Are optical vortices robust in twisted fibres?

Abstract

We study the question of robustness of optical vortices (OVs) in twisted anisotropic and elliptical fibres with respect to external perturbations which do not depend on the longitudinal coordinate. On the basis of the developed modification of the coupled mode theory we show that the topological charge carried by OVs in twisted elliptical and anisotropic fibres proves to be robust with respect to induced material anisotropy of the fibre. In contrast, OVs and the topological charge carried are unstable with respect to induced z-independent additional ellipticity of the transverse cross-section of the fibre. In this case OVs are found to invert their initial topological charge upon propagation. We also compare the robustness of OVs in twisted fibres with their robustness in ideal fibres.

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Are optical vortices robust in twisted fibres?
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IOP PUBLISHING JOURNAL OF OPTICS
J. Opt. 14 (2012) 085702 (6pp) doi:10.1088/2040-8978/14/8/085702
Are optical vortices robust in twisted
fibres?
C N Alexeyev
Taurida National V I Vernadsky University, Vernadsky Prospekt, 4, Simferopol, 95007, Crimea, Ukraine
E-mail: c.alexeyev@yandex.ua
Received 19 April 2012, accepted for publication 27 June 2012
Published 13 July 2012
Online at stacks.iop.org/JOpt/14/085702
Abstract
We study the question of robustness of optical vortices (OVs) in twisted anisotropic and
elliptical fibres with respect to external perturbations which do not depend on the longitudinal
coordinate. On the basis of the developed modification of the coupled mode theory we show
that the topological charge carried by OVs in twisted elliptical and anisotropic fibres proves to
be robust with respect to induced material anisotropy of the fibre. In contrast, OVs and the
topological charge carried are unstable with respect to induced z-independent additional
ellipticity of the transverse cross-section of the fibre. In this case OVs are found to invert their
initial topological charge upon propagation. We also compare the robustness of OVs in twisted
fibres with their robustness in ideal fibres.
Keywords: optical vortex, spun elliptical fibres, spun anisotropic fibres
(Some figures may appear in colour only in the online journal)
1. Introduction
The past few years of the development of singular optics [1]
have unveiled the extreme relevance of optical vortices
(OVs) [2] to the problem of increasing the ‘information
capacity’ of a signal. In a vast body of research it has
been established that using OVs as information carriers could
essentially increase the amount of information which can be
encoded in a photon [3–8]. Basically, this property of OVs is
closely related to the fact that OVs are eigenstates of orbital
angular momentum (OAM) operators, and states with intrinsic
OAM possess an additional degree of freedom. The possibility
to encode information in this extra dimension makes OVs
very promising for communications in both optical and radio
bands [9–11], and in other applications. In addition to this
property, the information encoded in OVs turns out to also
be more protected: any attempt at eavesdropping in the
free-space channel would inevitably raise the uncertainty
in the main channel [4]. This also applies to fibre-optics
communications, as has been demonstrated of late [12].
However, free-space OV-based communication, along
with other factors, suffers from aberrations due to atmospheric
turbulence [13–15], which results in scattering of the initial
OAM state into the whole set of OAM states. Such a
cross-talk between the channels makes this communication
liable to signal distortions. In this regard communication via
optical waveguides seems to be more protected from external
sources. Nevertheless, in optical fibres propagation of fields
with embedded phase singularities encounters another sort of
obstacle, connected with internal factors. As has been shown,
due to the spin–orbit interaction (SOI), which inevitably arises
in spatially inhomogeneous media, OVs with topological
charge 1 are unstable in the case where the signs of their
circular polarization and topological charge are opposite [16].
Higher-order OVs in ideal fibres are stable with respect to the
SOI; however, they invert their charge upon propagation in
the presence of arbitrary small external perturbations, such as
ellipticity of the fibre’s cross-section [17] or induced one-axis
anisotropy [18].
