Transcript Cerenkov radiation in photonic crystals

Color of shock waves in
photonic crystals
Reed, Soljacic, Joannopoulos, Phys. Rev. Lett., 2003
Miguel Antonio D. Sulangi
PS 175.2
What are photonic crystals?
• Photonic crystals = periodic dielectric media
• The periodicity of the dielectric constant
ensures that only certain modes (or
frequencies of light) propagate through the
crystal.
• Disallowed frequency ranges = bandgaps
Playing with photonic crystals.
• Idea of authors: why not introduce a shock
wave onto the photonic crystal?
• The shock wave propagates through crystal,
changing the characteristic spacing of the
dielectrics.
• What will happen?
• Onto the paper!
Introduction to the paper.
• “Unexpected and stunning new physical
phenomena result when light interacts with a
shock wave or shock-like dielectric modulation
propagating through a photonic crystal. These
new phenomena include the capture of light
at the shock wave front…and broadband
narrowing...”
Introduction to the paper.
• The effect of a “shock-like modulation of the
dielectric” is studied.
• Computational simulations were done, as
opposed to experiments.
• Some interesting phenomena are reported.
Results first.
The following phenomena were reported:
• The transfer of light frequency from the
bottom of the bandgap to the top.
• The capture of light of significant bandwidth
at the shock front for a controlled period of
time.
• The increase or decrease of the bandwidth of
light by orders of magnitude with 100%
energy conservation.
Methods.
• Maxwell’s equations were solved using finite
difference time domain (FDTD) simulations…
• …for one dimension, single polarization, and
normal incidence.
• FDTD involves discretizing the time;
otherwise, there are no other approximations.
Methods.
• The dielectric constant is described by
  (x,t)
and Maxwell’s equations are solved for a timeand space-dependent dielectric constant.
Methods.
• For the simulations, the dielectric constant
takes on the following functional form:
 
  v 
x ˆ ct
v ˆ   
( xˆ  , t  )  7  6sin  3xˆ  t  log2cosh xˆ  tˆ 
a
a
c  
  c 

 
Note that v is the shock speed and a is the
period of the pre-shocked crystal.
Methods.
• The dielectric function represents two
photonic crystals of periods a and a/2 which
meet at an interface.
• The shock wave compresses the lattice by a
factor of two.
• The interface moves at a speed v, which is set
for the simulations at 3.4 × 10-4c.
Results, pt. II.
Schematic of a shock wave moving to the right which compresses the lattice by a factor of
two. Light incident from the right (red arrow) will be converted up in frequency at the shock
front and escape to the right.
Results, pt. II.
Large frequency shift across the bandgap. Depicted are four moments in time during a
computer simulation of a shock moving to the right. The shock front location is indicated by
the dotted green line. The light begins the simulation below the gap in the unshocked
material as in the schematic above. As the light propagates to the left, most of it is trapped at
the shock front until it escapes to the right at a much higher frequency.
Results, pt. II.
Bandwidth narrowing. Depicted are two moments in time during computer simulation of the
shock. The shock front is indicated by the dotted green line. Light is confined between the
reflecting shock front on the left and a fixed reflecting surface on the right. As the shock
moves to the right, the bandwidth of the confined light is decreased by a factor of 4.
Results, pt. II.
• Frequency shift and light localization: when
light propagates in the opposite direction to
that of the shock wave, the frequency
becomes higher, and vice versa.
• Why? Because light bounces back and forth at
the shock front, a la “hall of mirrors.”
• Also, light escapes as pulses as the shock wave
propagates, which represent multiple discrete
frequencies.
Results, pt. II.
• Bandwidth broadening/narrowing: bandwidth
is narrowed by as much as 4%, and is done
with 100% efficiency.
• Note that bandwidth broadening is possible
using nonlinear materials, but narrowing is
very difficult to achieve using other methods.
• This is achieved by confining the light spatially
at the shock front, creating a narrow pulse.
Analysis.
• Frequency shift and light localization: Each time the shock
wave propagates through one lattice unit, the crystal on the
right is reduced in length by one lattice unit and the crystal on
the left is increased by one lattice unit.
• This means that the number of states in each band must
decrease by one in the pre-shocked crystal and increase by
one in the post-shocked crystal.
• it is necessary for a mode to move up through the overlapping
gap formed by the 2nd bandgap in the preshocked region and
the 1st bandgap in the postshocked region.
Analysis.
• Frequency shift and light localization: Light is
“trapped” in a cavity which is “squeezed” as
the shock compresses the lattice, thereby
increasing the frequency.
• This occurs once each time the shock
propagates through a lattice unit.
Analysis.
• Bandwidth narrowing: the light is spatially
confined at the shock front, leading to a
buildup of wavefronts at the shock front and
creating a pulse significantly narrower.
• Also, the process is independent of intensity.
Further work (i.e., experiments)
• How to generate shocks for experiments:
Physical shocks will do, and velocities work
even for nonrelativistic cases (~104 m/s).
• Also, the dielectric constant can be modulated
in a shock-like manner by non-mechanical
means using materials with nonlinear optical
response.
Applications.
• Solar power: by narrowing the bandwidth of
sunlight (which is very broadband), it is
possible to harness energy more efficiently.
• Quantum optics/telecommunications:
trapping light for controlled periods of time;
frequency conversion; signal modulation.
Conclusions.
By propagating a shock wave through a
photonic crystal, frequency changes, spatial
localization of light, and bandwidth
narrowing/broadening will result.
References.
• E.J. Reed, M. Soljacic, J.D. Joannopoulos,
"Color of shock waves in photonic crystals,"
Phys. Rev. Lett. 90, 203904 (2003).
• D.J. Griffiths, Introduction to Electrodynamics.
Englewood Cliffs, NJ: Prentice Hall (1999).