Vector vortex beams are specialized light beams characterized by phase singularities, topological charge, and non-uniform polarization distributions [1]. Due to their unique properties, these beams find applications across various fields such as optical trapping [2], optical communication [3], and computation [4]. Vector vortex beams are represented on the higher-order Poincaré sphere (HOPS) [5], wherein the pole denotes scalar vortex beams and the equator signifies pure vector vortex beams.

So far, different tools and techniques have been used to generate these beams [6]. Devices such as Q-plates [7], ring resonators [8], and exciton–polariton lasers [9] are used to make vector beams which are very simple but can only generate vector vortex beams with a fixed charge and polarization distribution. On the other hand, metamaterials [10] are simple and capable of arbitrary vector vortex generation but demand intricate nanofabrication and are time-consuming and expensive. Spatial light modulator (SLM)-/DMD-based systems [11] are also capable of arbitrary beam generation but demand a huge optical system assembled on an optical bench. Polarization holography has also been used for vector vortex generation but requires several exposures [12–14] or complex optical setups [15] and is not exploited for arbitrary vector vortex beam generation. Here, for the first time, we report a significantly simplified approach that demands i) only a double exposure on a polymer material (size: 5 mm $\times$ 5 mm) using an interferometric setup with SLM, hence no intricate fabrication required; ii) only illuminating the hologram by a collimated laser beam of diameter 7 mm for vector vortex generation, hence no complex optical setup is involved; and iii) only a change in the polarization state of the illuminating beam for the generation of arbitrary vector vortex beams on the HOPS. The simplicity and versatility of the proposed method make it better than the other methods mentioned above.

A polarization hologram is one that is recorded using a superposition of orthogonally polarized object and reference beams on a photo-birefringent material where the recorded pattern is a modulation in birefringence (not transmittance). Here, we record two such holograms sequentially on the same area of an azo-carbazole polymer film for multiplexing. The first hologram is recorded using a left circularly polarized (LCP) vortex beam of topological charge -l, as the object beam ($|-l\rangle |L\rangle$), and a right circularly polarized (RCP) Gaussian beam as the reference beam ($|0\rangle |R\rangle$) as shown in Fig. 1(a). In the second exposure, an RCP vortex beam of topological charge +l ($|+l\rangle |R\rangle$) and an LCP Gaussian beam ($|0\rangle |L\rangle$) are used as the object and reference beams, respectively (Fig. 1(b)). Now, on reconstructing the multiplexed hologram with a reading beam of different polarization states, different beams on the HOPS (including vector vortex beams) can be generated, as shown in Figs. 1(c)–1(e). When the reading beam is an LCP Gaussian beam ($|0\rangle |L\rangle$), the reconstructed beam will be an RCP +l charged scalar vortex beam ($|+l\rangle |R\rangle$) as shown in Fig. 1(c). When the hologram is reconstructed with a linearly polarized reading beam, a superposition of states that corresponds to a vector vortex beam ($|+l\rangle |R\rangle$ + $e^{i\phi }|-l\rangle |L\rangle$) will be generated as shown in Figs. 1(d1)–1(d3). These beams are known as pure vector vortex beams, and they lie on the equator of the HOPS, represented by points A, B, C, and D (in case of a first-order Poincaré sphere (FOPS)) as shown in Fig. 2. Now, when the hologram is read using an RCP Gaussian beam ($|+0\rangle |R\rangle$), the reconstructed beam will be an LCP -l charged scalar vortex beam ($|-l\rangle |L\rangle$) as shown in Fig. 1(e). The reconstructed beams that are shown in Figs. 1(c) and 1(e) represent the northern and southern poles of the HOPS, which are marked as points E and F, respectively (in FOPS), in Fig. 2. When the reading beam is set to any other elliptical polarization state, the beams that correspond to other arbitrary points on the HOPS (other than the ones at the poles and the equator) can be generated (e.g., point “G” on the FOPS shown in Fig. 2). The key points from the above discussion are as follows: i) this method follows the classical superposition-based approach for the generation (reconstruction) of arbitrary vector vortex beams, but does not require any beam splitting, beam combining, or precise collinear superposition arrangements, which significantly simplifies the system, and ii) by simply setting the reading beam to be linearly polarized and then gradually varying it (i.e., rotating a half-wave plate (HWP)), we can generate any pure vector vortex state that lies on the equator of the HOPS; and by setting the polarization state of the reading beam to be circular and then gradually varying it (i.e., rotating a quarter-wave plate (QWP)), we can generate vector vortex states that lie on a vertical path on the HOPS from one pole to the other.

 

Fig. 1. Schematic of (i) hologram recording showing the first exposure (a) and the second exposure (b); (ii) reconstruction schemes (c)–(e), using LCP (c), linearly polarized (d1)–(d3), and RCP (e) reading beams.

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Fig. 2. Schematic representation of the first-order Poincaré sphere, where the northern pole represents the vortex beam of topological charge (l) $+1$ and the southern pole represents the vortex beam of topological charge $-1$.

