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Measuring obscured OAM spectrum using Stokes
fluctuations
Tushar Sarkar,1 Reajmina Parvin,2 Maruthi M. Brundavanam,2 Rakesh Kumar
Singh,1,*
1 Department of Physics, Indian Institute of Technology (Banaras Hindu University), Varanasi,
221005, Uttar Pradesh, India
2 Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur – 721302, India
*Corresponding author: krakeshsingh.phy@iitbhu.ac.in
Abstract
A novel technique to measure the orbital angular momentum (OAM) spectrum of the beam obscured
by a complex random media is proposed and experimentally demonstrated. This is realized by
measuring the complex correlation polarization function (CPCF) with the help of a two-point Stokes
fluctuations correlation of the random light. The OAM spectrum analysis is implemented by
projecting a complex field of the CPCF into spiral harmonics. A detailed theoretical framework is
developed to measure complex amplitude and to decompose different integer OAM states of the
obscured beam. The developed theoretical framework is verified by numerical simulation and tested
by experimental demonstration of OAM mode compositions of the fractional optical vortex (FOV)
beam.
Introduction
A light beam carrying helical mode has attracted tremendous interest in recent years owing to its
unique features, exhibiting a spiral phase structure1-5. This is inscribed by a spiral phase factor ,
where denotes azimuthal angle and l the topological charge (TC) denotes an explicit amount of
OAM equal to ħ per photon1-4. The beam with integer TC is referred to as an integer vortex beam.
The integer vortex beam reveals a null in the heart of the amplitude due to the point singularity around
the spiral phase1-3. On the other hand, a beam with non-integer TC is referred to as a FOV. The
phase structure of the FOV contains a single mixed screw edge dislocation and the intensity
distribution is no longer circularly symmetric1-5. The FOV could be deemed as multiplexing integer
OAM beams with different intensity weights2, 3. The composition of OAM modes and FOV show
prominent applications in quantum digital spiral imaging4, free-space optical communication, and
sensing1-3. The measurement of the OAM mode spectrum is also central to others applications such as
cryptography, unconventional interferometer, etc.
Several techniques have been proposed over the years to detect OAM modes compositions. These
techniques include mode decomposition using digital hologram6, 7, probing the OAM spectrum using
the tilted lens8, analysis of OAM spectrum using the holographic approach, and spatial light
modulator9, measuring OAM spectrum by digital analysis of interference pattern10. In addition, the
OAM discrimination method was developed to measure FOV and identify the difference between
intrinsic OAM and total OAM for fractional OAM states11. A method was proposed to quantitatively
measure non-integer OAM with a cylindrical lens and a camera12. Recently, a method was proposed
for precision measurement of fractional OAM based on a two-dimensional multifocal array consisting
of different integer vortices13. The OAM sorting methods based on the ray optics coordinate
transformation have also been developed14, 15. However, these methods consider propagation in free
space or homogeneous media.
On the other hand, measuring the OAM spectrum of the target obscured by the complex random
media is a challenging task and yet a highly practical problem owing to the scrambling of the spatial
structure of the light5, 16. When the beam propagates through a scattering media, the inhomogeneity in
the optical path lengths scrambles the target beam5, 16, 20. The scrambled light plays a detrimental role
in optical communication, remote sensing, and wireless communication17, 18. Therefore, previously
mentioned techniques are not capable to measure the OAM spectrum of the obscured beam5-15. In a
separate investigation, sorting of spatially incoherent optical vortex mode is implemented using two-
phase masks such as for transformation and the correction in the two-shot intensity measurements19.
This article proposes a new method to measure the OAM spectrum of the beam from higher-order
Stokes Parameters (SPs) fluctuation correlations of the randomly light. A detailed theoretical
framework is established to measure the OAM spectrum using SPs fluctuations correlations. The
correlations of the fluctuations of all SPs of the random light are evaluated which imparts a 4 x 4
correlation matrix with a total of sixteen elements. Out of these sixteen elements, only four elements
help to extract CPCF from spatially fluctuating random light. Hereinafter, the complex field of the
CPCF projection over the helical basis is applied to examine the composition of the OAM spectrum.
The proposed analysis technique is free from using the specialized mask and additionally non-
interferometric, iteration-free, and free from the pre-calibration requirement of the scattering
medium17. Due to these unique features, our experimental technique offers high flexibility and
robustness. The proposed theoretical framework is verified by simulation results and also confirmed
by experimental results. The application of our technique is demonstrated in the recovery of the
quantitative information of the FOV obscured by the random scattering medium and measuring
decomposition of the FOV in different integer OAM states. The detailed theoretical explanation and
the corresponding experimental test are discussed below.
