A compact diffractive sorter for high-resolution demultiplexing of orbital angular momentum beams

The design and fabrication of a compact diffractive optical element is presented for the sorting of beams carrying orbital angular momentum (OAM) of light. The sorter combines a conformal mapping transformation with an optical fan-out, performing demultiplexing with unprecedented levels of miniaturization and OAM resolution. Moreover, an innovative configuration is proposed which simplifies alignment procedures and further improves the compactness of the optical device. Samples have been fabricated in the form of phase-only diffractive optics with high-resolution electron-beam lithography (EBL) over a glass substrate. A soft-lithography process has been optimized for fast and cheap replica production of the EBL masters. Optical tests with OAM beams confirm the designed performance, showing excellent efficiency and low cross-talk, with high fidelity even with multiplexed input beams. This work paves the way for practical OAM multiplexing and demultiplexing devices for use in classical and quantum communication.


S1. SOFT-LITHOGRAPHY PROCESS OPTIMIZATION
During the first tests of the replica process the replicas showed an increase in the background noise, i.e. the scattered radiation, if compared with the master. This effect has been attributed to some deviations in the replica process, since the original masters exhibited better values of crosstalk.
Since the PDMS mold was made following a standard process [1], the imperfect curing of the resin has been considered and investigated.
The average cross-talk has been chosen as control parameter for each replica, therefore a set of OAM beams in the range from -6 to +6 have been sorted for each sample, and the same data analysis of the master has been repeated. The sample chosen was a three-copies fan-out sorter. The choice was driven by the aligning time of the sample on the optical table.
The radiation dose has been changed by increasing the exposure time of the NOA sample under UV light, and the curve in Supplementary Figure 1 has been obtained. There is an almost abrupt discontinuity between the underexposed replica and the samples with an optimal curing time.

S2. OPTICAL CHARACTERISATION
The experimental setup used for characterization of the mode-sorter (MS) elements is given in Supplementary Figure 3(a). The same setup was used to study the performance of the sorters with regards to sorting multiplexed OAM modes. In this analysis, the fan-out unwrapper and the double phase-corrector were written on two separated glass slides. A HeNe beam (633nm) was expanded with an objective lens and collimated to overfill a spatial light modulator (SLM) used to generate vortex beams through phase and amplitude modulation. The 1 st order was then isolated, demagnified by a factor of 2 and imaged onto the first mode sorter element (unwrapper) through a 4-f telescope system with an aperture in the Fourier plane. Images of the unwrapper and phasecorrector elements can be seen for the 1-copy case in Suppl. Fig. 3(b). Due to the sensitivity of mode sorters to alignment, the 1 st element was mounted with double-sided tape onto a holder allowing for x-and y-transverse plane adjustment as well as tip-tilt degrees of freedom. Two opposing mirrors were included before the mode sorter so as to allow for walking the beam through the elements and thus allowing for easy control of the input beam. The second element requires an additional degree of freedom (DoF) in the z-axis or optical axis direction as the separation distance requires a great degree of precision in addition to the minute separation distance of 8.500 mm being difficult to position without the control afforded by a translation stage. The second element was thus also attached onto the same type of mount with double sided tape and that attached to a x-,y-and z-translation stage as depicted in Suppl. Fig. 3(a). Varying the z-axis DoF also allows for alignment to be streamlined. For the 5-copy mode sorter case, the additional copies increased the associated sensitivity of the alignment requiring translations stages that were more stable than for the previous examples. For given wavelength and focal length, the theoretical separation distance Δs is determined by the MS design parameter a, according to Δs=λf/(2πa). The expected value was calculated to be 63 μm compared to an average experimental value of 62.5 μm (red line gradient) in Suppl. Fig. 4(a) for the 1-copy mode sorter. For the 3-copy mode sorter, the theoretical separation distance was calculated to be 126.5 μm which compared to an average experimental value of 126.9 μm (red line gradient) is in excellent agreement. Similarly, good agreement was seen for the 5-copy sorter where an experimental average separation of 157.7 μm can be seen in comparison to the theoretical

 
where W theo (ℓ) is the theoretical or multiplexed weighting associated with the OAM mode ℓ and W exp (ℓ) is the detected equivalent of the mode. A significant increase in the accuracy of modal detection occurs as the copy numbers increase. More specifically, the similarity of 79.1% for the 1copy is increased by 22% when adding 2-copies which is further refined to a 97.1% similarity when 5-copy is utilized.