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For the discussions on experimental results, we use the following mathematical representation of an arbitrary polarization state ($\psi _l$) on the HOPS [5]:

The optical setup used for recording polarization holograms is shown in Fig. 3. The Gaussian beam from a laser of wavelength 532 nm is filtered, expanded, and then guided through a polarizer (P1) and a half-wave plate (HWP$_1$) to achieve an arbitrary linear polarization state. The resulting beam is then split into two orthogonal linear polarization states using a polarizing beam splitter (PBS), where the beam that passes through (p-polarized) serves as the reference beam. The reflected s-polarized beam is directed toward a spatial light modulator (SLM) (HOLOEYE Photonics AG, resolution of 1920 $\times$ 1200 pixels, pixel pitch of 8.1 µm, and frame rate of 60 Hz) which displays the computer-generated hologram (CGH) of a vortex beam. The first-order diffracted beam reflected from the SLM contains the vortex beam (the object beam), which is 2$\times$ demagnified and filtered using a 4f optical relay setup consisting of lenses $L_1$ and $L_2$ of focal lengths 200 mm and 100 mm, respectively. The quarter-wave plates QWP$_{1}$ and QWP$_{2}$ set the object and reference beams in an orthogonal circularly polarized configuration. The object and reference beams are then superposed at a 20° angle on an azo-carbazole polymer film, synthesized in our laboratory [16]. The first and second recordings were done using the beam configurations shown in Figs. 1(a) and 1(b), respectively. The recording beam power was 14 mW, with the first exposure lasting 6 s and the second exposure for 3 s. An equal exposure will result in the second recorded hologram exhibiting a higher diffraction efficiency than the first, and hence, an unequal exposure is used to restore the balance between both reconstructions. The maximum observed diffraction efficiency for the first recorded hologram (after the first exposure) was 14% and for the multiplexed hologram (after the second exposure) was 11%. A higher diffraction efficiency can be achieved if the hologram is recorded with a higher power ($\approx$50 mW) for the same exposure time. After the two exposures, the writing beams are turned off, and the sample is illuminated with a collimated reading laser beam of wavelength 640 nm. Though the recording and reconstruction setups are put together (Fig. 3), the recording setup (green laser and SLM) is not required to be present during reconstruction. In other words, it is enough to simply illuminate the recorded hologram with a collimated beam anywhere and anytime, which establishes the significance of the proposed method. It is to be noted that the reconstruction is limited to longer wavelengths ($\approx$600 nm) since the used azo-copolymer sample exhibits strong absorption in shorter wavelengths (400–600 nm). The reading beam illuminates the sample at the same angle subtended earlier by the writing reference beam. The linear polarization state of the reading beam is controlled by using the half-wave plate (HWP$_2$), and the circular polarization state is controlled by the quarter-wave plate (QWP$_3$). The resulting first-order diffracted beam is captured using a polarization camera (Sony Baumer VCXU-50MP), which enables the measurement of polarization distribution across the beam cross section.

 

Fig. 3. Optical setup used for sequential recording of the polarization hologram (a) and photograph of the azo-carbazole polymer film used for recording (b). Components used in the setup are P, polarizer; HWP, half-wave plate; QWP, quarter-wave plate; S, stopper; M, mirror; SH, shutter; BS, beam splitter; PBS, polarization beam splitter; L, lens; BE, beam expander; and SLM, spatial light modulator.

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Figure 4 shows the intensity profiles captured with and without a rotating polarizer when the hologram is illuminated using a linearly polarized beam (at four different orientations). We start by setting the initial linear polarization state of the reading beam (using HWP$_2$) such that it generates a radially polarized vector vortex beam and is shown in Fig. 4(a). This beam lies on the equator of the FOPS with $v = \pi /2$ and $\phi = 0$. Now, on rotating HWP$_2$ by 22.5° (the polarization state of the reading beam is rotated 45°), a vector beam that resides on a different axis of the FOPS at $v = \pi /2$ and $\phi = \pi /2$ is generated, which is shown in Fig. 4(b). A subsequent 22.5° rotation of HWP$_2$ yields an azimuthally polarized vector vortex beam, shown in Fig. 4(c), which is represented on the FOPS at $v = \pi /2$ and $\phi = \pi$. Finally, another 22.5° rotation of HWP$_2$ results in the fourth fundamental vector beam (Fig. 4(d)), and it lies on the FOPS at $v = \pi /2$ and $\phi = 3\pi /2$. We can conclude that by rotating the polarization state of a linearly polarized reading beam, we can traverse through the equator of the FOPS. It is to be noted that the exposure ratios of the two recorded holograms have an effect on the relationship between the polarization state of the reading beam and the generated vector vortex beam at the equator, which demands further investigation to understand this relationship.

 

Fig. 4. Intensity profiles (captured with and without polarizer) of vector vortex beams, situated on the equator of the FOPS ($v=\pi /2$) and at various longitudinal positions ($\phi$) = 0(a), $\pi /2$(b), $\pi$(c), and $3\pi /2$(d).

