# Probing topological charge of discrete vortices

Vasu Dev  Vishwa Pal<EMAIL_ADDRESS>Department of Physics, Indian
Institute of Technology Ropar, Rupnagar, Punjab 140001, India

###### Abstract

Discrete vortex, formed by a one-dimensional (1D) ring array of lasers,
contains high output power as compared to a conventional continuous vortex,
therefore, has attracted considerable interest due to widespread applications
in various fields. We present a method for probing the magnitude and sign of
the topological charge (TC) of an unknown discrete vortex, by analyzing the
interference pattern of a 1D ring array of lasers. The interference pattern of
an unknown discrete vortex with TC$\neq 0$ is averaged with the interference
pattern of TC$=0$, which gives rise to a variation in the fringe visibility as
a function of laser number ($j$) in a 1D ring array. The number of dips
observed in the fringe visibility curve is found to be proportional to the
magnitude of TC of a discrete vortex. The sign of TC is determined by
averaging the interference patterns of unknown discrete vortex (TC$\neq 0$)
with known TC$=+1$. The number of dips in the fringe visibility curve
decreases by one for a positive TC, and increases by one for a negative TC.
Further, we have verified our method against the phase disorder, and it is
found that the phase disorder does not influence an accurate determination of
TC of a discrete vortex. The working principle as well as numerical and
experimental results are presented for the discrete vortices with TC from
small to large values. An excellent agreement between the experimental results
and numerical simulations is found. Our method can be useful in the
applications of discrete vortices.

††preprint: APS/123-QED

##  I Introduction

Optical beams with orbital angular momentum (OAM) (also called optical vortex)
have attracted considerable attention due to their applications in a wide
range of fields- from scientific research to advanced technologies, such as
optical communications with increased data carrying capacity [1, 2], optical
trapping and manipulation [3, 4, 5], microscopy [6], quantum information
processing [7, 8], remote sensing [9], high-resolution imaging [10],
generation of magnetic fields [11], high-harmonic generation [12],
astrophysics for cornography [13], and probing the angular velocity of
spinning microparticles or objects [14]. An optical vortex possesses a helical
wavefront that has a phase singularity in the center, resulting a central dark
spot in the transverse field distribution. The optical vortex caries an
orbital angular momentum of $l\hbar$ per photon, due to an azimuthal phase
term $\exp(il\phi)$, where $l$ is an integer number and is known as the
topological charge (TC). Several investigations have been performed on optical
vortex beams such as Laguerre-Gaussian (LG) beam [15], Bessel-Gaussian beam
(BG) [16], and circular Airy vortex beam (CAB) [17]. These beams consist of
continuous intensity around the central dark spot (doughnut shape), and the
phase along a contour enclosing the singularity point varies from $0$ to
$2l\pi$. Due to continuous intensity and phase over the contour, these beams
fall in the category of continuous vortex. The continuous vortex has also been
investigated in multimode optical cavities [18], condensed matter spins [19],
superfluid helium [20], and Bose Einstein condensates (BECs) [21].

A new class of discrete vortices have also been investigated in different
systems, such as in optical lattices of BECs [22], photonic structure of light
[23, 24, 25, 26, 27], coupled lasers [28, 29, 30], optical parametric
oscillators [31], and fiber arrays [32]. Generally, the discrete vortex
consists of a finite number of sites (lasers/beamlets/waveguides) in a 1D ring
array, where intensity in the center is zero, and the phase circulates from
one site to the next either in clockwise (vortex) or anti-clockwise direction
(anti-vortex). These consist of step-like behavior of phase along the contour
encompassing the phase singularity [25, 30, 32]. The discrete vortices are
particularly interesting for high-power applications. These have been
generated by different means such as coherent combining of lasers beams [33,
34, 35, 36, 37], where the phases of laser beams are precisely controlled,
phase-locking of lasers [28, 29, 30], and nonlinear waveguides [25].

The applications of the optical vortex are usually associated with its orbital
angular momentum, thus accurately determining the TC values (magnitude and
sign) of optical vortex is very important. Therefore, exploring simple and
effective methods to accurately measure TC of an optical vortex has been a
highly challenging issue, and continuous efforts are growing in this
direction. Over the years, several methods have been investigated, which are
broadly classified in two categories: first type is based on the interference
approach, and the second type is based on the diffraction approach. The
information of topological charge is manifested in the intensity distribution
when vortex beam either undergoes diffraction through slits/apertures [38,
39], or interfere with another beam (plane wave/vortex beam/spherical wave)
[40, 41]. Therefore, from the diffraction pattern or interference pattern, the
topological charge of vortex beam can be inferred. The interference approaches
are mainly based on the Mach-Zehnder interferometer [40, 41, 42], Fizeau
inteferometer [43], Sagnac interferometer [44], double-slit interferometer
[45], multipoint interferometer [46], and Talbot interferometer [47]. Whereas,
the diffraction approaches involve the annular aperture [48], triangular
aperture [49], single-slit [38, 39], multi-pinhole plate [50], gratings [51],
and metasurfaces [52], etc. Further, some more methods for the detection of TC
of vortex have also been proposed, which are based on conformal mappings [53],
multiplane light conversion [54], mode converter [55], rotational Doppler
effect [56], and two-dimensional material [57]. The above methods rely on
detecting the topological charges indirectly by analyzing the changes in the
intensity distributions. However, some direct methods for determining TC of a
vortex have also been investigated. These are based on directly measuring the
phase distribution of a vortex, and the method includes phase-shifting digital
hologram [58] and Shack-Hartmann wavefront sensors [59].

Many of these methods pose limitations in various forms, such as complexity in
the setup, unable to precisely determine high-order TCs (magnitude and sign),
sensitive to aperture dimensions, and sensitive to aberrations in the system.
Further, most of these methods infer the TC of a vortex by propagation and
analysis of changes in its intensity distribution. These methods have been
applied to the continuous systems (continuous vortices). However, it is well-
known that discrete systems show different propagation behaviour, so a natural
question arises that whether the above methods can be applied or not for
identifying the TCs of discrete vortices.

Methods based on direct measurement of phase distribution of a vortex [58, 59]
can be useful for determining small TC values of discrete vortex, however,
given the aberrations as well as very small differences in the phases of
lasers for large TCs, these may not be able to precisely determine the high-
order TCs of discrete vortices. Here, we present a new method for precisely
determining magnitude and sign of TCs (from small to large values) of discrete
vortices, based on measuring the interference patterns of 1D ring array of
lasers. More specifically, we average the interference pattern of an unknown
discrete vortex (TC$\neq 0$) with the interference pattern of ring array with
known TC, which gives rise to a variation in the fringe visibility as a
function of laser number, and the number of dips observed in fringe visibility
provides an accurate information of magnitude and sign of an unknown TC. Our
method is also found to be robust against the phase disorder in a system.

