# Pancharatnam–Berry Optical Elements for Spin and Orbital Angular Momentum Division Demultiplexing

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Phase Pattern Calculation

_{j}} is given by the linear combination over a group of n orthogonal modes {ψ

_{j}= R

_{j}(ρ, ϑ)exp(iℓ

_{j}ϑ)} as it follows [20]:

_{j}, ϑ

_{j})} are the n vectors of carriers spatial frequencies in polar coordinates, c

_{j}are complex coefficients whose modulus is given arbitrarily, usually unitary, and the phases are free parameters of the task, fitted in such a manner that Equation (1) becomes an exact equality. The coefficients are given by the following relation:

_{j}, φ

_{j})} of the corresponding signal spots in the far-field are given by:

^{®}is used to calculate the phase pattern for the desired set of OAM values {ℓ

_{j}} and corresponding carrier spatial frequencies {(β

_{j}, ϑ

_{j})}. The implemented algorithm is based on a successive computation of the sum in Equation (1) and integrals in Equation (2), using the fast Fourier transform algorithm and considering definite limitations, as explained in Ruffato et al. [9], in particular the quantization of phase into 16 equally-spaced levels.

#### 2.2. Subwavelength Grating Design

#### 2.3. Fabrication

^{2}from previous dose matrix analyses. After development in an isopropyl alcohol (IPA): deionized water 3:7 solution for 60 s, the exposed area results to be removed.

_{2}stripping of the residual layer removal characterized by a 40-sccm O

_{2}flow rate, 200 W of radio-frequency (RF) power of coil for the generation of plasma, 4 mTorr of chamber pressure, and an RF platen power of 10 W for the acceleration of the ions on top of the sample. The etching step is performed in a three-gases mix of plasma (C

_{4}F

_{8}at a flow rate of 60 sccm, SF

_{6}at 30 sccm, and Ar at 10 sccm) with a coil RF power of 400 W, a RF platen power of 20 W (corresponding to an acceleration bias of 90 V), and a chamber pressure of 8 mTorr. Finally, an O

_{2}cleaning step was carried out, which was characterized by an O

_{2}flow rate of 50 sccm, a coil RF power of 800 W, an RF platen power of 20 W, and a chamber pressure of 20 mTorr. In order to have pattern transfer from the EBL-lithographed resist to the Si substrate, a 7-s stripping process in O

_{2}plasma has been performed, and successively, an etching time of 1 min and 12 s in fluorine-based plasma occurred.

_{2}treatment was performed for 13 s to remove the residual layer, a 10-nm Cr hard mask layer was deposited by e-gun evaporation, and the pattern transferring from the resist structure into the Cr mask was carried out by lift-off in a sonicated acetone bath for 3 min.

_{2}treatment was considered for 30 s to remove the last resist contaminants; then, pattern transfer in the fluorine-based plasma for 3 min and 9 s occurred. The etching time was finely tuned in order to obtain a final depth of around 535 nm, as prescribed by the numerical results in Figure 3 for a grating period of 290 nm and a duty cycle of around 0.5. In Figure 4, inspections of the final sample via scanning electron microscopy (SEM) are reported. In particular, the well-defined line profile can be appreciated as proof of the success of the fabrication recipe for pattern transfer onto a silicon substrate.

#### 2.4. Optical Characterization Setup

_{F}= 7.5 mm (A375TM-C, Thorlabs, Newton, NJ, USA), which is linearly polarized and expanded with a first telescope (f

_{1}= 3.5 cm, f

_{2}= 10.0 cm, Thorlabs, NJ, USA) before illuminating the display of the SLM. A 50:50 beam splitter (Thorlabs, Newton, NJ, USA) is inserted after the telescope in order to retain a coherent Gaussian beam for interferometric analysis. Then, a 4-f system (f

_{3}= 20.0 cm, f

_{4}= 12.5 cm, Thorlabs, Newton, NJ, USA) with an aperture in the Fourier plane isolates and images the first-order encoded mode. A second 50:50 beam splitter is used to split the beam and check the input beam profile with a first camera (pixel pitch 15 μm, WiDy SWIR 640U-S, NIT, Verrières-le-Buisson, France). Afterwards, the OAM beam illuminates the patterned zone of the silicon sample, which is mounted on an XY translator with micrometer drives (ST1XY-S/M, Thorlabs, Newton, NJ, USA). The far field is collected by a second camera (WiDy SWIR 640U-S, NIT, Verrières-le-Buisson, France) that is placed at the back focal plane of a lens with f

_{5}= 7.5 cm (Thorlabs, NJ, USA). A sequence of a linear polarizer (LPIREA100-C, Thorlabs, Newton, NJ, USA) and quarter-wave plate (WPQ10M-1310, Thorlabs, Newton, NJ, USA) is placed before and after the optical element, in reverse order, in order to generate and filter the desired circular polarization state. A Mach-Zehnder interferometric setup is added in order to analyze the phase pattern of the modes generated with the SLM.

