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Abstract
A hybrid polarization grating that can spatially separate orthogonal states of a vector beam with the same polarization topological charge is presented. The hybrid polarization grating is assembled using a fork-shaped polarization grating and a quarter-wave polarization grating and acts as a common pass interferometer for right- and left-circular polarization components of incident vector beams. The hybrid polarization grating can separate two vector beam states that have a 90 ° relative polarization azimuth angle difference. The number of detectable vector beams can be increased by replacing the hybrid polarization grating with a crossed-hybrid polarization grating. Device feasibility was demonstrated experimentally using hybrid polarization grating and crossed-hybrid polarization grating fabricated using the photoalignment method for photoreactive liquid crystals. This approach has the potential to demultiplex several vector beams stably and simultaneously using a compact optical system and should be applicable to vector beam division multiplexing and other applications requiring vector beam detection.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In recent years, the vector beam (VB) has been expected to be used as a new optical multiplexing parameter to increase the information transfer capacity [1–3] because VBs can have an infinite series of polarization topological charges (PTCs). The PTC represents the amount and direction of polarization azimuth rotation of the VB around the optical axis. To realize optical communications using VBs, a mode demultiplexer is required to decode the transferred information, and several different methods have been proposed using geometric phase elements, including the $q$-plate (QP) [4,5], the mode sorter [6–8], and fork-shaped polarization gratings (FPGs) [9–11]. Although the $q$-plate and the FPG can demultiplex VBs with low crosstalk and produce a compact optical system, the detectable VBs are limited to a single PTC for a single QP and FPG. In contrast, although the mode sorter can demultiplex several types of PTC using a single set of elements, it requires two geometric phase holograms with a 4-$f$ configuration scheme, which leads to an increase in the size of the optical system required. As an alternative, the present authors previously developed a method that used a crossed-fork-shaped polarization grating (CFPG), which has the functionality of several types of FPG [11]. The CFPG can demultiplex PTCs with single elements simultaneously, which means that it has the advantages of both the QP and the mode sorter.
We note here that a VB has two orthogonal states for the same PTC, such as the radial and azimuthal polarizations (RP and AP), with a relative polarization azimuth angle rotation of 90 ° between these states. However, the mode sorter, the FPG, and the CFPG cannot distinguish between these states because these methods only detect the number of PTCs. The difference between the two orthogonal states of a VB is the relative phase difference between right- and left-circular polarization (RCP and LCP) components of the VB. One approach to enable detection of this relative phase difference is to measure the interference signal between the RCP and LCP components of the VB. Ndagano et al. reported a method based on a Mach-Zehnder interferometer assembled with a polarization grating (PG) and a mode sorter to obtain the interference signal between the LCP and RCP components of VBs [12]. The PG is a space-variant polarization element whose optical axes are continuously rotated along with grating vector direction and has a function that it can spatially separate the RCP and LCP light components of incident polarized light as diffraction light [13–17]. Hence, the Mach-Zehnder interferometer using the PG can obtain interference signal between RCP and LCP components of the incident VB by recombining decomposed circular polarization components using mirror and beam splitter. However, in this method, the LCP and RCP components of the VB propagate into different optical paths, which meant that the interference signal obtained could be unstable with respect to environmental perturbation. Additionally, the optical system tended to be large because of the requirement for separate optical paths. Under these circumstances, in this paper, we propose an alternative approach that uses a hybrid polarization grating (HPG) assembled with an FPG and a quarter-wave polarization grating (QWPG). Two adhered PGs operate as a common-path optical interferometer between the RCP and LCP components of an incident light beam. In addition, the relative phase difference can be adjusted by varying the relative lateral positions of the FPG and the QWPG because of the geometric phase. Since our method works as a common-path interferometer, the signal obtained is immune to environmental perturbations. Furthermore, we also designed a crossed-HPG (CHPG) that is assembled with a crossed-FPG (CFPG) and a crossed-PG (CPG), which allows us to detect some VB types with different PTC values simultaneously.
