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- Advanced Photonics
- Vol. 5, Issue 5, 056002 (2023)

Abstract

1 Introduction

The pioneering works of Beth and Mechanical1 and Allen et al.2 illustrated that photons can possess two distinct forms of angular momentums (AMs), spin angular momentum (SAM), and orbital angular momentum (OAM). SAM has two eigenvalues $\sigma =\pm 1$, corresponding to the right and left circular polarizations of a macroscopic beam,1 while the eigenvalue $l$ of OAM, also known as topological charge, can be any integer. The eigenvalue $l$ determines the helical wavefront as $\mathrm{exp}(il\phi )$, where $\phi $ is the azimuthal angle.2^{,}3 Originally, researchers focused more on OAM beams, namely, optical vortices, due to their enormous potential in large-capacity data transmission,4^{–}^{–}^{,}15 Currently, TAM has inspired many advanced applications, ranging from classical to quantum physics such as laser processing16 and motion detection.17 Moreover, TAM provides 2 degrees of freedom (DoFs), SAM and OAM; thus it is of great significance in achieving higher-dimensional entanglements.18^{,}19

Processing TAM components on demand is a crucial basis for TAM applications. In other words, for a TAM beam, one can extract corresponding TAM photons and drop out irrelevant TAM photons selectively, which can be regarded as a TAM filter, analogous to a common wavelength/frequency filter. Such manipulation is of great significance, especially as the receiver for TAM-based communications, remote detection, and optical computing. TAM filtration can be accomplished through simultaneous SAM and OAM filtration. SAM filtration is simple and can be implemented through polarization elements such as wave plates, whereas OAM filtration is intricate. In the past two decades, much effort has been devoted to excavating more effective schemes for OAM probing and extraction, for instance, by probing and recognizing OAM modes through diffraction optical elements (DOEs),20^{–}^{,}28 and so on. Various OAM sorters have also been developed such as the Mach–Zehnder interferometer29^{–}^{–}^{–}

In this work, we demonstrate a TAM manipulator for photons that enables selective TAM extraction. The TAM filter has a quasi-symmetric structure, where the incident beam can be seen as propagating through TAM separation unit twice, in opposite directions. In the TAM separation section, a couple of DOEs are fabricated through polymerized liquid crystals (LCs), designed from wavefront-copy embedded log-polar coordinate mapping, leading to little cross talk of OAM separation. Meanwhile, these DOEs introduce conjugated phase modulations for opposite SAM components and thus separate each TAM state spatially. A spatial filter is employed to reserve or block single or multiple TAM states, and then the separated beams are recovered through the reverse TAM separation unit. Such a process maps the incident beam from spatial domain to “position-TAM domain” for easy filtering, and then converts it back into the spatial domain again. To prove the principle, 42 single TAM states are involved to evaluate the mode purity reduction for a nonfiltering test; 16 modes are used for selective blocking tests. The simulation and experiment are accomplished with a favorable performance.

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2 Concept of the TAM Manipulation

As mentioned above, a single TAM mode, also known as a separable state, is written in direct product form as $|\sigma \u27e9|l\u27e9$. A TAM beam $|\psi \u27e9$ could simultaneously contain multiple TAM states, which has the form of $|\psi \u27e9=\sum _{\xi}\sum _{\kappa}{\alpha}_{\xi}^{\kappa}|{\sigma}_{\xi}\u27e9|{l}_{\kappa}\u27e9$ with ${\alpha}_{\xi}^{\kappa}$ the complex coefficient, namely, multiplexed TAM states. Here the SAM component $|{\sigma}_{\xi}\u27e9$ has two eigenvalues, $\sigma =+1$ and $\sigma =-1$, corresponding to the right and left circular polarizations, and can be denoted as $|R\u27e9$ and $|L\u27e9$, respectively. The OAM component $|{l}_{\kappa}\u27e9\propto \mathrm{exp}(i{l}_{\kappa}\phi )$ with $\phi $ the azimuthal angle and $\kappa $ can be any integer. The TAM represents the AMs of each photon as $J=(\sigma +l)\hslash $ under paraxial approximation with $\hslash $ the reduced Planck constant.

