Compact and high-resolution optical orbital angular momentum sorter

Light with helical phase fronts is associated with orbital angular momentum (OAM) states denoted as $φ(r,ϕ)=exp(ilϕ)$⁠, where ϕ is the angular coordinate and l can take any integer value.^1,2 Each photon carries an angular momentum of $lℏ$⁠. Due to the intrinsic property of dynamic rotation and unbounded state space, these light beams have found their way into many disciplines such as optical tweezers,³ high-bandwidth free-space data transmission,⁴ quantum cryptography,⁵ and nonlinear interaction with matter.⁶ 

Efficient separation of light beams based on their OAM quantum number l is of great importance in classical and quantum information systems.⁷ A fork diagram can generate or remove a specific helical phase structure and can be utilized to detect one OAM state at a time⁸ or detect multiple states with a reduced efficiency.⁹ A Mach–Zehnder interferometer with Dove prisms can achieve the separation of OAM states, and it requires N − 1 cascaded interferometers to separate N modes.¹⁰ 

One effective method for measuring the orbital angular momentum states of light harnesses a log-polar optical transform based on conformal mapping.¹¹ Two diffractive optical elements are implemented to transform a spiral phase into a prism phase. A Fourier lens then discriminates different angular momentum states into distinct lateral positions, allowing for single photon detection and effective measurements of multiple states simultaneously. The sorting method is based on unitary transformations, and the theoretical efficiency is unity. However, the finite size of the unwrapped beam results in overlap between two neighboring OAM states. To reduce the overlap caused by finite size diffraction, a diffractive fan-out element and a corresponding phase corrector can be placed following the two-piece log-polar optics to increase the resolution of the OAM sorter. Therefore, four pieces of bespoke diffractive/refractive optical elements plus three lenses in between are implemented in previous experimental demonstrations as illustrated in Fig. 1(a).^7,12,13

In the current paper, we customize and utilize only two phase elements to achieve high-resolution OAM sorting as illustrated in Fig. 1(b). The optical system is greatly simplified and performs equally well as the more complex system in terms of resolution and efficiency. The first element in our scheme is a quadratic fan-out mapper that simultaneously achieves the log-polar transformation, fan-out beam copying, and beam focusing. The mathematical expression of the quadratic fan-out mapper is expressed as follows:

The log-polar transformation term is capable of transforming the ring-shape intensity distribution of an OAM state to an arc and eventually to a straight line as the beam propagates to the Fourier plane.^9,14 Different OAM states are thus converted from a set of concentric rings into a set of parallel lines. The parameters d and p determine the size and location of the transformed beam in the Fourier plane, respectively. The parameter f is the distance from the quadratic fan-out mapper to the dual-phase corrector. The fan-out term splits the unwrapped beam into several copies and positions several copies of the tilted planar wavefronts side by side.^5,10,11,15 (2N + 1) is the number of copies of the unwrapped beam, θ is the angular separation between the neighboring copies, y is the transverse direction along which the copies are made, and a_m and b_m are the optimization parameters to achieve highly efficient and uniform beam splitting. The lens term is employed to complete the Fraunhofer diffraction within a short distance without using a physical lens.

The second phase element is a dual-phase corrector that corrects both the distorted phase from the log-polar mapping and the phase jumps generated from the beam copying process. The mathematical expression of the dual-phase corrector is given by

where rect(x) ≡ 1 for |x| < 1/2 and 0 otherwise. The parameters x′ and y′ are the coordinates of the dual-phase corrector plane that is Fourier Transform (FT) related to the quadratic fan-out mapper plane. The phase profile of the dual-phase corrector is a summation of (2N + 1) segments where (2N + 1) is the number of copies of the unwrapped beam. The parameter L satisfies $L=θf/(2π)$ so that each segment can correct the distorted phase of the corresponding copy of the unwrapped beam. The distorted phase is calculated by the stationary phase approximation.^6,12 The phase jumps between the neighboring copies of the unwrapped beam are compensated via $ϕbc(m)$ to ensure a smooth wavefront of the enlarged beam.

