Theoretical study of coaxial fibers

Frederico Dias Nunes; Cláudia Adriana de Souza Melo; Humberto Filomeno da Silva Filho

doi:10.1364/AO.35.000388

Applied Optics
Vol. 35,
Issue 3,
pp. 388-399
(1996)
•https://doi.org/10.1364/AO.35.000388

Theoretical study of coaxial fibers

Frederico Dias Nunes, Cláudia Adriana de Souza Melo, and Humberto Filomeno da Silva Filho

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Frederico Dias Nunes, Cláudia Adriana de Souza Melo, and Humberto Filomeno da Silva Filho, "Theoretical study of coaxial fibers," Appl. Opt. 35, 388-399 (1996)

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Abstract

A mathematical solution of the wave equation for coaxial fibers having four different refractive-index profiles is presented. The transcendental equations are obtained under LP approximation and calculated for comparison of HE_mn-mode-dispersion characteristics. Attention is paid to the HE₁₁ and HE₁₂ modes because of their importance for modeling directional couplers, and calculations are carried out to obtain dispersion dependence on dimensional parameters of the fibers. The field expressions also are given, and the spatial distributions of the HE₁₁ and HE₁₂ modal fields of all coaxial structures for different conditions of propagation are calculated.

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Region	n ₄ < n_e < n ₂ (WI)	n ₂ < n_e < n ₃ (WI)	n ₃ < n_e < n ₁
Region	n ₄ < n_e < n ₂ (WI)	n ₄ < n_e < n ₃ (WII)	(WI and WII)
Rod	$\begin{array}{l} A J_{m} (\frac{U_{1} ρ}{a}) cos (m θ) \\ U_{1} = {({k_{0}}^{2} {n_{1}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} A J_{m} (\frac{U_{1} ρ}{a}) cos (m θ) \\ U_{1} = {({k_{0}}^{2} {n_{1}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} A J_{m} (\frac{U_{1} ρ}{a}) cos (m θ) \\ U_{1} = {({k_{0}}^{2} {n_{1}}^{2} - β^{2})}^{1 / 2} a \end{array}$
Gap	$\begin{array}{l} [B J_{m} (\frac{W_{2} ρ}{a}) + C Y_{m} (\frac{W_{2} ρ}{a})] cos (m θ) \\ W_{2} = {({k_{0}}^{2} {n_{2}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} [B I_{m} (\frac{W_{2} ρ}{a}) + C K_{m} (\frac{W_{2} ρ}{a})] cos (m θ) \\ W_{2} = {({k_{0}}^{2} {n_{2}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} [B I_{m} (\frac{W_{2} ρ}{a}) + C K_{m} (\frac{W_{2} ρ}{a})] cos (m θ) \\ W_{2} = {(β^{2} - {k_{0}}^{2} {n_{2}}^{2})}^{1 / 2} a \end{array}$
Tube	$\begin{array}{l} [D J_{m} (\frac{U_{3} ρ}{b}) + E Y_{m} (\frac{U_{3} ρ}{b})] cos (m θ), \\ U_{3} = {({k_{0}}^{2} {n_{3}}^{2} - β^{2})}^{1 / 2} b \end{array}$	$\begin{array}{l} [D J_{m} (\frac{U_{3} ρ}{b}) + E Y_{m} (\frac{U_{3} ρ}{b})] cos (m θ) \\ U_{3} = {({k_{0}}^{2} {n_{3}}^{2} - β^{2})}^{1 / 2} b \end{array}$	$\begin{array}{l} [D I_{m} (\frac{U_{3} ρ}{b}) + E K_{m} (\frac{U_{3} ρ}{b})] cos (m θ) \\ U_{3} = {(β^{2} - {k_{0}}^{2} {n_{3}}^{2})}^{1 / 2} b \end{array}$
Clad	$\begin{array}{l} F K_{m} (\frac{W_{4} ρ}{b}) cos (m θ) \\ W_{4} = {(β^{2} - {k_{0}}^{2} {n_{4}}^{2})}^{1 / 2} c \end{array}$	$\begin{array}{l} F K_{m} (\frac{W_{4} ρ}{b}) cos (m θ) \\ W_{4} = {(β^{2} - {k_{0}}^{2} {n_{4}}^{2})}^{1 / 2} c \end{array}$	$\begin{array}{l} F K_{m} (\frac{W_{4} ρ}{b}) cos (m θ) \\ W_{4} = {(β^{2} - {k_{0}}^{2} {n_{4}}^{2})}^{1 / 2} c \end{array}$