The proposed TAM manipulator is expected to retain or eliminate specific TAM components as needed. Figure 1 sketches the concept and principle of the TAM manipulator. For instance, an incident beam comprises four TAM states, while what we want is two of them. After passing through a spatial filter, only the desired TAM components are reserved, while others are removed. For this purpose, each TAM state must be exactly extracted, where a feasible way is to separate various TAM states with high accuracy. Specifically speaking, each single TAM state $|\sigma \u27e9|l\u27e9$ of the incident beam could be diffracted to different positions in the observation plane, in which each single TAM mode can be blocked or passed, enabling selective TAM manipulation. That is, some of the TAM states are filtered. The separated modes are then combined through a symmetric unit and reconverted to a TAM beam again to accomplish the TAM manipulator.

Figure 1.Concept of a TAM manipulator: structure of a TAM carried beam passes through the TAM manipulator for filtering. Beam propagation diagram of (a) the TAM separator and (b) the TAM reverser.

2.1 Manipulating TAM States on Demand: TAM Separation

TAM state separation could be viewed as simultaneous separation of SAM and OAM. The SAM separation can be accomplished through a polarization grating, the essence of which is a geometric phase modulation result from the photon spin Hall effect.41 As for OAM separation, its schemes are diversified,29^{–}^{,}33 and ${\mathrm{D}}^{2}\mathrm{NN}$.28^{,}42 Obviously, integrating polarization grating and log-polar transformation elements enable SAM and OAM separation simultaneously; thus, TAM separation is available. Actually, in the past decade, based on this idea, various TAM separation schemes have been proposed.38^{–}

High-performance SAM separation can be achieved easily through geometric phase optical elements (GPOEs). The conversion efficiency of a GPOE can reach 100% under half-wave conditions, and their thickness is in the wavelength magnitude. A GPOE with spatially varying main-axis orientations according to $\varphi $ can transform the incident circular polarization to its opposite; namely, the SAM state is reversed. Meanwhile, an additional phase is introduced as $\pm 2\varphi $ for incident LCP and RCP, respectively. Obviously, embedding a linear gradient phase denoted by $2\varphi =k\text{\hspace{0.17em}}\mathrm{sin}\text{\hspace{0.17em}}(\theta x)$, where $k$ and $\theta $ are the wavenumber and the period of linear gradient phase, respectively, the two SAM components will be diffracted to different spatial positions and thus be effectively separated.

For separating OAM DoF, the fan-out scheme, proposed by Mirhosseini et al.,33 is employed. Such a fan-out scheme is actually the wavefront-copy associated with log-polar coordinate transformation. It reduces the spatial spectral range of the converted beam, thus achieving a narrower pattern width after separation. Previous research has proved that OAM separation is still feasible when the unwrapper or corrector adopts an opposite modulation phase, while the separated states from positive and negative modulation are centrosymmetric on the separating plane.43 Therefore, separating OAM mode with little cross talk through the geometric phase modulation is available (see Note 1 in the Supplementary Material for details).

Based on the above fan-out OAM separation and geometric phase modulation, we propose a precise TAM separation scheme. The TAM separator is implemented by a couple of GPOEs working as an unwrapper and a phase corrector. As shown in Fig. 1, the mode separation and phase correction GPOEs, referred to as the unwrapper and the corrector for simplicity, associated with two lenses, form the TAM separator. The unwrapper (${\mathrm{U}}_{1}$), the first lens (${\mathrm{L}}_{1}$), the corrector (${\mathrm{C}}_{1}$), and the second lens (${\mathrm{L}}_{2}$) are placed in sequence along the $z$ axis. The unwrapper and the corrector are, respectively, located in the front and rear focal planes of the lens ${\mathrm{L}}_{1}$, whose corresponding coordinates are denoted as $(x,y)$ and $(u,v)$, separately. And the lens ${\mathrm{L}}_{2}$ is used to focus the light field passing through the corrector. Fusing the modulation phases corresponding to SAM and OAM separation, the additional geometric phase of the unwrapper element is described by