To neutralize the influence of intrinsic SOI on the
propagation of OVs it has been suggested that twisted optical
fibres of two types should be used for OV-based fibre
communications. In the fibres of the first type the director
of the induced transverse anisotropy regularly rotates with
zincreasing. The modes of such anisotropic twisted fibres
have been shown to be linearly polarized (LP) OVs, or
12040-8978/12/085702+06$33.00 c
2012 IOP Publishing Ltd Printed in the UK & the USA

J. Opt. 14 (2012) 085702 C N Alexeyev
LVs [19]. Another type of twisted fibres is obtained when
an elliptical fibre is drawn from a rotating preform at a
constant speed. In this case in the resulting fibre the major
axis of the deformation ellipse rotates in the transverse plane
as zincreases. As has been shown, for submillimetre pitch
values the modes of such elliptical twisted fibres are given
by circularly polarized OVs [20]. In this way, the problem of
effective suppression of the SOI is solved by twisting initially
perturbed translationally invariant optical fibres.
Those papers, however, did not adequately answer
the question of stability of OVs with respect to external
perturbation. In translationally invariant fibres the robustness
of modes is usually judged by the spacing of their propagation
constants βias compared to the corrections to phase velocities,
which are introduced by the perturbation. If the latter
are comparable with the difference between βi, then the
mode structure changes under the influence of perturbation.
Otherwise, the perturbation is mainly manifested in phase
corrections. It turned out that this simple reasoning proved to
be inapplicable for twisted fibres, whose modes were shown
to be represented by Bloch modes—certain combinations of
fields each travelling with their own phase velocities. The
spectral representation of such modes, where each mode was
ascribed a single ‘propagation constant’, was found to be
possible only in the rotating with zlocal coordinate frame.
In such a frame the spectral curves of the vortex modes
for vortex-maintaining fibres are widely spaced, which was
interpreted in favour of robustness of OVs in twisted elliptical
and anisotropic fibres.
Meanwhile, some recent results concerning propagation
of OVs in coupled fibres suggest that the robustness of
OVs in twisted vortex-preserving fibres has been somewhat
overestimated. Indeed, as has been shown in [21], in coupled
elliptical twisted fibres OVs get converted in each arm of the
coupler into vortices with the opposite topological charge.
In this way, if one treats the other fibre as a perturbation to
the first fibre, then OVs exhibit instability even in a strongly
twisted fibre, where the vortex mode regime is implemented.
Such a contradictory result calls for further investigation of
the problem of stability of OVs in twisted fibres.
The aim of this paper is to study in detail the question of
robustness of OVs in twisted elliptical and anisotropic fibres.
To this end we develop a modification of the coupled mode
theory so as to enable ourselves to study the influence of
translationally invariant perturbation on the structure of modes
of translationally non-invariant fibres. We demonstrate that
OVs in twisted fibres exhibit selective robustness to external
perturbations. So, in twisted anisotropic fibres they are robust
with respect to external constant material anisotropy (stress),
whereas arbitrary small anisotropy of their form makes them
change their charge upon propagation.
2. Modified coupled mode theory
Although application of the perturbation theory method
proved to be convenient and reliable for the solution of
the problem of regularly twisted fibres, in the case where
the refractive index is given by the sum of a periodic and
an z-independent function, it seems impossible to obtain
any eigenvalue equation of the type suggested in [19,20].
In such a situation it is natural to try to make use of
another powerful (and more conventional) method, namely
the coupled mode theory (CMT). However, its classical
variant, where the zero-order Hamiltonian is assumed to
be z-independent and the perturbation is given by some
periodic in zfunction [22], should be somewhat modified
to meet the needs of our particular problem. Indeed, in our
case the modes of the twisted fibres are formed by the
periodic refractive index n(x,y,z), whereas the perturbation
term 1n(x,y)is translationally invariant. As is known, in
the scalar approximation, which seems to be sufficient for
the description of the vortex mode regime, where the SOI is
suppressed by the twisting, the transverse electric field E
E(for
simplicity we omit the subscript ‘t’) satisfies the equation [22,
23]
{∇2+k2[n2(x,y,z)+1n2(x,y,z)]}E
E(x,y,z)=0,(1)
where ∇2is the Laplace operator, k=2π/λ and λis
the wavelength in vacuum. The regular refractive index n2
describes the effect of twisting combined with the refractive
index distribution ˜n2(x,y)of an ideal fibre. For anisotropic
twisted fibres one has [19]
n2(x,y,z)= ˜n2(x,y)+1n2 0 e−2iqz
e2iqz 0!,(2)
where 1n2=1
2(n2
e−n2
o),n2
eand n2
obeing the principal values
of the transverse refractive index tensor, q=2π /H,His the
twist pitch. As usual, ˜n2(x,y)=n2
co(1−21f(x,y)), where
nco is the refractive index in the core, 1is the refractive
index contrast, and fis the profile function [23]. Note that
in equation (2) we use a representation in the basis of circular
polarizations, where E±=(Ex∓iEy)√2. The refractive index
in elliptical twisted fibres is given by [20]
n2(x,y,z)= ˜n2(x,y)−2n2
co1δrdf
drcos 2(ϕ −qz), (3)
where cylindrical polar coordinates (r, ϕ, z)are implied, δ
1 is the ellipticity parameter.
