1. Maksim Skorobogatiy John Joannopoulos MIT, Department of Physics
The idea of using periodic dielectric materials (photonic crystals) to alter the dispersion relation of photons has received a widespread interest because of numerous potential applications. Such passive elements as waveguide bends, mirror surfaces, channel drop filters can be substantially improved if constructed on the basis of photonic crystals. Recently a strong interest has developed for the incorporation of nonlinear field dependent materials into photonic crystals, thus opening opportunities of constructing such active devices as optical switches second harmonic generators and so on. In all these studies, the photonic crystals are constrained to be in a static , or inertial, frame of reference. In this work we investigate what happens if a photonic crystal is considered in a noninertial frame of reference. This is of relevance, for example, if one starts rigidly shake a photonic crystal with some driving frequency. We show that it is possible to develop active photonic crystal elements even with linear materials by working with nonstationary photonic crystals.
Photonic Modes and Induced Iterband Transitions in the Photonic Crystals Undergoing Rigid Vibrations
Maksim Skorobogatiy
John Joannopoulos
MIT, Department of Physics

2. Stationary Photonic Crystal
We start our consideration with a case of a stationary photonic crystal. For simplicity we consider a one dimensional periodic structure of dielectric slabs with a period R. Such a structure is described by the usual set of Maxwell equations where dielectric constant e(r) possess an additional translational symmetry e(r+R) = e(r) . In the following we are going to consider only the modes propagating along the direction of periodicity.

3. Band Structure of a Stationary Photonic Crystal
It can be shown that the eigen modes propagating in the direction of periodicity of a one dimensional photonic crystal can be classified by only two parameters. One of them is the magnitude of the wavevector k and another one is the value of the frequency w at such a wavevector k. Moreover, these modes also possess a so called Bloch symmetry stating that the values of an eigen mode at spatial points separated by a vector of periodicity R are different from each other only by a certain phase. This from also suggests that the wavevector k can now be confined to an interval (-p/R,p/R) and for any chosen k there will be a discrete set of frequencies w.Thus, the well known band structure description of periodic structures is introduced.

4. Non-Stationary Photonic Crystal
Now consider a case of a non-stationary one dimensional photonic crystal which is shaken with a driving frequency W along the direction of spatial periodicity. In this case, dielectric constant becomes time dependent possessing two additional symmetries: spatial and temporal periodicity. Maxwell equations describing the evolution of the non-stationary modes are modified to include the time dependence of the dielectric constant.

5. Effective Band Structure of a Non-Stationary Photonic Crystal
It can be shown that as in the case of the stationary photonic crystals the non-stationary eigen modes can be classified by only two parameters, the wavevector k and the corresponding frequency w . Additionally, non-stationary eigen modes also possess spatial and temporal Bloch symmetries. As we saw in the case of the stationary photonic crystals spatial Bloch symmetry leads to the band structure representation stating that all the frequency bands can be folded into the first spatial Brillouin zone where the k wavevector is confined to the interval (-p/R,p/R). By analogy, temporal Bloch symmetry implies that there also exists a temporal Brillouin zone such that all the bands can be folded into the first temporal Brillouin zone along the frequency direction thus confining the frequency parameter w to the interval (-W/2,W/2). If during the folding into the first temporal Brillouin zone different bands start crossing each other then avoiding crossing occurs.This situation happens if before folding along the frequency axis bands were separated by the value of driving frequency W.

6. Vibrationally Induced Interband Transitions in a Non-Stationary Photonic Crystal
When the driving frequency W of the periodically shaken photonic crystal matches the bandgap between the stationary modes of the same stationary photonic crystal there exists another interesting interpretation to the nature of the non-stationary modes. In this case one can look at the non-stationary mode as consisted of the linear combination of the ground state stationary mode of frequency w0 and the first excited state stationary mode of frequency w0+W. Coefficients in the linear combination of such stationary modes will be proportional to the time dependent populations of these modes. Thus, an induced transition between the resonant stationary modes is initiated. Transition time between the stationary modes can be shown to be proportional to the period of vibrations of the photonic crystal devided by the interband coupling coefficient M.

7. Interband Transition Time Estimates
For a particular scenario when the crystal is shaken periodically with a constant velocity v in between the turning points one can calculate the coupling coefficient to be proportional (GD) which is, in turn, proportional to the ratio of the amplitude of vibrations to the periodicity of the crystal. If the bandgap is comparable to the frequency of the ground state mode then due to the fact that vibrational velocities are much smaller than the velocities of light in the media the inter-band coupling coefficient will be much smaller than unity and it will generally take many oscillation for a transition to occur. One such scenario is shown on the figure to the right. In the microwave regime, for example, where R~1 cm, v~10 m/sec and W~1GHz we estimate a transition time to be on the order of 10-3 sec.
We thus demonstrated the possibility of the interband transitions in a photonic crystal when the driving frequency matches the gap frequency between the corresponding stationary modes. This also shows that one can build active elements of the linear dielectric materials by working in a nonstationary frame.
Interband Transition Time Estimates