Optics Express
Vol. 12,
Issue 1,
pp. 198-207
(2004)
•https://doi.org/10.1364/OPEX.12.000198

Comparative analysis of Bragg fibers

Shangping Guo, Sacharia Albin, and Robert S. Rogowski

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Shangping Guo, Sacharia Albin, and Robert S. Rogowski, "Comparative analysis of Bragg fibers," Opt. Express 12, 198-207 (2004)

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About this Article
History
- Original Manuscript: November 25, 2003
- Revised Manuscript: December 31, 2003
- Manuscript Accepted: January 5, 2004
- Published: January 12, 2004

Abstract

In this paper, we compare three analysis methods for Bragg fibers, viz. the transfer matrix method, the asymptotic method and the Galerkin method. We also show that with minor modifications, the transfer matrix method is able to calculate exactly the leakage loss of Bragg fibers due to a finite number of H/L layers. This approach is more straightforward than the commonly used Chew’s method. It is shown that the asymptotic approximation condition should be satisfied in order to get accurate results. The TE and TM modes, and the band gap structures are analyzed using Galerkin method.

©2004 Optical Society of America

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Figures (6)

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.

Fig. 1.

Fig. 1. Bragg modes and band gap structures calculated by transfer matrix method. Left: TE, Right: TM.

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Fig. 2.

Fig. 2. TE₀₁ and TM₀₁ at k=1.2 in the Bragg fiber, calculated by transfer matrix method. Left: TE, Right: TM.

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Fig. 3.

Fig. 3. Mode field of TE₀₁, TM₀₁ at k=1.2 by asymptotic method. Left: TE, Right: TM.

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Fig. 4.

Fig. 4. The index profile of a Bragg fiber in Galerkin method.

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Fig. 5.

Fig. 5. Band gap and Bragg modes obtained by Galerkin method. Left: TE, Right: TM.

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Fig. 6.

Fig. 6. Mode fields of Bragg mode by Galerkin method. Left: TE, Right: TM.

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Tables (1)

Table 1. Comparison of calculated effective indices by three methods at k ₀ =1.2

Equations (16)

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[\begin{matrix} E_{z} \\ H_{ϕ} \\ H_{z} \\ E_{ϕ} \end{matrix}] = [\begin{matrix} J_{m} (k_{i} r) & Y_{m} (k_{i} r) & 0 & 0 \\ \frac{i ω ε}{k_{i}} J_{m}^{'} (k_{i} r) & \frac{i ω ε}{k_{i}} Y_{m}^{'} (k_{i} r) & - \frac{m β}{k_{i}^{2} r} J_{m} (k_{i} r) & - \frac{m β}{k_{i}^{2} r} Y_{m} (k_{i} r) \\ 0 & 0 & J_{m} (k_{i} r) & Y_{m} (k_{i} r) \\ - \frac{m β}{k_{i}^{2} r} J_{m} (k_{i} r) & - \frac{m β}{k_{i}^{2} r} Y_{m} (k_{i} r) & - \frac{i ω μ}{k_{i}} J_{m}^{'} (k_{i} r) & - \frac{i ω μ}{k_{i}} Y_{m}^{'} (k_{i} r) \end{matrix}] [\begin{matrix} A \\ B \\ C \\ D \end{matrix}]

[\begin{matrix} A_{1} \\ B_{1} \\ C_{1} \\ D_{1} \end{matrix}] = [\begin{matrix} T_{11} & T_{12} & T_{13} & T_{14} \\ T_{21} & T_{22} & T_{23} & T_{24} \\ T_{31} & T_{32} & T_{33} & T_{34} \\ T_{41} & T_{42} & T_{43} & T_{44} \end{matrix}] [\begin{matrix} A_{N} \\ B_{N} \\ C_{N} \\ D_{N} \end{matrix}]