The paper is organized as follows. In Sec. II, we present the working
principle of our method with the illustrative results. In Sec. III, we have
described the experimental generation of discrete vortex as well as measuring
its interference pattern. In Sec. IV, we have presented the numerical and
experimental results on precise determination of TCs of discrete vortices for
different system sizes. Further, results on finding the sign of TCs
(positive/negative) are also presented. The robustness of our method is also
verified against the phase disorder. Finally, in Sec. V, we present the
concluding remarks.

##  II Working principle

We consider the interference of two waves of intensities $I_{1}$ and $I_{2}$:
one propagating in the $z$ direction; the other propagating at an angle
$\theta$ with respect to the $z$ axis, in the $x-z$ plane. The fields of these
interfering waves can be written as [60]

$\displaystyle U_{1}$ $\displaystyle=$
$\displaystyle\sqrt{I_{1}}\exp(-ikz)\exp(i\phi_{1}),$ (1) $\displaystyle
U_{2}$ $\displaystyle=$ $\displaystyle\sqrt{I_{2}}\exp[-i(k\cos\theta
z+k\sin\theta x)]\exp(i\phi_{2}),$ (2)

where $\phi_{1}$ and $\phi_{2}$ are the phases of two waves. At the $z=0$
plane, the resultant intensity after superposition of these two waves can be
written as

$I=\left|U_{1}+U_{2}\right|^{2}=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}\cos(k\sin\theta
x-d\phi),$ (3)

where $d\phi=\phi_{2}-\phi_{1}$. For $I_{1}=I_{2}=I_{0}$, we get

$\displaystyle I=2I_{0}\left[1+\cos\left(k\sin\theta x-d\phi\right)\right].$
(4)

The interference intensity distribution consists of straight fringes with
maxima and minima positions vary with $x$, and the fringe spacing is
$\beta=2\pi/k\sin\theta=\lambda/\sin\theta$. The locations of minima and
maxima can be determined from Eq. (4). For maxima:

$x_{\mathrm{max}}=\frac{[2m\pi+d\phi]}{k\sin\theta},~{}~{}~{}m=0,\pm 1,\pm
2,\pm 3,...$ (5)

For minima:

$x_{\mathrm{min}}=\frac{[(2m+1)\pi+d\phi]}{k\sin\theta},~{}~{}~{}m=0,\pm 1,\pm
2,\pm 3,...$ (6)

The position of the maxima and minima in the interference fringe pattern
depends on the value of relative phase shift $d\phi$. By changing it, these
maxima and minima positions shift either on the left or right with respect to
the initial positions. To show this, we have considered two Gaussian beams
with equal intensities $I_{1}=I_{2}=I_{0}$, tilt angle $\theta=110^{o}$ and
$d\phi=0~{}\mathrm{and}~{}\pm\pi/2$. The interference expression for these
cases can be written as:
(i) For d$\phi=0$

$I_{T1}=2I_{0}\left[1+\cos\left(0.94kx\right)\right].$ (7)

(ii) For d$\phi=-\pi/2$

$I_{T2}=2I_{0}\left[1+\cos\left(0.94kx+\pi/2\right)\right].$ (8)

(iii) For d$\phi=\pi/2$

$I_{T3}=2I_{0}\left[1+\cos\left(0.94kx-\pi/2\right)\right].$ (9)

Figure 1:  Interference pattern between the two waves having a relative phase
shift of (a) $d\phi=-\pi/2$, (b) $d\phi=0$, and (c) $d\phi=\pi/2$. The
averaged interference pattern obtained with superposition of (d) $d\phi=0$ and
$-\pi/2$, and (e) $d\phi=0$ and $\pi/2$. (f) The intensity cross-sections
taken along the horizontal axis in (a), (b) and (d). (g) The intensity cross-
sections taken along the horizontal axis in (b), (c) and (e).

The interference patterns obtained from Eqs. (7)-(9) are shown in Figs.
1(a)-1(c). As evident, for all three cases, interference patterns consists of
straight fringes with maxima and minima at different positions. We consider
the case of $d\phi=0$ as the reference interference pattern. A vertical white
dashed line marks the position of fringe with central maximum (Fig.1(b)). For
$d\phi=-\pi/2$, the fringes in the interference pattern are shifted towards
the right with respect to the reference white dashed line (Fig. 1(a)).
Similarly, for $d\phi=\pi/2$, the fringes in the interference pattern are
shifted towards the left with respect to the reference white dashed line (Fig.
1(c)). The interference pattern is quantified by the fringe visibility
$V=(I_{\mathrm{max}}-I_{\mathrm{min}})/(I_{\mathrm{max}}+I_{\mathrm{min}})$
[60], where $I_{\mathrm{max}}$ and $I_{\mathrm{min}}$ are the maximal and
minimal values of the time-averaged intensities in the interference pattern.
In all three individual interference patterns (for
$d\phi=0,-\pi/2,\mathrm{and}\,\pi/2$) the fringe visibility is found to be
$V=1$ (Figs. 1(a)-1(c)).

Further, we have performed averaging of these interference patterns. The
expression of averaged interference pattern can be written as following: (i)
For averaging the interference patterns corresponding to Figs. 1(a) and 1(b),
we add Eqs.(7) and (8)

$I_{\mathrm{Sum1}}=I_{0}\left[2+\cos\left(0.94kx\right)+\cos\left(0.94kx+\pi/2\right)\right].$
(10)

Note, due to averaging the factor 2 is dropped. (ii) For averaging the
interference patterns corresponding to Figs. 1(b) and 1(c), we add Eqs.(7) and
(9)

$I_{\mathrm{Sum2}}=I_{0}\left[2+\cos\left(0.94kx\right)+\cos\left(0.94kx-\pi/2\right)\right].$
(11)

The averaged interference patterns corresponding to Eqs. (10) and (11) are
shown in Figs. 1(d) and 1(e), respectively. As evident, when two interference
patterns with $d\phi=0$ and $d\phi=\pm\pi/2$ (each with $V=1$) are averaged,
they produce a resultant interference pattern, where the fringe visibility is
found to be reduced ($V=0.7$) (Figs. 1(d) and 1(e)). This is more clearly
shown in Figs. 1(f) and 1(g). Figure 1(f) shows the intensity cross-sections
taken along the horizontal axis in Fig. 1(a) (blue dot dashed curve), Fig.
1(b) (red dashed curve) and Fig. 1(d) (black solid curve). Similarly, Fig.
1(g) shows the intensity cross-sections taken along the horizontal axis in
Fig. 1(b) (red dashed curve), Fig. 1(c) (blue dot dashed curve) and Fig. 1(e)
(black solid curve). It is clearly evidenced that the averaged intensity
cross-section (black solid curve) results a reduced value of the fringe
visibility. Note, averaging the interference patterns corresponding to
$d\phi=0$ and $d\phi=\pm\pi$, results the fringe visibility $V=0$.

