Analytic Expressions for Orbital Angular Momentum Modal Crosstalk in a
Slightly Elliptical Fiber

Ramesh Bhandari

Laboratory for Physical Sciences, 8050 Greenmead Drive, College Park, Maryland
20740, USA

<EMAIL_ADDRESS>
###### Abstract

Assuming weakly guiding approximation, we examine orbital angular momentum
(OAM) mode mixing on account of ellipticity in a fiber and derive a complete
set of analytic expressions for spatial crosstalk, using scalar perturbation
theory that incorporates fully the existing degeneracy between a spatial OAM
mode and its degenerate partner characterized by a topological charge of
opposite sign. These expressions, consequently, include an explicit formula
for calculating the $2\pi$ walk-off length over which an input OAM mode
converts into its degenerate partner, and back into itself. We further explore
the applicability of the derived expressions in the presence of spin-orbit
interaction. The expressions constitute a useful mathematical tool in the
analysis and design of fibers for spatially-multiplexed mode transmissions.
Their utility is demonstrated with application to a few mode and a multimode
step-index fiber.

Index Terms: Orbital angular momentum, fiber ellipticity, fiber optics,
waveguides, mode mixing, crosstalk, perturbation theory, analytic expressions,
spin orbit interaction

##  1 Introduction

Currently there exists a vast interest in exploring the possibility of
increasing traffic flow multifold on a fiber simply by exploiting the
orthogonality of the orbital angular momentum (OAM) modes [1]. However, any
such effort must take into account fiber imperfections, like ellipticity,
which can cause mode-mixing (crosstalk) and reduce the quality of the signal.
An analytic treatment of the effect of ellipticity on OAM modes was earlier
provided [2], but it was limited to a low topological charge $l$, primarily
$l=1$ or $2$ ($l$ is the OAM carried by a single photon in units of the
reduced Planck’s constant ($h/(2\pi)$); besides, the methodology as presented
could not incorporate mode mixing with the neighboring OAM modes differing in
topological charges by a unit or more. More recently, several authors have
carried out numerical studies of the impact of ellipticity on OAM modes in a
variety of fibers: ring fiber [3, 5, 7], multi-ring fiber with trench [4], and
a graded-index fiber [6] but the studies are based on computationally-
intensive finite-element approximation methods, where it is not always easy to
ascertain the inherent behavior of the impact of a perturbation like
ellipticity in any detailed manner due to the associated high demands of
accuracy.

In this work, we assume the weakly-guiding approximation, which permits us to
write the vector modes as products of the spatial OAM modes and the associated
polarization [9, 11]. Further, confining our attention here to the effect of
ellipticity on the spatial mode component only, we employ the scalar wave
equation and derive a complete set of analytic expressions for the spatial OAM
modal crosstalk within the framework of perturbation theory. The obtained
expressions include explicit formulas for the $2\pi$ walk-off length, and
provide deeper insight into mode-mixing and thus the crosstalk behavior as a
function of ellipticity and the topological charge carried by the OAM mode.
The $2\pi$ walk-off length of a given input OAM mode refers to the distance
over which the mode transforms itself into its degenerate partner
characterized by a topological charge of opposite sign, and back into itself,
under the continuous impact of the perturbation (ellipticity here). We further
investigate the effect of spin-orbit interaction and provide approximate
bounds on the applicability of the the derived expressions. The developed
theory is illustrated with application to step-index few mode and multimode
fibers.

##  2 Scalar Wave Equation for the Slightly Elliptical Fiber

We first write the scalar wave equation for the propagation of an OAM mode
through an unperturbed fiber as an eigenvalue equation

$HO_{l,m}(r,\theta)=\beta_{l,m}^{2}O_{l,m}(r,\theta),$ (1)

where the Hermitian operator $H=\vec{\nabla}_{t}^{2}+k^{2}n^{2}(r)$;
$k=2\pi/\lambda$, where $\lambda$ is the wavelength; $\vec{\nabla}_{t}^{2}$ is
the transverse Laplacian; $n^{2}(r)=n_{1}^{2}(1-2\Delta f(r))$; $f(r)$ is the
index profile as a function of the radial cylindrical coordinate $r$ only,
$\Delta=(n_{1}^{2}-n_{2}^{2})/(2n_{1}^{2})$ is the index profile height
parameter, with $n_{1}>n_{2}$ [8]; parameters $l$ and $m$ are respectively the
topological charge (azimuthal mode number) and the radial mode number of the
unperturbed OAM mode (denoted $OAM_{l,m}$) with an amplitude

$O_{l,m}(r,\theta)=\frac{1}{\sqrt{N_{l,m}}}F_{l,m}(r)e^{il\theta},$ (2)

where $N_{l,m}$ is the normalization constant; the eigenvalue
$\beta_{l,m}^{2}$ is the square of the propagation constant of the $OAM_{l,m}$
mode, which gives the mode its $z$ dependence:

$\psi_{l,m}(r,\theta,z)=O_{l,m}e^{i\beta_{l,m}z}$ (3)

($z$ axis coincides with the fiber’s axis). The amplitude, $O_{l,m}$,
originates in the reduction of
$HE_{l+1,m},HE_{-l-1,m},EH_{l-1,m},EH_{-l+1,m}(|l|>1)$ to
$O_{l,m}\epsilon_{+},O_{-l,m}\epsilon_{-},O_{l,m}\epsilon_{-},O_{-l,m}\epsilon_{+}$,
respectively in the weakly guiding approximation ($\Delta<<1$) [9, 10];
$\epsilon_{\pm}=(1/\sqrt{2})(\hat{x}\pm i\hat{y}$) are the left (+) and right
(-) circular polarizations corresponding to a photon spin $+1$ and $-1$,
respectively (in units of $h/(2\pi)$); for the $l=1$ case,
$HE_{2,m},HE_{-2,m},\frac{1}{\sqrt{2}}(TM_{0,m}+\color[rgb]{0,0,0}i\color[rgb]{0,0,0}TE_{0,m}),\frac{1}{\sqrt{2}}(TM_{0,m}-iTE_{0,m})$
reduce to the products:
$O_{1,m}\epsilon_{+},O_{-1,m}\epsilon_{-},O_{1,m}\epsilon_{-},O_{-1,m}\epsilon_{+},$
respectively[9, 11].

Fig. 1: a) Cross section of the multimode fiber with core radius $a$ and
cladding radius $b(>>a);r$ and $\theta$ along with the $z$ coordinate ($z$
axis coincident with the fiber axis) constitute the polar coordinates; the
mode is assumed propagating in the $+z$ direction (out of the plane of the
paper). b) A slightly elliptical core shown relatively enlarged for clarity;
semimajor axis $\rho_{1}=a(1+\epsilon)$ and semiminor axis
$\rho_{2}=a(1-\epsilon)$ where ellipticity $\epsilon=e^{2}/4<<1$; $e$ is the
eccentricity of the ellipse.

We model the slightly elliptical fiber (see Fig. 1) as a perfectly round fiber
with a modified refractive index given by

$n^{\prime 2}(r,\theta)=n^{2}(r)-2\epsilon\Delta
n_{1}^{2}(\partial{f(r)}/\partial{r})r\cos(2\theta);$ (4)

ellipticity $\epsilon(<<1)$ is defined as the ratio of the difference of the
semimajor axis and the semiminor axis to their sum, and equals $e^{2}/4$,
where $e$ is the eccentricity of the ellipse [8, 2]. Replacing $n$ with
$n^{\prime}$ in Eq. 1, the perturbed scalar wave equation is

$(H+\epsilon\delta H)O_{l,m}^{\prime}(r,\theta)=\beta_{l,m}^{\prime
2}O_{l,m}^{\prime}(r,\theta),$ (5)

where $O^{\prime}_{l,m}(r,\theta)$ and $\beta_{l,m}^{\prime}$ are the
corresponding perturbed mode amplitude and propagation constant, respectively,
and

$\delta H=-2k^{2}n_{1}^{2}\Delta\cos 2\theta(\partial{f(r)}/\partial{r})r.$
(6)

The scalar wave solution incorporating the z dependence is written as
$O^{\prime}_{l,m}(r,\theta)e^{i\beta^{\prime}_{l,m}z}$. The $O^{\prime}_{l,m}$
amplitudes, like the $O_{l,m}$’s, form a complete set.

The effect of the perturbation $\epsilon\delta H$ is to cause mixing of an
input OAM mode, $OAM_{l,m}$ with other (orthogonal) fiber OAM modes
characterized by different pairs of parameters,
$(l^{\prime},m^{\prime})\neq(l,m)$, as described below.

##  3 Perturbation Solution

Following [12, 9], we expand the perturbed amplitude $O_{l,m}^{\prime}$ and
the perturbed propagation constant $\beta_{l,m}^{\prime}$ as

$O_{l,m}^{\prime}=O_{l,m}+\sum^{\prime}_{n,k}a_{(l,m)(n,k)}^{(1)}O_{n,k}+\sum^{\prime}_{n,k}a_{(l,m)(n,k)}^{(2)}O_{n,k}+....,$
(7) $\beta_{l,m}^{\prime
2}=\beta_{l,m}^{2}+\beta_{l,m}^{2(1)}+\beta_{l,m}^{2(2)}+....,$ (8)

where the contributions in different orders of the perturbation parameter
$\epsilon(<<1)$ are indicated by the superscripts in parentheses, and the
prime on the summation implies $O_{n,k}\neq O_{l,m}$; all amplitudes are
assumed normalized.

After inserting the above series in Eq. 5, taking the necessary scalar
products to solve in different perturbation orders [13, 14, 15], we obtain the
analytic expressions for the mixing coefficients and the propagation constant
corrections:

$a_{(l,m)(n,k)}^{(1)}=\frac{\epsilon\delta
H_{(n,k)(l,m)}}{(\beta_{l,m}^{2}-\beta_{n,k}^{2})},$ (9)
$a_{(l,m)(n,k)}^{(2)}=\epsilon^{2}\bigg{(}\sum_{r,s}\frac{\delta
H_{(n,k)(r,s)}\delta
H_{(r,s)(l,m)}}{(\beta_{l,m}^{2}-\beta_{n,k}^{2})(\beta_{l,m}^{2}-\beta_{r,s}^{2})}-\frac{\delta
H_{(n,k)(l,m)}\delta
H_{(l,m)(l,m)}}{(\beta_{l,m}^{2}-\beta_{n,k}^{2})^{2}}\bigg{)},$ (10)
$\beta^{2(1)}_{l,m}=\epsilon\delta H_{(l,m)(l,m)},$ (11)
$\beta^{2(2)}_{l,m}=\epsilon^{2}\sum_{n,k}\frac{\delta H_{(l,m)(n,k)}\delta
H_{(n,k)(l,m)}}{(\beta_{l,m}^{2}-\beta_{n,k}^{2})}.$ (12)

Eqs. 9-12 are standard equations up to second order in the perturbation
parameter $\epsilon$ (higher order contributions, although more complicated in
form, can similarly be determined, if needed). The matrix element

$\begin{split}\delta H_{(l,m)(n,k)}=&<O_{l,m}|\delta
H|O_{n,k}>=\int_{0}^{\infty}\int_{0}^{2\pi}O_{l,m}^{*}(\delta
H)O_{n,k}rdrd\theta\\\
&=-2k^{2}n_{1}^{2}\Delta\int_{0}^{\infty}\int_{0}^{2\pi}O_{l,m}^{*}((\partial{f(r)}/\partial{r})\cos(2\theta)O_{n,k}r^{2}drd\theta;\end{split}$
(13)

the bra ($<$), ket ($>$) notation signifies a scalar product. This matrix
element times $\epsilon$ represents the ellipticity-induced interaction (or
coupling) between the fields of the $OAM_{l,m}$ and $OAM_{n,k}$ modes.
Inserting Eq. 2 and integrating over the azimuthal part, we immediately see
that the $\delta H$ matrix elements are symmetric $(\delta
H_{(l,m)(n,k)}=\delta H_{(n,k)(l,m)}$), and nonzero only when $|n-l|=2$, a
selection rule also pointed out in [7]. Invoking this selection rule, we
obtain simplified forms