[\begin{matrix} E_{z} \\ H_{ϕ} \\ H_{z} \\ E_{ϕ} \end{matrix}] = [\begin{matrix} H_{m}^{I} (k_{i} r) & H_{m}^{II} (k_{i} r) & 0 & 0 \\ \frac{i ω ε}{k_{i}} H_{m}^{' I} (k_{i} r) & \frac{i ω ε}{k_{i}} {H'}_{m}^{II} (k_{i} r) & - \frac{m β}{k_{i}^{2} r} H_{m}^{I} (k_{i} r) & - \frac{m β}{k_{i}^{2} r} H_{m}^{II} (k_{i} r) \\ 0 & 0 & H_{m}^{I} (k_{i} r) & H_{m}^{II} (k_{i} r) \\ - \frac{m β}{k_{i}^{2} r} H_{m}^{I} (k_{i} r) & - \frac{m β}{k_{i}^{2} r} H_{m}^{I} (k_{i} r) & - \frac{i ω μ}{k_{i}} H_{m}^{' i} (k_{i} r) & - \frac{i ω μ}{k_{i}} {H'}_{m}^{II} (k_{i} r) \end{matrix}] [\begin{matrix} A_{N} \\ B_{N} \\ C_{N} \\ D_{N} \end{matrix}]

[\begin{matrix} A_{1} \\ B_{1} \\ C_{1} \\ D_{1} \end{matrix}] = [\begin{matrix} T_{11}^{'} & T_{12}^{'} & T_{13}^{'} & T_{14}^{'} \\ T_{21}^{'} & T_{22}^{'} & T_{23}^{'} & T_{24}^{'} \\ T_{31}^{'} & T_{32}^{'} & T_{33}^{'} & T_{34}^{'} \\ T_{41}^{'} & T_{42}^{'} & T_{43}^{'} & T_{44}^{'} \end{matrix}] [\begin{matrix} A_{N} \\ B_{N} \\ C_{N} \\ D_{N} \end{matrix}]

[T'] = [T] \times [\begin{matrix} 1 & 1 & 0 & 0 \\ i & - i & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & i & - i \end{matrix}]

[\begin{matrix} T_{21}^{'} & T_{23}^{'} \\ T_{41}^{'} & T_{43}^{'} \end{matrix}] [\begin{matrix} A_{N} \\ C_{N} \end{matrix}] = 0

\det [\begin{matrix} T_{21}^{'} & T_{23}^{'} \\ T_{41}^{'} & T_{43}^{'} \end{matrix}] = 0 .

Loss = \frac{40 π}{λ \ln 10} Im (n_{eff})

\frac{d^{2} f}{{dr}^{2}} + \frac{1}{r} \frac{df}{dr} + (k_{0}^{2} n^{2} - β^{2} - \frac{1}{r^{2}}) f = 0

\frac{d^{2} g}{{dr}^{2}} + \frac{1}{r} \frac{dg}{dr} + (k_{0}^{2} n^{2} - β^{2} - \frac{1}{r^{2}}) g - \frac{d \ln n^{2}}{dr} (\frac{dg}{dr} + \frac{1}{r} g) = 0

x = σ r^{2} / a^{2}, h (r) = \frac{n^{2} (r) - n_{cl}^{2}}{n_{co}^{2} - n_{cl}^{2}},

V^{2} = k_{0}^{2} a^{2} (n_{co}^{2} - n_{cl}^{2}), b = \frac{{(β / k_{0})}^{2} - n_{cl}^{2}}{n_{co}^{2} - n_{cl}^{2}}

f (x) = \sum_{i = 0}^{N - 1} a_{i} φ_{i} (x), g (x) = \sum_{i = 0}^{N - 1} b_{i} φ_{i} (x)

φ_{i} (x) = \sqrt{\frac{i!}{(i + m)!}} e^{- x / 2} x^{m / 2} L_{i}^{(m)} (x)

L_{i}^{(m)} (x) = {\sum_{k = 0}^{i} \frac{(i + m)!}{(i - k)! (k + m)! k!} (- x)}^{k}

[M] [A] = b [A]

Comparison of calculated effective indices by three methods at k ₀ =1.2

	TE₀₁	Error	TM₀₁	Error
Transfer matrix method	0.7859080	-	0.5270	-
Asymptotic method	0.79935	1.7×10-2	0.5785	9.8×10-2
Galerkin method	0.7858	-1.4×10-4	0.5335	1.2×10-2

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