The first term of Eq. (1) represents the log-polar coordinate transformation,32 where $\lambda $ is the wavelength of the incident beam, ${f}_{1}$ is the focal length of lens ${\mathrm{L}}_{1}$, and $a$ and $b$ are the transformation scaling parameters. Then the arc tangent term refers to the wave copying with the maximum diffraction order $N$ of the copy process and $\theta $ is the separation angle. ${c}_{q}$ and ${\zeta}_{q}$ are the complex coefficients for adjusting different diffraction orders.33 The last term is the linearly gradient phase for SAM separation, where ${\delta}_{1}$ determines the separation distance along the horizontal.

Take an incident TAM beam $|\psi \u27e9=\sum _{\xi =-\mathrm{1,1}}\sum _{{l}_{\kappa}=-4,+4}{\alpha}_{\xi}^{\kappa}|{\sigma}_{\xi}\u27e9|{l}_{\kappa}\u27e9$ as an example; after passing through the unwrapper and focused by lens ${\mathrm{L}}_{1}$, it is unwrapped and two observed SAM components are centrosymmetric around the origin (see Note 1 in the Supplementary Material for details). However, an inevitable phase distortion is introduced by the unwrapper and also the propagation from the front to back focal planes of lens ${\mathrm{L}}_{1}$. Therefore, a correction phase calculated by the stationary phase approximation44 is necessary,

In Eq. (2), $d=2\pi a$ denotes the transformed fringe length from OAM modes. ${\phi}_{q}(q)$ is used to compensate for the additional phase introduced in the wave-copying process. The last term is essential to guarantee SAM separation with zero cross talk, enabling the processor to support as many TAM states as possible. ${f}_{2}$ is the focal length of lens ${\mathrm{L}}_{2}$; ${\delta}_{2}$ works similar to ${\delta}_{1}$. The $\mathrm{rect}()$ function is described as follows:

When passing through the TAM separator, the incident TAM beam $|\psi \u27e9$ is converted into plane waves with different linearly gradient phases. Finally, they are focused by lens ${\mathrm{L}}_{2}$, and photons from various single TAM components are located in different positions. The horizontal separation distance between two SAM components becomes further apart. Both of them contain their corresponding OAM modes, distributed at vertical intervals (see Note 1 in the Supplementary Material for details). So far, the TAM states are completely separated, providing a path for subsequent TAM operation.

To achieve the above functions, we fabricate the unwrapper and the corrector through polymerized LCs, as displayed in Fig. 2. The main-axis orientation distributions of the unwrapper and corrector are given by ${\alpha}_{1}(x,y)={\varphi}_{1}(x,y)/2$ and ${\alpha}_{2}(u,v)=-{\varphi}_{2}(u,v)/2$, as shown in Figs. 2(a) and 2(b). Figures 2(c)–2(f) present the inspection of the fabricated GPOEs under a polarized optical microscope.

Figure 2.Details of the fabricated GPOEs. Main-axis orientation angle distributions of the designed elements for (a) the unwrapper and (b) the phase corrector. (e) Inspection of the corrector-center via polarized optical microscope. Inspection detail of the (d) unwrapper central zone and (f), (c) the corrector left and right regions.

2.2 Manipulating TAM States on Demand: TAM Reverser

After blocking the redundant TAM states at the rear focal plane of lens ${\mathrm{L}}_{2}$, the selected states need to be reconverted into the initial incident form. One of the most straightforward approaches is making the beam pass through a structure symmetrical with the TAM separator proposed above, for instance, using an element to reflect the separated beam back to the TAM separator. Although this method has advantages such as a compact structure and element saving, it does not apply to the TAM situation due to the spatial separation of SAM modes.

