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September 2005. Slow light and resonance phenomena in photonic crystals. Alex Figotin & Ilya Vitebskiy University of California at Irvine Supported by AFOSR. 2D periodicity. 1D periodicity. n 1. n 2. What are photonic crystals? Simplest examples of periodic dielectric structures.

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September 2005

Slow light and resonance phenomena in photonic crystals

  • Alex Figotin & Ilya Vitebskiy

  • University of California at Irvine

  • Supported by AFOSR


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2D periodicity

1D periodicity

n1

n2

What are photonic crystals?

Simplest examples of periodic dielectric structures

Each constitutive component is perfectly transparent, while their periodic array may not transmit E.M. waves of certain frequencies (frequency gaps).


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(k)

(k)

1

1

k

2

2

k

Typical kdiagramof a uniform anisotropic medium for a given direction of k. 1and 2 denote two polarizations.

Typical kdiagram of a photonic crystal for a given direction of Bloch wave vector k


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Slow light in photonic crystals:stationary points of dispersion relations

ω

g

0

a

k

Fragment of dispersion relation with stationary points a,g and 0.

Every stationary point of the dispersion relation (k) is associated with slow light.But there are some important differences between these cases.


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Semi-infinite photonic crystal

Passed slow mode

Incident wave of frequency s

Reflected wave

What happens if the incident light frequency coincides with that of a slow mode?

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.


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ω

g

0

a

k

Fragment of dispersion relation with stationary points a,g and 0.

  • 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.


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Lossless semi-infinite photonic slab

ω

Transmitted slow mode

g

0

ST

Incident wave

SI

k

SR

Reflected wave

Slow mode amplitude at steady-state regime


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ω

Band edge

g

k


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ω

0

k

Stationary inflection point


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ω

d

k

Degenerate band edge


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Lossless semi-infinite photonic slab

Transmitted slow mode

ΨT

Incident wave

ΨR

ΨI

Reflected wave

Space structure of the frozen mode


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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|2diverges as (0)2/3and the resulting field distribution |T(z)|2 is described by quadratic parabola.


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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.


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The question semi-infinite photonic crystal:

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.


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A semi-infinite photonic crystal1A2 F

y

x

AB AB

AB AB

AB AB

z

z

L

L

Simplest periodic layered arrays supporting stationary inflection point of the dispersion relation

Non-magnetic periodic stack with oblique anisotropy in the A layers

Magnetic periodic stack with misaligned in-plane anisotropy in the A layers


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A semi-infinite photonic crystal1 A2B

A1 A2B

A1 A2B

A1 A2B

z

L

Simplest periodic layered array capable of supporting degenerate photonic band edge

There are three layers in a unite cell L. A pair of anisotropic layers A1and 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.


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The first band of the semi-infinite photonic crystalk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.


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


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Frozen mode regime in finite periodic stacks semi-infinite photonic crystals. How important is the thickness of the photonic slab?

N = 256

N = 64

EM field distribution inside plane-parallel photonic crystal of thickness D at the frequencyωdof degenerate band edge. The incident wave amplitude is unity. The leftmost portion of the curves is independent of the thicknessD.


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A semi-infinite photonic crystals. How important is the thickness of the photonic slab?B AB

AB AB

AB AB

D

ω

g

k

Transmission band edge resonance(Fabry-Perot cavity resonance in a finite periodic stacknear the edge of a transmission band)


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A semi-infinite photonic crystals. How important is the thickness of the photonic slab?B A B

AB AB

AB AB

D

ω

g

k

Fabry-Perot cavity resonance in finite periodic stacks:regular band edge


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Transmission band edge resonance: regular band edge semi-infinite photonic crystals. How important is the thickness of the photonic slab?

Finite stack transmission vs. frequency

Field intensity distribution at frequency of first transmission resonance


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ω semi-infinite photonic crystals. How important is the thickness of the photonic slab?

ω

g

d

k

k

Fabry-Perot cavity resonance in finite periodic stacks:regular band edge vs. degenerate band edge.


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Transmission band edge resonance: degenerate band edge semi-infinite photonic crystals. How important is the thickness of the photonic slab?

Finite stack transmission vs. frequency

Field intensity distribution at frequency of first transmission resonance


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Publications semi-infinite photonic crystals. How important is the thickness of the photonic slab?

[1] A. Figotin and I. Vitebsky. Nonreciprocal magnetic photonic crystals. Phys. Rev. E 63, 066609, (2001)

[2] A. Figotin and I. Vitebskiy. Electromagnetic unidirectionality in magnetic photonic crystals. Phys. Rev. B 67, 165210 (2003).

[3] A. Figotin and I. Vitebskiy. Oblique frozen modes in layered media. Phys. Rev. E 68, 036609 (2003).

[4] J. Ballato, A. Ballato, A. Figotin, and I. Vitebskiy. Frozen light in periodic stacks of anisotropic layers. Phys. Rev. E 71, 036612 (2005).

[5] A. Figotin and I. Vitebskiy. Slow light in photonic crystals. Subm. to Waves in Random and Complex Media.(arXiv:physics/0504112 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).


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Auxiliary Slides semi-infinite photonic crystals. How important is the thickness of the photonic slab?


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Photonic crystal semi-infinite photonic crystals. How important is the thickness of the photonic slab?

Incident pulse

D

Photonic crystal

Passed slow pulse

D

Pulse incident on a finite photonic crystal


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Dispersion relation semi-infinite photonic crystals. How important is the thickness of the photonic slab?ω(k)

g

0

a

k

k0

Eigenmodes composition at different frequencies

Regular frequencies:

ω < ωa : 4 ex.

ω > ωg : 4 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)


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Transfer matrix formalism semi-infinite photonic crystals. How important is the thickness of the photonic slab?


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Extended mode semi-infinite photonic crystals. How important is the thickness of the photonic slab?: Im k = 0

Evanescent mode: Im k > 0

Evanescent mode: Im k < 0

Floquet mode: 01 (z) ~ z

Bloch eigenmodes

Non-Bloch eigenmode