topological charge (TC). In the present study, the magnitude of TC has been assumed unity, such that . As the beam propagates, the formation of a vortex can be observed through the doughnut-type intensity profile, which indicates the phase singularity along the propagation direction of the optical field. Then, the complex field amplitude of the resultant field in the transverse imaging plane at a propagation distance z can be estimated by Fresnel diffraction solution⁴³, which is expressed as follows,

4

The output term represent vortex beam where is the wave number and the coordinates and denote the integration variables and the integral limits correspond to the source area . Equation 4 representing the vortex beam with helical wavefront can be simplified as follows,

5

Where the term denotes the complex amplitude term resulting from the Fresnel diffraction solution. Equation 5 denotes the vortex beam having phase singularity with positive TC. As indicated in Fig. 2, the phase singularity arises at a region around which the phase values change by a signed integral multiple of 2π. These integral values are the TC, which indicates the number of windings of the phase per revolution around the singular point. The phase increases in opposite directions around the singular point for different signs of TC.

Fig. 2.

Generation of vortex beam through phase modulation of Gaussian beam. (a) Gaussian beam (b) azimuthal phase distribution, and (c) vortex beam with phase singularity.

The simulation result for creating phase singularity through phase modulation of the Gaussian beam is presented in Fig. 2, here we have used a spiral phase plate for generating the phase singularity in the input beam³⁰. Simulations were performed in the MATLAB platform to obtain the intensity and phase distribution of the beam⁴⁴. The Gaussian beam profile is calculated using Eq. (2) as shown in Fig. 2a. Figure 2b shows the phase with azimuthal dependence. Such spatial distribution of phase creates phase singularity along the propagation direction of the beam. From Fig. 2c it can be observed that the phase singularity is a point within the cross-section of an optical field where the phase is indeterminate, and the gradient of the phase circulates around the singularity. An optical field with azimuthally varying phases results in the formation of phase singularity⁴⁵.

Conversion from phase to polarization singularity

The proposed technique for direct conversion of a vortex beam into the polarization singularity with polarization singularity is described using the Jones matrix⁴⁶, formalism and the schematic diagram is shown in Fig. 3. The input vortex beam is transmitted through a linear polarizer with its polarization axis at 45° to the x-axis. After transmitting through the polarizer, the vortex beam can be represented as two orthogonal components of the polarization states, the Jones vector of the resultant beam, is expressed as,

6

Fig. 3.

Schematic diagram illustrating the conversion of phase to polarization singularity.

The above equation indicates that the light beam is linearly polarized at from the x-axis. Also, the beam has a spatially uniform polarization distribution. The term is the complex field amplitude of the input vortex beams as illustrated in Eq. (5).

The proposed method for phase-to-polarization singularity conversion is based on the phase modulation of the selective orthogonal component of the input singular light beam. Liquid crystals on silicon-based spatial light modulators (SLMs) with parallel alignment of liquid crystals are commercially available devices for phase modulation. Since the liquid crystal is birefringent, it allows polarization selective phase encoding of the desired phase distribution as the requirement. SLM modulates the light component that is polarized along its slow axis, while the other component remains unmodulated. Using this feature, an optical set-up can be designed to selectively control the phase of the light component through single modulation. In the present discussion, we assume that the slow axis of SLM lies along the x-axis, to ensure that the phase distribution is encoded along the x-component of the beam. The Jones vector of the resultant beam is obtained by multiplication of the Jones matrix⁴⁶, of the SLM with the incident beam as follows,

7

The term indicate the complex field amplitude of the resultant modulated beam after passing through the SLM. The phase value distribution to be encoded onto the SLM is expressed as follows,

8

In the above equation, represents the azimuthal phase gradient, distributed over the range 0 to 2π. The term is the extra phase delay to provide the desired phase modulation and term represent TC values of azimuthal phase distribution to be encoded.

The orthogonal polarization component of the modulated beam are linearly polarized along the and axis, respectively, so to change their polarization basis from linear to circular, the beam is transmitted through a quarter wave plate (QWP) with its slow axis aligned at . A quarter wave plate provides a phase shift of (or one-quarter of a wavelength) between polarization components, which helps to produce a circularly polarized beam. The resultant beam obtained by multiplication of the Jones matrix of quarter wave plate with modulated beam expressed as follows.