Effective-Refractive-Index Range	Transcendental Equation
n ₄ < n_e < n ₂ (WI)	$\frac{J_{m \pm 1} (U_{4})}{Y_{m \pm 1} (U_{4})} [\frac{J (U_{4}) \mp K (W_{4})}{Y (U_{4}) \mp K (W_{4})}] {\frac{Y_{m \pm 1} (W_{3})}{Y_{m \pm 1} (W_{2})} [\frac{Y (U_{3}) - Y (W_{3})}{J (U_{1}) - Y (W_{2})}] - \frac{J_{m \pm 1} (W_{3})}{J_{m \pm 1} (W_{2})} [\frac{Y (U_{3}) - J (W_{3})}{J (U_{1}) - J (W_{2})}]}$
	$= \frac{J_{m \pm 1} (U_{3})}{Y_{m \pm 1} (U_{3})} {\frac{Y_{m \pm 1} (W_{3})}{Y_{m \pm 1} (W_{2})} [\frac{J (U_{3}) - Y (W_{3})}{J (U_{1}) - Y (W_{2})}] - \frac{J_{m \pm 1} (W_{3})}{J_{m \pm 1} (W_{2})} [\frac{J (U_{3}) - J (W_{3})}{J (U_{1}) - J (W_{2})}]}$
n ₂ < n_e < n ₃ (WI)	$\frac{J_{m \pm 1} (U_{4})}{Y_{m \pm 1} (U_{4})} [\frac{J (U_{4}) \mp K (W_{4})}{Y (U_{4}) \mp K (W_{4})}] {\frac{\pm K_{m \pm 1} (W_{3})}{\mp K_{m \pm 1} (W_{2})} [\frac{Y (U_{3}) \mp K (W_{3})}{J (U_{1}) \mp K (W_{2})}] - \frac{I_{m \pm 1} (W_{3})}{I_{m \pm 1} (W_{2})} [\frac{Y (U_{3}) \pm I (W_{3})}{J (U_{1}) \pm I (W_{2})}]}$
n ₄ < n_e < n ₃ (WII)	$= \frac{J_{m \pm 1} (U_{3})}{Y_{m \pm 1} (U_{3})} {\frac{K_{m \pm 1} (W_{3})}{K_{m \pm 1} (W_{2})} [\frac{\pm J (U_{3}) - K (W_{3})}{\mp J (U_{1}) + K (W_{2})}] - \frac{I_{m \pm 1} (W_{3})}{I_{m \pm 1} (W_{2})} [\frac{J (U_{3}) \pm I (W_{3})}{J (U_{1}) \pm I (W_{2})}]}$
n ₃ < n_e < n ₁ (WI and WII)	$\frac{I_{m \pm 1} (U_{4})}{K_{m \pm 1} (U_{4})} [\frac{I (U_{4}) + K (W_{4})}{K (U_{4}) - K (W_{4})}] {\frac{K_{m \pm 1} (W_{3})}{K_{m \pm 1} (W_{2})} [\frac{K (W_{3}) - K (U_{3})}{K (W_{2}) \mp J (U_{1})}] + \frac{I_{m \pm 1} (W_{3})}{I_{m \pm 1} (W_{2})} [\frac{I (W_{3}) - K (U_{3})}{I (W_{2}) \mp J (U_{1})}]}$
	$= \frac{I_{m \pm 1} (U_{3})}{K_{m \pm 1} (U_{3})} {\frac{K_{m \pm 1} (W_{3})}{K_{m \pm 1} (W_{2})} [\frac{K (W_{3}) - I (U_{3})}{K (W_{2}) \pm J (U_{1})}] + \frac{I_{m \pm 1} (W_{3})}{I_{m \pm 1} (W_{2})} [\frac{I (W_{3}) - I (U_{3})}{I (W_{2}) \pm J (U_{1})}]}$