As in the standard variant of the CMT, we start from the
notion that the solutions E
Emof the zero-order equation are
known,
[∇2+k2n2(x,y,z)]E
Em(x,y,z)=0.(4)
In contrast to the standard CMT, here the dependence of E
Em
on zdoes not reduce to a simple multiplication by a factor
exp(iβz). Analogously, we search for the solutions of the
equation for a perturbed fibre,
{∇2+k2[n2(x,y,z)+1n2(x,y)]}E
E(x,y,z)=0,(5)
of the form
E
E(x,y,z)=X
m
Am(z)E
Em(x,y,z), (6)
2

J. Opt. 14 (2012) 085702 C N Alexeyev
where Am(z)are the slowly varying amplitudes. Substituting
(5) into (6) and allowing for (4) one can obtain
∇2E
E≈X
m(Am(z)∇2E
Em(x,y,z)
+2∂Am(z)
∂z
∂E
Em(x,y,z)
∂z),(7)
where, as usual, we neglected the term A00
m(z). In the standard
variant of CMT the derivative E
E0
mis replaced by the term
iβE
Em. In our case the situation is more complicated. As
follows from the results of [19], the dependence of the modes
E
Emon zis more intricate. As will be shown further, this
derivative should be replaced by the sum of two modes. In
the first approximation, however, this circumstance could be
neglected for the sake of clarity. Then the first term on the
right of (7) when combined with the Amn2E
Emterm vanishes
due to (4) and one arrives at the standard equation
X
m2iβm
dAm(z)
dzE
Em+k21n2(x,y)Am(z)E
Em=0,(8)
from whence it follows that
hk|ki2iβk
dAk
dz= −k2X
mhk|1n2|miAm.(9)
In the last equation we used standard Dirac notations,
where the scalar product implies integration over the total
cross-section of the fibre. Note that we do not specify
normalization and the phase exponentials should be included
into the structure of modes |mi. In this way, the lowest-order
approximation of coupled mode equations leads to the
standard equations, except for the difference that the modes
|midepend on zin a, generally, more complicated manner
than just through phase exponentials. Now we are in a position
to study the effect of z-invariant perturbations on the mode
structure in twisted fibres.
3. Perturbed anisotropic twisted fibres
Let us apply the CMT equations (9) to study the effect
of external perturbations on mode coupling in twisted
fibres. There are two main types of perturbation which
affect the propagation of light in optical fibres: induced
(stress-applied) material anisotropy and ellipticity of the
transverse cross-section. We consider first uniaxial anisotropy
described in the linear basis by the perturbation operator
1n2(x,y)=δn2ˆσz,(10)
where ˆσiis the Pauli matrix and δn2characterizes the induced
birefringence.
As follows from the results of [19], the modes of twisted
anisotropic fibres in the case where the vortex mode regime is
implemented are given (for the set with orbital number l=1)
by LVs, whose polarization adiabatically traces the direction
of the local anisotropy axes,
E
91=eiϕ cos qz
sin qz!L
exp i˜
β+E
2˜
βz,
E
92=e−iϕ cos qz
sin qz!L
exp i˜
β+E
2˜
βz,
E
93=eiϕ −sin qz
cos qz !L
exp i˜
β+E
2˜
βz,
E
94=e−iϕ −sin qz
cos qz !L
exp i˜
β+E
2˜
βz,
(11)
where E=1n2k2describes the initial anisotropy of the fibre,
˜
βis the scalar propagation constant and the subscript ‘L’
denotes representation in the basis of linear polarizations.