We utilize this principle of reduction in visibility by averaging the
interference patterns, for determining the magnitude and sign of topological
charge of an unknown discrete vortex.

##  III Experimental arrangement

Figure 2:  (a) Experimental arrangement for the generation of discrete vortex,
and determining its topological charge (including magnitude and sign). SLM:
Spatial light modulator; BS1, BS2, BS3: 50:50 Beam splitters; L1 and L2:
Plano-convex lenses with focal lengths $f_{1}$ and $f_{2}$, respectively; M1
and M2: Mirrors. (b) The phase hologram corresponding to a discrete vortex
with TC$=1$. (c) The phase distribution of a discrete vortex with TC$=1$. (d)
The intensity distribution of a discrete vortex with N$=10$ lasers. (e) The
interference pattern where a single selected reference laser interfered with
itself and with all the other lasers.

A discrete vortex consists of a finite number of sites
(lasers/beamlets/waveguides) in a 1D ring array, where intensity in the center
is zero, and the phase circulates from one site to the next either in clock-
wise or anti-clockwise direction [28, 35, 32, 25, 31]. The field of a discrete
vortex can be given as [36]:

$U(x,y;z=0)=U_{0}\sum_{j=1}^{N}e^{-\frac{(x-\beta_{j})^{2}+(y-\gamma_{j})^{2}}{\omega^{2}_{0}}}e^{i\phi_{j}},$
(12)

where $(\beta_{j},\gamma_{j})=\alpha(\cos\delta_{j},\sin\delta_{j})$
represents the coordinates of each site, $\alpha=d/{\sqrt{1-\cos(2\pi/N)}}$,
$\delta_{j}=\pi(2j-1)/N$, and $\phi_{j}=\mathrm{TC}.(2\pi(j-1)/N)$, where
$j=1,2,3...N$. $d$ denotes the center-to-center separation between two
neighboring sites. All sites in a discrete vortex have the same amplitude
$U_{0}$, beam waist $\omega_{0}$, and different phase $\phi_{j}$, satisfying
the periodic boundary conditions: $U_{j+N}=U_{j}$. The topological charge $TC$
of a discrete vortex can be defined as

$\mathrm{TC}=\frac{1}{2\pi}\sum^{N}_{j=1}\mathrm{arg}(U^{*}_{j}U_{j+1}),$ (13)

where $U_{j}$ represents the complex field of site $j$ on a ring. As opposed
to a continuous vortex, a discrete vortex is formed by combining several
lasers/beamlets/waveguides, thus does not suffer the output power limitations.

In our previous works, we have shown the generation of discrete vortex by
phase-locking lasers in 1D ring array inside a degenerate cavity [28, 29]. The
lasers were coupled by the Talbot diffraction, which caused the phase-locking
of lasers in a vortex configuration. However, the discrete vortex with
precisely controlled TC could not be generated. More recently, a new approach
has been demonstrated for the controlled generation of discrete vortex with
any TC from small to large values, where a spatial Fourier filtering mechanism
inside a degenerate cavity phase-locks the lasers in a desired vortex
configuration [30]. The spatial Fourier filtering eliminates the undesired
phase distributions by introducing extra losses to them, and accordingly only
minimum loss solution with desired phase distribution lase inside the cavity.

In the present work, our main goal is to efficiently determine the topological
charge (magnitude and sign) of a discrete vortex. To show the proof of concept
of our method, we have generated a discrete vortex from a computer generated
hologram using a spatial light modulator (SLM). The schematic of experimental
setup as well as representative results are presented in Fig. 2. We use a
phase only SLM with the screen resolution $1920\times 1080$ and pixel size
$8~{}\mu$m. A collimated linearly polarized laser beam with fundamental
Gaussian distribution, wavelength $\lambda=1064$ nm and beam waist radius 10
mm incidents normally on the SLM with a beam splitter (BS1). The size of an
input Gaussian beam is chosen such that it illuminates whole screen of SLM. On
the SLM screen a computer generated phase hologram corresponding to a discrete
vortex is applied (Fig. 2(b)) (see Appendix A). Phase hologram on the SLM
modulates amplitude and phase of an input Gaussian laser beam, and accordingly
the light from an input Gaussian laser beam splits in the form of multiple
lasers arranged on a 1D ring array with discrete phase distributions in a
vortex configuration. After reflection from SLM, we obtain modulated light in
several orders (see Appendix A), which contains the desired discrete vortex.
The desired discrete vortex in the first order is isolated with a spatial
Fourier filter placed in the middle of a telescope made with lenses L1
($f_{1}=30$ cm) and L2 ($f_{2}=20$ cm). After spatial filtering, we obtain a
clean discrete vortex at the focal plane of L2. The experimental phase
distribution and intensity distribution of a generated discrete vortex are
shown in Figs. 2(c) and 2(d), respectively. As evident, the discrete vortex
consists of N=10 lasers (with TEM00 fundamental Gaussian mode profiles) in a
1D ring array (Fig. 2(d)), and the lasers possess discrete phase distribution
(Fig. 2(c)) in a vortex configuration. This is identical to a discrete vortex
obtained by phase-locking of lasers in a 1D ring array inside a degenerate
cavity [28, 30].

To determine the magnitude and sign of topological charge of a discrete
vortex, we measure the interference between the lasers using a Mach-Zhender
interferometer, as shown in Fig. 2(a). The discrete vortex (Figs. 2(c) and
2(d)) splits into two channels (Ch-1 and Ch-2) at the beam splitter BS2. In
one channel (Ch-1), the discrete vortex is imaged directly onto the camera. In
the other channel (Ch-2), a single reference laser is selected with a circular
pinhole of suitable size, and then its light is expanded such that it fully
overlaps and interferes with the light from all other lasers with a beam
splitter BS3 on the camera. This enables that a single selected reference
laser interferes with itself and with all the other lasers. A small tilt (Eq.
(2)) between the two channels provides a few interference fringes for each
laser (as shown in the interference pattern Fig. 2(e)) from which the fringe
visibility is analyzed.

##  IV Results and discussions

To show that our method can efficiently determine the magnitude and sign of TC
(from small to large values) of discrete vortex, we have performed several
experiments considering different system sizes N. The experimental results are
supported by the numerical simulations. First, we have demonstrated our
approach for determining TCs of discrete vortices, formed with a 1D ring array
of N=10 lasers. The results for a discrete vortex with TC=1 are shown in Fig.
3.