$a_{(l,m)(l\pm 2,n)}^{(1)}=\frac{\epsilon\delta H_{(l\pm
2,n)(l,m)}}{(\beta_{l,m}^{2}-\beta_{l\pm 2,n}^{2})},$ (14) $a_{(l,m)(l\pm
4,n)}^{(2)}=\epsilon^{2}\sum_{k}\frac{\delta H_{(l\pm 4,n)(l\pm 2,k)}\delta
H_{(l\pm 2,k)(l,m)}}{(\beta_{l,m}^{2}-\beta_{l\pm
2,k}^{2})(\beta_{l,m}^{2}-\beta_{l\pm 4,n}^{2})},$ (15)

and so on. The propagation constant corrections in Eqs. 11 and 12 become

$\beta^{2(1)}_{l,m}=0,$ (16)
$\begin{split}\beta^{2(2)}_{l,m}=&\epsilon^{2}\bigg{(}\sum_{k}\frac{\delta
H_{(l,m)(l+2,k)}\delta
H_{(l+2,k)(l,m)}}{(\beta_{l,m}^{2}-\beta_{l+2,k}^{2})}\\\
&+\sum_{k^{\prime}}\frac{\delta H_{(l,m)(l-2,k^{\prime})}\delta
H_{(l-2,k^{\prime})(l,m)}}{(\beta_{l,m}^{2}-\beta_{l-2,k^{\prime}}^{2})}\bigg{)}.\end{split}$
(17)

We note that the mixing coefficient in first order perturbation, Eq. 14, not
only depends upon $\epsilon\delta H_{(l\pm 2,n)(l,m)}$, the ellipticity-
induced interaction between $OAM_{l,m}$ and $OAM_{l\pm 2,n}$ modes, but also
on what we term as a _propagator_ , which is the inverse of the difference
between the squares of the propagation constant of the $OAM_{l,m}$ mode and
the coupled $OAM_{l\pm 2,\color[rgb]{0,0,0}n}$ mode; larger this difference,
smaller this coefficient.

Fig. 2: a) An input $OAM_{1,1}$ mode ($l=1,m=1$) couples to the fields of the
neighboring $OAM_{3,1}$ and $OAM_{3,2}$ modes in first order perturbation;
each arrow represents a coupling mediated by the ellipticity-induced
interaction, $\epsilon\delta H$, times a propagator, which is the inverse of
the difference of the propagation constant-squared of the input mode and the
connected mode, $OAM_{3,1}$ or $OAM_{3,2}$; the propagation constant increases
from bottom to up, so $\beta_{1,1}>\beta_{3,1}>\beta_{3,2}$ b) Illustration of
the second order perturbation effect, where input $OAM_{1,1}$ is connected to
$OAM_{5,1}$ through the intermediate $OAM_{3,1}$ mode as well as the
intermediate $OAM_{3,2}$ mode, each sequence of connections making a separate
contribution in Eq. 15 through the summation index $k$. Because two steps
(transitions) are involved, the mixing process is of order $\epsilon^{2}$;
note that, in general, there can be more than two intermediate states (with
the same $l=3$ value), depending upon the fiber; see text.

The second order coupling of the input mode $OAM_{l,m}$ to $OAM_{l+4,n}$ mode
takes place in two steps by first coupling to $OAM_{l+2,k}$, which in turn
couples to $OAM_{l+4,n}$; it is therefore of order $\epsilon^{2}$ (all these
couplings follow the $|\Delta l|=2$ rule discussed above). The expression, Eq.
15, has two matrix elements and two propagators within the summation. In
general, coefficients, $a_{(l,m)(l\pm 2,n)}^{(i)},i\geq 1$, will have
$i~{}\delta H$ matrix elements connected by the $\Delta l=\pm 2$ rule, and
accompanied by an equal number of propagators, along with an appropriate sum
over all the intermediate states, as exemplified in Eq. 15. These coefficients
are dimensionless because $\epsilon\delta H$ (Eq. 6) has the units of the
square of the propagation constant, which cancel the units of the propagators,
equal in number. Fig. 2 is a pictorial representation of the results in Eqs.
14 and 15 for an $l=1,m=1$ mode. The first order correction to the propagation
constant, Eq. 11, becomes zero due to the selection rule (see Eq. 16), while
the second order correction, Eq. 12, is now given by Eq. 17; the number of of
propagators is one less than the number of matrix elements in the numerator,
consistent with the dimensionality of the square of a propagation constant.

It is also important to note here that the selection rule $|\Delta l|=2$ also
allows higher order transitions, which have been ignored in the above results.
As an example, in first order perturbation, an input $OAM_{4,m}$ mode will
couple to both $OAM_{6,n}$ and $OAM_{2,n}$ modes ($n\geq 1)$ in a single step
(Eq. 14). But perturbation theory (governed by the selection rule) also allows
the following types of couplings: the input $OAM_{4,m}$ mode couples first to
to an $l=6$ mode ($OAM_{6,k},k\geq 1$), which then couples to
$l=4~{}(OAM_{4,k^{\prime}},k^{\prime}\geq 1)$, which in turn couples
(transitions) back to the $OAM_{2,n}$ mode. There are three steps involved,
each step contributing a factor of the perturbation parameter, $\epsilon$;
this mode of coupling is therefore weaker than the single step process (Eq.
14) by $\epsilon^{2}$ (recall $\epsilon<<1$). Such _weaker_ terms are ignored
in arriving at Eq. 14. Weaker terms are also generated in the case of coupling
of $OAM_{l,m}$ to $OAM_{l\pm 4}$ in Eq. 15, but they are similarly ignored.

In what follows, OAM modes with negative topological charge will be indicated
by placing an explicit minus sign. Now, $\beta_{-l,m}=\beta_{l,m}$, a
consequence of degeneracy between $OAM_{-l,m}$ and $OAM_{l,m}$ modes; this
follows from the $l^{2}$ dependence of the wave equation [8]. Furthermore,
$\delta H_{(-l,m)(-l^{\prime},n^{\prime})}=\delta
H_{(l,m)(l^{\prime},n^{\prime})}$, which is seen from Eq. 13. Thus, when $l$
is replaced by $-l$ in Eqs. 14-17, identical right-hand expressions result. In
other words, the perturbation series

$O_{-l,m}^{\prime}=O_{-l,m}+\sum^{\prime}_{n,k}a_{(-l,m)(n,k)}^{(1)}O_{n,k}+\sum^{\prime}_{n,k}a_{(-l,m)(n,k)}^{(2)}O_{n,k}+....,$
(18)

transforms to

$O_{-l,m}^{\prime}=O_{-l,m}+\sum^{\prime}_{n,k}a_{(l,m)(n,k)}^{(1)}O_{-n,k}+\sum^{\prime}_{n,k}a_{(l,m)(n,k)}^{(2)}O_{-n,k}+....,$
(19)

where $a_{(l,m)(n,k)}^{(1)}$ and $a_{(l,m)(n,k)}^{(2)}$ are given by Eqs. 14
and 15, as in the $+l$ case.

###  3.1 Breaking of Degeneracy between $OAM_{l,m}$ and $OAM_{-l,m}$ Modes

The modes $OAM_{l,m}$ and $OAM_{-l,m}$ are degenerate in the subspace spanned
by these two (orthogonal) modes. However, because of the symmetry of the wave
equation with respect to the parameters $l$ and $-l$, the diagonal elements of
the $2~{}x~{}2$ matrix in the $(l,-l)$ subspace will be equal to each other,
and similarly, the off-diagonal elements. Further, invoking the $|\Delta l|=2$
selection rule, we immediately see that the off diagonal element that connects
the two degenerate states will become nonzero in perturbation order $l$ only;
this is due to the fact that the selection rule requires any transition
between them to occur in $l$ steps: $l\rightarrow l-2\rightarrow
l-4\rightarrow....\rightarrow-l+4\rightarrow-l+2\rightarrow-l$; each step
provides a multiplying factor of $\epsilon$, resulting in a $\epsilon^{l}$
dependence of the off-diagonal element (see Fig. 3 as an illustration).

Fig. 3: Illustration of the transition from $l=2,m=1$ state to $l=-2,m=1$
state, which occurs in two steps due to the $|\Delta l|=2$ rule; each sequence
of two transitions contributes separately in the summation of Eq. 22 (see
text).

In effect, we have a symmetric matrix, which needs to be diagonalized. A
formal procedure in Appendix A yields the eigenamplitudes:

$O^{\pm}_{l,m}=1/\sqrt{2}(O_{l,m}\pm O_{-l,m}),$ (20)

with corresponding eigenvalues

$(\beta^{\prime\pm}_{l,m})^{2}=\beta^{2}_{l,m}+\beta_{l,m}^{2(2)}\pm\gamma_{l,m}.$
(21)

$\beta_{l,m}^{2(2)}$ is given by Eq. 17 and

$\begin{split}\gamma_{l,m}=&(\epsilon)^{l}\sum\frac{\delta
H_{(l,m)(l-2,n)}\delta
H_{(l-2,n)(l-4,p)}....}{(\beta_{l,m}^{2}-\beta_{l-2,n}^{2})(\beta_{l,m}^{2}-\beta_{l-4,p}^{2})....}\\\
&\frac{....\delta H_{(-l+4,q)(-l+2,j)}\delta
H_{(-l+2,j)(-l,m)}}{....(\beta_{l,m}^{2}-\beta_{-l+4,q}^{2})(\beta_{l,m}^{2}-\beta_{-l+2,j}^{2})}~{}(l>1);\end{split}$
(22) $\gamma_{l,m}=\epsilon\delta H_{(l,m)(-l,m)}~{}(l=1).$ (23)

Each $\delta H$ matrix element in the numerator of Eqs. 22 and 23 is a
$|\Delta l|=2$ transition, there being $l$ in all; the number of propagators
is $l-1$. The sum runs over all the radial mode solutions of the intermediate
OAM states (repeated indices in Eq. 22). Illustrating further, for the
$l=3,m=1$ case, $\gamma_{3,1}$ in Eq. 22 will have products of matrix elements
$\delta H_{(3,1)(1,n)},\delta H_{(1,n)(-1,p)},\delta H_{(-1,p)(-3,1)}$ and
propagators
$(\beta_{3,1}^{2}-\beta_{1,n}^{2})^{-1},(\beta_{3,1}^{2}-\beta_{-1,p}^{2})^{-1}$,
along with the $\epsilon^{3}$ multiplying factor (repeated indices imply
summation); this is a three-step transition. Note that
$\beta_{-1,p}^{2}=\beta_{1,p}^{2}$ and $\delta H_{(-1,p)(-3,1)}=\delta
H_{(1,p)(3,1)}=\delta H_{(3,1)(1,p)}$ due to the symmetries discussed earlier.