2. Principle
VBs have a space-variant polarization pattern around their optical axis, as illustrated in Fig. 1(a). Here, we assume that the polarization azimuth of the VB rotates by $2\pi p$ rad in the counterclockwise direction around the optical axis, and its Jones vector can then be given by:
Fig. 1. (a) Polarization patterns of VBs with $p=\pm 1$ and $p=\pm 2$ where the case of $\gamma$ = $0$ or $\pi /2$. (b) Decomposition of orthogonal states of VBs into LCP and RCP components and the corresponding relative phase differences.
To detect the two orthogonal states of VBs with the same PTC, we propose the use of the HPG. Figure 2(a) shows schematics of the configuration and the functions of the HPG. The HPG is assembled from two types of geometric phase elements: the FPG and the QWPG. Use of the combination of the FPG and the QWPG allows us to decompose $\textbf {V}_{p,0}$ and $\textbf {V}_{p,\pi /2}$. The FPG and the QWPG have space-variant optical axis distributions with optical axes that rotate continuously along a grating vector direction, as illustrated in Fig. 2(b). The FPG has a branch point at the contour line of the optical axis orientation located at the element’s center. We assume here that the number of this branch is $m$. In the HPG, the retardations of the FPG and the QWPG on the elements are uniformly $\pi$ and $\pi /2$, respectively. Additionally, the grating period of the QWPG is one-half of the grating period of the FPG, as illustrated in Fig. 2(c). In addition, the initial alignment direction of the QWPG can be shifted laterally along the grating vector direction by $\Delta x$.
Fig. 2. (a) Schematics of the HPG and its functions. (b) LC alignment pattern for the FPG and QWPG, where the ellipse indicates the LC alignment direction. (c) Structure of the HPG. The QWPG has a one-half grating pitch of the FPG. Additionally, the initial alignment direction of the QWPG is shifted laterally by $\Delta x$, as shown by the red filled ellipse.
Here, we describe the function of the HPG using Jones calculus. As shown in Fig. 3(a) and (b), the incident VB first passes through the FPG. The Jones matrix of the FPG is written as:
From Eq. (5), when the VB passes through the FPG, the LCP/RCP light components of the VB diffract toward the +1st/−1st directions when the value of the TC is increased/reduced by $m$. Diffraction angle is $\Psi =\sin ^{-1}\left (\lambda /\Lambda \right )$, where $\lambda$ is wavelength of incident VB. The handedness of the polarization is also flipped from its incident state. The functions of the FPG are illustrated in Fig. 3(c). The $\pm$1st order light beams are then incident on the QWPG, as shown in Fig. 3(a). The Jones matrix of the QWPG is written as:
We note again here that the grating period of the QWPG is one-half of the grating period of the FPG, which means that the linear phase lamp along the grating vector direction is doubled when compared with the FPG. To simplify, we consider the beam propagation separately here for the $\pm$1st order light beams from the FPG, i.e., $\textbf {E}^{\textrm {FPG}}_{+1}=1/\sqrt {2}e^{i\left (m-p\right )\theta }e^{-i\gamma }e^{i2\pi x/\Lambda }|{R}\rangle$ and $\textbf {E}^{\textrm {FPG}}_{-1}=1/\sqrt {2}e^{-i\left (m-p\right )\theta }e^{i\gamma }e^{-i2\pi x/\Lambda }|{L}\rangle$. For the +1st and −1st order incident light beams, the respective outputs from the QWPG are written as
Equations (8a) and (8b) show that the QWPG separates the incident light spatially into the $\pm$1st order directions of the FPG with equal intensity, as illustrated in Fig. 3(a). We here briefly describe the behavior of the QWPG for circular polarization incidence. Although quarter wave plate converts incident circular polarization to linear polarization, the converted linear polarization can be decomposed into left and right circular polarization components which has same amplitude each other. At this time, the geometric phase is caused into the one side circular polarization component whose polarization handedness is opposite from incident circular polarization’s one. As a result, in view of circular polarization basis, the QWPG diffracts LCP/RCP and RCP/LCP component along 0th and +1st/−1st order directions when the LCP/RCP light passing through the QWPG as shown in Fig. 3(d). Additionally, the grating period of the QWPG is half the length of that of the FPG, which means that it shows a double diffraction angle when compared with that of the FPG. As a result, the wavevectors of the diffracted $\mp$1st order light beams from the QWPG are the same as those of the $\pm$1st order light beams from the FPG, as shown in Fig. 3(a) and (b). Therefore, the Jones vectors of the +1st and −1st order light beams output from the QWPG can be written as:
