Layered media and photonic crystals

Cord Arnold / Anne L’Huillier

A photonic crystal is a periodic arrangement

of a dielectric material that exhibits strong

interaction with light

Variation of refractive index on the scale of the wavelength λ

Photonic Crystals; J.D. Joannopoulos

Definition

Artificial photonic crsytals

1D: Bragg Reflector

J. Serbin

LZH

2D: Si pillar crystal

3D: Colloidal crystal

3D: Wood pile

diameter of the rods 200 nm,

spacing in-between 250 nm

Natural Examples

Natural opalsNatural opals

http://www.viewsfromscience.com/documents/webpages/natural_photonics_p1.html

Multilayer Optics

How to analyze a multilayer system

Scattering MatrixWave-Transfer Matrix

Si � Mi MΣ � S Σ

Analysis of multilayer optical system

Conversion

Cascaded system - Airy formulas

Examples: Homogeneous medium

Examples: Partially reflective mirror (beamsplitter)

Examples: Single dielectric boundary

Off-axis waves in layered media

• Reflections become angle-dependent

• Reflections become polarisation dependent

• The phase is obtained from a projection on the optical axis

• The angle is different in different refractive index media

Off Axis Waves in Layered Media

Single boundary

Fabry-Perot Etalon (Mirror)

Free spectral range

Dielectric Slab as a Fabry-Perot Etalon

Off-Axis Transmittance of FPE

Bragg grating

Dielectric Bragg grating

( ) πϕϕ mdndnk =+=+ 2211021

Condition

The accumulated phase shift must add to π, λ/2

Off-axis high-reflection mirrors

Examples for 1-dimensional multilayer structures

Dielectric multi-layer mirror:

- Alternating stack of high and low refractive index materials (n1>n2) with

optical thickness of lambda/4 in each layer.

- Reflectivity >99.999% for narrow bands, >99% for very broad bands

Typical coating materials:

- Magnesium fluoride n~1.3

- Silicone dioxide n~1.45

- Tantalum pentoxide n~2

- Zinc sulfide n~2.3

- Titanium dioxide n~2.4

Image source: Wikipedia

n1 n2

Examples for 1-dimensional multilayer structures

Chirped dielectric mirror:

- The layer thickness changes as function of depth into the mirror. Blue

wave lengths are reflected at the surface, red in the depth.

- Chirped mirrors are used to cancel dispersion in ultrashort pulse

oscillators.

- They form the basis of today’s femtosecond laser technology.

- Reflectivity >99% for 500-1000nm.

Examples for 1-dimensional multilayer structures

Fiber Bragg grating (FBG):

- Wavelength filtering and multiplexing

- Single frequency fiber lasers

- Stretching, compression, dispersion compensation with chirped FBGs

- Sensing, e.g. temperature and pressure

Image source: Wikipedia

A short introduction into photonic crystals

Dispersion relation K(ω) and wave localization for a 1d photonic

crystal made from alternating dielectric layers

Dispersion relation K(ω) for a 1d photonic crystal made from

alternating dielectric layers

Phase and group velocity

eff

pn

cv 0=Phase velocity

Group velocity

eff

gN

cv 0=

Slo

w r

egio

n

Photonic band gaps at off-axis propagation

Internal off-axis waves External off-axis propagation coupled into

the structure

Two-dimensional photonic crystals

Rectangular lattice

Triangular lattice

Photonic Crystal in Two Dimensions

Band Gap for TM modes

2D PhC, Complete (!) 2D Band Gap

Triangular lattice of holes! , a , r ~λ

3D Photonic Crystals: Yablonovite

Historically 1st 3D Band

Gap Crystal

3D PhC Woodpile structure

Defect modes

Defect modes

By introducing a “defect” in the

highly periodic environment as

for example removing one rod

or simply changing the radius

of one or a few roods, the

periodicity is broken and

“defect modes” can exist in the

forbidden gap!

This e.g. allows for laser

cavities with dimensions of the

order of one wave length!

Bending light in 2D PhC

PhC Add-Drop Splitter

λ1 >λ2 >λ3 λ1 >λ3

λ2

NB: Add-drop filter with wavelength selectivity of ca 1% having dimensions of ca.

one wavelength

End Of Lecture