Methodology
Consider a transversely polarized beam with orthogonal polarization states x and y. The complex field
at the transverse plane z=0 is expressed as
êê (1)
where êx and êy are horizontal and vertical polarization states of the light respectively and represents
spatial position vector at the transverse plane. , denotes the amplitude of the target beam and
non vortex beam respectively, is the phase structure of the target beam.
Now the polarized light represented by Eq. (1) propagates through a random scattering medium and
further travels down to the observation plane located in the far-field. A conceptual representation of
the propagation and generation of the random field is shown in Fig. 1. The random scattering leads to
a speckle pattern at the detector which is represented as.
ℱδêê (2)
where indicates the scattered field at the far-field, δ denotes the spatial random phase
introduced by the non-birefringent random scatterer, represents the spatial position vector at the
observation plane, and ℱ denotes two dimensional Fourier transform.
Figure 1. Conceptual representation of the generation of the Speckle pattern by orthogonally polarized
light fields.
The fluctuations between the SPs of the speckle field provide the complex polarization correlation
function which carries the information of the target beam. The SPs can be expressed in terms of four
Pauli spin matrices as20, 21
(3)
where † represents the Hermitian conjugate, is the 2-by-2 identity matrix and are the 2-
by-2 three Pauli spin matrices, which are defined as
(4)
and
(5)
Hence,
(6)
The fluctuations of each SP around their average value can be defined as
(7)
whereis the SP pertaining to a single realization of the field at a specific spatial point and
denotes its ensemble average. Now ensemble averaging is replaced by spatial averaging by
considering spatial stationary and ergodicity in space at the observation plane and 22, 23.
The 4 x 4 matrix of two-point SPs correlations can be defined as follows:
(8)
Using the Gaussian moment theorem, 4 x 4 Stokes fluctuations correlations matrix is transformed as
(9)
The terms
in Eq. (9) will be defined later.
The elements of the 4 x 4 complex polarization correlation matrix are evaluated from Eq. (9) by
considering spatial averaging and these elements are.
(10)
The elements and of the matrix are represented as
(11)
(12)
(13)
(14)
The real and imaginary parts of the CPCF are evaluated by adding
and elements.
(15)
(16)
Now, the CPCF is represented as
(17)
(18)
The correlation between two orthogonal polarization components
is expressed as
(19)
whereindicates the target source structure at the diffuser plane and f is the focal
length of the Fourier transforming lens.
Equation (18) states that the complex amplitude of the target beam can be recovered from the speckle
pattern. Now the recovered complex field can be decomposed to examine the OAM states of the
incident target beam.
To demonstrate the application of the proposed technique in the measurement of the OAM spectrum,
we considered the FOV as a target. For fractional values of the phase factor in the Eq. (1) can be
represented as and characterized in terms of Fourier series or superposition of all the integer
OAM modes as13.
φ
∞
∞ (20)
where can be represented as
where n is an integer number. Therefore, the light beam with fractional OAM modes could be deemed
as multiplexing integer OAM beams with different weights.
Experiment and Results
The proposed experimental setup for measuring the OAM spectrum is shown in Fig. 2.
A spatially filtered He-Ne laser light beam of wavelength 633 nm is attenuated with a neutral density
filter (NDF) which reduces the unwanted power of the beam. The half-wave plate (HWP) is used to
orient the incoming beam at 450 with respect to the horizontal direction. The 50:50 beam splitter
divides the 450 polarized beam into two equal intensity beams. The beam transmitted from BS is used
to illuminate a phase-only spatial light modulator (SLM) with a resolution of 1920 x 1080 and a pixel
pitch of 8 μm (Pluto from Holoeye) working in the reflection mode. The SLM is used to load the FOV
phase structure. This SLM modulates only the x-polarization component which is loaded with the
FOV and the y-polarization component remains intact i.e. plane wave. The reflected beam from SLM
is directed towards ground glass (GG) by a reflection from BS. The GG is introduced in the path of
the beam as a scattering media. Further, the beam propagates through the GG and is randomly
scattered, the GG scrambles the incident light and generates a speckle pattern. The random field from
the GG is Fourier transformed by a lens (L) of focal length 150mm as described by Eq. (2).
Figure 2. Sketch of the experimental set-up of the proposed technique. He-Ne Laser, NDF is a neutral
density filter, HWP is a half-wave plate, BS is a beam splitter, SLM is a spatial light modulator, GG is
ground glass, L is a lens, QWP is a quarter-wave plate, LP is a linear polarizer, CCD is a charge-
coupled device.