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In order to verify the polarization state of the generated vector vortex beam, we have experimentally measured the Stokes parameters ($S_1$, $S_2$, and $S_3$) for each beam and have computed the respective state of polarization (SOP) [17,18]. The SOP can be determined by computing the parameters $\phi$ and $v$ using the expressions $\phi = \arctan (S_2/S_1)$ and $v = \arcsin (S_3/S_0)$, respectively, where $S_0 = \sqrt {S_1^2 + S_2^2 + S_3^2}$. The Stokes parameter and the SOP of all four fundamental vector vortex beams are shown in Fig. 5, where the first three columns show the Stokes parameter ($S_1$, $S_2$, and $S_3$) and the last column shows the calculated SOP. The results shown in Fig. 5 confirm that the generated beams are four pure vector vortex states that lie on the FOPS at the latitude $v=\pi /2$ and at the longitude ($\phi$) given by (a) 0, (b) $\pi /2$, (c) $\pi$, and (d) $3\pi /2$.

 

Fig. 5. Stokes parameter ($S_1$, $S_2$, and $S_3$) and state of polarization (SOP) for vector vortex beams that lie on the equator of FOPS at $v=\pi /2$ and $\phi$ = 0(a), $\pi /2$(b), $\pi$(c), and $3\pi /2$(d).

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We now turn to reconstructions using circularly/elliptically polarized reading beams by rotating the QWP$_3$. We start by setting the QWP$_3$ such that the reading beam assumes an LCP state. The reconstructed beam is an RCP scalar vortex beam of charge l = $+1$ and is shown in Fig. 6(a). This beam lies on the northern pole of the FOPS at $v=0$ and $\phi =0$. Next, we rotate the QWP by 22.5°, which makes the reading beam to acquire an elliptical polarization state. The reconstructed beam has a partial vector vortex state and is positioned between the pole and equator ($v \in (0,\pi /2)$ and $\phi \in [0,2\pi ]$) on the FOPS and is shown in Fig. 6(b). Further rotating the QWP$_3$ by another 22.5° gives rise to a linearly polarized reading beam, which results in a pure vector vortex beam being reconstructed, as shown in Fig. 6(c). This beam lies on the equator of the FOPS at $v =\pi /2$ and $\phi \in [0,2\pi ]$. Lastly, a further rotation of the QWP$_3$ by 45° sets the reading beam to the RCP state, which results in the generation of an LCP scalar vortex beam of charge l = $-1$ as shown in Fig. 6(d). This beam is located on the southern pole of the FOPS at $v=\pi$ and $\phi =0$ of the FOPS. Hence, a continuous rotation of the QWP (i.e., changing the polarization of the reading beam from LCP to RCP) lets us traverse the FOPS vertically from the north to the south pole. However, it is to be noted that the path is not along the longitudinal axis, but an “eight”-shaped path on the FOPS (similar to the one traversed by a QWP on the Poincaré sphere), shown as red dotted lines in Fig. 2. But traversing along the longitudinal axis can also be made possible by a combined rotation of the HWP and QWP in the reading beam. However, tracing the exact path warrants further investigation, considering the theory of polarization holography. In order to quantify the topological charge, the beam is passed through a cylindrical lens ($f$ = 200 mm), and the intensity distribution at the focal plane is captured. The resulting intensity profiles for each of the beams (Figs. 6(a)–6(d)) are shown in the last column of Fig. 6, which validates the topological charge to be properly generated. Now, considering the resolution, changes in the intensity profile of the pure vector vortex beam can be detected for a rotation of the input beam’s polarization as small as 1°. But, in the case of partial vector vortex beams (i.e., as the beam moves closer to the poles of the Poincaré sphere), the changes become progressively hard to detect. It should also be noted that a recording angle of 20° (between the object beam and the reference beam) violates paraxial approximation, and hence, the circularly polarized beams (RCP and LCP) assumed here are not ideally circular but with some ellipticity. For a more accurate analysis, one must adopt the non-paraxial polarization holography theory [19]. Another factor to consider is the deviation from the ideal polarization state caused by different wavelengths (532 nm and 640 nm) used for recording and reconstruction.

 

Fig. 6. Intensity profiles of a scalar vortex of topological charge $+1$ (a), partial vector vortex (b), pure vector vortex (c), and scalar vortex of topological charge $-1$ (d), captured with and without a polarizer and after passing through cylindrical lens ($f$ = 200 mm), obtained by varying the polarization of the reading beam from LCP to RCP.

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In conclusion, we have presented a straightforward and efficient technique for creating arbitrary vector vortex beams on the HOPS by employing a novel “double-exposure polarization-multiplexed hologram recording” method, which requires a simple interferometric setup. The two notable significance are i) simplicity, which requires simply irradiating the sample with a polarization-controlled collimated laser beam, and ii) versatility, traversing the HOPS by simply changing the polarization state of the reading beam. The limitations of the proposed method are i) high absorption at shorter wavelengths, ii) room temperature storage (<1 month), and iii) deviation from an ideal circular polarization due to different wavelengths being used for recording (532 nm) and reconstruction (640 nm), which are purely material-oriented and can be overcome by implementing the proposed method in other photo-birefringent materials such as liquid crystals and PQ/PMMA photopolymer. Hence, the advantage of the proposed method is preserved and has the potential to aid developments in optical information processing applications.