Equations (9a) and (9b) show that the +1st and −1st order light beams output from the QWPG are both superpositions of two circularly polarized (CP) beams with the same handedness as each other. The two superposed CP beams are converted from the RCP and LCP components of the incident VB, respectively, because these beams have oppositely signed $p$ values, as shown in Fig. 3(a). Therefore, the HPG operates as an interferometer between the RCP and LCP components of the incident VB. In addition, when the FPG and the QWPG are adhered together ($l=0$), the RCP and LCP components are coaxially superposed ($d=0$) and propagate along the same optical path. Therefore, the interference signals of the $\pm$1st order directions are immune to experimental perturbations. We also found that the relative phase difference between the RCP and LCP components of the incident VB can be adjusted via the lateral shift of the QWPG denoted by $\Delta x$ from Eqs. (9a) and (9b). Therefore, when the values of the relative phase difference $\delta \varphi$ between the two interfering beams for the +1st and −1st orders output from the HPG are set at $0$ and $\pi$, respectively, only the +1st/−1st order spots have the Gaussian beams for $\textbf {V}_{p,0}$/ $\textbf {V}_{p,\pi /2}$ incidence because of constructive and destructive interference, as shown in Fig. 3(a) and (b).
Fig. 3. Schematics of mode detection process between (a) $\textbf {V}_{p,0}$ and (b) $\textbf {V}_{p,\pi /2}$ using the QWPG and the FPG. (c) and (d) are the functions of the FPG and the QWPG for the incidence of circular polarizations.
From Eqs. (9a) and (9b), the intensities of the $\pm$1st order beams can be calculated as follows:
Figure 4(a) shows the intensities of the $\pm$1st order light beams plotted as functions of $\Delta x/\Lambda$ in the case where $m=p$. The red and blue solid lines represent the +1st and −1st order light beams, respectively. The upper and lower graphs in Fig. 4(c) correspond to the cases where $\gamma = 0$ and $\gamma = \pi /2$, respectively. We found that the intensity varies sinusoidally, depending on the value of $\Delta x/\Lambda$. In the case where $\Delta x/\Lambda =1/8$, the intensities of the +1st order and −1st order beams become 1 and 0, respectively, under the conditions that $m=p$ and $\gamma =0$. In contrast, the intensities of the +1st order and −1st order beams become 0 and 1 under the conditions where $m=p$ and $\gamma =\pi /2$. We can also analytically obtained $\Delta x/\Lambda = 1/8$ from the Eq. (10a) and (10b) by assuming [$I_{\textrm{+1}}=1$, $I_{\textrm{-1}}=0$, $m=p$, and $\gamma =0$] or [$I_{\textrm{+1}}=0$, $I_{\textrm{-1}}=1$, $m=p$, and $\gamma =\pi /2$]. Figure 4(b) and (c) show the numerically simulated diffraction patterns from the HPG in the cases where (b) the RP and (c) the AP are incident on the HPG. We assume that the amplitude distributions for the RP and the AP are both set to a pillbox shape. We found that the RP and the AP are spatially separated into the $\pm$1st order directions and converted into plane waves because the diffraction patterns have Airy discs. These results indicate that the HPG can detect the two orthogonal states of VBs with the same PTC by measuring the Gaussian components of the $\pm$1st order light beams. By varying the $m$ value of the FPG, we can also detect VBs with other PTCs. Note here that a similar approach was previously proposed by Chen et al. to split an orthogonal linear polarization, i.e., the case where $m=0$ [18]. The HPG proposed here can be regarded as an evolutionary development of this work.