_Physical Interpretation_ : The linear combinations given in Eq. 20 are the
field amplitudes of the corresponding _Linearly Polarized (LP)_ modes, the $+$
sign corresponding to the $LP^{(a)}_{l,m}$ spatial mode and the $-$ sign to
the $LP^{(b)}_{l,m}$ spatial mode, with intensities proportional to
$\cos^{2}(l\theta)$ and $\sin^{2}(l\theta)$, respectively; they are, however,
now slightly nondegenerate (see Eq. 21). If we further invert Eq. 20, we
obtain

$O_{\pm l,m}=1/\sqrt{2}(O^{+}_{l,m}\pm O^{-}_{l,m}).$ (24)

Consider now an $OAM_{l,m}$ mode,
$\psi_{l,m}\color[rgb]{0,0,0}(z)\color[rgb]{0,0,0}=O_{l,m}e^{i\beta_{l,m}z}$
(see Eq. 3), incident on the slightly elliptical fiber at $z=0$. In view of
the result, Eq. 24, this mode will travel down the fiber as an equal
superposition of two slightly nondegenerate $LP$ fields with propagation
constants, $\beta^{{}^{\prime}+}_{l,m}$ and $\beta^{{}^{\prime}-}_{l,m}$, as
given by Eq. 21. After traveling a distance $L$ within the fiber, its
composition will change to

$\psi^{(e)}_{l,m}(L)=\frac{1}{\sqrt{2}}\Big{(}O_{l,m}^{+}e^{i\beta^{\prime+}_{l,m}L}+O^{-}_{l,m}e^{i\beta^{\prime-}_{l,m}L}\Big{)},$
(25)

where the superscript $e$ refers to the slight ellipticity of the fiber (we
have ignored here the interaction with other modes, which will be done in the
next section). We now substitute Eqs. 20 and 21 into the above equation and
use the fact that
$\beta^{{}^{\prime}+}_{l,m}+\beta^{{}^{\prime}-}_{l,m}\approx 2\beta_{l,m}$,
since $\beta^{(2)}_{l,m}<<\beta_{l,m}$. This yields after some algebra

$\psi_{l,m}^{(e)}(L)=\Big{(}\cos(\pi L/L^{(2\pi)}_{l,m})O_{l,m}+i\sin(\pi
L/L^{(2\pi)}_{l,m})O_{-l,m}\Big{)}e^{i\beta_{l,m}L},$ (26)

where

$L^{(2\pi)}_{l,m}=\frac{2\pi}{\color[rgb]{0,0,0}\Delta\beta^{\prime}_{l,m}}\color[rgb]{0,0,0}\approx\frac{2\pi\beta_{l,m}}{\gamma_{l,m}};$
(27)

$\color[rgb]{0,0,0}\Delta\beta^{\prime}_{l,m}\color[rgb]{0,0,0}=\beta^{\prime+}_{l,m}-\beta^{\prime-}_{l,m}\approx((\beta^{{}^{\prime}+}_{l,m})^{2}-(\beta^{{}^{\prime}-}_{l,m})^{2}))/(2\beta_{l,m})=\gamma_{l,m}/\beta_{l,m}$
(see Eqs. 21). The entering $OAM_{l,m}$ mode oscillates into and out of its
degenerate partner under the impact of ellipticity; $L^{(2\pi)}_{l,m}$ is the
$2\pi$ walk-off length, which is the distance over which a given mode
transforms into its degenerate partner, and back into itself. From Eq. 26, we
see that at $L=L^{(2\pi)}_{l,m}/4$, the input $OAM_{l,m}$ has partly
transformed into its degenerate partner, $OAM_{-l,m}$, which now has the same
amplitude as the parent mode, $OAM_{l,m}$. At $L=L^{(2\pi)}_{l,m}/2$, the
input mode has completely transformed into its degenerate partner (see Fig.
4). This cycle of interconversion of the two degenerate modes continues as $L$
increases further.

From Eqs. 27, 22, and 23, we also find that for given topological charge $l$,
$L^{(2\pi)}_{l,m}\propto\epsilon^{-l}$. Thus, larger the ellipticity
$\epsilon$, smaller the $2\pi$ walk-off length. _Physically, this implies that
as the deviation from fiber circularity (ellipticity $\epsilon$) increases, an
ellipticity-induced torque acting on the given $OAM_{l,m}$ mode increases,
thus shortening the distance over which the $OAM_{l,m}$ mode converts into the
$OAM_{-l,m}$ mode as depicted in Fig. 4; this conversion, which occurs over
half the $2\pi$ walk-off length, involves an OAM transfer of $2l$ in magnitude
(such a transfer of OAM also has been discussed in connection with fiber bends
[9]). Similarly, for fixed $\epsilon$, as the required OAM transfer of $2l$
increases with $l$, we expect the transition to $OAM_{-l,m}$ mode to take
place over a longer fiber length._ In other words, the $2\pi$ walk-off length
$L^{(2\pi)}_{l,m}$ increases with $l$ for fixed $\epsilon$. These features of
the $2\pi$ walk-off length are further illustrated numerically in Section 6
with reference to a step-index fiber.

Fig. 4: Amplitude-squared, equal to $\cos^{2}(\pi L/L^{(2\pi)}_{l,m})$ and
$\sin^{2}(\pi L/L^{(2\pi)}_{l,m})$ for the $OAM_{l,m}$ and $OAM_{-l,m}$ modes,
respectively, as a function of $L$ (see Eq. 26); at $L=L^{(2\pi)}_{l,m}/2$,
the input $OAM_{l,m}$ mode has transformed completely into its degenerate
partner, the $OAM_{-l,m}$ mode.

###  3.2 New Perturbation Series

Perturbation fixes the appropriate linear combinations of the degenerate pairs
of OAM modes in accordance with Eq. 20 (see Appendix A). Therefore, we must
work with these linear combinations, and replace the perturbation series in
Eqs. 7 and 19 with the following series:

$O^{\prime\pm}_{l,m}=O^{\pm}_{l,m}+\sum_{i=1,k}a^{(i)}_{(l,m)(n,k)}O^{\pm}_{n,k}$
(28)

derived in Appendix A; $n=l\pm 2i$, i.e., in the above sum, $n$ takes on both
the values, $l+2i$ and $l-2i$, as in Eqs. 14 and 15 due to the $|\Delta l|=2$
rule. The mode amplitudes $O^{\pm}_{n,k}=1/\sqrt{2}(O_{n,k}\pm O_{-n,k})$ are
linear combinations akin to $O^{\pm}_{l,m}$ (Eq. 20). Like the latter, they
represent the $LP$ fields corresponding to the $OAM_{n,k}$ mode; they are also
slightly nondegenerate due to the slight fiber ellipticity, with
$\color[rgb]{0,0,0}(\color[rgb]{0,0,0}\beta^{{}^{\prime}+}_{n,k}\color[rgb]{0,0,0})^{2}\color[rgb]{0,0,0}-\color[rgb]{0,0,0}(\color[rgb]{0,0,0}\beta^{{}^{\prime}-}_{n,k}\color[rgb]{0,0,0})^{2}\color[rgb]{0,0,0}=2\gamma_{n,k}$
(see Eqs. 22 and 23). Because of the linearity of the wave equation,
$O^{\prime\pm}_{l,m}$ also satisfy the wave equation, Eq. 5, and form a
complete set. The $O^{{}^{\prime}+}_{l,m}$ amplitudes expand in terms of the
$O^{+}_{l^{\prime},m^{\prime}}$ amplitudes, and similarly the
$O^{{}^{\prime}-}_{l,m}$ amplitudes in terms of their corresponding
counterparts, $O^{-}_{l^{\prime},m^{\prime}}$ (Eq. 28). Conversely, the
$O^{+}_{l,m}$ amplitudes are expandable in terms of
$O^{{}^{\prime}+}_{l^{\prime},m^{\prime}}$ amplitudes and similarly
$O^{-}_{l,m}$ in terms of $O^{{}^{\prime}-}_{l^{\prime},m^{\prime}}$.
Hereafter, the eigenmodes pertaining to amplitudes, $O^{\pm}_{l,m}$ will
correspondingly be denoted by $OAM^{\pm}_{l,m}$.

In the next section, we use the above series to determine the complete
expression that includes the mixing of the $OAM_{l^{\prime},m^{\prime}}$
modes, $l^{\prime}\neq-l$, not considered in Section 3.1.

##  4 OAM Mode-Mixing and Crosstalk

For purposes of derivation, consider an input $OAM^{+}_{l,m}$ mode with
amplitude $O^{+}_{l,m}$ entering a straight, slightly elliptical fiber. Let
$\psi{}_{l,m}^{+(e)}\color[rgb]{0,0,0}(r,\theta,z)|_{z=L}$ denote the
amplitude within the perturbed fiber at a distance $L$ from its entry point
($z=0$). Suppressing the arguments $r,\theta$ (for brevity reasons), we can
then write

$\color[rgb]{0,0,0}\psi{}_{l,m}^{+(e)}(L)\color[rgb]{0,0,0}=\sum_{l^{\prime},m^{\prime}}{\eta^{+}_{(l,m)(l^{\prime},m^{\prime})}O^{{}^{\prime}+}_{l^{\prime},m^{\prime}}}e^{i\beta^{{}^{\prime}+}_{l^{\prime},m^{\prime}}L},$
(29)

since the perturbed solutions $O^{{}^{\prime}+}_{l^{\prime},m^{\prime}}$ also
form a complete set (see Section 3.2).
$\eta^{+}_{(l,m)(l^{\prime},m^{\prime})}$ are the mixing coefficients to be
determined. Using the boundary condition,
$\psi{}_{l,m}^{+(e)}(0)\color[rgb]{0,0,0}=O^{+}_{l,m}$, and assuming
orthonormality of $O^{{}^{\prime}+}_{l,m}$’s, we immediately obtain
$\eta^{+}_{(l,m)(l^{\prime},m^{\prime})}=\int^{\infty}_{0}\int^{2\pi}_{0}O^{*^{\prime}+}_{l^{\prime},m^{\prime}}O^{+}_{l,m}rdrd\theta$.
Inserting Eq. 28, we get

$\eta^{+}_{(l,m)(l^{\prime},m^{\prime})}=\delta_{ll^{\prime}}\delta_{mm^{\prime}}+\sum_{i=1}a^{(i)}_{(l^{\prime},m^{\prime})(l,m)}\delta_{l^{\prime},l\pm
2i}.$ (30)

In the second term of Eq. 30, $l^{\prime}$ is related to $l$ through the
relationship: $l^{\prime}=l\pm 2i$. Substituting Eqs. 30 and 28 into Eq. 29
yields

$\begin{split}\psi^{+(e)}_{l,m}(L)=O^{+}_{l,m}e^{i\beta^{{}^{\prime}+}_{l,m}L}&+\sum_{i=1,m^{\prime}}a^{(i)}_{(l,m)(l^{\prime},m^{\prime})}O^{+}_{l^{\prime},m^{\prime}}e^{i\beta^{{}^{\prime}+}_{l,m}L}+\sum_{i=1,m^{\prime}}a^{(i)}_{(l^{\prime},m^{\prime})(l,m)}O^{+}_{l^{\prime},m^{\prime}}e^{i\beta^{{}^{\prime}+}_{l^{\prime},m^{\prime}}L}\\\
&+\sum_{i=1,m^{\prime}}\sum_{i^{\prime}=1,m^{\prime\prime}}a^{(i)}_{(l^{\prime},m^{\prime})(l,m)}a^{(i^{\prime})}_{(l^{\prime},m^{\prime})(l^{\prime\prime},m^{\prime\prime})}O_{l^{\prime\prime},m^{\prime\prime}}e^{i\beta^{{}^{\prime}+}_{l^{\prime},m^{\prime}}L},\end{split}$
(31)

where $l^{\prime}=l\pm 2i$ and $l^{\prime\prime}=l^{\prime}\pm 2i^{\prime}$.
An identical expression results for an input $OAM^{-}_{l,m}$ mode, except that
the $-$ superscript replaces the $+$ superscript everywhere. For an incident
$OAM_{l,m}$ mode in which we are interested, we combine the two results:

$\color[rgb]{0,0,0}\psi{}_{l,m}^{(e)}(L)\color[rgb]{0,0,0}=\frac{1}{\sqrt{2}}\Big{(}\color[rgb]{0,0,0}\psi{}_{l,m}^{+(e)}(L)+\color[rgb]{0,0,0}\psi{}_{l,m}^{-(e)}(L)\Big{)},$
(32)

since $O_{l,m}=\frac{1}{\sqrt{2}}(O^{+}_{l,m}+O^{-}_{l,m}$) (see Eq. 24).

