Chiyan Luo Mihai Ibanescu Evan J. Reed Steven G. Johnson J. D. Joannopoulos MIT

1. Properties of Doppler Radiation in Photonic Crystals Chiyan Luo Mihai Ibanescu Evan J. Reed Steven G. Johnson J. D. Joannopoulos MIT

2. Motivation • Periodic optical modulations give rise to many unusual dispersion properties, e.g. PBGs, as well as negative refraction. • In particular, an oscillator at frequency ω0inside a PBG is forbidden to radiate • What happens to a moving oscillator? • Inside a uniform material, the radiation of a moving source follows the usual Doppler’s law.

3. What has been known? • Most prior work resides in microwave circuits or waveguide systems: • Backward-wave oscillators • Cyclotron resonance masers • Frequency harmonics generation determined by the phase-matching criteria between the source and the spatial grating structure. • The Cherenkov effect corresponds to the case of ω=0. • A shock front can be regarded as a special type of radiation source.

4. What we study • Properties of Doppler radiation in presence of strong optical modulations and PBGs in bulk photonic crystals. • Anomalous effects due to the photonic band structure.

5. The phase-matching condition Scenario #1 (nonrelativistic) In the long-wavelength limit, the usual frequency harmonics are indexed by different reciprocal lattice vector G’s. The strength of each harmonic decays with increasing order.

6. The phase-matching condition Scenario #2 (nonrelativistic) Near a PBG edge, anomalous effects begin to take place: Both the forward- and the backward-propagating waves are negatively shifted.

7. The phase-matching condition Scenario #3 (nonrelativistic) When ω0 falls within a PBG, the Doppler frequency shift is no longer proportional to the velocity but determined by the photonic band structure. When v << c, these anomalously-shifted radiation processes occur with a weak efficiency.

8. An example radiation pattern The anisotropic features are associated with the directional collimation properties of photonic crystals and can be analyzed using the group-velocity flow-map techniques v=0.2c, ω=0.5(2πc/a), in a metallic photonic crystal with r=0.2a

9. Other Possibilities • A finite-sized source whose coherence length is comparable to the lattice constant will eliminate many higher order emissions and give rise to a much simpler picture. • Slow-light propagating bands in photonic crystals might be used to realize an “optical boom” (the analog of sonic boom in acoustics) • Oscillating sources traveling along low-symmetry direction in a bulk crystal or a quasi-crystal can give rise to quasi-continuum emission.

10. Possible mechanisms for generation of fast-moving oscillators in experiments • Cyclotron resonances in electron beams, powerful at microwave frequencies (strong magnetic field needed). • Less straightforward in the optical regime: • Fast beams of gas ions with infrared transition frequencies? • Nonlinear polarizations effects? • Solid-state exciton condensates?