Solving characteristic equation of orbital angular momentum modes in a ring fiber

Yixiao Zhu; Fan Zhang

doi:10.3788/COL201513.030501

Journals >Chinese Optics Letters >Volume 13 >Issue 3 >Page 030501 > Article

Chinese Optics Letters
Vol. 13, Issue 3, 030501 (2015)

Solving characteristic equation of orbital angular momentum modes in a ring fiber

Yixiao Zhu and Fan Zhang^*

Author Affiliations

State Key Laboratory of Advanced Optical Communication Systems and Networks, Peking University, Beijing 100871, China

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DOI: 10.3788/COL201513.030501 Cite this Article Set citation alerts

Yixiao Zhu, Fan Zhang. Solving characteristic equation of orbital angular momentum modes in a ring fiber[J]. Chinese Optics Letters, 2015, 13(3): 030501 Copy Citation Text

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Abstract

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.

\nabla^{2} {\begin{matrix} E_{z} \\ H_{z} \end{matrix}} + k^{2} {\begin{matrix} E_{z} \\ H_{z} \end{matrix}} = 0,

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E_{z} = R (r) \exp (\pm i l θ) \exp (i β z),

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R (r) = {\begin{cases} a I_{l} (v r) & 0 \leq r < r_{1} \\ b J_{l} (u r) + c Y_{l} (u r) & r_{1} \leq r < r_{2} \\ d K_{l} (v r) & r_{2} \leq r \end{cases},

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u^{2} = n_{1}^{2} k_{0}^{2} - β^{2}, v^{2} = β^{2} - n_{2}^{2} k_{0}^{2},

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{\begin{cases} E_{r} = \frac{i β}{ω^{2} μ ε - β^{2}} (\frac{\partial E_{z}}{\partial r} + \frac{ω μ}{β r} \frac{\partial H_{z}}{\partial θ}) \\ E_{θ} = \frac{i β}{ω^{2} μ ε - β^{2}} (- \frac{ω μ}{β} \frac{\partial H_{z}}{\partial r} + \frac{1}{r} \frac{\partial E_{z}}{\partial θ}) \\ H_{r} = \frac{i β}{ω^{2} μ ε - β^{2}} (\frac{\partial H_{z}}{\partial r} - \frac{ω ε}{β r} \frac{\partial E_{z}}{\partial θ}) \\ H_{θ} = \frac{i β}{ω^{2} μ ε - β^{2}} (\frac{ω ε}{β} \frac{\partial E_{z}}{\partial r} + \frac{1}{r} \frac{\partial H_{z}}{\partial θ}) \end{cases},

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| \begin{array}{l} I_{l} (v r_{1}) & - J_{l} (u r_{1}) & - Y_{l} (u r_{1}) & 0 & 0 & 0 & 0 & 0 \\ \frac{i l β}{v^{2} r_{1}} I_{l} (v r_{1}) & \frac{i l β}{u^{2} r_{1}} J_{l} (u r_{1}) & \frac{i l β}{u^{2} r_{1}} Y_{l} (u r_{1}) & 0 & \frac{ω μ}{v} I_{1}^{'} (v r_{1}) & \frac{ω μ}{u} J_{1}^{'} (u r_{1}) & \frac{ω μ}{u} Y_{1}^{'} (u r_{1}) & 0 \\ 0 & 0 & 0 & 0 & I_{l} (v r_{1}) & - J_{l} (u r_{1}) & - Y_{l} (u r_{1}) & 0 \\ \frac{ω ε_{2}}{v} I_{1}^{'} (v r_{1}) & \frac{ω ε_{1}}{u} J_{1}^{'} (u r_{1}) & \frac{ω ε_{1}}{u} Y_{1}^{'} (u r_{1}) & 0 & - \frac{i l β}{v^{2} r_{1}} I_{l} (v r_{1}) & - \frac{i l β}{u^{2} r_{1}} J_{l} (u r_{1}) & - \frac{i l β}{u^{2} r_{1}} Y_{l} (u r_{1}) & 0 \\ 0 & J_{l} (u r_{2}) & Y_{l} (u r_{2}) & - K_{l} (v r_{2}) & 0 & 0 & 0 & 0 \\ 0 & \frac{i l β}{u^{2} r_{2}} J_{l} (u r_{2}) & \frac{i l β}{u^{2} r_{2}} Y_{l} (u r_{2}) & \frac{i l β}{v^{2} r_{2}} K_{l} (v r_{2}) & 0 & \frac{ω μ}{u} J_{1}^{'} (u r_{2}) & \frac{ω μ}{u} Y_{1}^{'} (u r_{2}) & \frac{ω μ}{v} K_{1}^{'} (v r_{2}) \\ 0 & 0 & 0 & 0 & 0 & J_{l} (u r_{2}) & Y_{l} (u r_{2}) & - K_{l} (v r_{2}) \\ 0 & \frac{ω ε_{1}}{u} J_{1}^{'} (u r_{2}) & \frac{ω ε_{1}}{u} Y_{1}^{'} (u r_{2}) & \frac{ω ε_{2}}{v} K_{1}^{'} (v r_{2}) & 0 & - \frac{i l β}{u^{2} r_{2}} J_{l} (u r_{2}) & - \frac{i l β}{u^{2} r_{2}} Y_{l} (u r_{2}) & - \frac{i l β}{v^{2} r_{2}} K_{l} (v r_{2}) \end{array} | = 0,