For the reflection schemes, lens ${\mathrm{L}}_{2}$ and the reflector constitute an equivalent $4\text{-}f$ system, whose input and output planes correspond to the $(u,v)$ planes of corrector and reversal corrector, respectively. It is obvious that the output beam is centrosymmetric with the input, and the linear gradient phases of it in $u$ axis will be reversed when passing through the equivalent $4\text{-}f$ system. As the main-axis orientation angle distribution of the corrector is asymmetrical in the $u$ axis, the reversed-corrector must be rotated 180 deg to achieve reversal phase modulation. While the SAM modes will overlap at the front focal plane of the reversal first lens, the reversed-unwrapper does not require rotation. Therefore, the TAM reverser cannot reuse the TAM separator (see Note 2 in the Supplementary Material for details), and a transmissive scheme must be employed here to recover TAM beams.

The right part of Fig. 1 gives the structure of the TAM reverser, and the order of these elements along the $z$ axis is the third lens (${\mathrm{L}}_{3}$), rotated-corrector (${\mathrm{C}}_{2}$), the fourth lens (${\mathrm{L}}_{4}$), and unwrapper (${\mathrm{U}}_{2}$). The TAM separation plane (Sp) and the rotated-corrector are located at the front and rear focal planes of lens ${\mathrm{L}}_{3}$, respectively. Meanwhile, the rotated-corrector and unwrapper are located at the front and back focal planes of lens ${\mathrm{L}}_{4}$, separately.

3 Results

As shown in Fig. 1, the photon TAM manipulator consists of a TAM separator and a reverser. The distributions of input TAM beams after each GPOE are given in Note 3 in the Supplementary Material. In our design, the range of TAM separation is SAM modes $\sigma =-\mathrm{1,}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}1$, OAM modes ${l}_{\kappa}=-10,-9,\dots ,+10$, totaling 42 single separable TAM states. The simulated and experimental incident beam is a horizontally polarized Gaussian beam carrying a spiral phase, while the Laguerre–Gauss (LG) beam is a theoretical situation. Obviously, the efficiency of the TAM separator is the basis of the photon TAM processor, and a concept of optical finesse is adopted for quantitative evaluation.21 In simulation cases, the optical finesse is calculated as 2.63 and 2.60 for $\sigma =1$ and $\sigma =-1$, while it drops down to 1.03 and 1.79 for $\sigma =1$ and $\sigma =-1$ in experiment. The deviation can be ascribed to the fabricating error of GPOEs and the slight misalignment between these GPOEs in experiment. More detail on the separation evaluation is presented in Note 4 in the Supplementary Material.

Though the experimental TAM separation efficiency is still nonideal, it is sufficient for further TAM processor experiments. To evaluate the performance of the processor, a series of numerical simulations and experiments are carried out. The complete optical details are given in Note 5 in the Supplementary Material.

Considering the quasi-symmetric structure of the concept, it is imperative to verify the consistency of the optical characterization of the input and output beams when there is no occlusion on the TAM separation plane. Figure 3(a) shows the results of single-mode TAM beams $|{\psi}_{1}\u27e9=|L\u27e9|-7\u27e9$ and $|{\psi}_{2}\u27e9=|R\u27e9|+8\u27e9$ passing through the manipulator (other results are given in Note 6 in the Supplementary Material). The numerical simulation indicates that the intensities of separation (filter) plane are in accordance with the former theory. However, though the phase distributions of the output beams meet the expectation, the intensities are imperfect. Therefore, evaluating the TAM mode purity loss passing through the processor without filtering is necessary. Figure 3(b) shows the output TAM spectrum for the inputs of those single TAM states. The simulation result is acceptable that the average loss of mode purity is 2.63%, whereas the experimental result increases it to 10.12%. With the application of high-efficiency GPOEs, the insertion loss of the system is estimated to be 0.83 dB when there is no filtering. In a word, for the straight-through cases, the outputs from the proposed TAM processor are nearly identical with inputs in simulation, while the experiment result is slightly deviated.

