Solving characteristic equation of orbital angular
momentum modes in a ring fiber
Yixiao Zhu (朱逸萧) and Fan Zhang (张 帆)*
State Key Laboratory of Advanced Optical Communication Systems and Networks,
Peking University, Beijing 100871, China
*Corresponding author: fzhang@pku.edu.cn
Received October 8, 2014; accepted December 22, 2014; posted online March 12, 2015
The characteristic equation of orbital angular momentum modes in a ring fiber is derived. By solving the
equation with the graphical method, mode distribution in a ring fiber can be precisely determined for arbitrary
fiber parameters without relying on simulation of the vector field. This will provide a useful method to determine
the separation between quasi-degenerate modes in a ring fiber.
OCIS codes: 050.4865, 060.2270, 060.2310.
doi: 10.3788/COL201513.030501.
Light beams carrying orbital angular momentum (OAM)
are characterized as a spiral phase structure of expðilθÞ,
where l is topological charge (an integer), and θ is the
azimuthal angle
[1,2]
. The OAM modes with different
topological charge number l are inherently orthogonal
to each other, and therefore can be considered as an addi-
tional available degree of freedom for multiplexing
information
[1,3]
. Combining OAM with other traditional
multiplexing technologies such as wavelength division
multiplexing (WDM), the capacity and spectral efficiency
of optical communication systems will be greatly
enhanced
[3–5]
.
Although there have been several reports for free-space
transmission based on OAM
[3,5,6]
, transmission in fiber
can avoid atmospheric disturbance
[7]
, and make long-
distance transmission feasible. However, OAM modes are
unstable in terms of propagation in a conventional step-
index fiber due to the mode coupling
[2,8]
. It is known that
hybrid modes (HE
lm
and EH
lm
) are Eigen modes in a fiber.
The combination of quasi-degenerate HE
lþ1;m
and EH
l−1;m
modes results in linearly polarized (LP) modes (i.e.,
LP
l;m
¼ HE
lþ1;m
þ EH
l−1;m
, l ≥ 1), while the combination
of intrinsic degenerate HE
odd
lm
and HE
even
lm
(EH
even
lm
and
EH
even
lm
) modes with π∕2 phase shift generates OAM modes
[i.e., OAM
ðl−1Þ
¼ HE
even
lm
i×HE
odd
lm
,OAM
ðlþ1Þ
¼
EH
even
lm
i×EH
odd
lm
]
[8–10]
. In a conventional multimode
fiber, LP modes are easily produced by coupling because
the effective refractive index (ERI) difference between
HE
lþ1;m
and EH
l−1;m
modes is too small
[2,4,8]
. To overcome
this problem, several schemes were proposed such as
coiling the fiber
[11]
, using spun elliptical and anisotropic fi-
bers
[12]
, and the intensely twisted elliptical fiber was based
on band-gap Bragg selection
[13]
. Nevertheless, recently
more attention has been paid to the structure of a ring
fiber, which splits the quasi-degenerate modes by increas-
ing the ERI difference. The generation of a higher-order
OAM mode in a ring fiber has been studied
[14]
and analysis
of the modes has been given
[15]
. Then generation and
multiplexing OAM modes in a ring fiber was proposed
[16]
.
To our best knowledge, theoretical analysis of the mode
properties in a ring fiber made before are all based on
the weakly guiding approximation (WGA), which focus
on LP modes and thus cannot show the difference between
quasi-degenerate modes
[17,18]
.
In this Letter, a modal characteristic equation is derived
by rigorously solving the Helmholtz equation. Based on
this characteristic equation, we investigate the influence
of the ring fiber structure parameters on the ERI of the
Eigenmodes. Besides, it should be noted that the exact
degeneracy of two optical vortices with opposing topologi-
cal charges and spin will interact each other due to infini-
tesimal ellipticity induced by stress
[19]
. However, our
theory is applicable to the ideal ring fiber, and the situa-
tion where fibers are slightly elliptical deserves further
study.
Figure
1 shows the cross section and the refractive
index (RI) profile of a ring fiber. It consists of three
concentric regions: the inner clad, the core, and the outer
clad. The RI of the core is n
1
while the RI of the inner and
outer clad are both n
2
which satisfies n
1
> n
2
. The inner
radius and outer radius of ring fiber are r
1
and r
2
, respec-
tively. A cylindrical coordinates is set due to the longitude
invariance and the angular symmetry of the geometry.
We first deal with the longitudinal components of the
electric and magnetic field by solving the Helmholtz
equation
[20]
.
∇
2
E
z
H
z
þ k
2
E
z
H
z
¼ 0; (1)
where ∇
2
¼
∂
2
∂r
2
þ
1
r
∂
∂r
þ
1
r
2
∂
2
∂θ
2
þ
∂
2
∂z
2
, and k is the wavenum-
ber in the corresponding regions. By applying the method
of variable separation, E
z
can be written as
E
z
¼ RðrÞ expðilθÞ expðiβzÞ; (2)
COL 13(3), 030501(2015) CHINESE OPTICS LETTERS March 10, 2015
1671-7694/2015/030501(5) 030501-1 © 2015 Chinese Optics Letters
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