Fig. 4. (a) Diffraction efficiencies of $\pm$1st order diffraction light beams plotted as functions of the grating shift $\Delta x$. (b) and (c) show the numerically simulated diffraction patterns when the RP and AP are incident on the HPG, respectively.
Given the proposed application of the device, it is essential to be able to detect several types of PTC simultaneously. Although the HPG described above can detect the difference between $\gamma =0$ and $\pi /2$, it cannot detect the differences between the PTC values. To overcome this issue, we extend our approach using the CFPG and the CPG. The CFPG and the CPG are complex geometric phase elements that integrate several types of PG function onto single elements [11,19,20]. Based on the principle described above, we simulated the diffraction properties of the CHPG when assembled with the CFPG and the CPG. Here, we define the optical axis orientations of the CFPG and the CPG as
Fig. 5. (a) Optical axis orientation distributions of the CFPG and the CPG. (b) Numerically simulated diffraction patterns output from the CHPG in the cases where the incident VBs are $\textbf {V}_{1,0}$, $\textbf {V}_{1,\pi /2}$, $\textbf {V}_{-1,0}$, and $\textbf {V}_{-1,\pi /2}$. (c) Numerically simulated normalized relative intensities at the center positions of each of the diffraction spots A$_+$, A$_-$, B$_+$, and B$_-$, plotted as functions of the lateral shift of the CPG. Red and blue lines correspond to the $\pm$ order spots. Solid and dashed lines correspond to the A$_{\pm 1}$ and B$_{\pm 1}$ spots. The parameters of the incident VBs are shown in the upper right-hand corner of each graph.
3. Experimental demonstration
To verify the feasibility of the approach that was described theoretically in the previous section, we performed experimental demonstrations. In these experiments, to fabricate the PGs, we used a photo-crosslinkable polymer liquid crystal (PPLC; Hayashi Telempu Co.) in which the LC director can be aligned based on the polarization azimuth angle of the illuminating ultra-violet (UV) linearly polarized beam. An FPG and a QWPG were fabricated by recording a polarization hologram between a Gaussian beam and an OV with parameter $m$ and a state of polarization where a Gaussian beam and the OV have opposite circular polarizations to each other. The fabrication process is described in detail in [11]. Additionally, the CFPG and the CPG were fabricated by applying polymerized LC coatings onto the photoalignment substrates, which are multiply-recorded polarization holograms with varying grating vector directions. Figure 6 shows polarization microscope images of the fabricated (a) FPG, (b) QWPG, (c) CFPG, and (d) CPG. Their grating pitches are approximately 7.4 $\mu$m, 3.7 $\mu$m, 120 $\mu$m, and 60 $\mu$m, respectively. The retardations of the FPG/CFPG and the QWPG/CPG are optimized for wavelengths of 532 /1550 nm. Tables 1–4 show the diffraction efficiencies that were measured experimentally by passing the LP/[RCP and LCP] through the [FPG and CFPG]/[QWPG and CPG], respectively. The wavelengths of the incident polarized beams are 532 nm/1550 nm for the [FPG and QWPG]/[CFPG and CPG]. As a light source, we used a frequency-doubled Nd:YAG laser ($\lambda$=532 nm) and a laser diode (LD) ($\lambda$=1550 nm) for the HPG and the CHPG, respectively. Note here that these diffraction efficiencies were calculated while neglecting the intensity losses due to higher-order diffraction spots and the Fresnel loss. For the FPG and the QWPG, the diffraction efficiencies for each spot are defined as $I_{\textrm{k}} = I_{\textrm{k}}/ \left (I_{\textrm{0th}}+I_{\textrm{1st}}+I_{\textrm{-1st}}\right )$, where k is the label of the diffraction spot. For the CFPG and the CPG, the diffraction efficiency of each spot is defined as $I_{\textrm{k}} = I_{\textrm{k}}/ \left (I_{\textrm{0th}}+I_{\textrm{A+}}+I_{\textrm{A-}}+I_{\textrm{B+}}+I_{\textrm{B-}}\right )$.
