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Microcavities based on multimodal interference

Björn Maes, Mihai Ibanescu, John D. Joannopoulos, Peter Bienstman, and Roel Baets

Open Access Open Access

Abstract

We describe intricate cavity mode structures, that are possible in waveguide devices with two or more guided modes. The main element is interference between the scattered fields of two modes at the facets, resulting in multipole or mode cancelations. Therefore, strong coupling between the modes, such as around zero group velocity points, is advantageous to obtain high quality factors. We discuss the mechanism in three different settings: a cylindrical structure with and without negative group velocity mode, and a surface plasmon device. A general semi-analytical expression for the cavity parameters describes the phenomenon, and it is validated with extensive numerical calculations.

©2007 Optical Society of America

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Figures (13)
Fig. 1.
Fig. 1. (a) General picture of a system with two circulating modes. (b) Schematic of the cylindrical structure. The dashed line is the axis of the cylinder. (b) Dispersion of the HE11 mode.

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Fig. 2.
Fig. 2. (a) Q versus L of the resonances. Dots are data points from mode expansion (CAMFR), crosses present results from FDTD (MEEP). (b) ω versus L for the same cavity modes.

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Fig. 3.
Fig. 3. (a) Field plot of some resonances. The electric field along ϕ is shown. L n and ωn are L/a and ω ×(a/2πc), respectively. (b) Far-field on- and off-resonance. The magnetic field along the direction of the axis is shown. The cavity is located to the left of these plots.

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Fig. 4.
Fig. 4. Magnitudes of the reflection matrix elements for the cylindrical structure with the negative group velocity mode.

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Fig. 5.
Fig. 5. Magnitude squared of the two eigenvalues for the cylindrical structure in the frequency range with a negative group velocity mode.

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Fig. 6.
Fig. 6. Dispersion of the TE modes with angular momentum zero in the cylindrical structure. The frequency region with two guided modes is indicated.

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Fig. 7.
Fig. 7. (a) Q versus L of resonances. Dots are data points from mode expansion (CAMFR), crosses present checks with FDTD (MEEP). (b) ω versus L for the cavity modes in the cylindrical structure with two positive v g waveguide modes.

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Fig. 8.
Fig. 8. (a) Field plot of some resonances. The electric field along ϕ is shown; only one half is presented. Ln and ωn are L/a and ω ×(a/2πc), respectively. (b) Magnitudes of the reflection matrix elements for the cylindrical structure in the frequency range with two positive group velocity modes.

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Fig. 9.
Fig. 9. Magnitude squared of the two eigenvalues for the cylindrical structure in the frequency range with positive group velocity modes.

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Fig. 10.
Fig. 10. (a) Schematic of the two-dimensional, non-cylindrical plasmon structure. (b) Dispersion of the central waveguide (black), and of the outside waveguides (red).

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Fig. 11.
Fig. 11. (a) Q versus L of a resonance. Data calculated with mode expansion (CAMFR). (b) ω versus L for the plasmonic cavity.

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Fig. 12.
Fig. 12. (a) Depiction of the magnetic field at the peak of the resonance shown in Fig. 11(a) (L = 0.238L p and ω = 0.601ωp). (b) Magnitudes of the reflection matrix elements for the plasmonic structure.

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Fig. 13.
Fig. 13. Numerically calculated magnitude squared of an eigenvalue for the plasmonic structure.

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Equations (10)

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Q = ω r L ∣ v g ∣ ( 1 − ∣ λ ∣ 2 ) .
P × R [ c 0 c 1 ] = λ [ c 0 c 1 ] ,
v g average = ∣ v g 0 ∣ ∣ c 0 ∣ 2 + ∣ v g 1 ∣ ∣ c 1 ∣ 2 .
P = [ exp ( − i k 0 L ) 0 0 exp ( − i k 1 L ) ] ,
R = [ r 00 r 01 r 10 r 11 ] ,
λ = exp ( − i Δ L ) [ d cos ⁡ ( kL ) − iδ sin ⁡ ( kL ) ± ( − id sin ⁡ ( kL ) + δ cos ⁡ ( kL ) ) 2 + r 01 2 ] .
∣ λ ∣ 2 ≈ ∣ r 01 ∣ 2 ± 2 cos ⁡ ( kL ) Re ( r 01 d * ) ∓ 2 sin ⁡ ( kL ) Im ( r 01 δ * ) ,
λ ≈ ± r 01 exp ( − i Δ L ) .
∣ λ ∣ 2 ≈ ∣ d ± δ ∣ 2 ± Re ( c 2 ( d * ± δ * ) exp ( ± i Δ L ) − id sin ⁡ ( Δ L ) + δ cos ⁡ ( Δ L ) ) .
∣ λ + ∣ 2 ≈ r 00 2 + 2 r 01 2 cos ⁡ ( 2 kL ) + r 01 4 r 00 2 .
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