###  4.1 Solution up to First Order

Considering $i=1$ only in Eq. 31 and discarding the fourth (quadratic) term on
the RHS, we obtain

$\color[rgb]{0,0,0}\psi{}_{l,m}^{+(e)}(L)\color[rgb]{0,0,0}=O^{+}_{l,m}e^{i\beta^{{}^{\prime}+}_{l,m}L}+\sum_{l^{\prime}=l\pm
2,m^{\prime}}a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}O^{+}_{l^{\prime},m^{\prime}}e^{i\beta^{{}^{\prime}+}_{l,m}L}-\sum_{l^{\prime}=l\pm
2,m^{\prime}}a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}O^{+}_{l^{\prime},m^{\prime}}e^{i\beta^{{}^{\prime}+}_{l^{\prime},m^{\prime}}L},$
(33)

where we have used the fact that
$a^{(1)}_{(l^{\prime},m^{\prime})(l,m)}=-a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}$
(see Eq. 14); each summation includes only two terms: $l^{\prime}=l\pm 2$. By
replacing the + sign of the superscript everywhere in the above equation with
the - sign, we obtain the expression for the incident mode $OAM^{-}_{l,m}$:

$\psi{}_{l,m}^{-(e)}(L)=O^{-}_{l,m}e^{i\beta^{{}^{\prime}-}_{l,m}L}+\sum_{l^{\prime}=l\pm
2,m^{\prime}}a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}O^{-}_{l^{\prime},m^{\prime}}e^{i\beta^{{}^{\prime}-}_{l,m}L}-\sum_{l^{\prime}=l\pm
2,m^{\prime}}a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}O^{-}_{l^{\prime},m^{\prime}}e^{i\beta^{{}^{\prime}-}_{l^{\prime},m^{\prime}}L}.$
(34)

Combining the two expressions in accordance with Eq. 32, we obtain, up to
first order in $\epsilon$,

$\begin{split}&\color[rgb]{0,0,0}\psi{}_{l,m}^{(e)}(L)\color[rgb]{0,0,0}=\Big{(}\cos(\pi
L/L^{(2\pi)}_{l,m})O_{l,m}+i\sin(\pi
L/L^{(2\pi)}_{l,m})O_{-l,m}\Big{)}e^{i\beta_{l,m}L}\\\
&+2i\sum_{l^{\prime}=l\pm
2,m^{\prime}}a_{(l,m)(l^{\prime},m^{\prime})}^{(1)}\Big{(}\cos(\pi
L/L^{(2\pi)}_{l^{\prime},m^{\prime}})O_{l^{\prime},m^{\prime}}\\\ &+i\sin(\pi
L/L^{(2\pi)}_{l^{\prime},m^{\prime}})O_{-l^{\prime},m^{\prime}}\Big{)}(\sin(\beta_{l,m}-\beta_{l^{\prime},m^{\prime}})L/2)e^{i(\beta_{l,m}+\beta_{l^{\prime},m^{\prime}})L/2}.\end{split}$
(35)

_Physical Interpretation_ : The first two terms correspond to the mixing of
the input $OAM_{l,m}$ mode with its degenerate partner, $O_{-l,m}$ mode, which
was discussed in detail in Section 3.1. Mixing with the other modes
$(l^{\prime}=l\pm 2)$ is captured in the second part of the right-hand-side of
Eq. 35, where the mixed modes also occur in degenerate pairs and the OAM modes
of a degenerate pair oscillate into each other. However, this mixing overall
is suppressed by the first order perturbation mixing coefficient. Physically,
the incoming $OAM_{l,m}$ mode couples first to the $OAM_{l\pm
2\color[rgb]{0,0,0},m^{\prime}\color[rgb]{0,0,0}}$ mode due to the $\Delta
l=\pm 2$ rule with an amplitude given by Eq. 14. But as this coupled
$OAM_{l\pm 2,m^{\prime}}$ mode travels down the slightly elliptical fiber, it
also continuously feels the impact of the ellipticity-induced torque, changing
gradually into its degenerate partner in an oscillatory manner, in accordance
with its characteristic $2\pi$ walk-off length (Eq. 27). A multiplicative
$sine$ term is a consequence of the interference of the input $OAM_{l,m}$ mode
and the coupled $OAM_{l\pm 2,m^{\prime}}$ modes.

For the input OAM modes defined by a $-l$ topological charge, we replace Eq.
32 with
$\psi{}_{l,m}^{(e)}(L)\color[rgb]{0,0,0}=\frac{1}{\sqrt{2}}(\color[rgb]{0,0,0}\psi{}_{l,m}^{+(e)}(L)-\color[rgb]{0,0,0}\psi{}_{l,m}^{-(e)}(L))$,
which is consistent with the requirement,
$\color[rgb]{0,0,0}\psi{}_{-l,m}^{(e)}(0)\color[rgb]{0,0,0}=O_{-l,m}=\frac{1}{\sqrt{2}}(O^{+}_{l,m}-O^{-}_{l,m})$,
the latter following from Eq. 24. Substituting Eqs. 33 and 34, we obtain the
same final expression as in Eq. 35, except that $l$ is replaced with $-l$ and
$l^{\prime}$ with $-l^{\prime}$ everywhere, and amplitudes, $O_{l,m}$ and
$O_{l^{\prime},m^{\prime}}$ are interchanged with their degenerate partners,
$O_{-l,m}$ and $O_{-l^{\prime},m^{\prime}}$, respectively, and $l\pm 2$ in the
the summation on the RHS is replaced with $-l\pm 2$.

_Special Cases_

1) For the $|l|=2$ case, we can have $l^{\prime}=0$ (see Eq. 35), which is a
nondegenerate case. Therefore, $\delta\beta^{{}^{\prime}\pm}_{0,m}=0$, which
implies from Eq. 27, $L^{(2\pi)}_{0,m^{\prime}}=\infty$. Substituting in Eq.
35, we correctly obtain $\sin(\pi L/L^{(2\pi)}_{0,m^{\prime}})=0$ and
$\cos(\pi L/L^{(2\pi)}_{0,m^{\prime}})=1$.

2) For the $l=0$ incident mode (the nondegenerate case), the cosine and the
sine terms in Eq. 35 involving the $O_{l,m}$ and the $O_{-l,m}$ modes,
respectively, become one and zero, as discussed above. The summation, however,
now includes two terms, one corresponding to $l^{\prime}=2$ and the other
corresponding to $l^{\prime}=-2$, yielding the expression,

$\color[rgb]{0,0,0}\psi{}^{(e)}_{0,m}(L)\color[rgb]{0,0,0}=O_{0,m}e^{i\beta_{0,m}L}+2ia^{(1)}_{(0,m)(2,m^{\prime})}e^{i\pi
L/L^{(2\pi)}_{2,m^{\prime}}}(O_{2,m^{\prime}}+O_{-2,m^{\prime}})\sin(\beta_{0,m}-\beta_{2,m^{\prime}})e^{i(\beta_{0,m}+\beta_{2,m^{\prime}})L/2}e^{i\beta_{2,m}L}.$
(36)

The expression is symmetric in the coupled modes $OAM_{2,m^{\prime}}$ and
$OAM_{-2,m^{\prime}}$. Physically, the input $OAM_{0,m}$ mode couples with the
$l^{\prime}=\pm 2$ modes with equal strength, each producing its own
$cosine,~{}sine$ oscillatory behavior, which cancels out with the net effect
of a common (extra) phase factor governed by the $2\pi$ walk-off length.

###  4.2 Crosstalk

Crosstalk (or charge weight)[3, 6, 9], $\chi_{(l,m)(n,k)}$ (expressed in dB)
for the various component $OAM_{n,k}$ modes of the output,
$\color[rgb]{0,0,0}\psi{}^{(e)}_{l,m}(L)$, is given by

$\chi_{(l,m)(n,k)}(L)=10log_{10}|<O_{n,k}|\color[rgb]{0,0,0}\psi{}^{(e)}_{l,m}(L)\color[rgb]{0,0,0}>|^{2}=10log_{10}\Big{|}\int_{0}^{\infty}\int_{0}^{2\pi}O_{n,k}^{*}\color[rgb]{0,0,0}\psi{}^{(e)}_{l,m}(L)rdrd\theta\Big{|}^{2}.$
(37)

Upon substitution of Eq. 35, we find the crosstalk with the input mode’s
degenerate partner, $OAM_{-l,m}$, is

$\ \chi_{(l,m)(-l,m)}(L)=10log_{10}\sin^{2}(\pi L/L^{(2\pi)}_{l,m}).$ (38)

For the content of the original $OAM_{l,m}$ mode remaining within the
admixture, Eq. 35, we find

$\chi_{(l,m)(l,m)}(L)=10log_{10}\cos^{2}(\pi L/L^{(2\pi)}_{l,m}).$ (39)

The crosstalk with the neighboring modes, $l^{\prime}=l\pm
2~{}(l^{\prime}>0)$, is from Eqs. 35 and 37

$\chi_{(l,m)(l^{\prime},m^{\prime})}=10log_{10}\big{(}4|a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}|^{2}\cos^{2}(\pi
L/L^{(2\pi)}_{l^{\prime},m^{\prime}})\sin^{2}((\beta_{l,m}-\beta_{l^{\prime},m^{\prime}})L/2)\big{)},$
(40)

and with the corresponding degenerate partners, $l^{\prime}=-l\mp
2~{}(l^{\prime}<0),$ is given by

$\chi_{(l,m)(l^{\prime},m^{\prime})}=10log_{10}\big{(}4|a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}|^{2}\sin^{2}(\pi
L/L^{(2\pi)}_{l^{\prime},m^{\prime}})\sin^{2}((\beta_{l,m}-\beta_{l^{\prime},m^{\prime}})L/2)\big{)}.$
(41)

For the case of $l^{\prime}=0$, which occurs when $|l|=2$, we simply set
$L^{(2\pi)}_{0,m^{\prime}}=\infty$ in Eqs. 40 and 41 to obtain $\chi_{(\pm
2,m)(0,m^{\prime})}=10log_{10}\big{(}4|a^{(1)}_{(2,m)(0,m^{\prime})}|^{2}\sin^{2}((\beta_{2,m}-\beta_{l0,m^{\prime}})L/2)\big{)}$.
The maximum possible value from Eqs. 40 and 41 is

$\chi^{(max)}_{(l,m)(l^{\prime},m^{\prime})}\approx
10log_{10}(4|a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}|^{2});$ (42)

this maximum corresponds essentially to setting the magnitude of the
sinusoidal factors in Eqs. 40 and 41 to unity; these amplitudes vary rapidly
with $L$ due to the small propagation constant differences in the argument of
the second sinusoidal factor in each of Eqs. 40 and 41. For the $l=0$
(nondegenerate) input mode, the crosstalk with its neighbors, the
$OAM_{2,m^{\prime}}$ and $OAM_{-2,m^{\prime}}$ modes, is given by
$\chi_{(0,m)(\pm
2,m^{\prime})}=10log_{10}(4|a^{(1)}_{(0,m)(2,m^{\prime})}|^{2}\sin^{2}((\beta_{0,m}-\beta_{2,m^{\prime}})L/2))$
(see Eq. 36).

##  5 Validity of the Scalar Theory

The crosstalk equations in Section 4.2 along with the expressions for the
$2\pi$ walk off length (Eqs. 27, 22, and 23) and the admixed neighbor mixing
amplitude of Eq. 14 (in first order perturbation), comprise the fundamental
set of equations derived on the basis of scalar perturbation theory. The
weakly guiding approximation permitted us to write the vector mode solutions
as products of the spatial modes and the polarization states, which then
facilitated the use of scalar wave equation to study the impact of fiber
ellipticity on the spatial modes alone and their evolution as a function of
propagation distance. Inherent in the use of the scalar wave equation is the
assumption of the smallness of spin-orbit ($SO$) interaction, which is then
neglected.

