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Rational design of an integrated directional coupler for wideband operation

Nicolás Passarelli, Stefano Palomba, Irina Kabakova, and C. Martijn de Sterke

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Abstract

We consider a design procedure for directional couplers for which the coupling length is approximately wavelength-independent over a wide bandwidth. We show analytically that two coupled planar waveguides exhibit a maximum in the coupling strength, which ensures both wideband transmission and minimal device footprint. This acts as a starting point for mapping out the relevant part of phase space. This analysis is then generalized to the fully three-dimensional geometry of rib waveguides using an effective medium approximation. This forms an excellent starting point for fully numerical calculations and leads to designs with unprecedented bandwidths and compactness.

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Supplementary Material (1)

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Supplement 1       Equation 2

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (9)
Fig. 1.
Fig. 1. Schematic of the effective medium approach. (a) The full TE-like problem of two coupled rib waveguides can be considered approximately as the product of two problems involving planar waveguides. (b) Each vertical slice acts as a planar waveguide with associated fundamental mode and effective index (Problem A). (c) In Problem B, these vertical structures are replaced by uniform media with refractive index following from Part A. Problem A involves the TE modes of planar waveguides, whereas Problem B involves TM modes.

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Fig. 2.
Fig. 2. Geometries considered in this work: (a) planar geometry; (b) rib-geometry. Each of the waveguides has a width $d$ and the edge-to-edge spacing is $g$.

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Fig. 3.
Fig. 3. (a) Effective index ${n_{{\rm eff}}} \equiv \beta c/\omega$ (blue) for the fundamental TE mode of a planar waveguide and the overlap $S$ (red) with an adjacent waveguide for $d = 100\;{\rm nm} $, $g = 100\;{\rm nm} $, core refractive index ${n_f} = 3.5$, and cladding refractive index ${n_c} = 1.5$. (b) Cross- (left-hand side) and self-coefficients (right-hand side) defined in Eq. (3).

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Fig. 4.
Fig. 4. (a) Coupling constant versus wavelength for two planar waveguides. Green curve: coupled mode theory; yellow curve: exact result; dots: numerical result. The vertical line indicates the cutoff of the odd mode. (b) Transmission versus wavelength for different lengths near ${L_c}$. All parameters are the same as in Fig. 3.

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Fig. 5.
Fig. 5. Coupling constant $\kappa$ versus $\lambda$ and the geometrical parameter (a) $d$ at fixed $g = 100\;{\rm nm} $, and (b) $g$ at fixed $d = 100\;{\rm nm} $. The gray curves track the maximum at given $\lambda$. (c) Wavelength of maximum coupling versus $g$ and $d$. (d) Magnitude of maximum $\kappa$ versus $g$ and $d$. The blue line curves $\lambda = 1.55\,\,\unicode{x00B5}{\rm m}$. The refractive indices are the same as in Figs. 2 and 3.

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Fig. 6.
Fig. 6. Effective refractive index and permittivity versus wavelength from Problem A. (a) Effective refractive indices versus wavelength for different widths $h$; (b) permittivity contrast versus wavelength for ${h_2} = 175\;{\rm nm} $ and different ${h_1}$. Parameters ${n_f} = 3.5$, ${n_c} = 1$, ${n_s} = 1.5$.

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Fig. 7.
Fig. 7. Rib waveguides result, Problem B. (a) ${\kappa _{{\max}}}$ versus $d$ and $g$. ${h_1} = 50\;{\rm nm} $, ${h_2} = 175\;{\rm nm} $. The blue line corresponds to ${\lambda _{{\max}}} = 1.55\,\,\unicode{x00B5}{\rm m}$. (b) $\kappa$ curves for different ${h_1}$, ${h_2} = 175\;{\rm nm} $ fixed. $g = 125\;{\rm nm} $, $d = 275\;{\rm nm} $ [star in (a)]. The dotted lines were calculated with FDE. The material properties are the same as in Fig. 6.

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Fig. 8.
Fig. 8. Transmission of the rib waveguide coupler from FDE calculations for different device lengths for ${h_1} = 50\;{\rm nm} $, ${h_2} = 175\;{\rm nm} $, $g = 125\;{\rm nm} $, and $d = 275\;{\rm nm} $.

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Fig. 9.
Fig. 9. Modal fields. (a) Single waveguide modes for three wavelengths. (b) Self- and cross- modal product for three wavelengths.

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Equations (7)

Equations on this page are rendered with MathJax. Learn more.

κ = β + − β − 2 .
κ = K c r o s s − S ⋅ K s e l f 1 − S 2 ,
K ij = 1 4 ω ε 0 ∫ ∞ ( ε ( x ) − ε 0 ( x ) ) e i ⋅ e j d A .
S = 1 4 ∫ ∞ [ e i × h j + e j × h i ] d A .
P 1 = P 0 cos 2 ⁡ ( κ z ) , P 2 = P 0 sin 2 ⁡ ( κ z )
tan ⁡ ( 2 π ε f − ε e f f d λ ) = R ( ε e f f , π ε e f f − ε c g λ ) ,
K s e l f ≈ 1 4 ω ε 0 Δ ε d A 2 e − 2 γ c g , K c r o s s ≈ 1 4 ω ε 0 Δ ε d A 2 e − γ c g .
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