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## Photonic properties of non-crystalline solids

Paul Steinhardt, Princeton

Photonic crystals are semi-conductor devices for light, that is, with them we can get allowed and forbidden frequency bands. They are not the topic of discussion of this talk. Instead we ask ourselves,

Q. What do we know about the photonic properties of non-crystalline solids?

Q. Why talk about non-crystalline solids?

Schroedinger – Maxwell analogy

Schroedinger equation Maxwell’s Equations
Quantum Classical
“Scalar” (spin neglected) Vector. TM + TE modes. Different band gaps for each mode.
Complicated by e-e interaction Linear. No $\gamma - \gamma$ interaction.
Fundamental scale No fundamental scale limit. If the problem is solved for one scale, it is solved for all. We can choose whichever scale is convenient to work with experimentally.( Ignoring absorption, which is frequency dependent and can’t be scaled any way we want.)
Massive quanta. Parabolic dispersion Massless quanta. Linear dispersion
Atomic/molecular structures Can design continuous structures

2 distinct scattering mechanisms:

• Bragg scattering: ‘large scale’ resonance of the array
• Mie scattering: ‘small scale’ resonance of scatterers
To get the largest band gap, we find conditions such that Bragg and Mie scattering reinforce each other. The experimental challenge is that the TE and TM modes prefer different dielectric patterns to achieve an optimum bandgap. The search is for “complete band gaps“, which are the overlap of TE and TM, so that they are band gaps for both polarization modes. The optimum configuration was found to be cylinders connected by a trivalent network of walls.
In general the band gap for TE < TM. The complete gap is, obviously, narrower than both. The figure of merit of the band gap is delta(f)/f, where delta(f) is the width of the band gap.
The major disadvantage found was that the band gap is anisotropic, which meant less symmetry. ( Symmetry beyond 6-fold is forbidden). There has been no theoretical framework to optimize band gaps. Most of the results were achieved by brute force simulations on computer or by running experiments.
Non-crystalline solids:
Until 1984 solids were thought to be either crystalline or disordered. Today we know of quasicrystals and many new classes of disordered solids.
Q. Why even consider non-crystalline solids?
• Higher symmetry may give wider band gaps, at least for some dielectric constant ratios.
• New types of structures, different modes and defects can find new applications.

Octogonal quasicrystal

In their pape rin 1984, Paul Steinhardt and D. Levine talk about quasicrystals,  the natural extension of the notion of a crystal to structures with quasiperiodic, rather than periodic, translational order. In their paper in Nature in 2005,  W. Man et al show that quasicrystals can have optical bandgaps rendering them useful to technology and also talk about the experimental measurement of the photonic properties of icosahedral quasicrystals.
It was found that quasicrystals beat diamond, the previous record holder, in terms of symmetry and wider band gaps. To find a more evenly distributed network, Delaunay tesselation was used.
Q. What if we applied it to isotropic disordered point pattern such as Poisson or hyperuniform?
Type Figure of merit
Crystal 19.3%
Quasicrystal 16.5%
Disordered 10.2% (isotropic rings – Bragg pattern)

We see that have a tradeoff between band gap and isotropy.