The polarization states of the speckle pattern is characterized by measuring the SPs using a quarter-
wave plate (QWP) and linear polarizer (LP) combination as shown in Fig. 2. The different
combinations of QWP and LP are used to record intensity distribution of the speckle using a charge-
coupled device (CCD) camera with a dynamic range 8-bit and resolution of 1280 x 1024 pixels and a
pixel pitch of 4.65 micron [Thorlab model No. DCU224M]. The CCD captures the intensity pattern
and the four SPs are determined from the captured speckle patterns by using the following
equations20.
°°°°, (21)
°°°° (22)
°°°° (23)
°°°° (24)
where is the intensity at the observation plane when the axes of the QWP and LP are at
and respectively as measured from the horizontal direction.
Figure 3. represent experimentally measured Stokes Parameters (a) S0 (b) S1 (c) S2 and (d) S3.
The experimentally measured SPs are used to evaluate the correlations of the fluctuations of the SPs.
The experimentally measured SPs are used to extract the real and imaginary parts of the two-point
CPCF using Eq. (15) and (16) respectively. The CPCF between two orthogonal polarization
components is evaluated using Eq. (17). To evaluate the proposed technique, we performed computer
simulation and experimental tests.
Simulation and experimental results of the amplitude and phase distribution of the incident FOV with
and are shown in Fig. 4 and Fig. 5. The CPCF encodes the complex FOV. Figs. 4,
Figure 4. Simulation results; (a)-(c) represent amplitude distribution of CPCF for three different cases
and (d)-(f) are the corresponding phase distribution.
(a)-(c) show amplitude distribution and (d)-(f) show phase distribution of CPCF for .
Figs. 5(a)-(c) and 5(d)-5(f) represent corresponding experimental results. Fig. 4, (a)-(c) represent
simulated results of amplitude distribution, and experimental results in Fig 5, (a)-(c) reveal a dark line
in the CPCF. The OAM mode in one of the orthogonal polarization component of incoming light
forms a dark line in the CPCF. To quantitatively investigate the different integer OAM modes of FOV
with , the spatial phase structure of the CPCF are shown in Figs. 4, (d)-(f) and 5, (d)-
(f) for simulation and experimental results respectively. The spatial phase distribution of CPCF in
Figs. 4 and Figs. 5 reveal the phase profile of FOV.
Figure 5. experiment results; (a)-(c) represent amplitude distribution of CPCF for three different
cases and (d)-(f) are the corresponding phase distribution.
OAM Spectrum Analysis
The orthogonal projection method is used to decompose the different integer OAM modes in terms of the
OAM power spectrum. To decompose the different integer OAM modes of the target FOV, an
experimentally measured complex field from the CPCF is projected onto spiral harmonics , where n is
the topological charge. The complex coefficient is evaluated by integrating the recovered complex field
with respect to the azimuthal angle. The complex coefficient carries each OAM value as a function of the
radial coordinates. Now, the OAM power spectrum of the beam is investigated by integrating the modulus
square of . Further, the OAM power spectrum is used to represents each OAM component in terms of
azimuthal modes20, 24.
The angular Fourier transform is applied over the recovered complex field to evaluate the complex
coefficient.
(25)
The complex coefficient is used to investigate the OAM power spectrum of the FOV beam by
integrating along with the radial coordinates.
∞
(26)
where,
∞
denotes beam power and denotes the OAM power spectrum.
Figure 6. Panels (a)-(c) and (d)-(f) represent simulation and experimental results for the OAM
distribution for three different OAM modes with .
Simulation and experimental results of the OAM distributions are shown in Fig. 6. Figs. 6, (a)-(c) and
6, (d)-(f) represent simulation and experimental results respectively. In Figs. 6, (a)-(c) and (d)-(f), the
red and green color bars reveal the OAM distribution for three different OAM modes with
respectively. Fig 6(a) and 6(d) show OAM mode with consisting of two
integer modes Fig. 6(b) and 6(e) show OAM mode with consisting of two integer
OAM modes and Fig. 6(c) and 6(f) show OAM mode with consisting of two integer
OAM modes
Conclusions
We have proposed, modeled, and experimentally demonstrated a quantitative technique to measure
the OAM spectrum of the complex field obscured by random media. The developed theoretical
framework is verified by simulation results and also tested by experimental demonstration. The
applicability of the developed technique has been demonstrated experimentally to measure the OAM
spectrum of the FOV for three different cases. The experimental results indicate that the proposed
technique shows high flexibility and robustness. This technique is expected to play a crucial role in
quantum-inspired imaging, cryptography, and optical communication.
Acknowledgment
T. S. would like to acknowledge the University Grant Commission, India for financial support as
Senior Research Fellowship. Supports from the Council of Scientific and Industrial Research (CSIR), India-
Grant No 80 (0092) /20/EMR-II, Science and Engineering Research Board (SERB): CORE/2019/000026 are
acknowledged in this work.
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