Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media •...

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Layered media and photonic crystals Cord Arnold / Anne L’Huillier

Transcript of Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media •...

Page 1: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Layered media and photonic crystals

Cord Arnold / Anne L’Huillier

Page 2: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 3: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 4: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Examples of natural photonic crystals 

Natural opalsNatural opals

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

Page 5: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Multilayer optics

Page 6: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

How to analyze a multilayer system

Scattering MatrixWave‐Transfer Matrix

SiMi MΣ S Σ

Analysis of multilayer optical system

Conversion

Page 7: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Cascaded system ‐ Airy formulas

Page 8: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Examples: Homogeneous medium

Page 9: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Examples: Partially reflective mirror (beamsplitter)

Page 10: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Examples: Single dielectric boundary

Page 11: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 12: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Off‐axis waves in layered media 

Single boundary

Page 13: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Fabry‐Perot etalon

Free spectral range

Page 14: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Fabry‐Perot example

Parameters: |r1|=|r2|=0.5, d=0.5µm

Page 15: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 16: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Off‐axis transmittance of Fabry‐Perot etalon

Page 17: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Bragg grating

Page 18: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Dielectric Bragg gratings

mdndnk 2211021

Resonance condition:

The accumulated phase shift must add to i.e. 

bb

c

02/

2 0

bb

c

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

Page 19: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Dielectric Bragg grating

mdndnk 2211021

Condition

The accumulated phase shift must add to

Page 20: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 21: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Off‐axis high‐reflection mirrors

Page 22: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 23: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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.

Page 24: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 25: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

A short introduction into photonic crystals

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More examples from nature

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

Page 27: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 28: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 29: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 30: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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.

Page 31: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Examples for 2d photonic crystals

Rectangular lattice

Triangular lattice

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Rectangular lattice 2d photonic crystal

Band Gap for TM modes

Photonic Crystals; J.D. Joannopoulos

Rectangular lattice of pillars.

Page 33: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Triangular lattice 2d photonic crystal

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

Page 34: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Example for a 3D photonic crystals: Yablonovite 

Historically first 3D Band Gap Crystal

Photonic Crystals; J.D. Joannopoulos

Page 35: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

3D woodpile structure photonic crystal 

Photonic Crystals; J.D. Joannopoulos

Page 36: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Defect modes

Photonic Crystals; J.D. Joannopoulos

Page 37: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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

Page 38: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

Bending light in 2d photonic crystals

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

Page 39: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

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.

Page 40: Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media • Reflections become angle-dependent • Reflections become polarisation dependent •

End of lecture