Polarization-insensitive asymmetric four-wave mixing using circularly polarized pumps in a twisted fiber

Takuo Tanemura; Kazuhiro Katoh; Kazuro Kikuchi

doi:10.1364/OPEX.13.007497

Optics Express
Vol. 13,
Issue 19,
pp. 7497-7505
(2005)
•https://doi.org/10.1364/OPEX.13.007497

Polarization-insensitive asymmetric four-wave mixing using circularly polarized pumps in a twisted fiber

Takuo Tanemura, Kazuhiro Katoh, and Kazuro Kikuchi

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Takuo Tanemura, Kazuhiro Katoh, and Kazuro Kikuchi, "Polarization-insensitive asymmetric four-wave mixing using circularly polarized pumps in a twisted fiber," Opt. Express 13, 7497-7505 (2005)

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Abstract

We show theoretically and experimentally that the polarization sensitivity of asymmetric nondegenerate fiber four-wave mixing can be eliminated by using circularly polarized pump waves in a twisted fiber. By twisting a fiber at 15 turns/m and aligning the pump waves to a circular state of polarization, we successfully reduce the polarization sensitivity from 5.8 dB to 0.9 dB. Although the polarization-mode dispersion (PMD) of the twisted fiber sets the limitation to the conversion bandwidth, its effect is relatively small owing to the small PMD of the twisted fiber. The demonstrated scheme should be a simple and efficient way of realizing all-optical tunable wavelength converters and wavelength-exchange devices without polarization dependence.

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Fig. 1. Wavelength allocation of the asymmetric FWM.

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Fig. 2. (a) Experimental setup for the P-OTDR measurement. PC: polarization controller. (b) Measured P-OTDR traces. Black and gray traces show two different cases as we varied PC.

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Fig. 3. Experimental setup of the polarization-insensitive FWM using a twisted fiber. MZ: LiNbO₃ Mach-Zehnder modulator, PM: LiNbO₃ phase modulator, ECL: external-cavity CW laser, P: polarizer, HWP: half-wave plate, PC: polarization controller.

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Fig. 4. Eye diagrams of the output idler wave with the polarization scrambler OFF (left) and ON (right), when we employ the non-twisted DSF (a) and twisted DSF (b).

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Fig. 5. Polarization sensitivity of the conversion efficiency measured with the non-twisted DSF (dots) and twisted DSF (circle).

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Fig. 6. Polarization sensitivity versus the pump SOP. Dots and circles are the maximum and minimum conversion efficiency, respectively, measured by varying the signal SOP, while triangles are the ratio of those two, representing the polarization sensitivity.

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Fig. 7. The maximum (dots) and minimum (circles) conversion efficiency and their ratios (triangles), representing the polarization sensitivity, as functions of the pump-wavelength separation. Theoretical curves are calculated from Eqs. (17) and (23).

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Equations (25)

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P^{(3)} (r, t) = ε_{0} χ^{(3)} [E (r, t) \cdot E (r, t)] E (r, t),

E (r, t) = Re [E_{1} (r) \exp (- i ω_{1} t) + E_{2} (r) \exp (- i ω_{2} t) + E_{3} (r) \exp (- i ω_{3} t) + E_{4} (r) \exp (- i ω_{4} t)],

P_{1}^{(3)} (r) = \frac{ε_{0} χ^{(3)}}{4} {{∣ E_{1} ∣}^{2} E_{1} [2 e_{1} + (e_{1} \cdot e_{1}) e_{1}^{*}] + 2 {∣ E_{2} ∣}^{2} E_{1} [e_{1} + (e_{1} \cdot e_{2}^{*}) e_{2} + (e_{1} \cdot e_{2}) e_{2}^{*}]},

P_{2}^{(3)} (r) = \frac{ε_{0} χ^{(3)}}{4} {{∣ E_{2} ∣}^{2} E_{2} [2 e_{2} + (e_{2} \cdot e_{2}) e_{2}^{*}] + 2 {∣ E_{1} ∣}^{2} E_{2} [e_{2} + (e_{1} \cdot e_{2}^{*}) e_{1} + (e_{1} \cdot e_{2}) e_{1}^{*}]},

P_{3}^{(3)} (r) = \frac{ε_{0} χ^{(3)}}{2} {{∣ E_{1} ∣}^{2} E_{3} [e_{3} + (e_{1}^{*} \cdot e_{3}) e_{1} + (e_{1} \cdot e_{3}) e_{1}^{*}] + {∣ E_{2} ∣}^{2} E_{3} [e_{3} + (e_{2}^{*} \cdot e_{3}) e_{2} + (e_{2} \cdot e_{3}) e_{2}^{*}]

