Guided modes on interfaces Surface states
Surface states in multilayer film (1D photonic crystal): States that are localized near the surface region Exponentially decaying into bulk and vacuum Compare with homogeneous slab (surface plasmon-polariton): Note here that the decaying solution is facilitated by the existence of a gap (for plasmon, it is the negative dielectric constant). Also note that in the bulk, the envelop is exponentially decaying, but it is oscillatory, as it must be, since it is not a homogeneous medium
Projected bands •
Projected band structure: Suppose we cut a crystal and expose a surface, k// is still a good quantum number, kz is not. Projected band structure is the projection of a 3D dispersion with a fixed k//, and all possible kz onto a 2D dispersion diagram Truly Guided modes (surface states etc) can only exist in “Gaps” of the projected band structure, otherwise can couple and decay into bulk states Truly guided modes should be “outside the light line”, which is the same as “outside the projected band structure of vacuum solutions”. Otherwise, couple and decay into air-side of the interface. Quasi-guided modes (e.g. surface resonances) can exist, if coupling is poor. Note the meaning of symmetry gap…
“Projected band structure” is familiar in electronic structure. Take Be (a HCP element) as example:
For this point we have to consider all the k-points in the bulk Brilloin zone with the same k// These points lie all along the direction of the bulk Brillouin zone.
Bulk band structure of Beryllium.
Projected bulk band structure and electronic surface states for Be(0001).
Surface Wave in 2D PBG system
Note that TM mode has a photonic band gap. The PBG is essential for the existence of the surface wave
Surface states for 3D photonic crystals
Note: Absolute gap in PBG implies gap in surface projected band structure Allow for possibility of surface state: surface guided mode
Existence of both TE-like and TMlike surface states Surface termination affects surface states. (This is not the case for electronic surface states).
Guided modes in Photonic Crystal Slabs
2D Photonic crystals
F ( x, y, z ) = F ( x, y )eikz
There is no band gap in the 3D sense (otherwise there is no need to think about 3D photonic crystals)
2D Photonic crystal slabs (air on both sides): Get the band gap back in some region of the BZ
Two ways to understand the dispersion
1. Start from a 2D PBG band structure (assume refractive index guide the wave in 2D for high enough k//)
1. Start from a homogenous high refractive index slab
2. Add in light cone (slab guided modes must be outside)
3. Note that the higher order modes start from finite frequencies, this allows for the possibility of a gap
3. But note that for thin slabs, dispersion can differ from 2D band structure 4. TE and TM classification no longer strictly valid. You do have even (~TE in lowest mode) and odd (~TM in lowest mode).
2. Add in periodicity: get BZ, band gaps…
Effect of slab thickness
1. Too thick: No kz quantization, no gap 2. Too thin: No material to guide the wave and scatter the wave to open band gap. You just get dispersion very close to the light line
PC slab with a background dielectric Light line pulled down:
c ω= k n
Loose phase space
If the refractive index of the substrate is higher than the effective index of the PC, waves no longer guided
Drill through the substrate: Reduce the effective refractive index of the substrate, Regain some phase space
Asymmetric substrate: (e.g. one side air, one side dielectric) Loose symmetry, symmetry gap no longer exists, loose gap However, if mode is strongly localized within the 2D PC slab, gap is still “approximately” there
Wave guides and light-bends in 2D universe: Guides modes exist and can turn sharp bends because of photonic band gap Is it still ok in 3D? Light cone of substrates, thin slab vs infinite crystal….
• Band gap is still there in thin PC slabs • Can still create a line defect and guide wave • and wave will not couple with slab modes even if it turns sharp corners
Mode guiding by linear defects (PC slab waveguide)s
Note: Index guided mode (strip has a higher refractive index than the effective index of PC) Gap guided mode is there Zone-folding
Real PC waveguides
Existence of Gap Index guided mode and gap guided mode and they couple Can tune width to change the coupling of these modes: modify dispersion and group velocity