1. Slow light and resonance phenomena in photonic crystals
September 2005
Slow light and resonance phenomena in photonic crystals
Alex Figotin & Ilya Vitebskiy
University of California at Irvine
Supported by AFOSR

2. What are photonic crystals?
Simplest examples of periodic dielectric structures
1D periodicity
2D periodicity n1 n2
Each constitutive component is perfectly transparent, while their periodic array may not transmit E.M. waves of certain frequencies (frequency gaps).

3. (k) k
(k) k 1 2
In uniform media, for any given direction of wave propagation we have exactly 4 Bloch waves with 2 different polarization and two opposite directions of propagation.
In PC we have totally different picture:
- strong space dispersion
- band gaps, implying that at certain frequencies, some of the Bloch solutions are evanescent.
Typical k   diagram of a uniform anisotropic medium for a given direction of k. 1 and 2 denote two polarizations.
Typical k   diagram of a photonic crystal for a given direction of Bloch wave vector k

4. Slow light in photonic crystals: stationary points of dispersion relations
Fragment of dispersion relation with stationary points a, g and 0. ω k g a
Every stationary point of the dispersion relation (k) is associated with slow light. But there are some important differences between these cases.

5. What happens if the incident light frequency coincides with that of a slow mode?
Reflected wave
Incident wave of frequency s
Passed slow mode
Semi-infinite photonic crystal
Will the incident light with the slow mode frequency s be converted into the slow mode inside the photonic crystal, or will it be reflected back to space? The answer depends on what kind of stationary point is associated with the slow mode.

6. In case g of a band edge, all incident light with  = g is reflected back to space. The fraction of the incident wave energy converted to the slow mode vanishes as  → g .
In case a of an extreme point, the incident light with  = a is partially reflected and partially transmitted inside in the form of the fast propagating mode. The fraction of the incident wave energy converted to the slow mode vanishes as  → a.
In case 0 of stationary inflection point a significant fraction of incident light can be converted to slow mode, constituting the so-called frozen mode regime.
Fragment of dispersion relation with stationary points a, g and 0. ω k g a

7. Slow mode amplitude at steady-state regime
Incident wave SI
Reflected wave SR
Lossless semi-infinite photonic slab
Transmitted slow mode ST ω k g

8. k ω g
Band edge

9. Stationary inflection pointk ω
Stationary inflection point

10. k ω d
Degenerate band edge

11. Space structure of the frozen mode
Incident wave ΨI
Reflected wave ΨR
Lossless semi-infinite photonic slab
Transmitted slow mode ΨT

12. Distribution of EM field and its propagating and evanescent components inside semi-infinite slab at frequency  close (but not equal) to 0 . The amplitude of the incident light is unity !!!
a) resulting field |T (z)|2 = |pr (z) + ev (z) |2, b) extended Bloch component |pr (z) |2 , c) evanescent Bloch component |ev (z) |2 .
As  approaches 0 , |pr |2 diverges as (0  )2/3 and the resulting field distribution |T (z) |2 is described by quadratic parabola.

13. Summary of the case of a plane EM wave incident on semi-infinite photonic crystal:
- If slow mode corresponds to a regular photonic band edge, the incident light of the respective frequency is totally reflected back to space without producing the slow mode in the periodic structure.
- The incident light can be linearly converted into a slow mode only in the vicinity of stationary inflection point (the frozen mode regime).
- If slow mode corresponds to degenerate photonic band edge, incident light of the respective frequency is totally reflected back to space. But in a steady-state regime it creates a diverging frozen mode inside the photonic crystal.

14. The question:
Can the electromagnetic dispersion relation of a periodic layered structure (1D photonic crystal) display a stationary inflection point or a degenerate band edge? In other words, can a 1D photonic crystal display the frozen mode regime?
The answer is:
Stationary inflection point and degenerate band edge, along with associated with them the frozen mode regime can only occur in stacks incorporating anisotropic layers.

