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Recirculation-enhanced switching in photonic crystal Mach-Zehnder interferometers

T. P. White, C. Martijn de Sterke, R. C. McPhedran, T. Huang, and L. C. Botten

Open Access Open Access

Abstract

We show that Mach-Zehnder interferometers (MZIs) formed from waveguides in a perfectly reflecting cladding can display manifestly different transmission characteristics to conventional MZIs due to mode recirculation and resonant reflection. Understanding and exploiting this behavior, rather than avoiding it, may lead to improved performance of photonic crystal (PC) based MZIs, for which cladding radiation is forbidden for frequencies within a photonic bandgap. Mode recirculation in such devices can result in a significantly sharper switching response than in conventional interferometers. A simple and accurate analytic model is presented and we propose specific PC structures with both high and low refractive index backgrounds that display these properties.

©2004 Optical Society of America

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Figures (5)
Fig. 1.
Fig. 1. (a) Schematic of a simple MZI with single input and output waveguides joined by arms A 1 and A 2 via Y-junctions Y 1 and Y 2. (b) Schematic of the propagating modes in each of the waveguide sections in (a). The green and red curves show the even and odd superpositions of the individual waveguide modes.

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Fig. 2.
Fig. 2. (a) Transmission contour plot as a function of λ and φ, calculated with Eq. (5) for a PC-based recirculating MZI with the geometry shown in (d), and parameters described in the text. The red and green lines correspond to the transmission plots in (b) and (c) at λ=1.5496µm and φ=7π/4, respectively. The dashed curve in (b) shows the transmission of a conventional MZI as a function of φ, given by Eq. (3). (d) Generic recirculating PC MZI formed in a square lattice of rods with the parameters given in the text. The red cylinders, with radius a′, are used to generate φ in Section 3.1. The magenta and blue dashed-dotted curves in (a) show φ as a function of λ when a′=0.20d for Lφ =8d and Lφ =16d respectively. (Video file showing transmission as a function of λ and φ 1.9Mb)

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Fig. 3.
Fig. 3. (a) Y-junction used in the PC MZI design of Section 3.2. (b) Transmission through the junction as a function of wavelength. The 98% transmission bandwidth is approximately 36nm.

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Fig. 4.
Fig. 4. Comparison of the transmission of the PC MZIs calculated with the rigorous numerical calculations (solid curves) and the semianalytic results of Eq. (5) (dashed curves). (a) TM transmission of the PC MZI shown in Fig. 2(d) with L=36d, Ly =5d, a′=0.20d and Lφ =8d (magenta) and Lφ =16d (blue). (b) TE transmission of the PC MZI shown in Fig. 5(d) with L=37d, Ly =11d, a′=0.30d and Lφ =3d (magenta) and Lφ =7d (blue).

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Fig. 5.
Fig. 5. (a) Transmission contour plot calculated with Eq. (5) for a PC MZI with the geometry shown in (d), and parameters described in the text. The red and green lines correspond to the transmission plots in (b) and (c) at λ=1.5595µm and φ=π/4, respectively. The dashed curve in (b) is the response of a conventional MZI, as given by Eq. (3). (d) Generic PC MZI formed in a triangular lattice of air holes with the parameters described in the text. The red cylinders, with radius a′ are used to generate φ. The magenta and blue dashed-dotted curves in (a) show φ as a function of λ when a′=0.30d for Lφ =3d and Lφ =7d respectively. (Video file showing transmission as a function of λ and φ 1.9Mb)

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Equations (9)

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| ψ ± 〉 = 1 2 ( | ψ 1 〉 ± | ψ 2 〉 ) .
| Ψ ( 0 ) 〉 ( 1 ) = | ψ + 〉
∣ Ψ ( L ) 〉 ( 1 ) = ( e i β L ∣ ψ 1 〉 + e i ( β L + φ ) ∣ ψ 2 〉 ) / 2 = e i χ ( cos ⁡ ( φ ⁄ 2 ) ∣ ψ + 〉 − i sin ⁡ ( φ ⁄ 2 ) ∣ ψ − 〉 ) ,
T = ∣ e i χ cos ⁡ ( φ ⁄ 2 ) ∣ 2 = cos 2 ( φ ⁄ 2 ) = 1 1 + tan 2 ( φ ⁄ 2 ) .
∣ Ψ ( 0 ) 〉 ( 2 ) = − i e i χ sin ( φ ⁄ 2 ) ( e i β L | ψ 1 〉 − e i ( β L + φ ) ∣ ψ 2 〉 ) ⁄ 2
= − i e 2 i χ sin ( φ ⁄ 2 ) ( cos ⁡ ( φ ⁄ 2 ) ∣ ψ − 〉 − i sin ( φ ⁄ 2 ) ∣ ψ + 〉 ) .
T = 4 sin 2 ( χ ) cos 2 ( φ ⁄ 2 ) ⁄ sin 4 ( φ ⁄ 2 ) 1 + 4 sin 2 ( χ ) cos 2 ( φ ⁄ 2 ) ⁄ sin 4 ( φ ⁄ 2 ) .
T = 4 cot 4 ( φ ⁄ 2 ) 1 + 4 cot 4 ( φ ⁄ 2 ) = 1 1 + tan 4 ( φ ⁄ 2 ) ⁄ 4 .
𝓕 = π ∣ cos ( φ ⁄ 2 ) ∣ 1 − ∣ cos ⁡ ( φ ⁄ 2 ) ∣ .
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