Here we briefly explore the impact of the presence of $SO$ interaction. The
$SO$ induced correction to the scalar propagation constant, denoted
$\delta\beta^{2(SO)}_{(l,m)}$, is of the order of $l\Delta/a^{2}$ [8, 16, 17],
being of opposite sign for the $-l$ state as compared to the $l$ state. This
then implies that the diagonal elements of the $2~{}x~{}2$ symmetric matrix in
the $(l,-l)$ subspace (see Section 3.1 and Eq. A. 20), hitherto equal, will
tend to become unequal, the change for one diagonal element being of the
opposite sign compared to the other. Consequently, this can affect the
accuracy of the derived results. In order for our results to be as accurate as
possible, we therefore impose the stringent condition that the splitting of
the degenerate spatial modes, $2\gamma_{l,m}$ due to ellipticity $\epsilon$
(Eqs. 21-23) far exceeds the splitting due to the $SO$ interaction, i.e., the
$(l,-l)$ degeneracy is still considered broken primarily through the elliptic
effects. A simple analysis based on the requirement
$\delta\beta^{2(SO)}_{(l,m)}<<\gamma_{l,m}$ as well as the sufficiency of the
first order perturbation theory considered in our derivations (see Appendix B,
[18]), yields, in an approximate manner, bounds on $\epsilon$

$\Big{(}\frac{lV}{2k^{2}a^{2}n_{1}^{2}}\Big{)}^{1/l}\Big{(}\frac{2}{V}\Big{)}<<\epsilon<<\Big{(}\frac{2}{V}\Big{)}.$
(43)

The lower bound, denoted $\epsilon_{lo}$, arises from the requirement of
$\delta\beta^{2(SO)}_{(l,m)}<<\gamma_{l,m}$, while the upper bound, denoted
$\epsilon_{u}$, originates in the requirement that
$|a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}|<<1$ (the lower bound is less
stringent than the upper bound, see Appendix B). $\epsilon_{lo}$ increases
with the topological charge $l$, implying a diminishing region of accuracy,
$\epsilon_{u}-\epsilon_{lo}$; this is consistent with increase in $SO$
interaction with $l$. If $V(=kan_{1}(2\Delta)^{1/2})$ were kept constant (say,
by decreasing $\Delta$ and increasing core radius $a$), transitioning to a
fiber with a larger value of core radius decreases the lower bound,
$\color[rgb]{0,0,0}\epsilon_{lo}\color[rgb]{0,0,0}$, thus expanding the
region, $\epsilon_{u}-\epsilon_{lo}$. In general, for large values of $V$, as
in a multimode fiber, the lower bound decreases but the upper bound becomes
tighter. Thus, the accuracy range, as determined by these bounds, depends upon
the fiber parameters and the value of the topological charge $l$. Conversely,
Eq. 43 can be a useful input in the design of fibers, which suffer slight
ellipticity and are governable by scalar perturbation theories due to the
generally low effects of spin-orbit orbit interaction. Note also that if the
above bounds are not strictly adhered to, the derived analytic expressions may
suffer in numerical accuracy, but the qualitative behavior (for example, the
rise in the $2\pi$ walk-off length with topological charge $l$ or its
decrement with increasing ellipticity, $\epsilon$) will likely hold; this is
also supported by physical reasoning (see Section 3.1). A detailed error
analysis is beyond the scope of the current work.

##  6 Application to a Step-Index Fiber

As an illustration, we now apply the derived expressions to a step-index
fiber, whose solutions are well known and well studied [8]:

$\displaystyle
O_{l,m}(r,\theta)=\frac{1}{\sqrt{N_{l,m}}}J_{l}(p_{l,m}r)e^{il\theta}\quad
for\ r\leq a$
$\displaystyle=\frac{1}{\sqrt{N_{l,m}}}\frac{J_{l}(p_{l,m}a)}{K_{l}(q_{l,m}a)}K_{l}(q_{l,m}r)e^{il\theta}\quad
for\ r\geq a,$ (44)

where $N_{l,m}$, the normalization constant, can be analytically determined
[14]; $J_{l}$ and $K_{l}$ are the Bessel and the modified Bessel functions,
respectively; $p_{l,m}=\sqrt{k^{2}n_{1}^{2}-\beta_{l,m}^{2}}$ and
$q_{l,m}=\sqrt{\beta_{l,m}^{2}-k^{2}n_{2}^{2}}$. The index profile $f(r)$ is a
step-function equal to zero for $r\leq a$ and equal to 1 for $r>a$. As a
result, $\partial{f(r)}/\partial{r}=\delta(r-a)$. The matrix element for the
$\delta H$ operator defined in Eq. 13 is now given by

$\delta H_{(l,m)(l\pm
2,m^{\prime})}=-(k^{2}n_{1}^{2}\Delta)\frac{J_{l}(p_{l,m}a)J_{l\pm 2}(p_{l\pm
2,m^{\prime}}a)}{\sqrt{N^{\prime}_{l,m}}\sqrt{N^{\prime}_{l\pm
2,m^{\prime}}}},$ (45)

where $N^{\prime}_{n,k}=N_{n,k}/2\pi a^{2}$ is dimensionless. Propagation
constants required in the calculations are determined using the well-known
characteristic equation for the scalar modes of a step-index fiber[8, 19]. All
numerical calculations are done in MatLab.

###  6.1 Few Mode Fiber

We assume $a=10\mu m,~{}n_{1}=1.45205,~{}n_{2}=1.44681$. The index height
profile parameter $\Delta=(n_{1}^{2}-n_{2}^{2})/(2n_{1}^{2})=0.0036<<1$. For a
wavelength, $\lambda=1.55\mu m$, normalized frequency $V=2\pi
a(n_{1}^{2}-n_{2}^{2})^{1/2}/\lambda=4.996$. As a result, only the
$OAM_{0,1},OAM_{1,1},OAM_{-1,1},OAM_{2,1},OAM_{-2,1}$ and $OAM_{0,2}$ modes
are supported. From Eq. 23 and Eq. 22,

$\gamma_{1,1}=\epsilon\delta H_{(1,1)(1,-1)}$ (46)

and

$\gamma_{2,1}=\epsilon^{2}\Big{(}\frac{(\delta
H_{(2,1)(0,1)})^{2}}{\beta^{2}_{2,1}-\beta^{2}_{0,1}}+\frac{(\delta
H_{(2,1)(0,2)})^{2}}{\beta^{2}_{2,1}-\beta^{2}_{0,2}}\Big{)}.$ (47)

Now using Eqs. 27, 22, and 23, we can calculate the $2\pi$ walk-off lengths,
which are displayed in Table 1 for various values of ellipticity.

Table 1: The $2\pi$ walk-off length, $L^{(2\pi)}_{l,m}$, specified in meters, as a function of ellipticity $\epsilon$ for different input modes, $OAM_{l,m}$; it varies as $\epsilon^{-l}$ for fixed $l,m$ (see Eqs. 27, 22, and 23); normalized frequency $V=4.996$. $l,m$ | $\epsilon=0.005$ | $\epsilon=0.010$ | $\epsilon=\color[rgb]{0,0,0}0\color[rgb]{0,0,0}.015$ | $\epsilon=0.025$ | $\epsilon=0.040$
---|---|---|---|---|---
1,1 | 0.372 | 0.186 | 0.124 | 0.074 | 0.047
2,1 | 29.18 | 7.30 | 3.24 | 1.17 | 0.456

Larger the ellipticity $\epsilon$, smaller the $2\pi$ walk-off length
$L^{(2\pi)}_{l,m}$ due to the $\epsilon^{-l}$ dependence. Likewise, larger the
value of topological charge $l$, for the same ellipticity $\epsilon$ value,
larger the $2\pi$ walk-off length (Section 3.1). This is what we observe in
Table 1. Note also that the ellipticity values for the $l=1$ case satisfy the
bounds of Eq. 43 ($\epsilon_{lo}=0.0003$ and $\epsilon_{u}=0.40$). For the
$l=2$ case, $\epsilon_{lo}=0.015$, implying that the results in the table
below this threshold may not be as accurate.

####  6.1.1 Mixing with Neighboring Modes

Input: $OAM_{1,1}$

The $OAM_{1,1}$ input mode can only mix with $OAM_{-1,1}$ due to the $\Delta
l=\pm 2$ rule (already considered above via Eq. 46 in connection with the
$2\pi$ walk-off length). Referring to Eq. 35, excluding $l^{\prime}=-1$, the
only other allowed value of $l^{\prime}$ is 3, which is not supported by the
fiber. So there is no other mode the input mode $OAM_{1,1}$ can mix with.

Input: $OAM_{2,1}$

Besides undergoing the transformation into the degenerate partner, the
$OAM_{-2,1}$ mode, the $OAM_{2,1}$ mode also mixes with the modes $OAM_{0,1}$
and $OAM_{0,2}$ with the amplitudes given by (see Eq. 35)

1)
$|<O_{0,1}|\color[rgb]{0,0,0}\psi{}^{(e)}_{2,1}>|=2|a^{(1)}_{(2,1)(0,1)}\sin((\beta_{2,1}-\beta_{0,1})L/2)|=0.246\epsilon|\sin(5735L)|$

2)
$|<O_{0,2}|\color[rgb]{0,0,0}\psi{}^{(e)}_{2,1}>|=2|a^{(1)}_{(2,1)(0,2)}\sin((\beta_{2,1}-\beta_{0,2})L/2)|=3.12\epsilon|\sin(915L)|$

The numerical coefficient of the amplitude is larger in 2) than in 1) due to
the closer proximity of the pair of modes $OAM_{2,1},OAM_{0,2}$ in their
propagation constant values as compared to the mode pair:
$OAM_{2,1},OAM_{0,1}$; these propagation constant differences are reflected in
the numerical part of the argument of the sine functions. The amplitudes vary
rapidly with $L$, but the maximum amplitude corresponds to setting the sine
term above to unity. These expressions are used in the crosstalk calculation
in Section 6.1.2.

Table 2:  Crosstalk, $\chi_{(2,1)(l^{\prime},m^{\prime})}$ (in dB) for the various component $OAM_{l^{\prime},m^{\prime}}$ modes within the $OAM_{2,1}$ output mode mixture, $\color[rgb]{0,0,0}\psi{}^{(e)}_{2,1}$ as a function of $L$ (see Section 4.2); the ellipticity $\epsilon$ is fixed at 0.025. The $2\pi$ walk-off length $L_{2,1}^{(2\pi)}=1.17m$ (see Table 1). $l^{\prime},m^{\prime}$ | L=0.10m | L=0.25m | L=0.50m | L=1.00m | L=1.50m
---|---|---|---|---|---
2,1 | -0.32 | -2.12 | -12.91 | -0.94 | -3.98
-2,1 | -11.53 | -4.12 | -0.23 | -7.12 | -2.22
0,1 | -44.33 | -44.89 | -47.33 | -44.22 | -46.95
0,2 | -30.48 | -27.30 | -22.87 | -25.07 | -30.86

####  6.1.2 Crosstalk

For $\epsilon=0.025$, we show in Table 2 crosstalk calculated in accordance
with Eqs. 37-40. At $L=0.10m$, we see that the output mixture is practically
all $OAM_{2,1}$ mode with little $OAM_{-2,1}$ mode content. However, at
$L=0.25m$, roughly one quarter of the $2\pi$ walk-off length value, the
contents of the $OAM_{2,1}$ and its degenerate partner, $OAM_{-2,1}$ mode are
approximately the same, with the latter becoming the major constituent of the
output mixture at $L=0.50m$ (roughly half the $2\pi$ walk-off length). At
$L=1m$, slightly less than the $2\pi$ walk-off length, the $OAM_{2,1}$ mode is
the main contributor to the mixture, and this cycle of mode conversion from
the $OAM_{2,1}$ mode into $OAM_{-2,1}$ mode, and back into the original
$OAM_{2,1}$ mode, continues as $L$ is increased beyond the $2\pi$ walk-off
length. The contents of the other admixed modes $l^{\prime}\neq\pm l$ remain
very low being (upper) bounded by values of $-44.22dB$ and $-22.16dB$
(calculated from Eq. 42) for $\chi^{(max)}_{(2,1)(0,1)}$ and
$\chi^{(max)}_{(2,1)(0,2)}$, respectively.

####  6.1.3 OAM Mode Propagation in a Multiplexed Environment

If we now adopt a crosstalk criterion of
$\chi_{(2,1)(l^{\prime},m^{\prime})}\leq-20dB$ for successful $OAM_{2,1}$
transmission, then the maximum propagation distance, denoted
$L^{(max)}_{2,1}$, that is allowed is given by setting
$10log_{10}(\sin^{2}(\pi
L^{(max)}_{2,1}/L^{(2\pi)}_{2,1})\color[rgb]{0,0,0})\color[rgb]{0,0,0}=-20$,
which is needed to suppress the admixed $-l$ degenerate $OAM$ component (see
Eq. 38). This gives rise to multiple solutions due to the multi-valued
$\sin^{-1}$ function: $L^{(max)}_{l,1}=0.037m,~{}L^{(max)}_{l,1}=(\pm
0.037+n)L^{(2\pi)}_{l,1}(m)$, where $n>0$ is an integer. In other words, the
propagation distance $L$ must satisfy the constraint $L\leq
L^{(max)}_{2,1}=0.037m,(nL^{(2\pi)}_{2,1}-0.037m)<L\leq(nL^{(2\pi)}_{2,1}+0.037m)$,
where $n(>0)$ is an integer. Because $L^{(2\pi)}_{2,1}=1.17m$, this implies
permitted transmissions at most of the order of a few meters suitable for
inter-shelf/rack distances within data centers. Furthermore, the $2\pi$ walk-
off length of the degenerate partner is identical and the nondegenerate modes,
$OAM_{0,1}$ and $OAM_{0,2}$, which can only mix with the $OAM_{2,1}$ and the
$OAM_{-2,1}$ modes due to the $\Delta l=\pm 2$ selection rule, satisfy the
suppression constraint of $-20dB$ (see Table 2). Consequently, under this
criterion we can have simultaneous transmission of the four spatial modes,
$OAM_{2,1},OAM_{-2,1},OAM_{0,1},$ and $OAM_{0,2}$ in a multiplexed
environment, thereby increasing the data transmission capacity four-fold.
Additional tables can be constructed for different sets of ellipticity and
fiber parameters to gain insight into the constraints imposed by ellipticity
in the design of such fibers.