9

The beam is then propagated to a short transmission distance . The resultant complex electric field is then expressed as follows⁴³,

10

In this equation, the term represents the Jones vector of the resultant output beam in the circular basis of polarization. It can represent a beam with polarization singularity on substitution of appropriate parameters in the term . Both C-point and V-point singularity can be realised through the encoding of appropriate phase distribution through the SLM without the need for any mechanical movement of optical elements. Figure 3 describes the sequential process of the proposed concept of direct conversion of phase to polarization singularity of the input beam through polarization-sensitive phase modulation⁴⁷.

Polarization analysis

Stokes parameters provide a convenient way to characterize the polarization singularity of the optical field. Stokes parameters , , , and are related to a component of the transverse electric field of light , which is mathematical expressed as^1,46–48,

11

Where a superscript star () indicates the complex conjugate operation and denotes the time average. The parameter indicates the total intensity of the light.

We can determine stroke parameters of the optical field through the intensity detector after passing across a linear polarizer and quarter wave plate (QWP) at various retardation values and polarizer alignment angles respectively.

The optical set-up for intensity measurement is represented in Fig. 7, and the resultant intensity of light as their function is given as⁴⁶,

12

Fig. 7.

- Schematic diagram for conversion process of phase to V-point polarization singularity.

Using Eq. (12), Stokes parameters can be obtained from the intensity measurements using the following relation,

13

Polarization distribution is crucial for measuring the Stokes parameter at each spatial point.

The angular parameter of the polarization ellipse can be obtained from the Stokes parameters⁴⁶, represented as follows, the angle of azimuth and angle of ellipticity are as follows,

14

where the range of azimuthal angle is and the other parameter is lies between . So, all these things are introduced to the characterization of polarization singularity. First, we evaluated Stokes parameters from the intensity using Eq. (13). Then, the angular parameters were obtained using Eq. (14), to plot the polarization ellipses at different points. Stokes parameters⁴⁶ have been utilized for simplicity when determining and calculating the polarization distribution of light from intensity records.

Conversion of phase to C-point singularity

This section discusses the direct conversion of input vortex beams with C-point polarization singularities, its schematic diagram indicated by Fig. 4. C-point singularity occurs in an in-homogeneously polarized ellipse field, where the state of polarization is circular³². At this point, the orientation of the major axis of the polarization ellipse is undefined, making it a singularity in the azimuth distribution of the field. Figure 4, shows the sequential diagram of the conversion of C-point singularity. General C-point polarization singularities include those designated as Type C1, C2, C3 and C4, shown in Figs. 5, 6.

Fig. 4.

Schematic diagram for conversion process of phase to C-point polarization singularity.

Fig. 5.

Results of simulated C - point polarization singularity of type C1 & C3. Column (i) shows total intensity. Columns (ii)–(vii), show intensity when beams are analyzed using a QWP and a polarizing filter at different sets of retardation and polarizer alignment angles. Column (viii) presents polarization distribution, (ix)–(x) presents corresponding polarization angular parameters and (xi)–(xiii) represents Stokes parameters, respectively.

Fig. 6.

Results of simulated C – point polarization singularity of type C2 & C4. Column (i) shows total intensity. Columns (ii)–(vii), show intensity when beams are analyzed using a QWP and a polarizing filter at different sets of retardation and polarizer alignment angles. Column (viii) presents polarization distribution, (ix)–(x) presents corresponding polarization angular parameters and (xi)–(xiii) represents Stokes parameters, respectively.

This article consists of such types of polarization singularity generated directly from phase singularity. For converting the C-point from the input vortex beam, a phase pattern with TC = − has been put in the phase modulator. During modulation, only one orthogonal component of the input vortex beam is modulated, and the other ones remain unchanged because the phase modulator modulates one component that is polarized along the slow axis of the phase modulator, which is shown in Fig. 4, in terms of and . The term is the phase given by Eq. (3). After phase modulation, input beams are converted into modulated beams with topological charges and . The C-point singularity can be attributed to these two modulated beams, which are linearly polarized. Finally, the modulated beams are transmitted through QWP1 at for an elliptical polarization basis. For Analysing that the C -point has spatially varying polarization³², the beam is transmitted through a linear polarizer and gets the various intensity distributions in different orientations angles.

Linear polarization components that are polarized along the horizontal, vertical, + 45°, and − 45° correspond to orientation angles 0°, 90°, 45°, and 135° with respect to the horizontal axis. The recorded intensity is displayed in Fig. 5 columns (i-v). Similarly, the right- and left-handed circular polarization components are verified through QWP2, which is inserted ahead of the analyser at transmission angles of 45° and 135° in Fig. 4. Simulation results of the right- and left-handed circular polarization components are recorded in Figs. 5, 6, from columns (vi-vii).