Figure 3:  Results for a discrete vortex with TC=1. (a) The near-field
intensity distribution of a discrete vortex formed by a 1D ring array of N=10
lasers. The near-field phase distribution with (b) TC=0, and (c) TC=1.
Interference pattern when a single laser (reference laser denoted by dashed
circle) interferes with itself and with all the other lasers for (d, g) TC=0,
and (e, h) TC=1. (f) The average of interference patterns of TC=0 (d) and TC=1
(e). (i) The average of interference patterns of TC=0 (g) and TC=1 (h). (j)
The variation of fringe visibility as a function of laser index.

Figure 3(a) shows the intensity distribution of a discrete vortex, indicating
that it consists of a 1D ring array of N=10 lasers with TEM00 fundamental
Gaussian mode profiles, and each laser has the beam waist $0.16$ mm and
separated with a distance of $0.84$ mm (center-to-center). Figure 3(b) shows
the phase distribution of 1D ring array of N=10 lasers with TC=0 (in-phase
distribution), indicating that there is no net phase circulation around the
center. Figure 3(c) shows the discrete phase distribution of 1D ring array of
N=10 lasers with TC=1, indicating the phase circulation around the center
(discrete vortex). Note, the intensity distribution of 1D ring array of lasers
for both TC=0 and TC=1 is the same. Figures 3(d)-3(f) show the simulated
results of individual interference patterns of 1D ring array of N=10 lasers
with TC=0 and TC=1, and their averaged interference pattern, respectively.
Figures 3(g)-3(i) show the experimental results of individual interference
patterns of 1D ring array of N=10 lasers with TC=0 and TC=1, and their
averaged interference pattern, respectively. The white dashed circles in Figs.
3(d), 3(e), 3(g) and 3(h) mark the location of a selected reference laser. The
individual interference patterns of 1D ring array of N=10 lasers with TC=0 and
TC=1 are obtained using Mach-Zhender interferometer (Fig. 2(a)), where a
selected reference laser interferes with itself and with all the other lasers.
As evident, in the individual interference patterns corresponding to TC=0 and
TC=1, the fringes are uniform at all the lasers and show no
distinguishabilities between TC=0 and TC=1 (Figs. 3(d)& 3(e) and Figs. 3(g) &
3(h)). However, in the averaged interference pattern the fringes show
variation at the lasers (Figs. 3(f) and 3(i)). Particularly, at the reference
laser $j=1$ the fringes appear with clear maxima and minima, and the fringes
degrade monotonically for the lasers $j=2-6$ and after that fringes again
improve monotonically for the lasers $j=6-10$.

We have analyzed the fringe visibility at all the lasers in these individual
and averaged interference patterns (see Appendix B). The quantified simulated
and experimental results are shown in Fig. 3(j), which shows the variation of
fringe visibility as a function of laser index $j$ ($j=1,~{}2,~{}3...10$). As
evident, for the individual interference patterns corresponding to TC$=0$
(solid black curve with squares (simulation) and dashed green curve with
diamonds (experiment)) and TC$=1$ (pink dashed curve with triangles
(simulation) and cyan dashed curve with cross (experiment)), the fringe
visibility does not show any variation, and is found to be maximum V$\approx
1$ at all the lasers. Note, due to small intensity differences between the
lasers, the experimental visibility is found to be a bit less than V$=1$.
However, in the averaged interference pattern the fringe visibility shows
variation as a function of laser index $j$ (shown by red dashed curve with
stars (experiment) and blue solid curve with circles (simulation)). For
example, the fringe visibility is found to be maximum (V$=1$) at laser $j=1$
and then it decreases monotonically until laser $j=6$ (minimum $V$), and after
that it again increases and becomes maximum again at laser $j=10$.

These variations in the fringe visibility as a function of laser index can be
understood by the working principle given in Sec. II (see also Appendix B). In
case of TC$=0$, all the lasers have zero phase ($\phi_{j}=0$ and
$d\phi_{1,j\geq 1}=0$, where $j=1,~{}2,~{}3...N$), whereas for TC$=1$, the
phases of the lasers grow in a clockwise direction ($\phi_{j}=(j-1)2\pi/N$ and
$d\phi_{1,j\geq 1}=(j-1)2\pi/N$, where $j=1,~{}2,~{}3...N$). So the reference
laser ($j=1$) in both cases will have the interference fringes with maxima and
minima occurring at the same positions, and hence averaging does not reduce
the fringe visibility. However, for laser $j\geq 2$, we have $d\phi_{1,j\geq
2}=0$ in case of TC=0, and $d\phi_{1,j\geq 2}=(j-1)2\pi/10$ ($N=10$ in the
present case) in case of TC=1. Thus, in the interference patterns of TC=0 and
TC=1, the fringes with maxima and minima will occur at different positions in
the same laser $j\geq 2$ (Eqs. (5) and (6)), and as a result of averaging the
fringe visibility is found to be reduced at that laser (Fig. 1, and see
Appendix B). The fringe visibility is found to be lowest at laser $j=6$, where
the phase shift $d\phi_{1,6}=\pi$ becomes maximum in TC=1 and the fringes are
shifted maximally, thus the averaging results a lowest value of fringe
visibility. For lasers $j=7-10$ the fringe visibility again increases
monotonically. For laser $j=7$ in TC=1, we have
$d\phi_{1,7}=6.(\frac{2\pi}{10})$, which can also be written as
[$2\pi-4.(\frac{2\pi}{10})$] or $-4.(\frac{2\pi}{10})$. This becomes
equivalent to $-d\phi_{1,5}$. The positive and negative values of the same
magnitude $d\phi$ produce the same shift of fringes in the interference
pattern, as shown in Fig. 1. The magnitude of $d\phi_{1,7}$ is found to be
less than $d\phi_{1,6}$, so fringe visibility in the averaged interference
pattern at laser $j=7$ is found to be increased, and it is the same as for
laser $j=5$. Similarly, the fringe visibility for lasers
$j=8,~{}9~{}\mathrm{and}~{}10$ can also be explained (for details, see
Appendix B). The variation in the fringe visibility as a function of laser
index is characteristics to the TC of a discrete vortex, so it serves as the
basis for an accurate determination of TC. In particular, the observation of
number dips in the fringe visibility curve provides the information of TC. For
example, a single minimum denotes the value of TC$=1$.

Figure 4:  For a discrete vortex of system size N=10 lasers, the determination
of TC=2, 3 and 4. The average of interference patterns of TC$=0$ and (a, d)
TC$=2$, (b, e) TC$=3$, and (e, f) TC$=4$. (g-i) The Fringe visibility as a
function of laser index, corresponding to (a, d), (b, e), and (e, f),
respectively. Blue solid curve with circles represents the simulation, and Red
dashed curve with stars represents the experiment. Dashed white circle on (a,
d) marks the location of a selected reference laser $j=1$. Figure 5:
Determination of higher values of TC of a discrete vortex with a large system
size N=30 lasers. The average of interference patterns of TC$=0$ and (a, e)
TC$=11$, (b, f) TC$=12$, (c, g) TC$=13$, and (d, h) TC$=14$. (e-h)The Fringe
visibility as a function of laser index, corresponding to (a, e), (b, f), and
(c, g), respectively. Blue solid curve with circles represents the simulation,
and red dashed curve with stars represents the experiment. Dashed white circle
on (a, e) marks the location of a selected reference laser $j=1$.