###  6.2 Multimode Fiber

Fig. 5: a) The effective refractive indices, $n_{eff}$’s shown with a cut-off
of 1.456, for most modes up to $l=6$; the values decrease with increasing $l$
and $m$, which is shown explicitly in parentheses for the $l=0$ mode b) The
$2\pi$ walk-off length, $L^{(2\pi)}_{(l,1)}$, as a function of ellipticity
$\epsilon$ for different values of $l$; for fixed $l$, $L^{(2\pi)}_{(l,1)}$
varies inversely as $\epsilon^{l}$ c) The $2\pi$ walk-off length,
$L^{(2\pi)}_{(l,1)}$, as a function of topological charge $l$ for different
values of $\epsilon$ d) $\chi_{(l,1)(l-2,2)}^{(max)}$, the maximum possible
crosstalk of $OAM_{l,1}$ mode with the neighboring mode, $OAM_{l-2,2}$, as a
function of $\epsilon$ for different $l$ values; see text.

We assume $a=25\mu m,n_{1}=1.461$, $n_{2}=1.444$; these parameters correspond
to a real _ThorLabs_ multimode fiber.
$\Delta=(n_{1}^{2}-n_{2}^{2})/(2n_{1}^{2})=0.012$. We use $\lambda=1.55\mu m$.
The normalized frequency $V$ is 22.3. Although the maximum allowed value of
$l$ is 18, we consider $l$ up to 6. Fig. 5a shows the effective refractive
indices for the various OAM modes. The values decrease with increasing $l$ and
$m$. For fixed $l$, the spacing between consecutive $m$ indices also
increases, the increase being larger for larger $l$ values. The effective
refractive index difference, $\Delta n_{eff}$, is the smallest for the
$OAM_{5,1}$ and the $OAM_{0,3}$ mode pair and equals $0.64~{}x~{}10^{-4}$.
However, the $\Delta l=\pm 2$ rule prevents any transition between these two
modes in any order of perturbation theory. The $OAM_{2,1}$ and the $OAM_{0,2}$
modes, which are very close in proximity ($\Delta n_{eff}=1.25~{}x~{}10^{-4}$)
but also satisfy the $\Delta l=\pm 2$ rule, will mix in first order; the
mixing coefficient, $a^{(1)}_{(2,1)(0,2)}$, is likely to be larger as compared
with the mixing coefficient of the modes $OAM_{2,1}$ and $OAM_{0,1}$ that also
satisfy the $\Delta l=\pm 2$ transition rule, but have a much larger value of
$\Delta n_{eff}=6.26~{}x~{}10^{-4}$. In general, the proximity of the
$OAM_{l,1}$ and the $OAM_{l-2,2}$ modes in their $n_{eff}$ values (see Fig.
5a) causes the latter to be the predominantly admixed neighboring mode.

Fig. 5b shows $L^{(2\pi)}_{(l,1)}$ as a function of ellipticity $\epsilon$ for
different values of topological charge $l$. It is calculated using Eqs. 27,
22, and 23. It decreases as ellipticity $\epsilon$ is decreased, the drop
being sharper for higher $l$ values; this is due to the $\epsilon^{-l}$
dependence. We note here that in the limit $\epsilon\rightarrow 0$, the $2\pi$
walk-off length approaches infinity, consistent with the absence of crosstalk
(see Eqs. 38, 41 and 14) for a perfectly round fiber ($\epsilon=0)$. In Fig.
5c, the ellipticity $\epsilon$ is fixed. The $2\pi$ walk-off length rises with
$l$, the rise being sharper with $l$, which implies that the higher $l$ value
modes will have relatively less crosstalk with their degenerate partners (see
Eq. 38).

Table 3: The lower bound, $\epsilon_{l\color[rgb]{0,0,0}o\color[rgb]{0,0,0}}$ and the upped bound $\epsilon_{u}$ for different values of $l;~{}\epsilon_{u}$ is independent of $l$ (see Eq. 43). $l,m$ | $\epsilon_{lo}$ | $\epsilon_{u}$
---|---|---
1,1 | $4.6X10^{-5}$ | 0.09
2,1 | 0.003 | 0.09
3,1 | 0.010 | 0.09
4,1 | 0.019 | 0.09
5,1 | 0.027 | 0.09

With respect to the impact of SO interaction on our results, we find that the
larger radius $a$ of the core of the multimode fiber compared to that of the
few mode fiber causes the $SO$ interaction to reduce; on the other hand, the
elliptic effects leading to splitting of the degenerate modes are enhanced due
to the relatively smaller propagation constant differences on account of a
larger value of $V$. The preferred region of $\epsilon$ (Eq. 43) expands to
accommodate lower values of $\epsilon$, but diminishes as $l$ increases; see
Table 3, where the lower bound $\epsilon_{lo}$ increases with $l$, consistent
with a rising value of the ”undesired” SO interaction (of order
$l\Delta/a^{2})$; the desired ellipticity range, $\epsilon_{u}-\epsilon_{lo}$,
therefore, reduces with $l$. If we were to increase the core radius to
$62.5\mu m$ corresponding to another commercially available fiber, according
to Eq. 43 and the fact that
$V=ka\color[rgb]{0,0,0}n_{1}(2\color[rgb]{0,0,0}\Delta\color[rgb]{0,0,0}){}^{1/2}$,
the lower bound $\epsilon_{lo}$ would decrease, for fixed $l$, by a factor
$(50/62.5)^{1/l}(50/62.5)$; the reduced $\epsilon_{lo}$ values then would be
0.074, 0.014, 0.020 for $l=3,4,5$, respectively. Thus, by increasing the core
radius, we can accommodate lower values of $\epsilon$ at higher values of $l$.
Such analyses can be useful in the design of fibers with low spin-orbit
interaction.

In Table 4, we show the crosstalk experienced by the input $OAM_{3,1}$ and
$OAM_{2,1}$ modes in a fiber with ellipticity $\epsilon=0.015,$ (which lies
within the bounds defined by Table 3). While these modes change into their
respective degenerate partners, they also mix with the neighboring modes,
$OAM_{l\pm 2,m^{\prime}}$ with amplitudes proportional to
$a^{(1)}_{(\color[rgb]{0,0,0}l\color[rgb]{0,0,0},m)(\color[rgb]{0,0,0}l\color[rgb]{0,0,0}\pm
2,m^{\prime})}$ determined from Eq. 14. The crosstalk values displayed in
Table 4 are all seen to be below the maximum possible values of
$\chi^{(max)}_{(3,1)(1,2)}=-16.14dB$ and $\chi^{(max)}_{(2,1)(0,2)}=-14.02dB$
(Eq. 42). The $2\pi$ walk-off length for the $l=3$ modes is $4.15m$. We note
from the table that at $L=0.5m$, the $OAM_{3,1}$ mode is essentially pure with
very little $OAM_{-3,1}$ content. However, at $L=1.0m$, which is approximately
one-quarter the $2\pi$ walk-off length, the contents of the $OAM_{3,1}$ mode
and its degenerate partner are roughly equal. At $L=2.0m$ (approximately half
the $2\pi$ walk-off length), the $OAM_{3,1}$ is basically converted fully into
the $OAM_{-3,1}$ mode. Similarly, at $L=4m$ (slightly less than the $2\pi$
walk-off length), most of the $OAM_{-3,1}$ mode has changed back into the
$OAM_{3,1}$ mode. This oscillatory behavior continues, with a period equal to
$4.15m$, the $2\pi$ walk-off length. For the
$\color[rgb]{0,0,0}l\color[rgb]{0,0,0}=2$ case, the $2\pi$ walk-off length is
$0.63m$; the changes here are more rapid on account of the smaller value of
the $2\pi$ walk-off length.

Table 4: Crosstalk, $\chi_{(l,m)(l^{\prime},m^{\prime})}$ (in dB) (see Section 4.2); $m=1$, $l=2$ and $3$. Ellipticity $\epsilon$ is fixed at $0.015$. $L$ is the length of the fiber traversed by the $OAM_{l,1}$ mode input at one end of the fiber; $l^{\prime}=l-2$. $l^{\prime},m^{\prime}$ | L=0.5m | L=1.0m | L=2.0m | L=4.0m | L=5.0m
---|---|---|---|---|---
3,1 | -0.64 | -2.77 | -24.92 | -0.06 | -1.94
-3,1 | -8.65 | -3.26 | -0.01 | -18.91 | -4.44
1,2 | -20.59 | -23.07 | -19.11 | -34.04 | -18.65
-1,2 | -16.18 | -17.55 | -15.75 | -25.99 | -17.28
2,1 | -1.97 | -11.35 | -1.38 | -6.82 | -0.17
-2,1 | -4.38 | -0.33 | -5.66 | -1.01 | -14.06
0,2 | -17.86 | -16.69 | -19.92 | -16.26 | -30.70

###  6.3 Propagation distance in an OAM multiplexed environment

If we now consider a fiber which is perfect, except for ellipticity, and adopt
a somewhat relaxed criterion of crosstalk $<-14dB$ with any admixed mode, then
this criterion translates into a requirement of a maximum permitted
propagation distance, $L^{(max)}_{l,1}$, given by $10log_{10}sin^{2}(\pi
L^{(max)}_{l,1}/L^{(2\pi)}_{l,1})=-14$; this implies multiple solutions (as in
Section 6.1.3) and leads to the constraint that the propagation distance $L$
must satisfy: $L\leq
0.064L^{(2\pi)}_{l,1},~{}(n-0.064)L^{(2\pi)}_{l,1}<L<(n+0.064)L^{(2\pi)}_{l,1},~{}l=2,3$.
For example, L=4.40m satisfies the inequality for both $l=2$ and
$l=3~{}(n=1,7$ for $l=3,2$, respectively). This has the implication that we
can multiplex $OAM_{2,1},OAM_{-2,1},OAM_{3,1},OAM_{-3,1}$ and transmit without
crosstalk exceeding $-14dB$ when $L=4.40m$; this corresponds to an inter-
shelf/rack distance in a data center.

##  7 Further Remarks

1) Because the weakly guiding approximation causes the vector modes to be
expressed as products of the spatial OAM mode considered here and the circular
polarization (right or left) (Section 2), the impact of fiber ellipticity is
then the product of the impact of ellipticity on the spatial mode and the
polarization mode, evaluated separately. In general, the initial circular
polarization would become a linear combination of the left and right circular
polarizations; the overall final state (including polarizations) would be the
product of the final spatial mode state as determined by our analysis and the
final polarization state not determined here; this would, for example, result
in a linear combination of the quartets:
$O_{l,m}\epsilon_{+},O_{-l,m}\epsilon_{-},O_{l,m}\epsilon_{-},O_{-l,m}\epsilon_{+}$,
translating, respectively to the vector modes:
$HE_{l+1,m},HE_{-l-1,m},EH_{l-1,m},EH_{-l+1,m}(|l|>1)$. Our intent here is to
treat the impact of ellipticity on the spatial mode only, meaning we are not
concerned with the final polarization state of the output modes. In the
experimental domain, this corresponds to separating out the spatial modes, as
is done, for example, in[24, 25] using a mode sorter, and measuring their
individual intensities, without any regard to their polarization state.

