Broadband mode converters in three-waveguide couplers based on quantumlike adiabatic transfer

We propose a method for robust mode conversion between the two outer waveguides in a three-waveguide coupler based on analogies with quantum population transfer. Depending on the presence or absence of detuning between the propagation constants of the intermediate waveguide, the mode conversion can be analogous to the adiabatic-elimination or stimulated Raman adiabatic passage techniques. Both configurations give broadband and robust mode converters.

Figure 1

Example of a structure of three coupled adiabatic waveguides used for mode conversion. The color code visualizes the refractive-index contrast $\mathrm{\Delta }n$ normalized to the smallest refractive-index contrast $\mathrm{\Delta }{n}_0$: $\alpha =\mathrm{\Delta }n/\mathrm{\Delta }{n}_0$ (in the illustrated design $\mathrm{\Delta }{n}_0\equiv \mathrm{\Delta }{n}_{\text{WG1}}$). The curvatures of WG1 and WG3 are identical, and so are the minimum separation distances $s_0$ between the lateral waveguides and WG2. The $z$ position of $s_0$ is different: WG2 and WG3 are initially closer than WG1 and WG2 [$C_P\left(0\right)\ll C_S\left(0\right)$ and $C_P\left(L\right)\gg C_S\left(L\right)$, with $L$ being the length of the device]. Injection in WG1 (at $z=0$) corresponds to a counterintuitive order of the coupling constants and injection in WG3 to in an intuitive order. This shape of waveguides is kept unchanged throughout this work. The only variable parameters are the three relative refractive-index contrasts ${\alpha }_1, {\alpha }_2$, and ${\alpha }_3$ used to choose the mode conversion and the regime (STIRAP or AE). In the following BPM simulations, $L$ = 30 mm, $\mathrm{\Delta }{n}_0=0.004, s_0=9.6$ µm, and the width of the waveguides is $w=8$ µm.

Figure 2

(a)–(c) show the propagation of light obtained by BPM simulations at the wavelength $\lambda =1.55$ µm, where the shape of the waveguides corresponds to Fig. 1 with ${\alpha }_I\equiv {\alpha }_1=1$ (CI case, with injection of mode 0) and ${\alpha }_2=1$. ${\alpha }_O\equiv {\alpha }_3$ is changed to match three different modes: (a) ${\alpha }_O=1$, case ST_000 in Table 1, (b) ${\alpha }_O=2.40$, case ST_001, and (c) ${\alpha }_O=5.33$, case ST_002. The color code visualizes the normalized light intensity going from 0 (blue) to 1 (yellow). This illustrates three different cases of STIRAP-like transfer with mode conversion for (b) and (c). (d) Evolution of the transfer efficiencies [Eq. (7)] with the parameter ${\alpha }_O$ (${\alpha }_I={\alpha }_2=1$). The blue solid curve shows the counterintuitive case (${\eta }_{\mathrm{CI}}$), and the red dashed curve gives the intuitive case (${\eta }_{\mathrm{Int}}$). The three maxima where ${\eta }_{\mathrm{CI}}=1$ correspond to propagations (a), (b), and (c). The small inset illustrates the beam propagation for ${\alpha }_O=4$, where all modes of the output waveguide are mismatched.

Figure 3

(a) Color plot of ${\eta }_{\mathrm{CI}}$ as a function of ${\alpha }_O$ and ${\alpha }_2$ for the counterintuitive case at $\lambda =1.55$ µm. (b) Corresponding ${\eta }_{\mathrm{Int}}$ for the intuitive case. All the others waveguide parameters are as in Fig. 1 (${\alpha }_I$ = 1). At the positions of the white dashed lines (${\alpha }_2=1$), the efficiencies correspond to the profiles in Fig. 2. The three vertical regions with $\eta \approx 1$ correspond to ${\beta }_{I_0}\approx {\beta }_{O_k}$ for the first three modes.

Figure 4

Examples of beam conversions from mode 0 (${\alpha }_I=1$) to mode 1 (${\alpha }_O=2.4$) illustrated by the evolution of light intensities at $\lambda =1.55$ µm in the waveguide design in Fig. 1, except for ${\alpha }_2$, which varies: (a) ${\alpha }_2=1$, ST_001 [the CI case corresponds to Fig. 2], (b) ${\alpha }_2=2.4$, ST_011, (c) ${\alpha }_2=5.6$, AE_021, and (d) ${\alpha }_2=7$, AE_021b. The left column shows the CI cases, and the right one shows the intuitive ones. (a) and (b) show two converters based on STIRAP, while (c) and (d) illustrate AE configurations where both CI and intuitive propagations lead to robust mode conversion. In (d) the AE condition of Eq. (3) is better verified, and therefore, almost no light transits to WG2, even in the intuitive case. The color code shows the normalized light intensity.

Figure 5

Examples of other possible conversions illustrated by the evolution of light intensities at $\lambda =1.55$ µm in the waveguide designs in Fig. 1, except for ${\alpha }_I$ and ${\alpha }_O$ (${\alpha }_2=5.6$ as in Fig. 1). (a) ${\alpha }_I=2.4$ and ${\alpha }_O=1$ (AE_120), realizing conversion from mode 1 to 0, and (b) ${\alpha }_I=1$ and ${\alpha }_O=5.33$ (AE_022), realizing conversion from mode 0 to 2. The left column shows the CI case, and the right one shows the intuitive case. The color code shows the normalized light intensity. The top profiles (bottom profiles) show the output (input) transverse intensity profiles.

Figure 6

Evolution of ${\eta }_{\mathrm{CI}}$ (solid blue curve) and ${\eta }_{\mathrm{Int}}$ (dashed red curve) as a function of wavelength $\lambda$. (a)–(d) are for conversion from mode 0 (${\alpha }_I=1$) to mode 1 (${\alpha }_O=2.4$). (e) is for conversion from mode 0 to mode 2 (${\alpha }_O=5.33$). The panels differ by the refractive-index contrasts in WG2: (a) ${\alpha }_2=1$ [ST_001, with the corresponding beam propagation at $\overline{\lambda }=1.55$ µm in Fig. 4], (b) ${\alpha }_2=2.4$ [ST_011, corresponding to Fig. 4], (c) ${\alpha }_2=5.6$ [AE_021, corresponding to Fig. 4), (d) ${\alpha }_2=7$ [AE_021b, corresponding to Fig. 4], and (e) ${\alpha }_2=5.6$ with ${\alpha }_O=5.33$ [AE_022, corresponding to Fig. 5]. (f) shows conversions based on standard directional couplers for comparison. The solid curve is for conversion from mode 0 to mode 1 (${\alpha }_O=2.4$, DC_01), and the dashed curve is for conversion to mode 2 (${\alpha }_O=5.33$, DC_02). The design of the two straight waveguides is shown in the inset with $L=6.5$ mm and $s_0=2w$ for all $z$; the others parameters are the same as before.