To characterize C- point singularity several recorded intensity images are required. These images help to determine the polarization distributions, stokes parameters and angular parameters of the polarization singularity which are shown in Figs. 5 and 6, from columns (viii-xiii). For this configuration, the resultant optical field is given by Eq. (9) with the appropriate selection of given by Eq. 8. C-points occur when one of the TCs remains zero. Four different polarization distributions have been generated using suitable phase modulations obtained from Eq. (11) by substituting values accordingly. For types C1 and C3, the charges and extra phase delay for right-handed lemon and star fields following as and with QWP oriented at + 45. By appropriately selecting the above parameters, C-points can be generated using the setup shown in Fig. 7. Corresponding simulation data are represented in Fig. 5 with proper colour scales.

In the same way to generate types C2 and C4, charges and extra phase delay of left-handed lemon and star fields are as follows and where QWP is oriented at -45. By appropriately selecting the above parameters for PVD preparation, beams with C-points can be generated using the same setup, which is shown in Fig. 7. Its Simulation results are indicated in Fig. 6 with appropriate colour maps.

Conversion of phase to V-point singularity

A V-point polarization singularity occurs in an in-homogeneously polarized vector field, where the predominant SOP distribution is that of linearly polarized states. The V-point singularity in vector fields refers to an intensity null point in which polarization azimuth is undefined^32,37.

This section describes the direct conversion of V-point polarization singularity from phase singularity, which involves precise modulation of the phase patterns using the phase modulator to control the polarization characteristics and create V-point singularities in input vortex fields. This method offers compactness and robustness in generating V-point polarization singularities, making it easily integrated into various V-point singularities by changing its topological charge and extra phase delay which are shown in Figs. 8 and 9.

Fig. 8.

Results of simulated type (a & c) C – point polarization singularity. Column (i) shows total intensity. Columns (ii)–(vii), show intensity when beams are analyzed using a QWP and a polarizing filter at different sets of retardation and polarizer alignment angles. Column (viii) presents polarization distribution, (ix)–(x) presents corresponding polarization angular parameters and (xi)–(xiii) represents Stokes parameters, respectively.

Fig. 9.

Results of simulated type (b & d) C – point polarization singularity. Column (i) shows total intensity. Columns (ii)–(vii), show intensity when beams are analyzed using a QWP and a polarizing filter at different sets of retardation and polarizer alignment angles. Column (viii) presents polarization distribution, (ix)–(x) presents corresponding polarization angular parameters and (xi)–(xiii) represents Stokes parameters, respectively.

In the previous section, we discussed the schematic diagram illustrating the conversion of phase singularity to polarization singularity for C-point, a process that is similarly applicable to V-point singularity as well, which is shown by Fig. 7. The primary difference in this case, is that for the phase modulation of input vortex beam, grayscale pattern encoded on the phase modulator with TC is -2. After modulation magnitude of the modulated beams is which are making sure to V-point singularity in output beams.

So, the conversion of V-point polarization singularity from phase singularity for this configuration, the resultant optical field is given by Eq. (9) with the appropriate selection of given by Eq. 8. For V-point polarization singularity, the magnitudes of TCs and remain equal but opposite sign. For azimuthal and radial distributions of polarization around V-points, the value of phase delay is and , respectively, whereas TC remain for all these cases. Some commonly used beams, such as type V1 and V3 for the simulated polarization distributions for azimuthal and radial with colour maps, are shown in Fig. 8, where colour maps indicate the range of given parameters. Similarly, for types V2 and V4, the simulated polarization distributions for azimuthal and radial are shown in Fig. 9.

Also, we can produce a different combination of topological charges of modulated beam components which helps the generation of bright C-points, dark C-points, and V-points. Here in the case of Bright C-points are generated when one of the superposing orthogonal components of the modulated beam is non-vortex (plane wavefront), with either , or . The handedness of the C-point matches the handedness of the non-vortex-modulated beam. In other cases dark C-points where superposing orthogonal components of modulated beams contain vortices with nonzero, unequal topological charges . Since both beams have dark vortex cores, the resulting C-point is dark. The handedness of the dark C-point is determined by the circular polarization component with the lower magnitude of the topological charge. Similarly, for the generation of a V-point singularity when always occur at the intensity null point³². Here we generate simulated bright and dark C-points and high order V-point singularity polarization distribution is shown in Fig. 10. Where for bright and dark C-points TCs are (),() and, () respectively are represented lower to high order Bright & Dark C-points. These points contain circular SOP in the centre of respective polarization distribution as well as V-point TCs are and where SOP is null in the centre of its polarization distribution, which is highlighted in Fig. 10. Table 1 presents key parameters characterizing the input scalar beams and output vector beams, including the TC and its polarization index ³². Each parameter value is essential for understanding the different types of vector beams which are shown below,

Fig. 10.