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| \begin{array}{l} \frac{i l β}{ω μ r_{1}} (\frac{1}{u^{2}} + \frac{1}{v^{2}}) & \frac{i l β}{ω μ r_{1}} (\frac{1}{u^{2}} + \frac{1}{v^{2}}) & E & F \\ ε_{0} W & ε_{0} X & - \frac{i l β}{ω r_{1}} (\frac{1}{u^{2}} + \frac{1}{v^{2}}) & - \frac{i l β}{ω r_{1}} (\frac{1}{u^{2}} + \frac{1}{v^{2}}) \\ \frac{i l β}{ω μ r_{2}} (\frac{1}{u^{2}} + \frac{1}{v^{2}}) & \frac{i l β}{ω μ r_{2}} (\frac{1}{u^{2}} + \frac{1}{v^{2}}) & G & H K \\ ε_{0} Y & ε_{0} Z K & - \frac{i l β}{ω μ r_{2}} (\frac{1}{u^{2}} + \frac{1}{v^{2}}) & - \frac{i l β}{ω μ r_{2}} (\frac{1}{u^{2}} + \frac{1}{v^{2}}) K \end{array} | = 0 .

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Step 1 : A_{1} = \frac{I_{l}^{'} (v r_{1})}{I_{l} (v r_{1})}, B_{i} = \frac{J_{l}^{'} (u r_{i})}{J_{l} (u r_{i})}, C_{i} = \frac{I_{l}^{'} (u r_{i})}{I_{l} (u r_{i})}, D_{2} = \frac{K_{l}^{'} (v r_{2})}{K_{l} (v r_{2})} (i = 1, 2),

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Step 2 : E = \frac{B_{1}}{u} + \frac{A_{1}}{v}, F = \frac{C_{1}}{u} + \frac{A_{1}}{v}, G = \frac{B_{2}}{u} + \frac{D_{2}}{v}, H = \frac{C_{2}}{u} + \frac{D_{2}}{v}, K = \frac{Y_{l} (u r_{2}) / Y_{l} (u r_{1})}{J_{l} (u r_{2}) / J_{l} (u r_{1})} W = \frac{n_{1}^{2} B_{1}}{u} + \frac{n_{2}^{2} A_{1}}{v}, X = \frac{n_{1}^{2} C_{1}}{u} + \frac{n_{2}^{2} A_{1}}{v}, Y = \frac{n_{1}^{2} B_{2}}{u} + \frac{n_{2}^{2} D_{2}}{v}, Z = \frac{n_{1}^{2} C_{2}}{u} + \frac{n_{2}^{2} D_{2}}{v}, M = \frac{l β}{k_{0}} (\frac{1}{u^{2}} + \frac{1}{v^{2}}) .

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\frac{M^{4}}{r_{1}^{2} r_{2}^{2}} {(K - 1)}^{2} + (K E H - F G) (K W Z - X Y) - M^{2} [\frac{(K H - G) (K Z - Y)}{r_{1}^{2}} + \frac{(K E - F) (K W - X)}{r_{2}^{2}} - K \frac{(E - F) (Y - Z) + (H - G) (X - W)}{r_{1} r_{2}}] = 0 .

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b = \frac{β^{2} - k_{0} n_{2}^{2}}{k_{0} n_{1}^{2} - k_{0} n_{2}^{2}},

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V^{2} = k_{0}^{2} {(r_{2} - r_{1})}^{2} (n_{1}^{2} - n_{2}^{2}),

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r_{ave} = (r_{1} + r_{2}) / 2 .

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Yixiao Zhu, Fan Zhang. Solving characteristic equation of orbital angular momentum modes in a ring fiber[J]. Chinese Optics Letters, 2015, 13(3): 030501

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