Further, we have determined the large values of TC$>1$ in a discrete vortex
formed with a 1D ring array of N=10 lasers. The results are shown in Fig. 4.
Figures 4(a)-4(c) and 4(d)-4(f) show the simulated and experimental results
obtained by averaging the interference patterns of TC=2, TC=3 and TC=4 with
TC=0 (TC$=0$ \+ TC$=2$, TC$=0$ \+ TC$=3$, and TC$=0$ \+ TC$=4$), respectively.
The white dashed circles in Figs. 4(a) and 4(d) mark the location of a
selected reference laser $j=1$. The location of reference laser remains the
same in other averaged interference patterns (Figs. 4(b) $\&$ 4(c) and Figs.
4(e) $\&$ 4(f)). As evident, in the averaged interference patterns the fringes
at the lasers exhibit different behaviour. The analyzed fringe visibility as a
function of laser index ($j$) corresponding to TC$=2$, $3$ and $4$ are
presented in Figs. 4(g)-4(i), respectively, indicating different variations
related to the values of TCs. This is due to the fact that the phase
distribution of discrete vortex depends on the value of TC as
$\phi_{j}=\mathrm{TC}.(2\pi(j-1)/10)$ ($j=1,~{}2,~{}3...10$), and accordingly
the fringes at the lasers are shifted by different amounts with respect to the
fringes in TC=0. Thus, averaging of interference patterns of TC$\neq 0$ and
TC=0 results a different variation in the fringe visibility with the laser
index (see Appendix B), and can be used to determine the value of TC. For
example, the averaging of TC$=0$+TC$=2$ produces two dips in the fringe
visibility curve, confirming the value of TC$=2$ (Fig. 4(g)). Similarly, the
observation of three and four dips in the fringe visibility curve (Figs. 4(h)
and 4(i)) confirms the value of TC$=3$ and $4$, respectively. The simulation
and experimental results show an excellent agreement.

Further, to show an accurate determination of higher order TC of a discrete
vortex, we have considered a 1D ring array of N=30 lasers. Using the
experimental arrangement shown in Fig. 2(a), we have generated discrete
vortices with TC$=11$ to 14, and determined their values by averaging their
interference patterns with TC$=0$. The results are shown in Fig. 5. Figures
5(a)-5(d) and 5(e)-5(h) show the simulated and experimental results obtained
by averaging the interference patterns of TC=11, 12, 13 and 14 with TC=0
(TC=0+TC=11, TC=0+TC=12, TC=0+TC=13 and TC=0+TC=14), respectively. The white
dashed circles in Figs. 5(a) and 5(e) mark the location of a selected
reference laser $j=1$. The location of reference laser remains the same in
other averaged interference patterns. As evident, in the averaged interference
patterns, corresponding to different TCs, the fringes at the lasers exhibit
different behaviour, and enables an accurate identification of corresponding
TC of discrete vortices. The analyzed fringe visibility as a function of laser
index ($j$) corresponding to TC$=11-14$ is shown in Figs. 5(i)-5(l),
respectively. The phase distribution of discrete vortex with different TCs,
formed with a 1D ring array of N=30 lasers cane be given as
$\phi_{j}=\mathrm{TC}.(2\pi(j-1)/30)$, where $j=1,~{}2,~{}3,...30$. The
observed variation in the fringe visibility as a function of laser index ($j$)
can be explained with the same reasons as explained above for discrete vortex
with N=10 lasers. It is clearly evidenced that the averaging of interference
patterns TC=0+TC=11, TC=0+TC=12, TC=0+TC=13 and TC=0+TC=14 leads to distinct
variations in the fringe visibility as a function of laser index, and produces
different number of dips corresponding to different values of TCs. For
example, averaged interference pattern of TC=0+TC=11 produces eleven number of
dips, which confirms the value of TC=11 (Fig. 5(i)). Similarly, the
observation of twelve, thirteen and fourteen dips corresponding to TC=0+TC=12,
TC=0+TC=13, and TC=0+TC=14, confirms the values of TC$=12-14$, respectively.
We have observed an excellent agreement between the numerical and experimental
results, which indicates an accurate determination of high order TCs of
discrete vortices.

Figure 6:  Simulation results for determining the sign of TC of a discrete
vortex with N=10 lasers. The averaging of interference patterns of TC$=1$ and
(a) TC=$1$, (b) TC$=2$, (c) TC$=3$, and (d) TC$=4$. The averaging of
interference patterns of TC$=1$ and (e) TC$=-1$, (f) TC$=-2$, (g) TC$=-3$, and
(h) TC$=-4$. (i-l) The fringe visibility as a function of laser index,
corresponding to (a,e), (b, f), (c, g), and (d, h), respectively. A dashed
white circle in (a) marks the location of a selected reference laser $j=1$.
Figure 7:  Experimental results for determining the sign of TC of a discrete
vortex with N=10 lasers. The averaging of interference patterns of TC$=1$ and
(a) TC=$1$, (b) TC$=2$, (c) TC$=3$, and (d) TC$=4$. The averaging of
interference patterns of TC$=1$ and (e) TC$=-1$, (f) TC$=-2$, (g) TC$=-3$, and
(h) TC$=-4$. (i-l) The fringe visibility as a function of laser index,
corresponding to (a,e), (b, f), (c, g), and (d, h), respectively. A dashed
white circle in (a) marks the location of a selected reference laser $j=1$.

So far, we have successfully shown an accurate determination of maginutude of
TC of a discrete vortex by averaging the interference patterns of TC$\neq 0$
with TC$=0$. However, with this averaging the same results are obtained for
both positive and negative values of TC, and can not distinguish the sign of
TC$\neq 0$. To determine the sign of TC of a discrete vortex, we average the
interference pattern of unknown discrete vortex having TC$\neq 0$ with the
interference pattern of known discrete vortex having TC$=1$ (instead of TC$=0$
as earlier). To show it, we have considered discrete vortices with different
TCs in a 1D ring array of N=10 lasers, as shown in Fig. 6.

