Complete achromatic optical switching between two waveguides with a sign flip of the phase mismatch
Abstract
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.
pacs:
42.82.Et, 42.81.Qb, 42.79.Gn, 32.80.XxI Introduction
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.
Ii Model of two coupled waveguides
We consider two evanescently-coupled planar optical waveguides as shown in Fig. 1. In the paraxial approximation, the propagation of a monochromatic light beam in the waveguides can be analyzed in the framework of the coupled mode theory (CMT) (1); (2); (3). The corresponding evolution of the wave amplitudes can be described by a set of two coupled differential equations (in matrix form),
(1) |
which has the form of a Schrödinger equation (4) with being the longitudinal coordinate. The components of the vector are the amplitudes of the fundamental modes in the two waveguides and are the corresponding normalized light intensities. The operator describes the interaction between the waveguide modes and is explicitly given as,
(2) |
Here, with is the constant propagation coefficient of the -th waveguide and is the variable coupling coefficient between the waveguides. We note that only the difference between the diagonal terms is important and it is called phase mismatch. We remove and from Eq. (2) by incorporating them as phases in the amplitudes and . Hence, Eq. (1) obtains the following form,
(3) |
In the next section we shall derive the solution to Eq. (3) for the step-sech model for which the coupling and phase mismatch are given by
(4a) | |||||
(4b) | |||||
Here, is the full width at half maximum for the coupling and we have also chosen the point to be the middle of the waveguides. Without loss of generality, the constants and are assumed positive. |
(5) |
with . Then, we can decouple from by taking a second derivative in , which gives
(6) |
The next step is to change the independent variable from to tanh noting that , and . We thus rewrite Eq. (6) for using Eqs. (4) as,
(7) |
This equation has the same form as the Gauss Hypergeometric equation (11); (12):
(8) |
where the overdot denotes a derivative in . The solution to Eq. (8) is given in terms of a linear combination of two Gauss Hypergeometric functions, and (11); (12); (9); (10). For the system considered here and described by Eq. (7) the corresponding parameters are
(9) |
We then find that the solution to Eq. (7) is given by,
(10) | |||||
where and are integration constants. Furthermore, using for the second amplitude we obtain,
(11) |
The integration constants and depend on the initial conditions and are given by
(12) | |||||
(13) |
Hence, for (), the wave amplitudes evolve according to with
(14) | |||
(15) |
That is, the propagator from () to () can be simply expressed as,
(16) |
where
(17) | |||||
(18) | |||||
with the exact form of the parameters
(19) | |||||
(20) |
Using the time-symmetry of equation (3) and taking into account that the only change for is the sign of , it is a simple matter to show that the propagator for () reads
(21) |
The total evolution propagator is
(22) |
Then the normalized light intensity in the second waveguide is given by
(23) |
Finally, we obtain the analytical expression for the light transfer between waveguides by substituting , , and in Eq. (23)
(24) |
where
(25) |
Recalling equation (9) and using the asymptotic expansions of (11); (12) in the limit of large coupling () the light intensity in the second waveguide turns to
(26) |
hence complete light switching between the two waveguides occurs.
Iv Adiabatic solution
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.
First we write Eq. (1) in the adiabatic basis (13); (14); (15)
(27) |
where the overdot denotes a derivative in the longitudinal coordinate and
(28) | |||||
(29) |
The amplitudes and in the adiabatic basis are connected with the diabatic (original) ones, and , via the rotation matrix
(30) |
as . When the evolution of the system is adiabatic, and remain constant (13); (14); (15). Mathematically, adiabatic evolution means that the non-diagonal terms in Eq. (27) are small compared to the diagonal terms and can be neglected. This restriction amounts to the following adiabatic condition on the interaction parameters (13); (14); (15):
(31) |
When the evolution is adiabatic the solution for the propagator in the adiabatic basis from an initial coordinate to a final coordinate reads
(32) |
where . The full propagator in the original basis for the model given in Eq. (4) reads
(33) |
Therefore if we take into account that and then the light intensity transfer to the second waveguide is
(34) |
Thus, tends to one in the case when 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 () that fulfills 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.