Since perturbation (10) is ϕ-independent, it couples only
the states with the same azimuthal dependence and different
polarization. Such fields for anisotropic twisted fibres are E
91
and E
93, as well as E
92and E
94. Due to (9) the coupled mode
equations for slow amplitudes A1and A3, which describe
amplitude corrections to E
91,3modes, can be shown to be
2i ˜
β
k2
dA1
dz= −δn2cos 2qzA1+δn2sin 2qze−iEz
˜
βA3,
2i ˜
β
k2
dA3
dz=δn2cos 2qzA3+δn2sin 2qzeiEz
˜
βA1.
(12)
Following the standard procedure we introduce new
amplitudes a1and a3as
A1(z)=a1(z)e−iEz
2˜
β,A3(z)=a3(z)eiEz
2˜
β,(13)
in which the set (12) becomes
ida1
dz=V11 −E
2˜
βa1+V12a3,
ida1
dz=V21a1+V22 +E
2˜
βa2,
(14)
where V12 =V21 =sin(2qz)k2δn2
2˜
β,−V11 =V22 =cos(2qz)k2δn2
2˜
β.
Generally speaking, for arbitrary values of the parameters
of (14) solving it might present a challenge. However, in
our case we have a strict hierarchy of characteristic inverse
lengths, qE
2˜
β˜
β. In this case the matrix elements Vij in
the first approximation can be treated as constants. Then if one
searches for the solutions in the form a1(z)=Aexp(−iλz)and
a3(z)=Bexp(−iλz), one obtains for the vector Ex=col(A,B)
the standard eigenvalue problem ˆ
VEx=λEα, where ˆ
V=
V11 −E
2˜
βV12
V21 V22 +E
2˜
β!. As is evident now, only if the perturbation
elements, which are proportional to k2δn2
2˜
β, are comparable
with E
2˜
βdoes the presence of perturbation lead to change
of the mode structure. Otherwise, the eigenvectors are given
by Ex1≈(1,0),Ex2≈(0,1), which means robustness of the
initial mode structure (11). In this way, the vortices are robust
with respect to induced material anisotropy in the twisted
anisotropic fibre. The value of the perturbation δnmust be
3

J. Opt. 14 (2012) 085702 C N Alexeyev
comparable with 1nto affect the vortex mode structure. This
conclusion is in agreement with the corresponding statement
of [19].
A quite different situation arises if one exposes the
twisted fibre to deformation of its transverse cross-section,
which is described by the perturbation term
1n2= −2n2
coδ1rf 0
rcos 2ϕ(15)
(see equation (13)). It is easily established that such an
operator couples the modes with the same polarization and
opposite topological charges within the mode group |l| = 1,
that is E
91and E
92, or E
93and E
94modes. The coupled mode
equations for slow amplitudes A1and A2look like
idA1
dz=δVA2,idA2
dz=δVA1,(16)
where δV=˜
βδ1
2R∞
0RF2
1(R)dR.
From (16) it immediately follows that the modes are
E
81=E
91+E
92
=cos ϕ cos qz
sin qz!L
exp i˜
β+E
2˜
β−δVz,
E
82=E
91−E
92
=sin ϕ cos qz
sin qz!L
exp i˜
βE
2˜
β+δVz.
(17)
In this way a small ellipticity drastically changes the
mode structure, and the modes are given in this case by LP
modes that trace the orientation of the local axes.
Such a mode structure results in OV conversion in a
perturbed fibre. Indeed, if the input end of such a fibre is
excited with an x-polarized charge-one OV then its evolution
in the fibre is given by
|8(z)i ∝ [−i|1,−1isin(δVz)e−iqz + |−1,1icos(δVz)eiqz]
+ [|1,1icos(δVz)e−iqz −i|−1,−1isin(δVz)eiqz],(18)
where we omit a phase multiplier. Here |σ, mi =
(1
iσ)exp(imϕ)Fm(r),σ= ±1 determines the circular polar-
ization, m=0,±1,±2, . . . is the topological charge and the
radial function satisfies the equation ∂2
∂r2+1
r
∂
∂r+k2˜n2−m2
r2−
˜
β2
mFm(r)=0. The evolution of the field (18) can be mapped
onto newly devised higher-order Poincar´
e spheres (PS) [20].