Table 1. Diffraction Efficiencies of the FPG
Table 2. Diffraction Efficiencies of the QWPG
Table 3. Diffraction Efficiencies of the CFPG
Table 4. Diffraction Efficiencies of the CPG
Figure 7 shows the experimental setup for mode detection of the VBs when using the HPG and the CHPG. As a light source, we used the Nd:YAG laser ($\lambda$=532 nm) and the LD ($\lambda$=1550 nm) for the HPG and the CHPG, respectively. A laser beam emitted by the light source is converted into a VB by passing it successively through the polarizer (P), the $q$-plate (QP; Thorlabs Inc.), and the half-wave plate (HWP), and it is then incident on the HGP or the CHPG. We prepared two kinds of QPs whose retardations respectively optimized at 532nm and 1550nm, and used them correctly according to the wavelength of the laser beam. The lateral positioning between the FPG/CFPG and the QWPG/CPG consisting of the HPG/CHPG was adjusted using a micro-stage. The normal distance between the FPG/CFPG and the QWPG/CPG was set at approximately 1 mm. Each diffraction spot was focused using a convex lens. At the focal plane, each diffraction spot was observed using a complementary metal-oxide-semiconductor (CMOS)/InGaAs camera for the HPG/CHPG.
Figure 8 shows the experimental intensity patterns for the $\pm$1st order diffraction spots obtained by passing four types of VBs through the HPG in an identical manner. We found that $\textbf {V}_{1,0}$ and $\textbf {V}_{1,\pi /2}$ are spatially separated and are converted into Gaussian beams by passing them through the HPG with $m=1$. In contrast, an HPG with $m=-1$ spatially separates the incident $\textbf {V}_{-1,0}$ and $\textbf {V}_{-1,\pi /2}$. These results agree with the numerically simulated functions of the HPG. Given the intended mode detection application, the contrast between the $\pm$1st order spots is important, and thus we also calculated the contrasts. We here define the contrasts as $\left (I_{\textrm{high}}-I_{\textrm{low}}\right )/\left (I_{\textrm{high}}+I_{\textrm{low}}\right )$, where Ihigh and Ilow represent the highest and lowest intensities between $\pm$1st order diffraction spots. In the cases of the $\textbf {V}_{1,0}$ and $\textbf {V}_{1,\pi /2}$ incidences, the contrasts between the $\pm$1st orders are 0.69 and 0.70, respectively. Besides, in the cases of the $\textbf {V}_{-1,0}$ and $\textbf {V}_{-1,\pi /2}$ incidences, the contrasts between the $\pm$1st orders are 0.72 and 0.72, respectively. Therefore, we successfully demonstrated mode detection of the orthogonal states of VBs with the same PTC using the HPG.
Fig. 8. Experimental diffraction patterns output by the HPG that were acquired by identically passing VBs of $\textbf {V}_{1,0}$, $\textbf {V}_{1,\pi /2}$, $\textbf {V}_{-1,0}$, and $\textbf {V}_{-1,\pi /2}$ through the HPG.
Figure 9 shows the intensity patterns of the A$_{+}$, A$_{-}$, B$_{+}$, and B$_{-}$ diffraction spots that were obtained experimentally by passing four types of VB through the CHPG in an identical manner. The results show diffraction patterns that are similar to the numerically simulated results, which are shown in Fig. 5(b). Note here that, for convenience, the contrast of each picture is enhanced to ensure that the brightness and the intensity of each diffraction spot do not correspond. The intensity of each diffraction spot is shown in each picture. In the cases of the $\textbf {V}_{1,0}$ and $\textbf {V}_{1,\pi /2}$ incidences, the contrasts between A$_{+}$ and A$_{-}$ are 0.61 and 0.51, respectively. In contrast, in the cases of the $\textbf {V}_{-1,0}$ and $\textbf {V}_{-1,\pi /2}$ incidences, the contrasts between B$_{+}$ and B$_{-}$ are 0.47 and 0.47, respectively. Although these contrasts have not matched the theoretically simulated results, we successfully demonstrated the functions of the CHPG, which can detect the four types of VB simultaneously.
