Layered media and photonic crystals

Cord Arnold / Anne L’Huillier

A photonic crystal is a periodic arrangementof 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 crystals

1D: Bragg Reflector

J. Serbin LZH

2D: Si pillar crystal

3D: Colloidal crystal

3D: Wood pilediameter of the rods 200 nm,spacing in-between 250 nm

Examples of natural photonic crystals 

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

SiMi 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

Free spectral range

Fabry‐Perot example

Parameters: |r1|=|r2|=0.5, d=0.5µm

Dielectric slab as a Fabry‐Perot etalon Parameters: n1=1.5, n2=3.5

Parameters: n1=1, n2=3.5, n3=1.5, dn2=0.5µm

Off‐axis transmittance of Fabry‐Perot etalon

Bragg grating

Dielectric Bragg gratings

mdndnk 2211021Resonance condition:

The accumulated phase shift must add to i.e. 

bb

c

02/

2 0b

bc

3/3 0

bb

c

4.2,45.12.0,3.0 :plotLower

4/ :plotUpper

21

2211

2211

nnndnd

ndnd

bb

b

n1 n1 n1n2 n2 n2 n1

Dielectric Bragg grating

mdndnk 2211021Condition

The accumulated phase shift must add to

Dielectric mirror – reflectivity vs. number of layer pairs

From medium n1 From air

n1 n1 n1n2 n2 n2n=1

n1=1.45n2=2.4

n1 n1 n1n2 n2 n2 n1 n1

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~1.38- Silicone dioxide n~1.49- Tantalum pentoxide n~2- Zinc sulfide n~2.32- 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

More examples from nature

McPhedran and Parker, Physics Today, 68:32 (2015).

Dispersion relation K() and wave localization for a 1d photonic crystal made from alternating dielectric layers

Correspondsto Λ=λ/2

First resonance

Correspondsto Λ=λ

Second resonance

In the theory of photonic crystals, a periodic structure is analized in a way to find solutions to the Maxwell equations making use of the periodicity of the structure, very much alike the analisys of travelling electron waves in a solid state material.

22

2

2

kg

g

Dispersion relation K(w) for a 1d photonic crystal made from alternating dielectric layers

Correspondsto Λ=λ/2

First resonance

Correspondsto Λ=λ

Second resonance

Matrix analysis as layer stack

Analysis as 1d photonic crystal

Phase and group velocity for a 1d photonic crystal made from alternating dielectric layers

effp n

cv 0Phase velocity:

Group velocity:eff

g Nc

dKdv 0

Slo

w re

gion

Photonic band gaps at off‐axis propagation for a 1d photonic crystal made from alternating dielectric layers 

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

The refractive index difference was chosen extreme in this example, i.e. n1=1.5 and n2=3.5.

Examples for 2d photonic crystals

Rectangular lattice

Triangular lattice

Rectangular lattice 2d photonic crystal

Band Gap for TM modes

Photonic Crystals; J.D. Joannopoulos

Rectangular lattice of pillars.

Triangular lattice 2d photonic crystal

Triangular lattice of holes! , a , r ~λPhotonic Crystals; J.D. Joannopoulos

Example for a 3D photonic crystals: Yablonovite 

Historically first 3D Band Gap Crystal

Photonic Crystals; J.D. Joannopoulos

3D woodpile structure photonic crystal 

Photonic Crystals; J.D. Joannopoulos

Defect modes

Photonic Crystals; J.D. Joannopoulos

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! 

Photonic Crystals; J.D. Joannopoulos

Bending light in 2d photonic crystals

Source: http://emt-photoniccrystal.blogspot.se/

Photonic crystal 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 

S. Fan et al., Phys. Rev. Lett., 80:960 1998.

End of lecture