2) The solutions, $O^{{}^{\prime}\pm}_{l,m}$, Eq. 28, are the eigenmodes of
the scalar wave equation: $(H+\epsilon\delta
H)O^{{}^{\prime}\pm}_{l,m}=(\beta^{{}^{\prime}\pm}_{l,m})^{2}O^{{}^{\prime}\pm}_{l,m}$.
Since these field amplitudes are dominated by the $O^{\pm}_{l,m}$ amplitudes,
these eigenmodes would display the fields of the $LP$ modes (Section 3.1).
Indeed such $LP$ mode-like fields have been observed in numerical simulations
using the finite -element solver COMSOL for the slightly elliptical ring-core
fibers in [5]. This qualitative agreement gives further credence to our
analytic results based on perturbation theory, and suggests additionally that
our results, Eqs. 20 and 35, are of a general nature applicable to other
fibers as well. Furthermore, the rapid rise in the value of the $2\pi$ walk-
off length, predicted by our analytic expressions, as the topological charge
$l$ of the incoming mode is increased (keeping ellipticity $\epsilon$
constant) or the ellipticity $\epsilon$ is decreased (keeping $l$ constant),
is also seen in the finite-element calculations in [3, 6].

3) Because the output intensity pattern of the final state is dominated by the
presence of the $OAM_{l,m}$ and $OAM_{-l,m}$ modes in the output mixture, the
output intensity pattern is essentially given by $I(r,\theta)=|(\cos(\pi
L/L^{(2\pi)}_{l,m})O_{l,m}+i\sin(\pi L/L^{(2\pi)}_{l,m})O_{-l,m})|^{2}$ (see
Eq. 35), which, upon insertion of
$O_{l,m}(r,\theta)=(1/\sqrt{N}_{l,m})F_{l,m}(r)e^{il\theta}$ and
$O_{-l,m}(r,\theta)=(1/\sqrt{N}_{l,m})F_{l,m}(r)e^{-il\theta}$, reduces to
$(1+\sin(2\pi L/L^{(2\pi)}_{l,m})\sin(2l\theta))F^{2}_{l,m}(r)/N_{l,m}$. This
result represents _tilted_ $2l$-lobed $LP$ mode patterns ([9, 21, 22]). For
example, at $L=L^{(2\pi)}_{l,m}/4$, intensity
$I\propto\cos^{2}\color[rgb]{0,0,0}(l\color[rgb]{0,0,0}(\theta-\pi/4l)\color[rgb]{0,0,0})$,
implying tilt angles of $\pi/(4l)$. For $l=1$, the tilt angle is $\pi/4$, a
result also predicted in [2].

##  8 Summary and Discussion

Working in the weakly guiding approximation, in which the vector modes reduce
to products of the spatial OAM modes and the associated polarizations, we have
focused on the spatial modes and used the scalar wave equation to develop
perturbation theory for spatial OAM mode mixing and the ensuing OAM modal
crosstalk due to slight ellipticity in a fiber. The developed perturbation
technique provides insight into the mechanism for mode mixing. A fundamental
mixing selection rule ($\Delta l=\pm 2$) makes the problem tractable and leads
to the derivation of analytic expressions for mode-mixing and consequently the
crosstalk. The expressions include an explicit formula for the $2\pi$ walk-off
length, which is essential in determining the degenerate OAM mode pair mixing.
The derived expressions embody complete mathematical dependence on topological
charge, ellipticity, and the length of the traversed fiber, enabling detailed
quantitative analyses of the ellipticity-induced crosstalk, and thereby
facilitating the study of optimal fiber parameters for minimal crosstalk.

Scalar perturbation theory is a well-established theory to study perturbations
in a given system [13]. This theory, as developed and presented here, provides
a convenient and useful framework to explore the impact of ellipticity. A
generic but vital feature in the use of this theory is the negligibility of
the polarization effects, especially the spin-orbit interaction that gives
rise to differences in the propagation constants of the vector modes,
$HE_{l+1,m}$ and $EH_{l-1,m}$. In this work, we subsequently explore the
effect of spin-orbit interaction to estimate bounds on the values of
ellipticity for which the derived results are most valid (accurate). The
validity range expands with increasing values of normalized frequency, $V$, as
in a multimode fiber (larger core radius implies relatively smaller spin-orbit
interaction), and dwindles as the topological charge is increased, consistent
with the fact that the spin-orbit interaction becomes significant at higher
values of $l$ and thus cannot be ignored. As a result, the numerical results
based on the developed theory are most accurate for low values of topological
charge (as determined by the fiber parameters). We have illustrated the
utility of the analytic expressions and their validity with application to a
step-index few mode fiber and a multimode fiber.

Prior work has focused on the use of computationally intensive finite-element
methods that would normally require high levels of precision to obtain
detailed relationships with fiber parameters. Therefore, the availability of
analytic expressions based on the presented scalar theory, along with their
utility constraints and the provided insight, serve as a valuable tool in the
analysis and design of fibers for OAM mode propagation. The technique, as
applied to slightly elliptical fibers, is novel and detailed, and, to our
knowledge, has not been presented before. This technique also serves as a
precursor to techniques incorporating groups of degenerate modes as in graded-
index fibers.

Appendix

##  Appendix A Derivation of the Mixing of the $OAM_{l,m}$ and $OAM_{-l,m}$
Modes due to Fiber Ellipticity

The derivation here follows closely that for the fiber bend in [9]. The states
$OAM_{l,m}$ and $OAM_{-l,m}$ are degenerate. Therefore, there are an infinite
number of combinations of these states that satisfy the unperturbed scalar
wave equation (Eq. 1). However, there is a unique pair of these linear
combinations, which are orthonormal, and appropriate to the perturbation under
consideration. We now seek these. Due to the Hermitian nature of $H$ and
$\delta H$, these linear combinations are related to the above two degenerate
states via a unitary transformation in the $(l,-l)$ subspace. Consequently, we
write

$O^{+}_{l,m}=U_{11}O_{l,m}+U_{12}O_{-l,m},$ (A.1)
$O^{-}_{l,m}=U_{21}O_{l,m}+U_{22}O_{-l,m},$ (A.2)

where $U_{i,j}$ are the elements of the $2~{}x~{}2$ unitary matrix $U$. We now
replace $O_{l,m}$ in Eq. 7 with $O^{+}_{l,m}$, and $O_{-l,m}$ in Eq. 19 with
$O^{-}_{l,m}$, and rewrite the two perturbation series in a standard procedure
[13, 14, 15] as

$O^{\prime\pm}_{l,m}=O^{\pm}_{l,m}+\sum_{i=1}\sum_{n\neq\pm
l,k}a^{\pm(i)}_{(l,m)(n,k)}O_{n,k},$ (A.3)

where index $i$ signifies the perturbation order. The two individual series
are labeled by $+$ and $-$ signs, with $n\neq+l$ applying to the former and
$n\neq-l$ applying to the latter. The coefficients, $a^{\pm(i)}_{(l,m)(n,k)}$,
are defined in the same way as $a_{(l,m)(n,k)}^{(i)}$ (see Eqs. 14 and 15),
with the difference, however, that the matrix elements, $\delta H_{(n,k)(\pm
l,m)}=<O_{n,k}|\delta H|O_{\pm l,m}>$ (see Eq. 13) are replaced with $\delta
H^{\pm}_{(n,k)(l,m)}=<O_{n,k}|\delta H|O^{\pm}_{l,m}>$. Substituting Eqs. A.1
and A.2, we obtain

$\delta H^{+}_{(n,k)(l,m)}=<O_{n,k}|\delta H|O^{+}_{l,m}>=U_{11}\delta
H_{(n,k)(l,m)}\delta_{n,l\pm 2}+U_{12}\delta H_{(n,k)(-l,m)}\delta_{n,-l\pm
2}$ (A.4)

and

$\delta H^{-}_{(n,k)(l,m)}=<O_{n,k}|\delta H|O^{-}_{l,m}>=U_{21}\delta
H_{(n,k)(l,m)}\delta_{n,l\pm 2}+U_{22}\delta H_{(n,k)(-l,m)}\delta_{n,-l\pm
2}.$ (A.5)

Replacing $\delta H_{(n,k)(l,m)}$ with Eqs. A.4 and A.5 in the perturbation
series, Eq. A.3, transforms it to

$O^{\prime\pm}_{l,m}=O^{\pm}_{l,m}+\sum_{i=1}\sum_{k}a^{(i)}_{(l,m)(n,k)}O^{\pm}_{n,k}$
(A.6)

where

$O^{+}_{n,k}=U_{11}O_{n,k}+U_{12}O_{-n,k},$ (A.7)
$O^{-}_{n,k}=U_{21}O_{n,k}+U_{22}O_{-n,k},$ (A.8)

and $n=l\pm 2i$ (both these values are included in the sum in Eq. A.6).
Starting from Eq. A.3, we have thus obtained a new form of the series,
expressed entirely in terms of linear combinations of the degenerate states.
Furthermore, these linear combinations, regardless of the topological charge,
are described uniquely by the elements of a single unitary matrix $U$ (to be
determined later). The mixing coefficients of $O^{\pm}_{n,k}$ are the same as
those in the original perturbation series for $O^{\prime}_{\pm l,m}$.

Consider now

$(H+\epsilon\delta
H)O^{+^{\prime}}_{l,m}=(\beta^{\prime+}_{l,m})^{2}O^{+^{\prime}}_{l,m}.$ (A.9)

Writing $(\beta^{\prime+}_{l,m})^{2}=\beta^{2}_{l,m}+\delta\beta^{\prime
2}_{l,m}$, where $\delta\beta^{\prime 2}_{l,m}$ is to be determined along with
matrix $U$, and substituting this expression in Eq. A. 9, we obtain

$(H-\beta_{l,m}^{2})O^{+^{\prime}}_{l,m}=(\delta\beta^{\prime
2}_{l,m}-\epsilon\delta H)O^{+^{\prime}}_{l,m}.$ (A.10)

Taking the scalar product on the left with $O_{l^{\prime},m}(l^{\prime}=\pm
l)$, we have

$<O_{l^{\prime},m}|H-\beta_{l,m}^{2}|O^{+^{\prime}}_{l,m}>=<O_{l^{\prime},m}|\delta\beta^{\prime
2}_{l,m}-\epsilon\delta H|O^{+^{\prime}}_{l,m}>.$ (A.11)

Now
$<O_{l^{\prime},m}|H-\beta_{l,m}^{2}|O^{+^{\prime}}_{l,m}>=<(H-\beta_{l,m}^{2})O_{l^{\prime},m}|O^{+^{\prime}}_{l,m}>$
due to the Hermiticity of $H-\beta_{l,m}^{2}$. But
$(H-\beta_{l,m}^{2})O_{l^{\prime},m}=0$. Therefore, the LHS of Eq. A.11 equals
zero. It follows then

$<O_{l^{\prime},m}|\delta\beta^{\prime 2}_{l,m}-\epsilon\delta
H|O^{+^{\prime}}_{l,m}>=0.$ (A.12)

Inserting Eqs. A. 6, A.1, and A.7 into Eq. A.12, we obtain

$\begin{split}&<O_{l^{\prime},m}|\epsilon\delta H-\delta\beta^{\prime
2}_{l,m}|(U_{11}O_{l,m}+U_{12}O_{-l,m})>\\\
&+\sum_{i=1}^{\infty}\sum_{k}a^{(i)}_{(l,m)(l\pm
2i,k)}<O_{l^{\prime},m}|\epsilon\delta H-\delta\beta^{\prime
2}_{l,m}|U_{11}O_{l\pm 2i,k}+U_{12}O_{-(l\pm 2i),k}>=0.\end{split}$ (A.13)

Now $l^{\prime}=\pm l$ leads to two linear equations in $U_{11}$ and $U_{12}$:

1) $l^{\prime}=l$

$\epsilon\sum_{i=1}^{\infty}\sum_{k}a^{(i)}_{(l,m)(l\pm 2i,k)}(U_{11}\delta
H_{(l,m)(l\pm 2i,k)}+U_{12}\delta H_{(l,m)(-(l\pm
2i),k)})-U_{11}\delta\beta^{\prime 2}_{l,m}=0.$ (A.14)