V Achromatic beam splitter
In this section we consider a symmetric array consisting of three coupled optical waveguides, as shown in Fig. 4. We assume that the middle waveguide is equally coupled to the two outer ones with coupling strength which is a function of the longitudinal coordinate . Furthermore, the propagation coefficients of the outer waveguides are assumed to be equal, that is, both have an equal refractive index and hence equal propagation coefficients , while the middle waveguide’s refractive index changes from to at the maximum of the coupling which also changes it’s propagation coefficient . The evolution of the light propagating in such a waveguide array is described by,
(35) |
where the phase mismatch , is the light amplitude in one of outer waveguides (the system is completely symmetric), and 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 using the transformation.
(36) |
where is the unchanged amplitude of the middle waveguide, and stands for the “bright” equal superposition of the amplitudes . Rewriting Eq. (35) in the new basis we obtain
(37) |
Indeed, we find that the state is decoupled from states and and the three-state problem is reduced to a two-state one involving states and only.
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 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 , 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 and phase mismatch ; 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.
Vi Conclusions
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).
Acknowledgements
We acknowledge financial support by SUTD start-up Grant No. SRG-EPD-2012-029 and SUTD-MIT International Design Centre (IDC) Grant No. IDG31300102.
References
- A. Yariv, IEEE J. Quantum Electron. QE-9, 919 (1973).
- H. A. Haus, C.G. Fonstad, IEEE J. Quant. Electron. 17, 2321 (1981).
- A. Yariv, Quantum Electronics (Wiley, New York, 1990).
- S. Longhi, Laser Photonics Rev. 3, 243 (2009).
- S. Longhi, Phys. Rev. A 71, 065801 (2005).
- S. Longhi, J. Opt. B. Quatum Semiclass. Opt. 7, L9 (2005).
- M. Ornigotti, G Della Valle, T. Toney Fernandez, A. Coppa, V. Foglietti, P Laporta and S Longhi, J. Phys. B: At. Mol. Opt. Phys. 41, 085402 (2008).
- F. Dreisow, A. Szameit, M. Heinrich, S. Nolte, A. Tünnermann, M. Ornigotti, and S. Longhi, Phys. Rev. A 79, 055802 (2009).
- N. V. Vitanov, New J. Phys. 9, 58 (2007).
- B. T. Torosov and N. V. Vitanov, Phys. Rev. A 76, 053404 (2007).
- A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions (New York, McGraw-Hill, 1953).
- M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (NewYork, Dover, 1964).
- L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Dover, New York, 1987).
- B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, New York, 1990).
- N. Vitanov, M. Fleischhauer, B. Shore, and K. Bergmann, Adv. At. Mol. Opt. Phys. 46, 55 (2001).
- N. V. Vitanov, J. Phys. B: At. Mol. Opt. Phys. 31, 709 (1998).
- F. Dreisow, M. Ornigotti, A. Szameit, M. Heinrich, R. Keil, S. Nolte, A. Tünnermann, and S. Longhi, Appl. Phys. Lett. 95, 261102 (2009).
- C. Ciret, V. Coda, A. A. Rangelov, D. N. Neshev, and G. Montemezzani, Opt. Lett. 37, 3789 (2012).
- A. A. Rangelov and N. V. Vitanov, Phys. Rev. A 85, 055803 (2012).
- Ph. Dittrich, G. Montemezzani, P. Bernasconi, and P. Günter, Opt. Lett. 24, 1508 (1999).
- M. Gorram, V. Coda, P. Thévenin, and G. Montemezzani, Appl. Phys. B 95, 565 (2009).
- C. Ciret, V. Coda, A. A. Rangelov, D. N. Neshev, and G. Montemezzani, Phys. Rev. A 87, 013806 (2013).