Finite element characterization of chromatic dispersion in nonlinear holey fibers

Takeshi Fujisawa; Masanori Koshiba

doi:10.1364/OE.11.001481

Optics Express
Vol. 11,
Issue 13,
pp. 1481-1489
(2003)
•https://doi.org/10.1364/OE.11.001481

Finite element characterization of chromatic dispersion in nonlinear holey fibers

Takeshi Fujisawa and Masanori Koshiba

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Takeshi Fujisawa and Masanori Koshiba, "Finite element characterization of chromatic dispersion in nonlinear holey fibers," Opt. Express 11, 1481-1489 (2003)

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Abstract

Chromatic dispersion characteristics of nonlinear photonic crystal fibers are, for the first time to our knowledge, theoretically investigated. A self-consistent numerical approach based on the full-vector finite-element method in terms of all the components of electric fields is described for the steady-state analysis of axially-nonsymmetrical nonlinear optical fibers. Electric fields obtained with this approach can be directly utilized for evaluating nonlinear refractive index distributions. To eliminate nonphysical, spurious solutions and to accurately model curved boundaries of circular air holes, curvilinear hybrid edge/nodal elements are introduced. It is found from the numerical results that under high optical intensity, chromatic dispersion characteristics become different from those of the linear state due to optical Kerr-effect nonlinearity, especially in short wavelength region.

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Fig. 1. Curvilinear high-order hybrid edge/nodal element.

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Fig. 2. Group velocity dispersion of circular-core optical fiber with core nonlinearlity.

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Fig. 3. Holey fiber.

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Fig. 4. Chromatic dispersion characteristics of nonlinear holey fibers with d/Λ=0.9 for different hole pitches of (a) Λ=1.0 µm, (b) Λ=1.5 µm, (c) Λ=2.0 µm, (d) Λ=2.5 µm.

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Fig. 5. Effective core areas of nonlinear holey fibers with Λ=1.5 µm and d/Λ=0.9.

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Fig. 6. Chromatic dispersion characteristics of nonlinear holey fibers with Λ=1.5 µm for different hole pitches of (a) d/Λ=0.5, (b) d/Λ=0.6, (c) d/Λ=0.7, (d) d/Λ=0.8.

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Fig. 7. Zero-dispersion wavelength of nonlinear holey fibers as a function of (a) hole pitch Λ and (b) of ratio of diameter to hole pitch d/Λ.

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Equations (23)

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\nabla \times (\nabla \times E) - k_{0}^{2} n^{2} E = 0

E (x, y, z) = e (x, y) \exp (- j βz)

n = n (x, y; {∣ e ∣}^{2}) .

E = (i_{x} e_{x} + i_{y} e_{y} + i_{z} e_{z}) \exp (- j βz)

= (i_{x} {U}^{T} + i_{y} {V}^{T}) {e_{t}}_{e} \exp (- j βz) + i_{z} j β {N}^{T} {e_{z}}_{e} \exp (- j βz)

[K (e)] {e} - β^{2} [M (e)] {e} = {0}

{e} = [\begin{matrix} {e_{t}} \\ {e_{z}} \end{matrix}]

[K (e)] = [\begin{matrix} [K_{tt} (e)] & [0] \\ [0] & [0] \end{matrix}]

[K_{tt} (e)] = \sum_{e} \iint_{e} [- \frac{\partial {U}}{\partial y} \frac{\partial {U}^{T}}{\partial y} + \frac{\partial {U}}{\partial y} \frac{\partial {V}^{T}}{\partial x} + \frac{\partial {V}}{\partial x} \frac{\partial {V}^{T}}{\partial y}

- \frac{\partial {V}}{\partial x} \frac{\partial {V}^{T}}{\partial x} + k_{0}^{2} n^{2} {U} {U}^{T} + k_{0}^{2} n^{2} {V} {V}^{T}]

[M (e)] = [\begin{matrix} [M_{tt}] & [M_{tz}] \\ [M_{zt}] & [M_{zz} (e)] \end{matrix}]

[M_{tt}] = \sum_{e} \iint_{e} [{U} {U}^{T} + {V} {V}^{T}] dx dy

[M_{tz}] = [M_{zt}]

= \sum_{e} \iint_{e} [{U} \frac{\partial {N}^{T}}{\partial x} + {V} \frac{\partial {N}^{T}}{\partial y}] dx dy

[M_{zz} (e)] = \sum_{e} \iint_{e} [\frac{\partial {N}}{\partial x} \frac{\partial {N}^{T}}{\partial x} + \frac{\partial {N}}{\partial y} \frac{\partial {N}^{T}}{\partial y} - k_{0}^{2} n^{2} {N} {N}^{T}] dx dy

P = \frac{1}{2} \iint (E \times H^{*}) \cdot i_{z} dx dy

= \frac{β}{2 k_{0} Z_{0}} ({e_{t}}^{T} [M_{tt}] {e_{t}} + {e_{t}}^{T} [M_{tz}] {e_{z}})

{e} = {e'} \sqrt{\frac{P}{P'}}

P' = \frac{β}{2 k_{0} Z_{0}} ({{e'}_{t}}^{T} [M_{tt}] {{e'}_{t}} + {{e'}_{t}}^{T} [M_{tz}] {{e'}_{z}})

g (V) = V \frac{d^{2} (V b)}{d V^{2}}

n^{2} = n_{co}^{2} + \frac{n_{co}^{2} n_{2}}{Z_{0}} {∣ e ∣}^{2} .

n^{2} = n_{L}^{2} + \frac{n_{L}^{2} n_{2}}{Z_{0}} {∣ e ∣}^{2}

A_{eff} = \frac{{(\iint {∣ E ∣}^{2} dx dy)}^{2}}{\iint {∣ E ∣}^{4} dx dy}

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Equations (23)

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