Figures 6(a)-6(d) show the simulated averaged interference patterns of
positive TC=1, 2, 3 and 4 with TC=1 (TC$=1+$TC$=1$, TC$=1+$TC$=2$,
TC$=1+$TC$=3$ and TC$=1+$TC$=4$), respectively. Figures 6(e)-6(h) show the
simulated averaged interference patterns of negative TC$=-1,~{}-2,~{}-3,$ and
$-4$ with TC=1 (TC$=1+$TC$=-1$, TC$=1+$TC$=-2$, TC$=1+$TC$=-3$ and
TC$=1+$TC$=-4$, respectively. A white dashed circle in Fig. 6(a) represents
the location of a selected reference laser $j=1$, which also remains the same
in all other interference patterns (Figs. 6(b)-6(h)). As evident, in the
averaged interference patterns of positive and negative TCs with TC=1, the
fringes at the lasers appear differently, thus enabling to distinguish the
sign of TC. These different interference patterns are again anticipated by the
different phase distribution of discrete vortices having different TCs. The
analyzed fringe visibility as a function of laser index ($j$), corresponding
to Figs. 6(a)-6(h), is shown in Figs. 6(i)-6(l). The red dashed curve with
circles shows the visibility curve when interference patterns of positively
charged TC$=+1$ to $+4$ are averaged with TC$=+1$, whereas, solid blue curve
with squares denote the case when interference patterns of negatively charged
TC$=-1$ to $-4$ are averaged with TC$=+1$. As evident, the variation of fringe
visibility with laser index ($j$) for the positive and negative TCs is found
to be different, and thus enables to identify the sign of TC. In particular,
for TC$=+1$, the fringe visibility is found to be 1 for all the lasers (no
variation), and no dip is observed (dashed red curve with circles in Fig.
6(i)). Whereas, for TC$=-1$, the fringe visibility shows variation, and two
dips are observed (solid blue curve with squares in Fig. 6(i)).

As discussed earlier in Figs. 3-5, when the interference patterns of positive
and negative TCs are averaged with the interference pattern of TC$=0$, the
number of dips in the fringe visibility curve is found to be proportional to
the magnitude of TC. For example, for TC$=\pm 1$ only a single dip is observed
in the fringe visibility curve (Fig. 3(j)). However, when the interference
patterns of positive and negative TCs are averaged with the interference
pattern of TC$=+1$, the number of dips in the fringe visibility curve
decreases by one for the positive TCs, and increases by one for the negative
TCs. For example, in Fig. 6(i), no dip is observed for TC$=+1$ and two dips
are observed for TC$=-1$. Similarly, for TC$=+2$, $+3$ and $+4$, the number of
dips are observed to be one, two and three, respectively (red dashed curve
with circles in Figs. 6(j)-6(l)). Whereas, for TC$=-2$, $-3$ and $-4$, the
number of dips are observed to three, four and five, respectively (solid blue
curve with squares in Figs. 6(j)-6(l)). This clearly indicates that the
observation of decrease/increase in the number of dips in the fringe
visibility curve provides an accurate information of positive/negative sign of
TC of a discrete vortex.

We have also verified these findings experimentally, the results are shown in
Fig. 7. The experimental results show excellent agreement with the
simulations. Thus, averaging the interference patterns of positive and
negative TCs with the interference pattern of TC$=+1$ provides an accurate
information of the sign of TC of a discrete vortex. This approach can be used
for identifying the sign of TC of any order.

Next, we have checked the robustness of our method against the phase disorder
(for example, aberrations due to misalignment of optical components). We have
generated phase disorder in a range [-$\pi$ to $\pi$] through a random phase
screen using Monte Carlo method, that behaves like a disorder media of desired
length and having disorder strength $C^{2}_{n}$ [61]. We have considered a
phase disorder with a strong strength $C^{2}_{n}=10^{-12}$. The simulated
results are shown in Fig. 8. Figures 8(a)-8(d) show the ideal phase
distribution of discrete vortices with TC=1 to 4 in a 1D ring array of N=10
lasers. After multiplying the phase disorder, the distorted phase
distributions of discrete vortices with TC=1 to 4 are shown in Figs.
8(e)-8(l). It is clearly evident that most of the lasers in the array consist
of more than a single phase.

To show whether our method can still determine accurately the information of
TC, we have measured the interference patterns of 1D ring array of lasers with
TC=1 to 4 and TC$=0$ under the same phase disorder, using Fig. 2(a). The
averaged interference patterns of TC=1 to 4 with TC=0 are shown in Fig.
8(i)-8(l). It is evident that in the averaged interference patterns of
different TCs, the fringes at the lasers are distributed differently with
different fringe visibility. Further, the orientations of the fringes at the
lasers are found to be different (unlike ideal discrete vortices in Figs. 2
and 3). This is anticipated by the multiple phase structure in each laser. The
analyzed fringe visibility as a function of laser index ($j$) for the all TC=1
to 4 (blue solid curve with circle for TC$=1$, red dashed curve with squares
for TC$=2$, black dot-dashed curve with star for TC$=3$ and pink dotted curve
with triangles for TC$=4$), are shown in Fig. 8(m). As evident, for different
values of TC, the variation in the fringe visibility consists of different
number of dips. In particular, corresponding to TC=1 to 4, the number of dips
in the fringe visibility are found to be one, two, three and four,
respectively, which clearly identifies the magnitude of TCs.

Figure 8:  Simulation results showing the effect of phase disorder on the
determination of topological charge of a discrete vortex with N=10 lasers. The
discrete vortex with topological charges TC=1 to 4 having (a-d) ideal phase
distribution, and (e-h) disordered phase distribution. The average of
interference patterns of TC$=0$ and (i) TC$=1$, (j) TC$=2$, (k) TC$=3$, and
(l) TC$=4$. (m) The fringe visibility as a function of laser index
corresponding to (i-j). A white dashed circle in (a) marks the location of a
selected reference laser. Figure 9:  Experimental results showing the effect
of phase disorder on the determination of topological charge of a discrete
vortex with N=10 lasers. The average of interference patterns of TC$=0$ and
(a) TC$=1$, (b) TC$=2$, (c) TC$=3$, and (d) TC$=4$. (m) The fringe visibility
as a function of laser index corresponding to (a-d). Note, the phase disorder
is kept the same as in Fig. 8. A white dashed circle in (a) marks the location
of a selected reference laser.

We have also verified experimentally the effect of phase disorder on the
determination of TC. The results for different values of TC=1 to 4, under the
same phase disorder strength $C^{2}_{n}=10^{-12}$ are presented in Fig. 9.
Experimentally, the phase disorder on the discrete vortex is realized by
imposing random phase on hologram along with the phase distribution of
discrete vortex [61]. Figures 9(a)-9(d) show the averaged interference
patterns of TC= 1 to 4 with TC=0 (TC=0+TC=1, TC=0+TC=2, TC=0+TC=3 and
TC=0+TC=4), respectively, indicating the different fringe distribution at the
lasers in each case. Corresponding to Figs. 9(a)-9(d), the analyzed fringe
visibility as a function of laser number ($j$) is shown in Fig. 9(e). As
evidenced, corresponding to different values of TC=1 to 4, the number of dips
in the fringe visibility variation is found to be one (blue solid curve with
circles), two (red dashed curve with squares), three (black dot-dashed curve
with stars) and four (pink dotted curve with triangles), respectively. This
clearly depicts that our approach is highly robust against the phase disorder,
as it does not affect the accurate determination of magnitude of TC of a
discrete vortex. Similarly, the sign of these disordered TCs can be determined
by averaging their interference patterns with the interference pattern of
disordered TC=1.