Polarization distributions of dark and bright C-point and higher order V-point singularities.

Table 1.

Generation of polarization singularity from phase singularity.

TC (l) of phase singularity in input beam	TC () in PVD encoded in SLM	Output vector beam		Type of singularity
		TCs of modulated beam components	Polarization index
1	−1	(0, 1)	−1/2	Star bright C-point
2	−2	(0, 2)	−1	Bright C-point
2	−3	(−1, 2)	−3/2	Dark C-point
1	−2	(−1, 1)	−1	Azimuthal V-point
2	−4	(−2, 2)	−2	Spider web V-point

is a topological charge of the left circularly polarized component and is the topological charge of the right circularly polarized component.

Topological conversion from C-point to V-point

Polarization-selective phase modulation can be also utilized for the topological transformation of C-point to V-point polarization singularity or otherwise. Figure 11 presents a schematic diagram that explains the mechanism of singularity conversion. Additionally, the simulated beam profile provides a detailed visualization of the conversion. This approach effectively manipulates the polarization distribution of the beam to achieve the desired topological mode conversion of light from C-point to V-point. It can be implemented through a single SLM that allows polarization selective phase modulation.

Fig. 11.

Schematic diagram of transformation between C-point to V-point singularity through polarization-selective phase modulation.

Experimental implementation

This section describes the experiment set-up for the conversion of Phase to Polarization singularity and is designed to investigate the formation of different C & V point singularities. Figure 12 represents the schematic diagram explaining the singularity transformation with simulated beam profiles. The experimental setup used to obtain the desired polarization singularity is shown in Fig. 13. High stability DPSS Laser has been used as the coherent light source, having a wavelength of 532 nm and a maximum power of 300 mW. The spiral phase plate (SPP) is used to manipulate the phase of light³⁰. Furthermore, Phase-only liquid crystal-based Spatial Light Modulators (SLMs) have generated polarization singularity with C & V points. The SLMs (X10468-04, Hamamatsu Japan) had a pixel pitch of 20 μm and a resolution of 792 × 600 pixels. Its wavelength operation range is 510 ± 50 nm which is used for the experiment are reflective types with 255 levels for addressing phase modulation from 0 to 2π. These modulated beams are transmitted through quarter-wave plates and polarizers to provide linear to circularly polarized beams by changing the basis of input beams. The laser beam is linearly polarized along the x-axis (laboratory horizontal). The slow axis of both the SLMs is also kept along the x-axis to achieve phase-only modulation²⁸. The intensity distribution of the stokes polarimetry is recorded through the camera (CCD sensor, Beam Gage Standard). The camera has a resolution of 1928 × 1448 pixels and a 3.69 μm pixel pitch.

Fig. 12.

Schematic diagram explaining the conversion from phase to C- and V-type polarization singularity.

Fig. 13.

Experimental set-up for directly converting phase to polarization singularity in the optical field. Where SPP is a spiral phase plate, SLM represents the spatial light modulator, P1 & P2 represent the polarizer, and QWP indicates the quarter-wave plate.

SPP is a specialized optical device that alters the phase of light to generate a vortex beam. The SLM is polarization sensitive and modulates only the light polarized along its slow axis²⁸. Its slow axis is kept along the -axis. The collimated laser beam is directed towards the SPP display for the generation of the input vortex beam. After transmission from the SPP display, the beam acquires the phase in one of the beam components. After that beam passes through a linear polarizer aligned at 45º and differentiates into two orthogonal parts with phase . Then the beam is projected onto the SLM display to modulate the phase in one of the beam components that is polarized along the slow axis of the SLM. Finally, the modulated beam is brought to an elliptical polarization basis using a QWP whose slow axis is aligned at 45º with the x-axis. The output beam can be represented by Eq. (9).

