Optical transitions and Rabi oscillations in waveguide arrays

: It is theoretically demonstrated that Rabi interband oscillations are possible in waveguide arrays. Such transitions can take place in optical lattices when the unit-cell is periodically modulated along the propagation direction. Under phase-matching conditions, direct power transfer between two Floquet-Bloch modes can occur. In the nonlinear domain, periodic oscillations between two different lattice solitons are also possible.


Introduction
It is a well known fact that when an electron "drops" from the conduction band to the valence band, the excess energy is converted into electromagnetic radiation (photons) or into lattice vibrations (phonons). This emission can take place spontaneously from zero-point quantum fluctuations or it can be induced through stimulated emission. On the other hand, light can also be absorbed in crystals, resulting in an electronic transition to a higher energy state (conduction band). What actually facilitates this process is in fact a periodic time perturbation added to the electron Hamiltonian due to the presence of an external optical field. In the absence of any phonon or defect interactions, such transitions obey selection rules and as a result they can only take place if the Bloch momentum is conserved (direct transitions). In addition, the energy must also be conserved. Therefore, the period of the harmonic perturbation should be equal to the energy difference between the two levels. Rabi oscillations in the framework of atom optics have been considered at both the theoretical [1] and experimental front [2].
In this context it is natural to ask if these same processes can also be observed in the optical domain, in particular in waveguide lattices. Already, there have been suggestions of observing such transitions in photonic crystal systems using either ) 2 ( χ nonlinear or ultrasonic time/space harmonic perturbations [3,4]. However, thus far such all-optical transitions have never been observed experimentally. In this paper, we show that optical Rabi inter-band transitions and nonlinear power exchange between two different gap solitons are possible in periodically modulated array structures (inset of Fig.1) [5,6]. These transitions can occur among bands or gaps in the Bloch momentum k-space, as shown in Fig.1. Note that the normalized value of the propagation constant in Fig. 1 can be either positive or negative, when compared to the propagation constant of the background refractive index of the periodic potential. The array is modulated in the propagation direction z with a period Λ , in such a way that the "energy difference" 2π Λ spans the "energy/eigenvector"-difference between the allowed bands. This of course occurs provided that the transition selection rules are respected. Fig.1 shows such a possible direct transition between the first and the third band of this structure. In the case shown, this transition is allowed because the Floquet-Bloch (FB) modes involved possess the same parity. We note that these transitions occur as a result of parametric mixing and thus are different from the recently reported Zener tunneling [7,8], as well as, from the Bragg-resonance induced transitions [9] and spatial four-wave-mixing effects [10].

Coupled mode theory of FB mode transitions
In order to understand the dynamics of inter-band optical transitions we derive the corresponding coupled-mode equations. This is done by using time-dependent perturbation techniques in the paraxial equation of diffraction. In this case, wave propagation in a transversely-periodic potential (unperturbed in z ) is governed by the following normalized equation If on the other hand, the lattice is perturbed by a z-dependent periodic potential, then the system obeys: has the same periodicity as the original lattice , V x z ′ have the same period D , the latter inner product can then be reduced to . This last result indicates that only direct transitions are allowed, i.e., the transverse momentum k must be conserved in order for a transition to occur. By using the orthogonality condition

Selection rule diagram and examples of FB mode transitions
As an example, we consider optical transitions in periodically modulated AlGaAs semiconductor lattices (waveguide arrays) when is excited, is shown in Fig. 3(a). The power distribution between the second and third band is also depicted in Fig. 3(b) as a function of propagation distance. As we can see, complete power exchange from the second to the third band takes place after 1.2 cm of propagation. Before this point, the two involved FB modes interfere, leading to secondary power oscillations. A second example of a weak FB mode transition is that of 1 3 ↔ at 0 k = . The beam propagation (when the FB mode 0,1 φ is excited), as well as the projected power distribution to the two bands, are given in Fig. 4(a), (b), respectively. The conversion efficiency is lower and the oscillation period longer (~6cm), since this transition is weaker than the previous one, as expected from the selection rule diagram of Fig. 2. In both cases the agreement between coupled mode theory and BPM simulations is excellent.

Nonlinear energy exchange in modulated lattices
Another exciting possibility is to examine whether nonlinear power exchange from gap to gap can also occur in such systems. Unlike the cases described above, power exchange between self-trapped states is a direct outcome of nonlinearity. This is because lattice solitons reside in the gaps of an otherwise defect-free lattice [5,6,[11][12][13]. In essence, this may take place while the wave itself maintains its particle nature. In this case, the wave propagation in the zmodulated periodic potential is governed by the nonlinear Schrödinger equation A numerical simulation predicting such a nonlinear power exchange from the semi-infinite gap to the second gap in a periodically modulated semiconductor waveguide array is examined. In particular, the beam diffracts under linear conditions, as shown in Fig. 5(a). For higher powers and the same excitation, the field self-localizes and forms a lattice soliton with propagation eigenvalue in one of the two gaps, (Fig. 5(b)). Here, the eigenvalue difference of the two solitons involved is taken to be 2π/Λ. In other words, the process is properly phasematched. However, such Rabi oscillations between lattice solitons survive only for a few oscillation periods: in every cycle the soliton becomes wider and some power escapes to extended states.

Summary
In conclusion, we have demonstrated theoretically that inter-band FB oscillations are possible in waveguide arrays. Such transitions can take place in optical lattices when the channels are periodically modulated along the propagation direction. In the nonlinear domain, oscillations between two different lattice solitons are also possible. Recent experimental observation of such transitions in modulated LiNbO 3 waveguide arrays [14] is in complete agreement with the presented theoretical results.