Broadband photonic transport between waveguides by adiabatic elimination

We propose an adiabatic method for the robust transfer of light between the two outer waveguides in a three-waveguide directional coupler. Unlike the established technique inherited from stimulated Raman adiabatic passage (STIRAP), the method proposed here is symmetric with respect to an exchange of the left and right waveguides in the structure and permits the transfer in both directions. The technique uses the adiabatic elimination of the middle waveguide together with level crossing and adiabatic passage in an effective two-state system involving only the external waveguides. It requires a strong detuning between the outer and the middle waveguide and does not rely on the adiabatic transfer state (dark state) underlying the STIRAP process. The suggested technique is generalized to an array of $N$ waveguides and verified by numerical beam propagation calculations.

Figure 1

Waveguide structure for adiabatic elimination in an array of (a) three and (b) six waveguides. The color code visualizes the normalized refractive index contrast $\alpha$. $\alpha =1$ is the maximum refractive index change of the outer waveguides and $\alpha =6$ the one for the central one(s), leading to a mismatch $\mathrm{\Delta }\beta$ of the propagation constants. Injection into the left waveguide 1 at $z=0$ corresponds to an intuitive order of coupling while injection into the right waveguide (waveguide 3, respectively 6) corresponds to a counterintuitive order of coupling.

Figure 2

Wave propagation in a three-waveguide structure calculated by the BPM method at the wavelength $\lambda =1550$ nm for light injection into the right waveguide (left column) and into the left waveguide (right column). (a) and (b) are for the case $\alpha =1$, for which all waveguides have the same index contrast and therefore $\mathrm{\Delta }\beta =0$. (c) and (d) are for $\alpha =6$ [refractive index design corresponding to the one of Fig. 1]. This gives the RAP-like light switching between the outer waveguides. (e) and (f) are for a fully symmetric waveguide structure for which ${\mathrm{\Delta }}_{\text{eff}}=0$ in (5) and for which the RAP-like transfer cannot take place ($\alpha =6$). The small top diagrams represent the intensity profiles at the end of the 30-mm propagation. The intensity color scale is the same for all the surface plots.

Figure 3

Visualization of the adiabatic parameters for the three-waveguide case ($N=3$) for the structure of Fig. 1 at $\lambda =1550$ nm, corresponding to the symmetric light transfer of Figs. 2 and 2. The different “$0i$” curves consider a possible coupling between the fundamental mode ($m=0$) of the outer waveguides (1 or 3) and the $i\mathrm{th}$ mode ($m=i$, $i=0$, 1, or 2) of the central waveguide 2. The panels show (a) the $z$ dependence of the adiabatic elimination criterion of Eq. (3), (b) ${\mathrm{\Delta }}_{\text{eff}}\left(z\right)$ [Eq. (6)], (c) $|C_{\text{eff}}\left(z\right)|$ [Eq. (7)], (d) the adiabatic criterion of Eq. (12) [deduced from (b) and (c)], and (e) the corresponding RAP-like two-waveguide structure underlying Eq. (5). In (e), the color scale illustrates the variation of the waveguide refractive index contrast with respect to the constant refractive index contrast $\mathrm{\Delta }{n}_0=0.004$ of the left waveguide.

Figure 4

Wave propagation in a three-waveguide structure as in Figs. 2 and 2 ($\alpha =6$) but for different wavelengths. (a) and (b) are at $\lambda =1300$ nm, and (c) and (d) are at $\lambda =1137$ nm.

Figure 5

Wave propagation in a six-waveguide structure calculated by the BPM method for light injection into the right waveguide (left column) and into the left waveguide (right column). The wavelength is $\lambda =1550$ nm. (a) and (b) are for the case $\alpha =1$. (c) and (d) are for $\alpha =6$ [refractive index design corresponding to the one of Fig. 1]. This gives the RAP-like light switching between the outer waveguides.

Figure 6

Evolution of the transfer efficiencies with wavelength for (a) $N=3$ and (b) $N=6$. The refractive index profiles correspond to the ones of Fig. 1. ${\eta }_{CI}$ (${\eta }_I$) is the transfer efficiency for the counterintuitive case (the intuitive case). The efficiency spectrum is compared to the one of a directional coupler (${\eta }_{\text{coupler}}$) composed of two straight waveguides and tuned to a wavelength of 1.55 $\mu \mathrm{m}$ (see text).

Figure 7

Dependence of the transfer efficiencies ${\eta }_{CI}$ and ${\eta }_I$ on the total length $L$ of the structure for $\lambda =1.55$ $\mu \mathrm{m}$ and $N=3$. The geometrical functions $\delta x_1\left(z\right)$ and $\delta x_N\left(z\right)$ are given by the parabolas of Eqs. (19). The waveguide's shape still correspond to the one of Fig. 1 if $z$ is replaced by a normalized propagation distance $z/L$ and the same $x$ dimension in micrometers is kept.