15. Simplest periodic layered arrays supporting stationary inflection point of the dispersion relationz L
A B A B
Non-magnetic periodic stack with oblique anisotropy in the A layers z y x L
A1 A2 F
Magnetic periodic stack with misaligned in-plane anisotropy in the A layers

16. Simplest periodic layered array capable of supporting degenerate photonic band edgez
A1 A2 B
There are three layers in a unite cell L. A pair of anisotropic layers A1 and A2 have misaligned in-plane anisotropy. The misalignment angle must be different from 0 and π/2. B – layers can be made of isotropic material, for example, they can be empty gaps. The k   diagram of the periodic stack is shown in the next slide.

17. The first band of the k   diagram of the 3-layered periodic stack for four different values of the B - layer thickness. In the case (b) the upper dispersion
curve develops degenerate band edge d. In the case (d) of B - layers absent, the two intersecting dispersion curves correspond to the Bloch waves with different symmetries; the respective eigenmodes are decoupled.

18. Up to this point we considered the frozen mode regime in semi-infinite photonic crystals. How important is the thickness of the photonic slab?

19. Frozen mode regime in finite periodic stacks
EM field distribution inside plane-parallel photonic crystal of thickness D at the frequency ωd of degenerate band edge. The incident wave amplitude is unity. The leftmost portion of the curves is independent of the thickness D.

20. Transmission band edge resonance (Fabry-Perot cavity resonance in a finite periodic stack near the edge of a transmission band) k ω g D
A B A B

21. Fabry-Perot cavity resonance in finite periodic stacks: regular band edgeω g D
A B A B
A B A B
A B A B

22. Transmission band edge resonance: regular band edge
Finite stack transmission vs. frequency
Field intensity distribution at frequency of first transmission resonance

23. Fabry-Perot cavity resonance in finite periodic stacks: regular band edge vs. degenerate band edge.ω g k ω d

24. Transmission band edge resonance: degenerate band edge
Finite stack transmission vs. frequency
Field intensity distribution at frequency of first transmission resonance

25. Publications
[1] A. Figotin and I. Vitebsky. Nonreciprocal magnetic photonic crystals Phys. Rev. E 63, , (2001)
[2] A. Figotin and I. Vitebskiy. Electromagnetic unidirectionality in magnetic photonic crystals. Phys. Rev. B 67, (2003).
[3] A. Figotin and I. Vitebskiy. Oblique frozen modes in layered media Phys. Rev. E 68, (2003).
[4] J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy. Frozen light in periodic stacks of anisotropic layers. Phys. Rev. E 71, (2005).
[5] A. Figotin and I. Vitebskiy. Slow light in photonic crystals. Subm. to Waves in Random and Complex Media.(arXiv:physics/ v2 19 Apr 2005).
[6] A. Figotin and I. Vitebskiy. Gigantic transmission band edge resonance in periodic stacks of anisotropic layers. Phys. Rev. E 72 (2005).

26. Auxiliary Slides

27. Pulse incident on a finite photonic crystal
Incident pulse
Photonic crystal D D
Passed slow pulse
Photonic crystal

28. Eigenmodes composition at different frequencies
Regular frequencies:
ω < ωa : ex.
ω > ωg : ev. (gap)
ωa < ω < ωg : 2 ex. + 2 ev.
Stationary points:
ω = ωa : 3 ex. + 1 Floq.
ω = ωg : 2 ev. + 1 ex. + 1 Floq.
ω = ω0 : 2 ex. + 2 Floq.
ω = ωd : 4 Floq. (not shown) k k0 g a 0
Dispersion relation ω(k)

31. Transfer matrix formalism

38. Evanescent mode: Im k > 0
Extended mode: Im k = 0
Evanescent mode: Im k < 0
Floquet mode: 01 (z) ~ z
Bloch eigenmodes
Non-Bloch eigenmode