The numerical and experimental results show an excellent agreement, and
clearly depicts that our method accurately determines the magnitude and sign
of TC of a discrete vortex. The determination of TC is not affected by the
imperfections such as the phase disorder caused by aberrations in the system.
Further, the method can be used for determining any small to large values of
TC.

##  V Conclusions

We have presented a new method for probing the magnitude and sign of an
unknown TC of a discrete vortex, which is formed by an array of lasers in 1D
ring geometry. The method relies on measuring the interference pattern of a
discrete vortex, which is obtained by interfering a single selected reference
laser with itself and with all the other lasers in a 1D ring array. The
discrete laser arrays with TC$=0$ and TC$\neq 0$ have different phase
distributions, thus produce interference patterns with shifted fringes. The
averaging of these phase shifted interference patterns give rise to a
variation in the fringe visibility as a function of laser number $(j)$ in the
array, thus enables identification of TC. The magnitude of TC of a discrete
vortex is found to be proportional to the number of dips observed in the
fringe visibility curve. The sign of TC is determined by averaging the
interference patterns of unknown discrete vortices (TC$\neq 0$) with known
TC$=+1$. The number of dips in the visibility curve decreases by one for a
positive TC, and increases by one for a negative TC. Our method works well for
any values of TC of a discrete vortex.

Further, we have also verified our method against the phase disorder that may
occur due to the presence of aberrations in the system. It is found that phase
disorder does not influence an accurate determination of TC of a discrete
vortex. We have found an excellent agreement between the experimental results
and numerical simulations. Our method can be useful in the applications of
discrete vortices.

###### Acknowledgements.

We acknowledge the funding support from the Indian Institute of Technology
(IIT) Ropar (ISIRD Grant No. 9-230/2018/IITRPR/3255), Science and Engineering
Research Board (Grant no. CRG/2021/003060). Vasu Dev acknowledges the
fellowship support by IIT Ropar.

##  Appendix A Phase hologram for the formation of a discrete vortex in a 1D
ring array of lasers

In order to generate discrete vortex with desired TC, the phase and amplitude
of the incident Gaussian beam is modulated by phase only holograms [62]. The
complex electric field of discrete vortex can be expressed as

$U(x,y)=A(x,y)\exp(i\phi(x,y)),$ (14)

where the amplitude $A(x,y)$ and the phase $\phi(x,y)$ take values in the
intervals [0,1] and [$-\pi$,$\pi$]. The aim is to encode the complex field
$U(x,y)$ by means of a phase transmittance function (phase hologram) to
incorporate amplitude variations as phase variations, that is, a function
$h(x,y)$ must be given by

$h(x,y)=\exp\left[i\psi\left(A,\phi\right)\right],$ (15)

where $\psi(A,\phi)$ accounts both amplitude and phase variations. To find the
desired form of phase function $\psi(A,\phi)$, $h(x,y)$ can be expressed as
Fourier series in the domain of $\psi$ as [62]

$h(x,y)=\sum_{p=-\infty}^{\infty}c^{A}_{p}\exp(ip\phi),$ (16)

where

$c^{A}_{p}=\frac{1}{2\pi}\int_{-\pi}^{\pi}\exp[i\psi(A,\phi)]\exp(-ip\phi)d\phi.$
(17)

The field $E(x,y)$ can be recovered from only the first-order term of Eq.
(17), provided that the following identity is fulfilled

$c^{A}_{1}=Ac,$ (18)

where $c$ is a positive constant. Further, the phase function $\psi(A,\phi)$
with odd symmetry can be expressed as

$\psi(A,\phi)=f(A)\sin(\phi).$ (19)

In this case, the phase transmittance function (Eq. 15) becomes
$h(x,y)=\exp[i.f(A)\sin(\phi)]$. Expressing it in the Fourier series using
Jacobi-Anger identity, we get

$\exp[i.f(A)\sin(\phi)]=\sum_{q=-\infty}^{\infty}J_{q}[f(A)]\exp(iq\phi),$
(20)

where $J_{q}$ represents Bessel function of $q$-th order. Using Eqs. (16),
(18) and (20), we get

$c^{A}_{1}=J_{1}[f(A)].$ (21)

Therefore, from Eqs. 18 and (21)

$Ac=J_{1}[f(A)].$ (22)

The function $f(A)$ can be determined by numerical inversion of the above Eq.
(22). The maximum value of c for which Eq. (22) can be fulfilled is $\sim
0.58$, which corresponds to the maximum value of the first-order Bessel
function $J_{1}(x)$, which occurs at $x\approx 1.84$. This restricts the
$f(A)$ in the interval [0,1.84]. The modulated field consists of first and
higher orders, thus to separate the first order, a blazed grating is added to
the phase of hologram. Thus the resultant phase will have the following form

Figure 10: Phase holograms corresponding to (a) TC=0, (b) TC=1, (c) TC=2, and
(d) TC=3. The phase distributions of lasers in 1D ring array corresponding to
these holograms for (e) TC=0, (f) TC=1, (g) TC=2, and (h) TC=3.

$\psi=\psi(A,\phi+2\pi N_{x}x+2\pi N_{y}y),$ (23)

where $N_{x}$ and $N_{y}$ denote the grating frequencies along $x$ and $y$
directions, respectively. As per the requirement of SLM, the phase of hologram
is divided into 256 levels. The spatial frequencies are chosen as $N_{x}=120$
and $N_{y}=60$. The holograms corresponding to TC$=0$ to 3 and system size
N$=10$ are shown in Fig. 10(a)-10(d), respectively. These holograms are used
on the SLM to produce the desired phase distributions of lasers in 1D ring
array with TC$=0$ to 3, as shown in Fig. 10(e)-10(h), respectively.

The similar procedure for generating discrete vortex from large system size
and higher TCs is used.

##  Appendix B Fringe visibility analysis

Here, we have presented the analysis of fringe visibility in the interference
patterns obtained for TC=0 (Fig. 3(b)) and TC=1 (Fig. 3(c)) as well as the
averaged interference pattern of these two. Note, the phase distribution of
discrete vortex in a ring array of N=10 lasers, is given as
$\phi_{j}=\mathrm{TC}.(j-1).(2\pi/10)$, where $j=1,~{}2,~{}3...$ denote the
laser number in the array. The interference patterns are obtained by using
Mach-Zhender interferometer (Fig. 2(a)), where a single selected reference
laser interferes with itself and with all the other lasers in the array. The
results of interference patterns are shown in Figs. 3(d)-3(f) (simulation) and
Figs. 3(g)-3(i) (experiment). The analysis of fringe visibility at all lasers
in the array is shown in Fig. 11 (simulation) and Fig. 12 (experiment).

