Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media •...
Transcript of Layered media and photonic crystals - Atomic Physics · Off‐axis waves in layered media •...
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 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
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2211
2211
nnndnd
ndnd
bb
b
n1 n1 n1n2 n2 n2 n1
Dielectric Bragg grating
mdndnk 2211021
Condition
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