Color of shock waves in photonic crystals

Reed, Soljacic, Joannopoulos, Phys. Rev. Lett., 2003

Miguel Antonio D. SulangiPS 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

and Maxwell’s equations are solved for a time- and space-dependent dielectric constant.

€

ε =ε(x, t)

Methods.

• For the simulations, the dielectric constant takes on the following functional form:

Note that v is the shock speed and a is the period of the pre-shocked crystal.€

ε( ˆ x =x

a, ˆ t =

ct

a) = 7 + 6sin π 3ˆ x −

v

cˆ t

⎛

⎝ ⎜

⎞

⎠ ⎟−

π

γlog 2cosh γ ˆ x −

v

cˆ t

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

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).