Figure 3.Performance of the TAM manipulator in straight-through cases. (a) The numerical simulation of various observing planes for incident beams

Further experiments of multi-TAM states incident are carried out. These input TAM states are $|{\mathrm{\Psi}}_{l}\u27e9=(|L\u27e9+|R\u27e9)(|l\u27e9+|-l\u27e9)$, $l=5,8,9,10$, namely, a horizontally linearly polarized twofold multiplexed OAM beam. The cases of both straight-through and selective-blocking are given for better comparison. Figure 4(a) shows the intensities’ distributions of the input, separation (filter), and the output plane for the two cases. Obviously, the intensities of outputs are consistent with the inputs at the straight-through cases, and the distributions of the separated plane conform to the previous theoretical analysis. To quantifiably evaluate the consistency of the TAM spectrum after and before passing through the processor, TAM spectrum fidelity $F$, analogous to quantum conception, is introduced and denoted by

Figure 4.Performance of the manipulator in straight-through and selective-blocking cases when multi-TAM states are incident. (a) Experimental results for incident beams

In Eq. (4), $|{\psi}_{i}\u27e9={(\begin{array}{cc}{p}_{L}& {p}_{R}\end{array})}^{\mathrm{T}}$ is the TAM state vector, where ${p}_{L}$ and ${p}_{R}$ are the relative OAM intensity spectrum vector for the left and right circular polarizations, respectively. $i$ has three symbols: “theo,” “sim,” and “exp,” corresponding to theoretical, simulated, and experimental cases, separately. $a,b$ denote the two cases to be compared. The greater $F$ value indicates that the experimental result is closer to the simulation result, and $F$ will be equal to 1 if $|{\psi}_{a}\u27e9=|{\psi}_{b}\u27e9$. Here ${F}_{\mathrm{sim},\mathrm{theo}}$ of $|{\mathrm{\Psi}}_{5}\u27e9,|{\mathrm{\Psi}}_{8}\u27e9,|{\mathrm{\Psi}}_{9}\u27e9,|{\mathrm{\Psi}}_{10}\u27e9$ are calculated as 0.998, 0.987, 0.992, and 0.986, respectively; ${F}_{\mathrm{exp},\mathrm{theo}}$ of them are measured as 0.832, 0.855, 0.875, and 0.902, separately. The simulation is in precise agreement with theoretical values, while the experimental and theoretical values are basically consistent.

As for the selective-blocking cases, the experimental results are shown in Fig. 4(a) with the states selection of $|{\mathrm{\Psi}}_{k}\u27e9$, where the spatial filter placed at the separation plane is presented as ${\mathrm{Sp}}_{2}$. In other words, the bottom half of the separation plane is blocked.

The experimental intensities are generally as expected: the petaloid input patterns are converted into doughnut patterns after the blocking. The calculated ${F}_{\mathrm{sim}/\mathrm{theo}}$ of $|{\mathrm{\Psi}}_{5}\u27e9,|{\mathrm{\Psi}}_{8}\u27e9,|{\mathrm{\Psi}}_{9}\u27e9,|{\mathrm{\Psi}}_{10}\u27e9$ are 0.998, 0.988, 0.991, and 0.988; their measured ${F}_{\mathrm{exp}/\mathrm{theo}}$ are 0.804, 0.876, 0.838, and 0.880. These experimental performances also meet expectations. Figure 4(b) gives the experimentally measured TAM spectrum of the input and output beam, illustrating that the TAM processor has the ability to choose any TAM states on demand (see Fig. S18 in the Supplementary Material for simulated results).