Fig. 9. Experimental diffraction patterns output by the CHPG that were acquired by identically passing VBs of $\textbf {V}_{1,0}$, $\textbf {V}_{1,\pi /2}$, $\textbf {V}_{-1,0}$, and $\textbf {V}_{-1,\pi /2}$ through the CHPG.
4. Discussions and prospects
In our experiments, the contrasts of the diffraction efficiencies among the $\pm$1st, A$_{\pm }$, and B$_{\pm }$ spots have not matched the simulated theoretical results. Several possible reasons for this outcome are considered here. The first is the fabrication accuracy of each of the PGs. As shown in Tables 1–4, the fabricated PGs do not show the ideal diffraction properties described in Section 2. These deviations in the diffraction properties lead to deterioration of the interference signal contrast. Additionally, mismatching of the grating periods between FPG/CPFG and the QWPG/CPG will also affect the interference signals. In this case, the two interfering beams converted from the RCP and LCP components of the incident VB will not have the same wave vectors, which means that the diffraction spots will have a fringe-like pattern caused by the interference between the two beams that have crossed each other at an angle. In our experiments, we compensated for the mismatching of the grating periods through slight adjustments of the tilt angles of the FPG and the CPFG for the grating vector direction. To improve the interference signal contrast, it will be necessary to fabricate ideal PGs.
The non-zero distance between the FPG/CFPG and QWPG/CPG degrades the performance of the HPG/CHPG because the two interference beams are a little bit shifted along the lateral direction. To avoid this effect, in our experiment, the two PG is closed to each other as far as possible. In the future, we adhere two PGs to each other with zero-distance after adjustment of lateral position and treated as a single optical element.
In this paper, we have demonstrated mode detection between $\textbf {V}_{1,0}$, $\textbf {V}_{1,\pi /2}$, $\textbf {V}_{-1,0}$, and $\textbf {V}_{-1,\pi /2}$. To apply our device as a mode-demultiplexer for VBs, it will be essential to increase the number of detectable modes for VBs to more than four states. In our previous work, we successfully fabricated a CFPG in which four FPGs with $m=\pm 1$ and $m=\pm 2$ were integrated by multiply recording four polarization holograms [11]. Therefore, the CHPG has the potential to increase the number of detectable VB states by using the CFPG and the CPG, which have numerous grating vectors. In this case, it will be necessary to design a diffraction efficiency balance between each of the diffraction spots for the CPG. It will also be necessary to design the magnitude and the direction of the lateral shift for the CPG.
As with the methods that use the QP, the FPG, and the CFPG, the basic principle for VB mode detection using the HPG and the CHPG is that the specific VB is converted into a Gaussian beam by applying a geometric phase to cancel the helical phase terms of the OV components of the incident VB. This type of approach must use single-mode fibers set at the required spots to filter the Gaussian beam components of the diffracted light beams to reduce crosstalk [4,5,9–11]. Because the purpose of our experiments is to demonstrate the HPG and CHPG diffraction properties that can be applied to VB-demultiplexing, we do not measure the crosstalk between each of the diffraction spots using the single-mode fiber. These measurements will be conducted after high-precision HPGs and CHPGs are realized in future work.
5. Conclusion
In conclusion, we have proposed an HPG and a CHPG to detect the orthogonal states of VBs with the same PTC. The diffraction properties of these gratings have been analyzed theoretically using Jones calculus. The HPG and the CHPG operate as a common-path interferometer between the RCP and LCP components of the incident VB, thus meaning that the diffraction patterns formed by the interference remain stable when subject to environmental perturbations. The feasibility of the proposed principle was demonstrated experimentally using an HPG and a CHPG that were fabricated using a photoalignment technique on a photoreactive liquid crystal, and the results showed good agreement with those obtained from the numerical simulations. Our approach can detect VBs simultaneously, including their initial polarization angle differences, using a compact optical system and would thus be applicable to optical communications and other applications in which a VB demultiplexer is required.
Funding
Japan Society for the Promotion of Science (20K14789); Support Center for Advanced Telecommunications Technology Research Foundation; Murata Science Foundation.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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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