## 3. Results

## 4. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

_{j}, ϑ

_{j})} (see Equations (1) and (5)). Therefore, after substituting the definition (A1) and the sorter phase function, Equation (A2) becomes:

_{j}}. After performing, inside each integral of the summation, the coordinate change:

**r’**

_{j}=

**r**− f/k

**β**

_{j}, we get:

_{n}the modified Bessel functions of the first kind, and applying the relation [34]:

_{j}, φ

_{j})} = {(f/kβ

_{j}, ϑ

_{j})} and azimuthal indices in the shifted set {(ℓ − ℓ

_{j})}. Only for m = 0 the Bessel function of the first kind J

_{m}is endowed with a central peak, otherwise the intensity profile presents a central null surrounded by a first ring whose radius increases with the azimuthal index m. Therefore, as shown in Figure 6 and in Figure A1, for an input vortex with charge ℓ a bright spot appears only when a zero-order Bessel-Gaussian beam is generated, i.e., for ℓ

_{j}= ℓ, while annular beams with different radii appear at the other positions. If the input OAM beam carries an OAM value which is not included in the sorting set of the designed sorter, e.g., ℓ = +2, there is no possibility to obtain a zero-order Bessel-Gaussian beam, therefore no bright peak appears (see Figure A1). As a matter of fact, for the three cases depicted in figure, we get:

- input: ℓ = 0 -> output: {0 + 9, 0 + 6, 0 + 3, 0 + 0, 0 − 3, 0 − 6, 0 − 9} = {+9, +6, +3, 0, −3, −6, −9}
- input: ℓ = +3 -> output: {+3 + 9, +3 + 6, +3 + 3, +3 + 0, +3 − 3, +3 − 6, +3 − 9} = {+12, +9, +6, +3, 0, −3, −6}
- input: ℓ = +2 -> output: {+2 + 9, +2 + 6, +2+3, +2 + 0, +2 − 3, +2 − 6, +2 − 9} = {+11, +8, +5, +2, −1, −4, −7}.

**Figure A1.**(

**a**) Interference patterns of the OAM beams exploited for the sorter characterization, for ℓ = 0, +2, +3. The number and twist-handedness of the spirals is indicative of the carried OAM. (

**b**) Output of the fabricated OAM for input right-handed circular polarization states. A bright spot appears when the azimuthal index of the generated far-field optical vortex is equal to zero. If the input beam carries an OAM value which is not included in the sorter set, as in the case of ℓ = +2, no peak appears in far-field.

## Appendix B

## References

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**Figure 1.**Scheme of the Pancharatnam–Berry optics working principle for an orbital angular momentum (OAM) beam sorting with the method of optical beam projection. When a circularly polarized OAM beam illuminates the optical element, a bright spot appears in the far field at a position depending on the carried OAM and on the polarization handedness. In the presented work, the Pancharatnam–Berry optical element (PBOE) is implemented in the form of a pixelated metasurface of rotated subwavelength grating.

**Figure 2.**(

**a**) Numerical phase pattern for the sorting of OAM beams in the range {−9, −6, −3, 0, +3, +6, +9}, with 16 phase levels. Pixel size: 6.125 μm × 6.125 μm. Radius size: 256 pixels. (

**b**) Far-field channel constellation for the given OAM set and circular polarization states.

**Figure 3.**Configurations of duty cycle and period for a silicon subwavelength grating, providing a π-delay retardation between TE and TM polarizations, for a wavelength of 1310 nm at normal incidence. Numerical data obtained with rigorous coupled-wave analysis (RCWA) for a grating thickness of 535 nm.

**Figure 4.**SEM inspections of the fabricated PBOE on a silicon substrate. Grating period Λ = 290 nm, duty cycle 0.5, thickness: 535 nm, pixel size: 6.125 μm, 16 rotation angles.

**Figure 5.**Scheme of the experimental setup used for the optical characterization of the fabricated sorter. The DFB laser output (λ = 1310 nm) is collimated at the end of the single mode fiber by means of an aspheric lens with a focal length f

_{F}= 7.5 mm, is linearly polarized (P

_{1}) and is expanded (f

_{1}= 3.5 cm, f

_{2}= 10.0 cm). The SLM first order is filtered (D

_{1}) and resized (f

_{3}= 20.0 cm, f

_{4}= 12.5 cm) before illuminating the sorter. A beam splitter (BS) is used both to check the input beam and collect the sorter output at the back focal plane of a fifth Fourier lens (f

_{5}= 7.5 cm). A sequence of linear polarizers and quarter-wave plates is placed before (P

_{2}, Q

_{1}) and after (Q

_{2}, P

_{3}) in the optical element, in reverse order, in order to generate and filter the desired circular polarization states. A Mach-Zehnder interferometric setup is added in order to analyze the singularity order of the generated OAM beams.

**Figure 6.**(

**a**) Interference patterns of the OAM beams exploited for the sorter characterization. The number and twist handedness of the spirals is indicative of the carried OAM. Output of the fabricated OAM for input right-handed (

**b**) and left-handed (

**c**) circular polarization states. As expected, the far-field bright spot position is in accordance with the far-field channel constellation in Figure 2b.

**Figure 7.**Total intensities in all of the detector regions for OAM mode input in the range {−9, −6, −3, 0, +3, +6, +9} and circular polarization of the experimental data. For each channel, the detection regions have the same size and are chosen to cover the central intensity peak area. Intensities are normalized to the total collected energy. The polarization labels σ

^{±}refers to the polarization state of the input channel (the output is cross-polarized).

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**MDPI and ACS Style**

Ruffato, G.; Capaldo, P.; Massari, M.; Mezzadrelli, A.; Romanato, F.
Pancharatnam–Berry Optical Elements for Spin and Orbital Angular Momentum Division Demultiplexing. *Photonics* **2018**, *5*, 46.
https://doi.org/10.3390/photonics5040046

**AMA Style**

Ruffato G, Capaldo P, Massari M, Mezzadrelli A, Romanato F.
Pancharatnam–Berry Optical Elements for Spin and Orbital Angular Momentum Division Demultiplexing. *Photonics*. 2018; 5(4):46.
https://doi.org/10.3390/photonics5040046

**Chicago/Turabian Style**

Ruffato, Gianluca, Pietro Capaldo, Michele Massari, Alessia Mezzadrelli, and Filippo Romanato.
2018. "Pancharatnam–Berry Optical Elements for Spin and Orbital Angular Momentum Division Demultiplexing" *Photonics* 5, no. 4: 46.
https://doi.org/10.3390/photonics5040046