Theoretically, the incident beam $|{\mathrm{\Psi}}_{l}\u27e9$ converts into a classical nonseparable state, denoted as $|{\mathrm{\Psi}}_{l}^{\prime}\u27e9=|L\u27e9|-l\u27e9+|R\u27e9|l\u27e9$ in the selective-blocking cases using a spatial filter, represented as ${\mathrm{Sp}}_{2}$ in Fig. 4. Such a process could be regarded macroscopically as a mode transformation from a twofold multiplexed OAM mode with horizontal linear polarization into a cylindrical vector mode with anisotropic polarization distributions. Therefore, an additional rotating polarizer is placed before the CCD camera for an anisotropic polarization measurement, whose experimental results are given in Fig. 5. For the input beam $|{\mathrm{\Psi}}_{6}\u27e9$, its intensity conforms to Marius’ law. Meanwhile, the theoretical rotation angle of the output pattern should be $\theta /l$, where $\theta $ is the rotation angle of the polarizer.45 Therefore, the rotation angle ${\theta}_{1}$ of the output pattern should be 7.5 deg and 15 deg when the polarizer rotates 45 deg and 90 deg, separately, corresponding to typical anisotropic polarization distribution. The measured results are ${\theta}_{1}=7.87\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$, ${\theta}_{2}=14.83\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$ and conform with the theoretical value.

Figure 5.Polarization distributions of experimental input and output beams. The input row shows the intensities of input beam. Output 1 is the intensity of the output beam at pass-through cases. Output 2 is the intensity of the output beam at selective-blocking cases. The arrow represents the direction of the polarizer, and the first column is the result with no polarizer.

The multi-TAM states experiment demonstrates the capacity of the system to simultaneously manipulate SAM and OAM modes, while the ability of OAM filtering with adjacent modes is the basic requirement for satisfying diversified manipulation. A horizontally polarized incident beam carrying adjacent OAM modes, denoted as $|{\mathrm{\Psi}}_{\mathrm{\Omega}}\u27e9=(|L\u27e9+|R\u27e9)(|+9\u27e9+|+10\u27e9)$, is employed. Figure 6(a) displays the intensity distribution of the input, separation (filter), and the output plane, for the straight-through and selective-blocking cases. The output pattern is similar with the inputs at the straight-through cases, and the distribution of the separated plane conforms to the analysis in Note 4 in the Supplementary Material. For the selective-blocking cases, adopting a spatial filter shown as ${\mathrm{Sp}}_{2}$ in Fig. 6 for concurrently filtering the OAM mode $|+10\u27e9$ and the two SAM modes, the output pattern converts from a notched ring into a complete ring.

Figure 6.Performance of the manipulator in straight-through and selective-blocking cases when adjacent OAM states are incident. (a) Experimental results for incident beams

Figure 6(b) gives the experimentally measured TAM spectrum of the output beam for the two cases. For straight-through cases, the calculated ${F}_{\mathrm{sim}/\mathrm{theo}}$ is 0.865 and ${F}_{\mathrm{exp}/\mathrm{theo}}$ is measured as 0.852, while ${F}_{\mathrm{sim}/\mathrm{theo}}$ is calculated as 0.823 and the measured ${F}_{\mathrm{exp}/\mathrm{theo}}$ is 0.772 for selective-blocking cases. The indicators and the patterns at the separation plane reveal that the filtered pattern and the filtered TAM spectrum are nonideal, which could be associated with the reduction in the focal length of ${\mathrm{L}}_{2}$ by half and the incident OAM beam without amplitude modulation (the Laguerre–Gaussian mode incident will get the ideal result). Therefore, the difference between the simulation and theory should be taken into account. The calculated ${F}_{\mathrm{exp}/\mathrm{sim}}$ is 0.898 for straight-through cases, and it is 0.827 for selective-blocking cases, illustrating that the experimental performance and the simulation are analogous, but both of them have gaps from the theory. Unlike the previous set of experiments, there will be a nonnegligible deviation when the incident beam without Laguerre–Gaussian amplitude modulation in the case of the carrying adjacent OAM modes.