The experimental setup of the compact and high-resolution sorter is illustrated in Fig. 2. The light source is a He-Ne laser with different OAM states, and the wavelength λ is 632.8 nm. The incident light beam is reflected by a cube beam splitter to the left half of a reflective spatial light modulator (SLM). The SLM panel is manufactured by Holoeye Photonics featuring a resolution of 1920 × 1080 with a pixel pitch of 8 μm and a fill factor of 87%. The resolution of each half of the SLM is 960 × 1080. The phase profile $Ψ1$ of the quadratic fan-out mapper given by Eq. (1) is implemented on the left half of the SLM. In our experimental demonstration, the parameters d, p, and f are 0.1592 mm, 1 mm, and 191 mm, respectively. Three copies are generated via the fan-out term to increase the sorting resolution with b_m = (1.329, 1, 1.329) and a_m = (⁠$−π/2$⁠, 0, $−π/2$⁠) where m = −1, 0, and 1.

The unwrapped fan-out beam is then redirected with a right-angle prism to the right half of the SLM. The right half of the SLM contains the phase structure $Ψ2$ of the dual-phase corrector given by Eq. (2). The dual-phase corrector corrects the distorted phase in each copy of the unwrapped beam as well as the abrupt phase jumps between the neighboring copies. The compensating phases $ϕbc(m)$ for the case of three copies are (0, $3π/2$⁠, 0) where m = −1, 0, and 1. The corrected beam is then reflected by another cube beam splitter and focuses to a charge-coupled device (CCD) camera by a Fourier lens L1. Note that a piece of black paper is sandwiched between the two cube beam splitters to prevent from beam propagation in unwanted directions. It is worthy of noting that the current experimental setup is not great in efficiency since two beam splitters are being used. Replacing the reflective SLM with transmissive ones and removing the beam splitters can easily solve the efficiency problem.

The simulation and experimental results for the one-copy fan-out case are presented in Figs. 3(a) and 3(b), respectively. The images display the intensity distributions captured by the CCD camera in the x″-y″ plane. It is clearly shown that the circular intensity distribution of an OAM mode is unwrapped to a straight line, and the helical phase gradient is transformed to the transverse phase gradient. Different OAM modes (l = −2, −1, 0, 1, and 2) are sorted into a set of parallel lines with various vertical positions. Figs. 4(a) and 4(b) render the simulation and experimental results for the three-copy fan-out case, respectively. Compared to the one-copy fan-out case, the parallel lines for the three-copy fan-out case become thinner indicating an increase in the resolution of OAM sorting. The resolution enhancement is proportional to the number of copies of the unwrapped beam.

The overlaid line scans of the intensity distributions in the center along the vertical direction are plotted in Fig. 5(a) for the one-copy fan-out sorting and in Fig. 5(b) for the three-copy fan-out sorting. The spacing between the neighboring OAM modes is identical in Figs. 5(a) and 5(b) but the width of the main lobe of each mode is compressed, i.e., the resolution of OAM sorting is enhanced. To evaluate the crosstalk effect of overlapped sidelobes, we calculate the optical finesse defined as the lateral distance between neighboring OAM states divided by the average FWHM of individual OAM modes. The optical finesse is improved from 1.37 as in Fig. 5(a) to 4.85 as in Fig. 5(b). Continuously increasing the number of fan-out copies will further enhance the sorting resolution.

In conclusions, we propose and demonstrate a compact and high-resolution sorter of OAM states of light. The sorter consists of a quadratic fan-out mapper and a dual-phase corrector placed in the pupil plane and the Fourier plane, respectively. The optical system is greatly simplified compared to previous demonstrations while the sorting performance is unchanged. The compact and high-resolution OAM sorter may be very beneficial to the scientific research on OAM states of light.

The authors acknowledge support by the National Natural Science Foundation of China (NSFC) (Grant Nos. 61505062 and 91438108). C.W. and J.C. have been supported by the China Scholarship Council (CSC).