Coaxial periodic optical waveguide

Tetsuya Kawanishi; Masayuki Izutsu

doi:10.1364/OE.7.000010

Optics Express
Vol. 7,
Issue 1,
pp. 10-22
(2000)
•https://doi.org/10.1364/OE.7.000010

Coaxial periodic optical waveguide

Tetsuya Kawanishi and Masayuki Izutsu

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Tetsuya Kawanishi and Masayuki Izutsu, "Coaxial periodic optical waveguide," Opt. Express 7, 10-22 (2000)

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Abstract

Guided modes in a dielectric waveguide structure with a coaxial periodic multi-layer are investigated by using a matrix formula with Bessel functions. We show that guided modes exist in the structure, and that the field is confined in the core which consists of the optically thinner medium. The dispersion curves are discontinuous, so that the modes can exist only in particular wavelength bands corresponding to the stop bands of the periodic structure of the clad. It is possible that the waveguide structure can be applied to filters or optical fibers to reduce nonlinear effects.

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Fig. 1. A waveguide structure with one-dimensional periodic structures.

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Fig. 2. Definition of propagation direction of a wave in a periodic structure.

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Fig. 3. Coaxial periodic optical waveguide.

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Fig. 4. Cross-section.

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Fig. 5. Dispersion curves for k ₀ c=1.0

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Fig. 6. Dispersion curves for k ₀ c=5.0

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Fig. 7. Stopband of a coaxial periodic optical waveguide.

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Fig. 8. Stopband of a one-dimensonal photonic band gap structure.

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Fig. 9. z-direction components of magnetic field of TE-polarized mode and electric field of TM-polarized mode, where k=1.2k ₀ and β=0.7859[TE], 0.5270[TM].

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Fig. 10. z-direction components of magnetic field of TE-polarized mode and electric field of TM-polarized mode, where k=3.0k ₀ and β=0.9672[TE], 0.9672[TM].

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Fig. 11. Intensity of EM wave field in CPOW, excited by TE₀₁ mode. |T|=0.986, |R|=0.007.

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Fig. 12. Intensity of EM wave field in CPOW, excited by TE₂₁ mode. |T|=0.479, |R|=0.049.

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Fig. 13. Intensity of EM wave field in CPOW, excited by TM₀₁ mode. |T|=0.440, |R|=0.106.

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Fig. 14. Intensity of EM wave field in CPOW, excited by TM₂₁ mode. |T|=0.049, |R|=0.493.

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Equations (16)

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\cos k_{TE} (a + b) = - \frac{n_{1}^{2} \cos^{2} θ_{1} + n_{2}^{2} \cos^{2} θ_{2}}{2 n_{1} n_{2} \cos θ_{1} \cos θ_{2}} \sin (n_{1} \cos θ_{1} k a) \sin (n_{2} \cos θ_{2} k b)

+ \cos (n_{1} \cos θ_{1} k a) \cos (n_{2} \cos θ_{2} k b)

\cos k_{TM} (a + b) = - \frac{n_{2}^{2} \cos^{2} θ_{1} + n_{1}^{2} \cos^{2} θ_{2}}{2 n_{1} n_{2} \cos θ_{1} \cos θ_{2}} \sin (n_{1} \cos θ_{1} k a) \sin (n_{2} \cos θ_{2} k b)

+ \cos (n_{1} \cos θ_{1} k a) \cos (n_{2} \cos θ_{2} k b),

∣ \cos k_{TE} (a + b) ∣ > 1,

∣ \cos k_{TM} (a + b) ∣ > 1,

r_{i} = {\begin{matrix} n (b + a) + c & i = 2 n \\ n (b + a) + b + c & i = 2 n + 1 \end{matrix} (n = 0, 1, 2, 3, \dots) .

\in_{i} = {\begin{matrix} \in_{I} & i = 2 n \\ \in_{II} & i = 2 n + 1 \end{matrix} (n = 0, 1, 2, 3, \dots) .

E_{z} (r) = {A_{i} J_{m} (q_{i} r) + B_{i} Y_{m} (q_{i} r)} \sin (m ϕ + θ_{m})

H_{z} (r) = {C_{i} J_{m} (q_{i} r) + D_{i} Y_{m} (q_{i} r)} \cos (m ϕ + θ_{m}),

[\begin{matrix} E_{z} (r) \\ H_{z} (r) \\ E_{ϕ} (r) \\ H_{ϕ} (r) \end{matrix}] = U \cdot T (r, \in_{i}) \cdot u_{i}

U \equiv [\begin{matrix} \sin (m ϕ + θ_{m}) & 0 & 0 & 0 \\ 0 & \cos (m ϕ + θ_{m}) & 0 & 0 \\ 0 & 0 & \cos (m ϕ + θ_{m}) & 0 \\ 0 & 0 & 0 & \sin (m ϕ + θ_{m}) \end{matrix}]

T (r, \in_{i}) \equiv [\begin{matrix} J_{m} (q_{i} r) & Y_{m} (q_{i} r) & 0 & 0 \\ 0 & 0 & J_{m} (q_{i} r) & Y_{m} (q_{i} r) \\ - \frac{j {β m J}_{m} (q_{i} r)}{q_{i}^{2} r} & - \frac{j {β m Y}_{m} (q_{i} r)}{q_{i}^{2} r} & + \frac{j ω μ_{0} {J'}_{m} (q_{i} r)}{q_{i}} & + \frac{j ω μ_{0} {Y'}_{m} (q_{i} r)}{q_{i}} \\ - \frac{j ω \in_{i} {J'}_{m} (q_{i} r)}{q_{i}} & - \frac{j ω \in_{i} {Y'}_{m} (q_{i} r)}{q_{i}} & + \frac{j β m J_{m} (q_{i} r)}{q_{i}^{2} r} & + \frac{j β m Y_{m} (q_{i} r)}{q_{i}^{2} r} \end{matrix}]

u_{i} = {[A_{i}, B_{i}, C_{i}, D_{i}]}^{t} .

T (r_{i}, \in_{i}) \cdot u_{i} = T (r_{i}, \in_{i + 1}) \cdot u_{i + 1} .

u_{0} = \prod_{i = 0}^{n} R_{i} \cdot u_{n + 1}, R_{i} \equiv T^{- 1} (r_{i}, \in_{i}) \cdot T (r_{i}, \in_{i + 1}),

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Equations (16)

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