4 Discussion and Conclusions

In summary, we have proposed a scheme for manipulating TAM states of photons. It is carried out by symmetrically cascading two analogous units. The TAM separator achieving a separation of arbitrary input TAM states could be understood as a transformation from the spatial domain into the position-TAM domain, where the incident beams are distributed as a set of nonseparable single TAM states in the position-TAM domain (the separation plane); the TAM reverser performs an opposite transformation, converting the beam back to the spatial domain. Our concept is supported by simulated and observed data.

The number of supported TAM states is determined by the following factors. Limited by the fabricated technique of GPOEs, the phase modulation exhibits deviation, which is shown in Note 7 in the Supplementary Material. Since the diameter of OAM beams increases with the absolute value of topological charge $l$, the area illuminated on the fabricated elements is limited for the incidence of low-order OAM modes, resulting in more insufficient phase modulation than that of the high-order modes. As for the high-order OAM mode, it is restricted by the Fresnel number of the designed elements.35 In other words, the inadequate phase modulation of low-order OAM modes and OAM cross talk of high-order modes limit the range of supporting states. The number of supported TAM states can be improved by increasing the resolution of OAM separation, such as increasing the radius of the incident beam and reducing the distance between the unwrapper and corrector. As for other nonideal results in experiments, the input beam should change into the Laguerre–Gaussian mode; these four GPOEs need to be aligned more precisely or be integrated.

This work provides an idea for simultaneously tailoring OAM and SAM modes, extending the way for many state-of-the-art explorations. To date, a series of computing schemes have been proposed in the OAM domain, such as OAM multiplier,46 OAM divider,47 and unitary operation.12 These operations can also be expanded to the TAM domain, which will enhance the versatility of optical computing. In conclusion, this work provides a scheme to manipulate TAM states on demand and attains the transformation from separable states to nonseparable states. The experiment demonstrates four-modes input, with two modes filtered, fitting well with the simulation. The TAM processor could be used in optical interconnects, optical computing, and quantum technique. Facing the potentially complex manipulation requirements, the spatial filter implemented in the separation plane should be programmable. Using a polarization-insensitive and programmable device like digital micromirror devices as the filter and folding the optical path may be a future effort.

5 Appendices

5.1 Numerical Simulation

The diffraction fields when TAM beams passing through the TAM processor were calculated through the diffraction angular spectrum theory.48 The modulation phase of unwrapper and corrector was defined on a mesh of $2000\text{\hspace{0.17em}\hspace{0.17em}}\text{pixels}\times 2000\text{\hspace{0.17em}\hspace{0.17em}}\text{pixels}$. And the theoretical TAM spectrum analysis of the output beams was implemented on the basis of spiral harmonic decomposition.26

5.2 Elements Fabrication

As the non-contact photoalignment method is implemented to produce the GPOEs, it can avoid mechanical damage, dust smirch, and static electricity.49 It has the advantages of high quality and high resolution when processing multidomain orientation LCs. The substrates of LCs are spin-coated with a layer of dye molecules, which have the characteristics of photo-isomerization and dichroic absorption. When irradiated by a polarized ultraviolet beam, the molecules tend to reorient their absorption oscillators perpendicular to the polarization. They will subsequently determine the orientation of the LCs. However, the reorientation of each LC molecule is also influenced by adjacent molecules, resulting in difficulty in achieving a perpendicular orientation between adjacent molecules. The orientation deviation is $\sim 2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{deg}$, causing the imperfect phase modulation of GPOEs.