The laser $j=1$ is a selected reference laser, where light interferes with
itself, and has initial phase difference d$\phi_{1,1}=0$ (Eq. (4)) for both
TC$=0$ and TC$=1$. The interference expressions at laser 1 in TC=0 and TC=1
can be written as following. Using Eq. 4)

$I_{T1}=I_{T2}=2I_{0}\left[1+\cos\left(0.94kx\right)\right],$ (24)

The expression of averaged interference pattern is given as

$I_{Sum}=\frac{I_{T1}+I_{T2}}{2}=2I_{0}\left[1+\cos\left(0.94kx\right)\right].$
(25)

As d$\phi_{1,1}=0$ for both cases of TC=0 and TC=1, so maxima and minima in
the fringe pattern (Eq. (24)) occur at the same locations (red dashed curve
and blue dot-dashed curve in Figs. 11(a) and 12(a)). In both cases, the fringe
visibility is found to be $V=1$. The average of these two interference
patterns gives no change in the fringe pattern (Eq. (25)), and the fringe
visibility remains unchanged $V=1$ (black solid curve in Figs. 11(a) and
12(a)).

Figure 11:  Simulated results. The intensity cross-section in the interference
patterns of TC$=0$ , TC$=1$ and averaged interference pattern TC$=0+$TC$=1$ at
(a) laser 1, (b) laser 2, (c) laser 3, (d) laser 4, (e) laser 5, (f) laser 6,
(g) laser 7, (h) laser 8, (i) laser 9, and (j) laser 10. The blue dot-dashed
curve denotes TC$=0$, red dashed curve denotes TC$=1$, and black solid curve
denotes TC$=0+$TC$=1$. Note, the intensity cross-sections in TC=0 and TC=1 are
normalized to the maximum value of 1. Figure 12: Experimental results with the
details same as given in Fig. 10.

The interference fringes at laser 2 in TC=0 and TC=1 are obtained when the
light from a selected reference laser $j=1$ interferes with the light from
laser $j=2$. In TC=0, the initial phase difference between the laser $j=1$ and
laser $j=2$ is $d\phi_{1,2}=0$, whereas, in TC=1 it is $d\phi_{1,2}=2\pi/10$.
The interference expressions $I_{T1}$ (in TC=0) and $I_{T2}$ (in TC=1) can be
given as

$\displaystyle I_{T1}$ $\displaystyle=$ $\displaystyle
2I_{0}\left[1+\cos\left(0.94kx\right)\right],$ (26) $\displaystyle I_{T2}$
$\displaystyle=$ $\displaystyle
2I_{0}\left[1+\cos\left(0.94kx-1.(2\pi/10)\right)\right].$ (27)

The averaged interference pattern is given as

$I_{Sum}=I_{0}\left[2+\cos\left(0.94kx\right)+\cos\left(0.94kx-1.(2\pi/10)\right)\right].$
(28)

The interference patterns (Eqs. (26) and (27)) consist of fringes with clear
maxima and minima, which give rise to the fringe visibility of $V=1$ (red
dashed curve and blue dot-dashed curve in Figs. 11(b) and 12(b)). As evident,
in these interference patterns the fringes (maxima and minima) are shifted
relative to each other, which occurs due to the different values of initial
phase difference $d\phi_{1,2}$ in TC=0 and TC=1. When these interference
patterns with shifted fringes are averaged, the resultant interference pattern
(Eq. 28) consists of fringes with reduced fringe visibility $V=0.94$ (black
solid curve in Figs. 11(b) and 12(b)).

Similarly the interference fringes at other lasers are obtained by interfering
the light from a selected reference laser $j=1$ and the light from other
lasers $j>2$. In general, in TC$=0$, the phase difference between the selected
reference laser $j=1$ and other lasers is given as d$\phi_{1,j\geq 1}=0$,
whereas, in case of TC$\neq 0$, it is given as d$\phi_{1,j\geq
1}=\mathrm{TC}.(j-1).(2\pi/N)$, where $N$ denotes the number of lasers in a 1D
ring array. Therefore, in case of TC=0 the interference fringes at lasers
$j=3,4,...10$ occur exactly at the same locations (red dashed curve in Figs.
11(c)-11(j) and Figs. 12(c)-12(j)). However, in case of TC=1 the interference
fringes at $j=3,4,...10$ occur at different locations due to different values
of d$\phi_{1,j\geq 3}$ (blue dot-dashed curve in Figs. 11(c)-11(j) and Figs.
12(c)-12(j)). Because of this when the interference patterns at the same
lasers ($j=3,4,...10$) for TC=0 and TC=1 are averaged, the resultant
interference pattern consists of reduced visibility (black solid curve in
Figs. 11(c)-11(j) and Figs. 12(c)-12(j)). For example, at lasers $j=3-6$ the
values of reduced fringe visibility are found to be 0.8 (Figs. 11(c) and
12(c)), 0.58 (Figs. 11(d) and 12(d)), 0.3 (Figs. 11(e) and 12(e)), and 0
(Figs. 11(f) and 12(f)), respectively. For the lasers $j=7-10$, the values of
fringe visibility are found to be 0.3 (Figs. 11(g) and 12(g)), 0.58 (Figs.
11(h) and 12(h)), 0.80 (Figs. 11(i) and 12(i)) and 0.94 (Figs. 11(j) and
12(j)), respectively. Note, a small discrepancy between simulated and
experimental fringe visibility is anticipated by a small intensity difference
between the lasers in the experiment. As evident, the fringe visibility
decreases monotonically for lasers $j=1-6$ and after that it again increases
from $j=7-10$. The increase in the fringe visibility for $j=7-10$ can be
explained as follows. For laser $j=7$, we have
d$\phi_{1,7}=1.6.(\frac{2\pi}{10})$, which can also be written as
$\left[2\pi-(4.(\frac{2\pi}{10}))\right]$ or $-4.(\frac{2\pi}{10})$. This
becomes equivalent to the phase difference between $j=1$ and 5 with the
negative sign i.e, -d$\phi_{1,5}$. Note, the positive and negative values of
the same magnitude d$\phi$ produce the same shift of fringes in the
interference pattern, as shown in Figs. 1(a) and 1(c). The phase difference
$|d\phi_{1,7}|<|d\phi_{1,6}|$, and due to this the fringe visibility in the
averaged interference at laser $j=7$ is found to be higher than laser $j=6$
(and same as for $j=5$). Similarly, the increased values of $V$ at laser
$j=8,~{}9,~{}\mathrm{and}~{}10$ can also be explained.

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