Results and discussion

The experimental results, as shown in Fig. 14, have a good correlation with the simulated data shown in Fig. 6. In this section we experimentally generated two sets of C-point singularity as illustrated in Fig. 14. Where Obtained intensity profiles of resultant beams are presented. This figure also shows the angular parameter distributions and corresponding to these beams. In all cases, the C-point singularity lies at the center of the beam cross-section. For star type C-point (C-point index = − 1/2), azimuth angle increases in the negative direction (clockwise) around the singular point whereas for lemon type C-point (C-point index = 1/2), it is along a positive direction (anticlockwise). Figure 14 shows the generated optical fields, denoted by C2 and C4, which are classified as left-handed star and left-handed lemons fields, respectively. In the same figure, columns (i-v) show the intensity profiles of optical fields, and columns (vi-viii) present the polarization distribution and spatial distribution of polarization parameters. The stokes parameters S1, S2 and S3 are shown in Columns (ix), (x) and (xi) for the determination of intensity and polarization distribution of the output beam⁴⁶. In Fig. 14, only one of the components is modulated. The dark points that appear in the intensity profiles are due to the optical vortex, which indicates the presence of phase singularity. The singularity in polarization can be observed in polarization distributions as shown in column (viii), which lies approximately at their center.

Fig. 14.

Experiment data of C-Ppints represented where Column (i) shows total intensity. Columns (ii)–(v), show intensity when beams are analyzed using a QWP and a polarizing filter at different sets of retardation and polarizer alignment angles. Column (vi) presents polarization distribution, (vii)–(viii) presents corresponding polarization angular parameters and (ix)–(xi) represents Stokes parameters, respectively.

In Fig. 15, the experimental data for the V-point singularity shows a strong agreement with the simulated results which is presented in Fig. 8. Intensity profiles for V-point singularity in the optical field are presented in columns (i) to (v) of Fig. 15. To further analyse the polarization distribution profile of resultant beams represented by column (vi). Angular polarization parameters and have also been shown in the column (vii) and (viii) of this figure. For V- point singularity stokes parameters are shown in columns (ix), (x) and (xi). It can be observed that the experimentally obtained intensity distribution of different vector fields is in good agreement with the simulated results.

Fig. 15.

Experiment data of V-Points represented, where Column (i) shows total intensity. Columns (ii)–(v), show intensity when beams are analyzed using a QWP and a polarizing filter at different sets of retardation and polarizer alignment angles. Column (vi) presents polarization distribution, (vii)–(viii) presents corresponding polarization angular parameters and (ix)–(xi) represents Stokes parameters, respectively.

The azimuth angle distribution identifies the polarization singularity index. The azimuth angle around the singularity points increases in the positive direction (counterclockwise) in all the cases. Therefore, for the vector fields represented in rows (a-b) in Fig. 15, the corresponding Poincare-Hopf (PH) index are + 1, and + 1, respectively. It should be noted that vector fields with azimuth and radial distributions are classified as Type-I and Type-II, respectively. Their polarization distribution is different, but they have the same PH index⁴⁹.

Prior research has shown that polarization singularity beams can be produced with a single SLM^3,50 and using a spiral phase plate³⁰. Nevertheless, most of these investigations use two-phase patterns to modulate the beam’s orthogonal components, usually splitting the SLM display into two regions for each component⁵¹. Furthermore, several methods use interferometric configurations for beam superposition, which may present alignment difficulties^33,52. On the other hand, this method’s use of a single-phase pattern and non-interferometric technique streamlines the process of converting phase singularity to polarization singularity or polarization singularity beams. Furthermore, this technique states that polarization singularities can be directly converted from input phase singularities.

Conclusion

In conclusion, the present work demonstrates a technique for direct conversion of phase to polarization singularity through polarization-selective single-pass phase modulation. This method is distinct from traditional approaches that rely on the conversion of scalar beams to polarization singularity beams. The present technique offers a straightforward way of singularity transformation that can be implemented with a device, such as SLM with parallel aligned liquid crystal configuration, and without any mechanical change in the setup for tunability. The setup does not require any critical experimental arrangements that make optical alignment easier and enhance robustness against external vibrations.

Singular optics has become a fast-growing research area owing to its emerging applications in crucial areas of science and technology. As a result, the creation, manipulation, and transformations of optical polarization singularity have become prerequisites for further investigations and applications of optical singularity. The present work offers a practical and straightforward way for singularity transformation which can be useful in all such applications. Therefore, the present work would be of significant interest to the scientific community. It holds potential applications in classical and quantum information processing, including communication and quantum state manipulation³⁹.

Author contributions

L. wrote the main manuscript text and prepared figures. All authors reviewed the manuscript.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the corresponding author, Praveen Kumar upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.