Region	n ₄ < n_e < n ₂ (MI)	n ₂ < n_e < n ₁ − (MI)	n ₁ < n_e < n ₃
Region	n ₄ < n_e < n ₂ (MI)	n ₄ < n_e < n ₁ − (MII)	(MI and MII)
Rod	$\begin{array}{l} A J_{m} (\frac{U_{1} ρ}{a}) cos (m θ) \\ U_{1} = {({k_{0}}^{2} {n_{1}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} A J_{m} (\frac{U_{1} ρ}{a}) cos (m θ) \\ U_{1} = {({k_{0}}^{2} {n_{1}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} A I_{m} (\frac{U_{1} ρ}{a}) cos (m θ) \\ U_{1} = {(β^{2} - {k_{0}}^{2} {n_{1}}^{2})}^{1 / 2} a \end{array}$
Gap	$\begin{array}{l} [B J_{m} (\frac{W_{2} ρ}{a}) + C Y_{m} (\frac{W_{2} ρ}{a})] cos (m θ) \\ W_{2} = {({k_{0}}^{2} {n_{2}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} [B I_{m} (\frac{W_{2} ρ}{a}) + C K_{m} (\frac{W_{2} ρ}{a})] cos (m θ) \\ W_{2} = {(β^{2} - {k_{0}}^{2} {n_{2}}^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} [B I_{m} (\frac{W_{2} ρ}{a}) + C K_{m} (\frac{W_{2} ρ}{a})] cos (m θ) \\ W_{2} = {(β^{2} - {k_{0}}^{2} {n_{2}}^{2})}^{1 / 2} a \end{array}$
Tube	$\begin{array}{l} [D J_{m} (\frac{U_{3} ρ}{b}) + E Y_{m} (\frac{U_{3} ρ}{b})] cos (m θ), \\ U_{3} = {({k_{0}}^{2} {n_{3}}^{2} - β^{2})}^{1 / 2} b \end{array}$	$\begin{array}{l} [D J_{m} (\frac{U_{3} ρ}{b}) + E Y_{m} (\frac{U_{3} ρ}{b})] cos (m θ) \\ U_{3} = {({k_{0}}^{2} {n_{3}}^{2} - β^{2})}^{1 / 2} b \end{array}$	$\begin{array}{l} [D J_{m} (\frac{U_{3} ρ}{b}) + E Y_{m} (\frac{U_{3} ρ}{b})] cos (m θ) \\ U_{3} = {({k_{0}}^{2} {n_{3}}^{2} - β^{2})}^{1 / 2} b \end{array}$
Clad	$\begin{array}{l} F K_{m} (\frac{W_{4} ρ}{c}) cos (m θ) \\ W_{4} = {(β^{2} - {k_{0}}^{2} {n_{4}}^{2})}^{1 / 2} c \end{array}$	$\begin{array}{l} F K_{m} (\frac{W_{4} ρ}{c}) cos (m θ) \\ W_{4} = {(β^{2} - {k_{0}}^{2} {n_{4}}^{2})}^{1 / 2} c \end{array}$	$\begin{array}{l} F K_{m} (\frac{W_{4} ρ}{c}) cos (m θ) \\ W_{4} = {(β^{2} - {k_{0}}^{2} {n_{4}}^{2})}^{1 / 2} c \end{array}$

Effective-Refractive-Index Range	Transcendental Equation
n ₄ < n_e < n ₂ (MI)	$\frac{J_{m \pm 1} (U_{4})}{Y_{m \pm 1} (U_{4})} [\frac{J (U_{4}) \mp K (W_{4})}{Y (U_{4}) \mp K (W_{4})}] {\frac{Y_{m \pm 1} (W_{3})}{Y_{m \pm 1} (W_{2})} [\frac{Y (U_{3}) - Y (W_{3})}{J (U_{1}) - Y (W_{2})}] - \frac{J_{m \pm 1} (W_{3})}{J_{m \pm 1} (W_{2})} [\frac{Y (U_{3}) - J (W_{3})}{J (U_{1}) - J (W_{2})}]}$
	$= \frac{J_{m \pm 1} (U_{3})}{Y_{m \pm 1} (U_{3})} {\frac{Y_{m \pm 1} (W_{3})}{Y_{m \pm 1} (W_{2})} [\frac{J (U_{3}) - Y (W_{3})}{J (U_{1}) - Y (W_{2})}] - \frac{J_{m \pm 1} (W_{3})}{J_{m \pm 1} (W_{2})} [\frac{J (U_{3}) - J (W_{3})}{J (U_{1}) - J (W_{2})}]}$
n ₂ < n_e < n ₁ (MI)	$\frac{J_{m \pm 1} (U_{4})}{Y_{m \pm 1} (U_{4})} [\frac{J (U_{4}) \mp K (W_{4})}{Y (U_{4}) \mp K (W_{4})}] {\frac{\pm K_{m \pm 1} (W_{3})}{\mp K_{m \pm 1} (W_{2})} [\frac{Y (U_{3}) \mp K (W_{3})}{J (U_{1}) \mp K (W_{2})}] - \frac{I_{m \pm 1} (W_{3})}{I_{m \pm 1} (W_{2})} [\frac{Y (U_{3}) \pm I (W_{3})}{J (U_{1}) \pm I (W_{2})}]}$
n ₄ < n_e < n ₁ (MII)	$= \frac{J_{m \pm 1} (U_{3})}{Y_{m \pm 1} (U_{3})} {\frac{K_{m \pm 1} (W_{3})}{K_{m \pm 1} (W_{2})} [\frac{\pm J (U_{3}) - K (W_{3})}{\mp J (U_{1}) + K (W_{2})}] - \frac{I_{m \pm 1} (W_{3})}{I_{m \pm 1} (W_{2})} [\frac{J (U_{3}) \pm I (W_{3})}{J (U_{1}) \pm I (W_{2})}]}$
n ₁ < n_e < n ₃ (MI)	$\frac{J_{m \pm 1} (U_{4})}{Y_{m \pm 1} (U_{4})} [\frac{J (U_{4}) \mp K (W_{4})}{Y (U_{4}) \mp K (W_{4})}] {\frac{K_{m \pm 1} (W_{3})}{K_{m \pm 1} (W_{2})} [\frac{\pm Y (U_{3}) - K (W_{3})}{I (U_{1}) + K (W_{1})}] - \frac{I_{m \pm 1} (W_{3})}{I_{m \pm 1} (W_{2})} [\frac{\pm Y (U_{3}) + I (W_{3})}{I (U_{1}) - I (W_{2})}]}$
n ₁ < n_e < n ₃ (MII)	$= \frac{J_{m \pm 1} (U_{3})}{Y_{m \pm 1} (U_{3})} {\frac{K_{m \pm 1} (W_{3})}{K_{m \pm 1} (W_{2})} [\frac{\pm J (U_{3}) - K (W_{3})}{I (U_{1}) + K (W_{2})}] - \frac{I_{m \pm 1} (W_{3})}{I_{m \pm 1} (W_{2})} [\frac{\pm J (U_{3}) + I (W_{3})}{I (U_{1}) - I (W_{2})}]}$