+ E_{1}^{*} E_{2} E_{4} [(e_{1}^{*} \cdot e_{2}) e_{4} + (e_{1}^{*} \cdot e_{4}) e_{2} + (e_{2} \cdot e_{4}) e_{1}^{*}]},

P_{4}^{(3)} (r) = \frac{ε_{0} χ^{(3)}}{2} {{∣ E_{1} ∣}^{2} E_{4} [e_{4} + (e_{1}^{*} \cdot e_{4}) e_{1} + (e_{1} \cdot e_{4}) e_{1}^{*}] + {∣ E_{2} ∣}^{2} E_{4} [e_{4} + (e_{2}^{*} \cdot e_{4}) e_{2} + (e_{2} \cdot e_{4}) e_{2}^{*}]

+ E_{1} E_{2}^{*} E_{3} [(e_{1} \cdot e_{2}^{*}) e_{3} + (e_{2}^{*} \cdot e_{3}) e_{1} + (e_{1} \cdot e_{3}) e_{2}^{*}]},

P_{1}^{(3)} (r) = \frac{ε_{0} χ^{(3)}}{2} ({∣ E_{1} ∣}^{2} + 2 {∣ E_{2} ∣}^{2}) E_{1},

P_{2}^{(3)} (r) = \frac{ε_{0} χ^{(3)}}{2} ({∣ E_{2} ∣}^{2} + 2 {∣ E_{1} ∣}^{2}) E_{2},

P_{3}^{(3)} (r) = ε_{0} χ^{(3)} ⌊ ({∣ E_{1} ∣}^{2} + {∣ E_{2} ∣}^{2}) E_{3} + E_{1}^{*} E_{2} E_{4} ⌋,

P_{4}^{(3)} (r) = ε_{0} χ^{(3)} ⌊ ({∣ E_{1} ∣}^{2} + {∣ E_{2} ∣}^{2}) E_{4} + E_{1} E_{2}^{*} E_{3} ⌋ .

\frac{d A_{1}}{dz} = \frac{2 i γ}{3} ({∣ A_{1} ∣}^{2} + 2 {∣ A_{2} ∣}^{2}) A_{1},

\frac{d A_{2}}{dz} = \frac{2 i γ}{3} ({∣ A_{2} ∣}^{2} + 2 {∣ A_{1} ∣}^{2}) A_{2},

\frac{d A_{3}}{dz} = \frac{4 i γ}{3} [({∣ A_{1} ∣}^{2} + {∣ A_{2} ∣}^{2}) A_{3} + A_{1}^{*} A_{2} A_{4} \exp (- i Δ kz)],

\frac{d A_{4}}{dz} = \frac{4 i γ}{3} [({∣ A_{1} ∣}^{2} + {∣ A_{2} ∣}^{2}) A_{4} + A_{2}^{*} A_{1} A_{3} \exp (i Δkz)],

{∣ \frac{A_{4} (L)}{A_{3} (0)} ∣}^{2} = \frac{16 γ^{2} P_{1} P_{2}}{9 g^{2}} \sin^{2} (gL),

g^{2} = {[\frac{2 γ (P_{1} - P_{2}) / 3 - Δ k}{2}]}^{2} + \frac{16}{9} γ^{2} P_{1} P_{2} .

{∣ \frac{A_{4} (L)}{A_{3} (0)} ∣}^{2} = \frac{16 γ^{2} P^{2} L^{2}}{9} {[\frac{\sin (Δ kL / 2)}{Δ kL / 2}]}^{2} .

Δk = β_{2} (Δ {ω_{1}}^{2} - {Δ ω}_{2}^{2}),

k_{1} = k_{0 R} - β_{1 R} Δ ω_{1} + β_{2} Δ ω_{1}^{2} / 2,

k_{2} = k_{0 R} - β_{1 R} Δ ω_{2} + β_{2} Δ ω_{2}^{2} / 2,

k_{3} = k_{0 L} + β_{1 L} Δ ω_{1} + β_{2} Δ ω_{1}^{2} / 2,

k_{4} = k_{0 L} + β_{1 L} Δ ω_{2} + β_{2} Δ ω_{2}^{2} / 2,

Δ k = β_{2} (Δ {ω_{1}}^{2} - Δ {ω_{2}}^{2}) - τ (Δ ω_{1} - Δ ω_{2}),

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Equations (25)

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