The first group of terms in square brackets in (18) determines
the trajectory on the PS with `= +1, which is described by
the following higher-order Stokes parameters:
S1
1= −sin(2qz)sin(2δVz),
S1
2=cos(2qz)sin(2δVz), S1
3= −cos(2qz). (19)
The second group in square brackets in (18) relates to the
trajectory on the PS with `= −1 and lead to the following
Stokes parameters:
S−1
1=sin(2qz)sin(2δVz),
S−1
2= −cos(2qz)sin(2δVz), S−1
3=cos(2qz). (20)
Figure 1. Schematic evolution of the state in an anisotropic twisted
fibre excited with a linearly polarized charge −1 OV on
higher-order Poincar´
e spheres with `= +1 (a) and `= −1 (b).
Inversion of the topological charge (half-period of evolution) is
shown. The arrows indicate the direction of evolution, which for the
`= +1 sphere (a) starts from the south pole and for the `= −1
sphere (b) from the north pole.
These equations define the trajectories of the evolution of the
field (18), which in the spherical coordinates r, ϕ, ϑ , naturally
associated with each of the PSs, read as
r=1, ϕ =2qz −π/2, ϑ =2δVz +π(21)
and
r=1, ϕ =2qz −π/2, ϑ =2δVz,(22)
correspondingly. The plots of these trajectories are schemat-
ically shown in figure 1. As follows, the state of the system
changes with zand, finally, the OV changes its topological
sign.
In this situation of instability of OVs it is necessary to
study the question more meticulously. A way of improving
the basic equations (9) is by taking into consideration that,
4

J. Opt. 14 (2012) 085702 C N Alexeyev
as a matter of fact, the z-derivatives of the modes (11)
are expressed through eigenmodes of the system in a more
elaborate manner than was assumed while deriving (9).
Indeed, as follows from (11), these relations are
∂E
91
∂z=iβ+E
91+qei1βzE
93,
∂E
92
∂z=iβ+E
92+qei1βzE
94,
∂E
93
∂z=iβ−E
93−qe−i1βzE
91,
∂E
94
∂z=iβ−E
94−qe−i1βzE
92,
(23)
where β±=˜
β±1β
2and 1β =E
˜
β. This leads to the following
set of equations in Ai(z):
iA0
1= ˜qA0
3e−i1βz+δVA2,
iA0
2= ˜qA0
4e−i1βz+δVA1,
iA0
3= −˜qA0
1ei1βz+δVA4,
iA0
4= −˜qA0
2ei1βz+δVA3,
(24)
where ˜q≡q/˜
β. By substituting A1=a1exp(−i1βz/2),
A2=a2exp(−i1βz/2),A3=a3exp(i1βz/2)and A4=
a4exp(i1βz/2)one can bring (24) to a translationally
invariant in zform. The resulting set can be quite easily solved
analytically. For example, for small values of the perturbation
δVthe first mode of the perturbed fibre looks like
E
81∝E
ψ1+E
ψ2−iqδV
˜
β1β (E
ψ3+E
ψ4), (25)
where we have omitted a phase factor. As is seen, allowing
for nonzero qin the equations for the field derivatives does
not lead to competition between the perturbation δVand qin
the expressions for the modes and results only in insignificant
changes in the mode structure. In this way spinning does
not improve the stability of OVs with respect to elliptical
perturbation of the cross-section.
4. Discussion
In the same manner one can treat the problem of OV stability
in twisted elliptical fibres. In this case, however, the analysis
is simplified due to the fact that in such fibres the OVs are
maintained in the form of circularly polarized fields only in
fibres with submillimetre pitches [20]. In this area of pitch
values the OVs have almost coincident values of propagation
constants, and the l=1 modes can be written as
Eα1–4 ≈e±iϕ 1
±i!L
Fl(R)ei˜
βz.(26)
As is evident, perturbation of the material anisotropy (10)
leads to coupling between modes with orthogonal polarization
creating hybrid modes ∝e±iϕ1
0Land e±iϕ0
1L. This type of
perturbation, therefore, does not affect the topological charge
of the transmitted field. In this way topological charge of
OVs, proves to be robust to material anisotropy perturbations.