To achieve effective resolution of TAM separation, the introduced wave-copying number is set to 3. Then the unwrapper and corrector are fabricated under the parameters of $\lambda =1617\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$, $d=2\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, $b=4\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, $c=[\mathrm{1.32,}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}\mathrm{1,}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}1.32]$, $N=1$, $\zeta =[-\pi /\mathrm{2,}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}0,\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}\pi /2]$, ${\delta}_{1}=0.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, ${f}_{1}=50\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, ${\delta}_{2}=2.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, ${f}_{2}=400\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{mm}$, $\phi =[\mathrm{0,}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}3\pi /\mathrm{2,}\phantom{\rule[-0.0ex]{0.2em}{0.0ex}}0]$, and the resolution is $2000\times 2000$. The fabricated elements consist of a large number of LC cells with the pixel pitch $5\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$, whose main axes are arranged as our design. To introduce the geometric phase, each cell must act as a sub-half-wave plate, and $2\pi ({n}_{\mathrm{eff}}-{n}_{o})d/\lambda =\pi $ must be satisfied, where ${n}_{\mathrm{eff}}$ denotes the effective extraordinary refractive index determined by a molecule’s tilting. ${n}_{o}$ denotes the ordinary refractive index, $d$ is the LC cell gap, and $\lambda $ is the wavelength.

5.3 Experimental Characterization

As for the TAM states preparation, a self-developed distributed feedback laser diode with the wavelength 1617 nm was adopted as the source to produce a laser beam. Then the beam was shaped by a single-mode fiber and was coupled into free space from a collimator with a diameter of 3 mm. The TAM states were generated by encoding specially designed holograms on a spatial light modulator (Holoeye, PLUTO-TELCO-013-C), associated with a series of polarization elements.

The experimental setup of the TAM processor essentially was consistent with the theoretical analysis, while the focal lengths of the second and third lenses ${\mathrm{L}}_{2}$ and ${\mathrm{L}}_{3}$ were changed to 200 mm, due to the size limitations of the optical platform. The four GPOEs were mounted on the six-axis locking kinematic mounts (AMM6-1A, Lbtek).

Infrared CCD cameras (Cobra2000-CL1280-130vt-00, LUSTER) were employed to capture patterns of input beams, separated beams, and output beams. Finally, the projection measurement26 was implemented to analyze the TAM spectra of output beams.

**Lang Li** is a PhD student at the Beijing Institute of Technology. He received his BS degree in optoelectronic information science and engineering from the Beijing Institute of Technology, Beijing, China, in 2020. He is currently working toward his PhD following Prof. Shiyao Fu in optical engineering at the School of Optics and Photonics, Beijing Institute of Technology, Beijing, China. His research interest concerns laser field tailoring, focusing on the related research of photon orbital angular momentum.

**Chunqing Gao** has been a professor at the School of Optics and Photonics, Beijing Institute of Technology, China, since 2001. He received his BS and MS degrees in optics from the Beijing Institute of Technology and his PhD in physics from the Technical University, Berlin, Germany. The current research in his group mainly focuses on innovative laser systems, laser beam manipulation, and related applications. He has published four books and more than 260 scientific papers in journals and conferences. He has 30 authorized patents in the fields of optics and lasers.

**Shiyao Fu** is currently a professor at the Beijing Institute of Technology. He received his PhD in electronic science and technology from the Beijing Institute of Technology in 2019. His current research interests cover complex structured field manipulation, innovative solid-state lasers, and their applications in critical engineering. He has published two books and more than 50 peer-reviewed papers. He has 20 authorized patents. He was selected for the national postdoctoral program for innovative talents. He is the recipient of the second prize of the Beijing Natural Science Award. He won the Wang Da-Heng Optics Prize, and his doctoral dissertation was recognized by the Chinese Institute of Electronics. He is a senior member of the Chinese Optical Society.

Biographies of the other authors are not available.

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Lang Li, Yingchi Guo, Zhichao Zhang, Zijun Shang, Chen Li, Jiaqi Wang, Liliang Gao, Lan Hai, Chunqing Gao, Shiyao Fu. Photon total angular momentum manipulation[J]. Advanced Photonics, 2023, 5(5): 056002

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