A compact single channel interferometer to study vortex beam propagation through scattering layers

We propose and demonstrate a single channel interferometer that can be used to study how vortex beams propagate through a scatterer. The interferometer consists of a multifunctional diffractive optical element (MDOE) synthesized by the spatial random multiplexing of a Fresnel zone plate and a spiral Fresnel zone plate with different focal lengths. The MDOE generates two co-propagating beams, such that only the beam carrying orbital angular momentum is modulated by an annular stack of thin scatterers located at the focal plane of the Fresnel zone plate, while the other beam passes through the centre of the annulus without any modulation. The interference pattern is recorded at the focal plane of the spiral Fresnel zone plate. The scattering of vortex beams through stacks consisting of different number of thin scatterers was studied using the proposed optical setup. Conflicting results have been reported earlier on whether higher or lower charge beams suffer more deterioration. The proposed interferometer provides a relatively simple and compact means of experimentally studying propagation of vortex beams through scattering medium.


S. 1 Fabrication of MDOEs
The design patterns were transferred to chromium coated mask plates using laser fabrication method in a conventional mask writer. The images of the mask patterns are shown in supplementary figure S1. The amplitude masks were used for the fabrication of two level binary phase elements using UV lithography on

S.2 Simulation results
Simulation was done using Fresnel approximations at λ = 632 nm, other parameters used were a sampling period of 4 µm, and a sampling space of 1000 × 1000 pixels, ( 1 ) = 25 cm and ( 2 ) = 30 cm. In the first step, the variation in the scattering ratio due to increasing number of layers in the stack of scatterers was studied. A scatterer is designed using Gerchberg Saxton algorithm (GSA) [1][2][3] with a scattering ratio given by b/B, where B is the length of the spectrum domain and b is the length outside which the intensity is zero.
The process is shown in figure S4. A complex amplitude C comprising a constant amplitude (White window at the input) and random phase distribution is Fourier transformed and the resulting amplitude is constrained to only have values within the scattering window of length b while the phase is retained. The process is iterated to obtain the random phase distribution, which will have a scattering ratio of σ = b/B. The procedure is repeated  figure S8. In this case, the scattering ratio is varied but the maximum phase retardation was maintained constant and it is seen that with an increase in the scattering ratio, the results improve contrary to the belief that strong scatterers distort more compared to weaker counterparts. However, when the phase retardation was increased along with the scattering ratio, the behavior reversed. The interference patterns for the case of 0.6π phase retardation for p = 1, 2 and 4 with a maximum phase retardation of 0.6π, 1.2π and 1.8π are shown in figure S9. In this case both the phase retardation as well as the scattering ratio were increased. Figure S8: Interference patterns for different scattering ratios and topological charges L=1 to 5. Figure S10: Plot of the correlation results for L=3 between different phase retardations (0.2π-π in steps of 0.2π) and the case in the absence of a scatterer. Blue -0.2π, Yellow -0.4π, Meroon -0.6π, Violet -0.8π and Green -π.

S.3 Study of scattering characteristics of scattering stack
The scattering characteristics of a stack of a scatterer was studied using an experimental setup as shown in figure   S11. Light from a He-Ne laser (λ= 632.8 nm) is passed through a neutral density filter and a stack of scatterers. The stack of scatterers was created by stacking one scatterer over the other. The maximum scattering degree of the scatterer is measured using trigonometry as = (x/2L). From the scattering degree, the maximum period of the scatterer can be approximated as Λ = (λ/sin ). The values of the scattering degree, period, etc., for the different number of scatterers that make the stack are given in Table -S1. With an increase in the number of layers, the scattering degree increased, while the effective scattering period decreased as described in the previous section.
Supplementary Figure S11: Experimental setup for studying the scattering characteristics of a stack of scatterer.