However, they are unstable upon changing the form of the
transverse cross-section: any perturbation of the type (15)
would lead to the appearance of a novel mode structure,
where the modes are given by Hermite–Gaussian-like fields
∝cos ϕ
0Land the like.
The situation is analogous to the case of strongly twisted
anisotropic fibres, where the twisting perturbation exceeds
the anisotropy-induced one. As can be derived from the
corresponding result of [25], the modes of such fibres are
also represented by (26) and the above discussion applies.
It turns out that as information carriers OVs are robust in a
narrow sense (that is they preserve their topological charge
and not necessarily their polarization) to external perturbation
of the material anisotropy. However, they prove to be unstable
to perturbation of the form of the fibre’s cross-section. This
statement does not agree with the corresponding conclusion
of [20] on the stability of OVs in strongly spun elliptical fibres.
Finally, let us compare the stability of OVs in
twisted fibres and the stability of OVs in ideal fibres. As
follows from the previous results [16,23], the higher-order
l>1 modes of ideal fibres can be represented in an
alternate form through OVs. So, OVs ∝eilϕ1
i≡ |1,liand
∝e−ilϕ1
−i≡ |−1,−lipropagate with a propagation constant
βh, whereas OVs ∝eilϕ1
−i≡ |−1,liand ∝e−ilϕ1
i≡
|1,liare characterized by a propagation constant βi. The
difference between these groups of fields is due to the
presence of SOI and in weakly guiding fibres is of the order
of 1 m−1. The behaviour of such a mode system under
perturbation can be easily established either by the standard
CMT or through the technique developed in [17,18]. As is
evident, perturbation of the material anisotropy would tend
to couple the OVs with opposite circular polarization and
the same topological charge l, that is |1,li,|−1,li, and
|−1,−li,|1,−liOVs. These fields have a difference of
δβ =βh−βibetween the propagation constants. Therefore,
to change the mode structure the external perturbation must
exert a comparable action, that is k1n∼βh−βi. Such
robustness is much less pronounced than the robustness for
OVs in twisted anisotropic fibres. In contrast, OVs in ideal
fibres exhibit greater robustness to perturbation of the form
as compared to vortices in twisted fibres. Indeed, violation of
the circular form of the cross-section would tend to couple
OVs with the same polarization but opposite topological
charges, that is |1,li,|1,−liand |−1,−li,| − 1,li. To
couple such states the perturbation operator must comprise
a term ∝cos 2lϕ. As is evident, the elliptical deformation
described by (15) cannot provide such coupling, and OVs
appear to be robust to elliptical deformations of the fibre’s
cross-section. Of course, for a realistic deformation in the
Fourier expansion of the cross-section’s form all harmonics
will be present. Their weights, however, will be less than that
of the elliptic-deformation term. In this way ideal fibres allow
robust propagation of OVs with respect to perturbation of the
form. One should remember, of course, that l=1 OVs cannot
propagate in ideal fibres at arbitrary polarization [16].
5

J. Opt. 14 (2012) 085702 C N Alexeyev
5. Conclusion
In this paper we have studied the question of robustness
of OVs in twisted anisotropic and elliptical fibres with
respect to external perturbations which do not depend on
the longitudinal coordinate. We showed on the basis of the
developed modification of the coupled mode theory that
the topological charge carried by OVs in twisted elliptical
and anisotropic fibres proves to be robust with respect to
induced material anisotropy of the fibre. In contrast, OVs and
the topological charge carried are unstable with respect to
induced z-independent additional ellipticity of the transverse
cross-section of the fibre. That is, in this case OVs would
invert their initial topological charge upon propagation. We
have also compared the robustness of OVs in twisted fibres
with the robustness in ideal fibres.
Acknowledgments
The author is grateful to M A Yavorsky and A V Volyar for
useful discussions and to the reviewer for bringing [24] to his
attention.
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