Source: https://www.groundai.com/project/complete-achromatic-optical-switching-between-two-waveguides-with-a-sign-flip-of-the-phase-mismatch/
Timestamp: 2019-04-18 20:31:55+00:00

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We present a two-waveguide coupler which, realizes complete achromatic all-optical switching. The coupling of the waveguides has a hyperbolic-secant shape while the phase mismatch has a sign flip at the maximum of the coupling. We derive an analytic solution for the electric field propagation using coupled mode theory and show that the light switching is robust again small-to-moderate variations in the coupling and phase mismatch. Thus, we realize an achromatic light switching between the two waveguides. We further consider the extended case of three coupled waveguides in an array and pay special attention to the case of equal achromatic light beam splitting.
The spatial light propagation in engineered coupled waveguide arrays is of fundamental importance to wave optics (1); (2); (3) . The electric field propagation in waveguide arrays can be accurately described within the coupled mode theory (1); (2); (3) and the resulting optical wave equation describing the spatial propagation of monochromatic light in dielectric structures is remarkably similar to the temporal Schrödinger equation describing a quantum-optical system driven by an external electromagnetic field (4) . The simplest realization of a waveguide array consists of two identical evanescently-coupled parallel waveguides. In this case, light is periodically switched between the waveguides throughout the evolution (3) in analogy to the quantum-optical Rabi oscillations (5) . Consequently, more complex waveguide configurations were designed to realize rich physical phenomena and for several of them analytical solutions for the light propagation have been described in the literature (6); (7); (8) .
In this work, we study the optical switching between two evanescently coupled planar waveguides whose coupling has a hyperbolic-secant shape and the phase mismatch is constant with a sign flip at the coupling maximum. We derive an analytic solution for complete light transfer (CLT) and we show that CLT is robust against variations in the experimental parameters; therefore, the technique is expected to find applications in achromatic light switching. Furthermore, we extend the model to three evanescently coupled waveguides in planar array and show that starting from the middle waveguide light can be equally split between the outer ones. We show that the light splitting is insensitive to fluctuations in the coupling and phase mismatch of the waveguides. Hence, this set-up may find an important technological application as an achromatic light beam-splitter. It is important to note, that the coupling model, which we consider here bears a close connection with the phase jump models from quantum optics (9); (10) , where the phase jump is instead in the coupling rather than the detuning. Such a model would also realize CLT in the system of two coupled waveguides but engineering a sign flip in the coupling would be a significant technological challenge.
Figure 1: (Color online) Evanescently coupled two waveguides made of two slabs with refractive indexes n2 and n3, embedded in a medium with an index of refraction n1. Gaussian shaped beam light is injected initially in the left waveguide and at the end of the adiabatic evolution the light is achromatically switched to the right waveguide.
Here, L is the full width at half maximum for the coupling Ω(z) and we have also chosen the point z=0 to be the middle of the waveguides. Without loss of generality, the constants Ω0 and Δ0 are assumed positive.
hence complete light switching between the two waveguides occurs.
We shall now derive the adiabatic solution for the general model where the waveguides’ coupling is a symmetric pulse-shaped smooth function and the phase mismatch has a sign flip at the coupling maximum.
Figure 2: (Color online) Demonstration of complete light switching between two waveguides. We assume sech-shaped coupling Ω(z)=Ω0sech(z/L) and step-phase mismatch Δ(z)=Δ0(1−2θ(z)), θ(z) is being the Heaviside step function. We select adiabatic parameters Ω0=50/L and Δ0=2/L.
Thus, I2(zf) tends to one in the case when Ω0>>Δ0 and the light is completely transferred between the waveguides. We note that Eq. (34) is valid not only if the coupling is given as hyperbolic-secant shape, but apply universally to every symmetric pulse-shaped smooth coupling (Ω(z)=Ω(−z)) that fulfills Ω(zi)=Ω(zf)=0 together with a sign flip of the phase mismatch at the coupling maximum. An example of complete adiabatic light switching between the two waveguides is shown in Figs. 2 and 3. In the simulations shown in Fig. 2 and Fig. 3 we have assumed hyperbolic-secant couplings, but any other smooth pulse-shaped coupling may be used. The contour plot in Fig. 3 demonstrates the robustness of the CLT against parameter variations.
Figure 3: (Color online) Contour plot of the final intensity in the right waveguide (waveguide two) from Fig. 1, if initially the light is injected only into the left waveguide (waveguide one). Numerically simulated contour plot is made for Eq. (3) using the coupling Ω0 and the phase mismatch Δ0 from Eq. (4), with the longitudinal coordinate z that change from −10L to 10L.
where the phase mismatch Δ(z)=β1−β2, c(1,3)(z) is the light amplitude in one of outer waveguides (the system is completely symmetric), and c2(z) is the amplitude in the middle waveguide.
Notably, Eq. (35) describing the light evolution in a system of three evanescently coupled waveguides is analogous to the Schrödinger equation describing a three-state quantum system subject to external electromagnetic field. It is well-known that the Hamiltonian of Eq. (35) has a zero eigenvalue whose eigenvector is a so-called “dark state” of the system, that is, it does not evolve under the evolution described by the Hamiltonian (16) . We introduce a new basis states including the dark state amplitude cd using the transformation.
Indeed, we find that the state cd is decoupled from states cb and c2 and the three-state problem is reduced to a two-state one involving states cb and c2 only.
Figure 4: (Color online) Evanescently coupled three waveguides made of three slabs with refractive indexes n2, n3 and n4, embedded in a medium of refraction index n1. Gaussian shaped beam light is injected initially in the middle waveguide and at the end it is achromatically divided equally between the right and the left waveguide.
In order to realize an achromatic beam splitter we take the following steps. Initially, we input the light in the middle waveguide with state amplitude c2 and following the evolution described by Eq. (37) which is similar to that described in Sections III and IV, the light is completely and robustly transferred into state cb, which is an equal superposition of the states of waveguides 1 and 3. Thus, the light at the end of the waveguides will be split equally between waveguides 1 and 3 (outer waveguides) as shown in Fig. 4. The light switching, as shown on Fig. 3, is robust against variations in the coupling Ω(z) and phase mismatch Δ(z); therefore, the technique is expected to be achromatic. In contrast to previously suggested achromatic adiabatic multiple beam splitters, which are based on an analog of stimulated Raman adiabatic passage from quantum optics and are unidirectional (17); (18); (19) , the above proposed beam splitting device works in forward and backward directions of light propagation equally well. Hence, the above described achromatic beam splitter is also bidirectional.
In conclusion, we presented a two waveguide coupler configuration which realizes complete achromatic all-optical switching robust to parameter fluctuations. We showed that the light propagation in the proposed waveguide coupler has an exact analytic solution which has the advantage of being valid for any values of the interaction parameters. In the limit of large coupling, complete light switching is achieved, which is insensitive to parameter fluctuations and is therefore achromatic. We furthermore showed that such a waveguide coupler can also be used for complete adiabatic light switching. An extension of this system to three coupled planar waveguides can be used as an achromatic beam splitter. We note that the achromaticity of the light transfer is guaranteed by the adiabatic nature of the process. Finally, the proposed waveguides coupler and beam splitter are experimentally feasible using photoinduced reconfigurable planar waveguides. The shapes and constants of propagation of such waveguides can be freely controlled by changing the local refractive index of the crystal with illuminating control light (20); (21); (18); (22) .
We acknowledge financial support by SUTD start-up Grant No. SRG-EPD-2012-029 and SUTD-MIT International Design Centre (IDC) Grant No. IDG31300102.
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