Region	n ₄ < n_e < n ₂ (WI)	n ₂ < n_e < n ₃ (WI)	n ₃ < n_e < n ₁
Region	n ₄ < n_e < n ₂ (WI)	n ₄ < n_e < n ₃ (WII)	(WI and WII)
Rod	$\begin{array}{l} A J_{m} (\frac{U_{1} ρ}{a}) cos (m θ) \\ U_{1} = {({k_{0}}^{2} {n_{1}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} A J_{m} (\frac{U_{1} ρ}{a}) cos (m θ) \\ U_{1} = {({k_{0}}^{2} {n_{1}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} A J_{m} (\frac{U_{1} ρ}{a}) cos (m θ) \\ U_{1} = {({k_{0}}^{2} {n_{1}}^{2} - β^{2})}^{1 / 2} a \end{array}$
Gap	$\begin{array}{l} [B J_{m} (\frac{W_{2} ρ}{a}) + C Y_{m} (\frac{W_{2} ρ}{a})] cos (m θ) \\ W_{2} = {({k_{0}}^{2} {n_{2}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} [B I_{m} (\frac{W_{2} ρ}{a}) + C K_{m} (\frac{W_{2} ρ}{a})] cos (m θ) \\ W_{2} = {({k_{0}}^{2} {n_{2}}^{2} - β^{2})}^{1 / 2} a \end{array}$	$\begin{array}{l} [B I_{m} (\frac{W_{2} ρ}{a}) + C K_{m} (\frac{W_{2} ρ}{a})] cos (m θ) \\ W_{2} = {(β^{2} - {k_{0}}^{2} {n_{2}}^{2})}^{1 / 2} a \end{array}$
Tube	$\begin{array}{l} [D J_{m} (\frac{U_{3} ρ}{b}) + E Y_{m} (\frac{U_{3} ρ}{b})] cos (m θ), \\ U_{3} = {({k_{0}}^{2} {n_{3}}^{2} - β^{2})}^{1 / 2} b \end{array}$	$\begin{array}{l} [D J_{m} (\frac{U_{3} ρ}{b}) + E Y_{m} (\frac{U_{3} ρ}{b})] cos (m θ) \\ U_{3} = {({k_{0}}^{2} {n_{3}}^{2} - β^{2})}^{1 / 2} b \end{array}$	$\begin{array}{l} [D I_{m} (\frac{U_{3} ρ}{b}) + E K_{m} (\frac{U_{3} ρ}{b})] cos (m θ) \\ U_{3} = {(β^{2} - {k_{0}}^{2} {n_{3}}^{2})}^{1 / 2} b \end{array}$
Clad	$\begin{array}{l} F K_{m} (\frac{W_{4} ρ}{b}) cos (m θ) \\ W_{4} = {(β^{2} - {k_{0}}^{2} {n_{4}}^{2})}^{1 / 2} c \end{array}$	$\begin{array}{l} F K_{m} (\frac{W_{4} ρ}{b}) cos (m θ) \\ W_{4} = {(β^{2} - {k_{0}}^{2} {n_{4}}^{2})}^{1 / 2} c \end{array}$	$\begin{array}{l} F K_{m} (\frac{W_{4} ρ}{b}) cos (m θ) \\ W_{4} = {(β^{2} - {k_{0}}^{2} {n_{4}}^{2})}^{1 / 2} c \end{array}$

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Tables (6)

Equations (48)

Applied Optics