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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

and EH

) 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.,

l;m

¼ HE

lþ1;m

þ EH

l−1;m

, l ≥ 1), while the combination

of intrinsic degenerate HE

odd

and HE

even

(EH

even

and

even

) modes with π∕2 phase shift generates OAM modes

[i.e., OAM

ðl−1Þ

¼ HE

even

 i×HE

odd

,OAM

ðlþ1Þ

even

 i×EH

odd

]

[8–10]

. In a conventional multimode

fiber, LP modes are easily produced by coupling because

the effective refractive index (ERI) difference between

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

while the RI of the inner and

outer clad are both n

which satisfies n

> n

. The inner

radius and outer radius of ring fiber are r

and r

, 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]

∇





þ k





¼ 0; (1)

where ∇

∂

∂r

∂

∂r

∂

∂θ

∂

∂z

, and k is the wavenum-

ber in the corresponding regions. By applying the method

of variable separation, E

can be written as

¼ RðrÞ expðilθÞ expðiβzÞ; (2)

COL 13(3), 030501(2015) CHINESE OPTICS LETTERS March 10, 2015

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