2) $l^{\prime}=-l$

$\epsilon\sum_{i=1}^{\infty}\sum_{k}a^{(i)}_{(-l,m)(l\pm 2i,k)}(U_{11}\delta
H_{(-l,m)(l\pm 2i,k)}+U_{12}\delta H_{(-l,m)(-(l\pm
2i),k)})-U_{12}\delta\beta^{\prime 2}_{l,m}=0.$ (A.15)

The above homogeneous equations have a non-trivial solutions if and only if
the determinant of the coefficients of $U_{11}$ and $U_{12}$ is zero. Invoking
the real and symmetric nature of $\delta H$ matrix elements and the fact that
the mixing coefficients for $l$ and $-l$ cases are identical in values (see
Section 3), it is easy to see that the corresponding matrix is of the form,
$M-\delta\beta^{\prime 2}_{l,m}I$, where matrix $M$ is symmetric (with
diagonal elements equal) and $I$ is the $2~{}x~{}2$ identity matrix. For
degeneracy to be broken between the $OAM_{l,m}$ and $OAM_{-l,m}$ modes, the
off diagonal elements of the $M$ matrix must be non-zero. Examining the
coefficient of $U_{12}$ in Eq. A.14, which is an off diagonal element of the
$2~{}x~{}2$ M matrix, we can set $\delta H_{(l,m)(-(l\pm 2i),k)}=\delta
H_{(-l,m)(l\pm 2i,k)}=\delta H_{(l\pm 2i,k)(-l,m))}$, using the symmetric
properties of the $\delta H$ matrix elements (see Section 3). Invoking the
$|\Delta l|=2$ rule, this element is non zero only if $i=l-1$ (this follows
from choosing the - sign in the index $l\pm 2i$; the + sign gives an
infeasible solution). The value of this off diagonal term, denoted
$\gamma_{l,m}$, is given by

$\gamma_{l,m}=\sum_{k}\epsilon~{}a^{(l-1)}_{(l,m)(-l+2,k)}\delta
H_{(-l+2,k)(-l,m)}.$ (A.16)

Expanding out the $a^{(l-1)}_{(l,m)(-l+2,k)}$ coefficient as a product of
$l-1~{}\delta H$ elements and the $l-1$ accompanying propagators (see Section
3), we obtain the form

$\begin{split}\gamma_{l,m}=&(\epsilon)^{l}\sum\frac{\delta
H_{(l,m)(l-2,n)}\delta
H_{(l-2,n)(l-4,p)}....}{(\beta_{l,m}^{2}-\beta_{l-2,n}^{2})(\beta_{l,m}^{2}-\beta_{l-4,p}^{2})....}\\\
&\frac{....\delta
H_{(-l+4,q)(-l+2,\color[rgb]{0,0,0}k\color[rgb]{0,0,0})}\delta
H_{(-l+2,k)(-l,m)}}{....(\beta_{l,m}^{2}-\beta_{-l+4,q}^{2})(\beta_{l,m}^{2}-\beta_{-l+2,k}^{2})}~{}(l>1).\end{split}$
(A.17)

Identical result follows if we examine the other off diagonal term of the M
matrix (coefficient of $U_{11}$ in Eq. A.15). For $l=1$
$\gamma_{l,m}=\epsilon\delta H_{(l,m)(-l,m)}~{}(l=1)$. Furthermore, we observe
from Eq. A.14 that in the lowest order, which corresponds to $i=1$, the
coefficient of $U_{11}$, denoted $\kappa_{l,m}$, is given by

$\kappa_{l,m}=\sum_{k}\epsilon a^{(1)}_{(l,m)(l+2,k)}\delta
H_{(l,m)(l+2,k)}+\sum_{k^{\prime}}a^{(1)}_{(l,m)(l-2,k^{\prime})}\delta
H_{(l,m)(l-2,k^{\prime})}.$ (A.18)

Substituting Eq. 14, we find

$\begin{split}\kappa_{l,m}=&\epsilon^{2}\bigg{(}\sum_{k}\frac{\delta
H_{(l,m)(l+2,k)}\delta
H_{(l+2,k)(l,m)}}{(\beta_{l,m}^{2}-\beta_{l+2,k}^{2})}\\\
&+\sum_{k^{\prime}}\frac{\delta H_{(l,m)(l-2,k^{\prime})}\delta
H_{(l-2,k^{\prime})(l,m)}}{(\beta_{l,m}^{2}-\beta_{l-2,k^{\prime}}^{2})}\bigg{)},\end{split}$
(A.19)

which is identified with $\beta^{2(2)}_{l,m}$ (see Eq. 17); this is not
surprising, since $\kappa_{l,m}$ is the diagonal term of the $2~{}x~{}2$
matrix $M$ in the $(l,-l)$ subspace. It is the nonzero nature of the off
diagonal term, $\gamma_{l,m}$, which breaks the degeneracy. Matrix M can now
be written as

$M=\begin{bmatrix}\beta^{2(2)}_{l,m}&\gamma_{l,m}\\\
\gamma_{l,m}&\beta^{2(2)}_{l,m}\end{bmatrix}.$ (A.20)

The eigenvalue equation is $(M-\delta\beta^{\prime 2}_{l,m}I)\phi=0$, where
$\delta\beta^{\prime 2}_{l,m}$ is the eigenvalue of matrix $M$, and $\phi$ (a
$2~{}x~{}1$ column vector) the corresponding eigenvector. Setting the
determinant of ($M-\delta\beta^{\prime 2}_{l,m}I$) to zero then yields the two
eigenvalues:

$\delta\beta^{\prime\pm 2}_{l,m}=\beta^{2(2)}_{l,m}\pm\gamma_{l,m}.$ (A.21)

That is,

$(\beta^{{}^{\prime}\pm}_{l,m})^{2}=\beta^{2}_{l,m}+\beta^{2(2)}_{l,m}\pm\gamma_{l,m}.$
(A.22)

The corresponding eigenvectors are $\frac{1}{\sqrt{2}}\begin{bmatrix}1\\\
1\end{bmatrix}$ and $\frac{1}{\sqrt{2}}\begin{bmatrix}1\\\ -1\end{bmatrix}$,
which are identified with the two column vectors of the $U$ matrix. That is,
$U_{11}=\frac{1}{\sqrt{2}}$, $U_{12}=\frac{1}{\sqrt{2}}$,
$U_{21}=\frac{1}{\sqrt{2}}$, and $U_{22}=-\frac{1}{\sqrt{2}}$. The unitary
matrix $U$ is independent of $l$. Inserting the numerical values of the
appropriate $U_{ij}$ elements in Eqs. A.1 and A.2, we obtain
$O^{+}_{l,m}=\frac{1}{\sqrt{2}}(O_{l,m}+O_{-l,m})$ and
$O^{-}_{l,m}=\frac{1}{\sqrt{2}}(O_{l,m}-O_{-l,m})$. Similar insertions in Eqs.
A.7 and A.8 yield $O^{+}_{n,k}=\frac{1}{\sqrt{2}}(O_{n,k}+O_{-n,k})$ and
$O^{-}_{n,k}=\frac{1}{\sqrt{2}}(O_{n,k}-O_{-n,k})$, where $n=l+2i,i>0$. As a
result, we now know the entire perturbation series, Eq. A.6.

Below, we indicate a few example forms for $\gamma_{l,m}$, using Eq. A.17. For
$l=2$,

$\gamma_{2,m}=\epsilon^{2}\sum_{i}\frac{(\delta H_{(2,m)(0,i)})(\delta
H_{(0,i)(-2,m)})}{\beta^{2}_{2,m}-\beta^{2}_{0,i}}.$ (A.23)

For $l=3$

$\gamma_{3,m}=\epsilon^{3}\sum_{i,j}\frac{(\delta H_{(3,m)(1,i)})(\delta
H_{(1,i)(-1,j)})(\delta
H_{(-1,j)(-3,m)})}{(\beta^{2}_{3,m}-\beta^{2}_{1,i})(\beta^{2}_{3,m}-\beta^{2}_{1,j})},$
(A.24)

where we have used the fact that $\beta^{2}_{-1,j}=\beta^{2}_{1,j}$.

We can similarly write down expressions for higher values of $l$.

##  Appendix B Derivation of the Bounds on Ellipticity

The spin-orbit ($SO$) interaction correction to the scalar propagation
constant, denoted $\delta\beta^{2(SO)}_{(l,m)}$, is of the order of
$l\Delta/a^{2}$ [8, 16, 17]. These corrections are of opposite sign for the
spin-orbit aligned case (e.g., $+l,S=+1$) and spin-orbit anti-aligned case
(e.g.,$-l,S=+1$), where $S$ denotes the spin of the OAM mode, with $S=\pm 1$
corresponding to left/right circular polarization. Consequently, we expect
these differences to manifest as unequal diagonal elements in the $2~{}x~{}2$
matrix considered in Section 3.1 and Appendix A. Therefore, to maintain the
validity of our results, we must ensure that these differences due to the SO
effects are much smaller than the splitting of the degenerate spatial modes,
$2\gamma_{l,m}$ due to ellipticity $\epsilon$ (Eq. 22), i.e., the $(l,-l)$
degeneracy is broken primarily through the ellipticity effects. By examining
the analytic form for $\gamma_{l,m}$ ( Eq. 22), we estimate the condition
under which this may hold.

From Eq. 13, we see that $\epsilon\delta H$ is of the order of $\epsilon\Delta
k^{2}n_{1}^{2}$. The magnitude of $\gamma_{l,m}$ in Eq. 22 is determined by
$l$ such factors in the numerator and $l-1$ factors involving propagation
constant differences:
$\beta^{2}_{l^{\prime},m^{\prime}}-\beta^{2}_{l^{\prime}\pm
2,m^{\prime\prime}}\approx
2kn_{1}(\beta_{l^{\prime},m^{\prime}}-\beta_{l^{\prime}\pm
2,m^{\prime\prime}})$. We now recall that the total number of $(l,m)$ pairs in
a fiber (not counting polarization states and the degenerate counterparts) is
approximately $\alpha V^{2}/8$ (for large $V$) [8, 19], where $\alpha$ varies
from 0.5 for a parabolic fiber to 1 for a step-index fiber. This implies an
average mode spacing of $k(n_{1}-n_{2})/(\alpha V^{2}/8)$ that can for
practical purposes for few mode (small $V$) as well as multimode (large $V$)
fibers be represented approximately by $k(n_{1}-n_{2})/V$; $V$ is the
normalized frequency equal to
$ka(n_{1}^{2}-n_{2}^{2})^{1/2}=kan_{1}(2\Delta)^{1/2}$. Thus,
$\gamma_{l,m}\sim(\epsilon\Delta
k^{2}n_{1}^{2})^{l}/(2k{}^{2}\color[rgb]{0,0,0}n_{1}(n_{1}-n_{2})/V)^{l-1}=\Delta
k^{2}n_{1}^{2}\epsilon^{l}(V/2)^{l-1}$. Imposing the condition
$\delta\beta^{2(SO)}_{(l,m)}<<\gamma_{l,m}$ now leads to the inequality

$\Big{(}\frac{l}{k^{2}a^{2}n_{1}^{2}(V/2)^{l-1}}\Big{)}<<\epsilon^{l}.$ (B.1)

We further derive an upper bound from the requirement that the magnitude of
the first order amplitudes $a^{(1)}_{(l,m)(l^{\prime},m^{\prime})}$ (Eq. 14)
be much smaller than unity[13] for results to be valid in first order
perturbation. From the above discussion, this implies $\epsilon\Delta
k^{2}n_{1}^{2}<<2k^{2}n_{1}(n_{1}-n_{2})/V$, yielding

$\epsilon<<2/V.$ (B.2)

Combining Eqs. B.1 and B.2, we obtain

$\Big{(}\frac{lV}{2k^{2}a^{2}n_{1}^{2}}\Big{)}^{1/l}\Big{(}\frac{2}{V}\Big{)}<<\epsilon<<\frac{2}{V}.$
(B.3)

We note here that the left hand side inequality becomes less stringent
(compared to Eq. B.1) for higher values of $l$.

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