Reversed Doppler Effect in Photonic Crystals

Abstract

In 1843, Johann Christian Doppler proposed an effect whereby the frequency of waves emitted from a moving object is shifted from the source frequency. The Doppler shift phenomenon has since realized applications ranging from weather and aircraft radar systems to satellite global positioning systems to the measurement of blood flow in unborn fetal vessels to the detection of extrasolar planets. The Doppler effect predicts that light shined by an ob- server onto an object moving toward him will be reflected with a higher frequency. In this Letter, we show that the established theory behind the Doppler shift breaks down for light reflected from a shock wave propagating in a photonic crystal (1,2), or material with a periodic modu- lation of the dielectric. We employ detailed numerical simulations and analytical theory to prove that anoma- lous Doppler shifts, both in sign and magnitude, can be observed. These effects are realizable under readily ex- perimentally accessible conditions. Anomalous Doppler effects have been observed in plasmas that propagate at near-relativistic speeds (3) and have been predicted to occur in pathological systems with simultaneously nega- tive effective permittivity and permeability (4,5). The anomalous Doppler effects presented in this Letter have a fundamentally different physical origin and can be observed at nonrelativistic speeds in systems with linear optical materials. In the shocked photonic crystal system, 100% of the incident light energy can be transferred into the anomalous Doppler shift. To explore the phenomena associated with light scat- tering from a shock wave in a photonic crystal, we perform finite-difference time-domain (FDTD) (6) simu- lations of Maxwell's equations in one dimension, single polarization, and normal incidence for a system described by a time-dependent dielectric � � x; t� . These simulations solve Maxwell's equations with no approximations except for the discretization, and are known to excellently re- produce experiments. The effects of a shock wave propagating in a 1D photonic crystal are shown in Fig. 1. The preshocked crystal (on the right) is comprised of two materials with identical elastic moduli and sound speeds, but differing dielectric. One layer has length d1 � 0:2a and the other has length d2 � 0:8a, where a is the preshock lattice constant. The compression of the lattice by the shock wave has two key effects on the photonic crystal: The lattice is compressed and the dielectric is changed through a strain dependence. If we focus on materials where the dielectric constant is increased with compres- sion, these two main effects affect the band gap frequency in opposing ways in the shock-compressed material: Decrease of the lattice constant increases the band gap frequency, but increasing the dielectric lowers the band gap frequency. The band gap can be made to decrease in frequency upon compression if materials with a suffi- ciently large dielectric dependence on strain are employed, dds , where material strain is given by s.