0. introduction 1. reminder: 2. photonic...
TRANSCRIPT
0. Introduction
1. Reminder:E-Dynamics in homogenous media and at interfaces
2. Photonic Crystals2.1 Introduction2.2 1D Photonic Crystals2.3 2D and 3D Photonic Crystals2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers
3. Metamaterials and Plasmonics3.1 Introduction3.2 Background3.2 Fabrication3.3 Experiments
Bibliography
• “Optik“, E. Hecht, Addison-Wesley(just as a reminder)
• “Nanophotonics“, P.N. Prasad, John Wiley & Sons (2004)(recent comprehensive overview, nothing in depth, good for finding further references and original work)
• “Photonic Crystals“, J.D. Joannopoulos, R.D. Meade, J.N. Winn,Princeton University Press(nice textbook introduction into the theory, mostly 2D)
• “Photonic Crystals“, K. Busch et al., eds., Wiley-VCH (2004)(collection of recent review papers, incl. experimental ones)
• “Optical Properties of Photonic Crystals”, K. Sakoda, Springer (2001)(advanced theory, mostly 2D, good introduction into symmetry properties)
0. Introduction
1. Reminder:E-Dynamics in homogenous media and at interfaces
2. Photonic Crystals2.1 Introduction2.2 1D Photonic Crystals2.3 2D and 3D Photonic Crystals2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers
3. Metamaterials and Plasmonics3.1 Introduction3.2 Background3.2 Fabrication3.3 Experiments
Optical properties of periodic structures
Lattice constant a
Wavelength λ
Optical properties of periodic structures
Lattice constant a
Wavelength λ
Optical properties of periodic structures
Lattice constant a
Wavelength λ
Theoretical framework
Relevant parameter: wavelength λ / lattice constant a
Geometrical optics …
… is only valid in the limit λ / a << 1.
… neglects wave effects (diffraction, interference).
… treats light propagation in terms of rays.
… is employed, e.g., in raytracing programs.
... have lattice constants much smaller than the wavelength of light (λ / a >> 1).
… can be treated as homogeneous media (Q.M. → ε,µ,n,Z).
... are common optical materials.
… have a refractive index n > 0.
“Normal” crystals …
... have lattice constants comparable to the wavelength
of light (λ / a ≈ 1).
… are (in most cases) artificial materials.
… exhibit a photonic band structure (Maxwell).
... can have a complete photonic bandgap.
Photonic crystals …
Metamaterials …
... have lattice constants smaller than the wavelength
of light (λ / a > 1).
… are artificial materials.
… can be treated as homogeneous media
(Maxwell → ε,µ,n,Z).
... can have a negative index of refraction n < 0.
Structure made at UCSD by David Smith
Photonics
“Photonics is the science and technology of generating and controlling photons, particularly in the visible and near infra-red light spectrum.
The science of photonics includes the emission, transmission, amplification, detection, modulation, and switching of light. Photonic devices include optoelectronic devices such as lasers and photodetectors, as well as optical fibers, photonic crystals, planar waveguides and other passive optical elements.”
http://en.wikipedia.org/wiki/Photonics
Example I
• DWDM (Dense Wavelength Division Multiplexing)
Can we fabricate these devices on a micron scale?
see Photonic Crystals
Example II
• Conventional optical fibers guide the light inside a glass core, thus showing dispersion. After a certain travel distance, information sent in the form of short laser pulses, smears out. Therefore, repeaters and amplifiers are needed.
Can we transmit light without dispersion?
see Photonic Crystal fibres
Example III
• With the further downscaling of conventional electronic components, quantum effects become important.
What can we expect from photonic structures on a wavelength or even smaller scales?
see Photonic Crystals, Quantumoptics
Example IV
• All known natural materials exhibit a positive index of refraction.
Can we design and fabricate artificial materials with a negative index of refraction?
see Metamaterials
0. Introduction
1. Reminder:E-Dynamics in homogenous media and at interfaces
2. Photonic Crystals2.1 Introduction2.2 1D Photonic Crystals2.3 2D and 3D Photonic Crystals2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers
3. Metamaterials and Plasmonics3.1 Introduction3.2 Background3.2 Fabrication3.3 Experiments
All of macroscopic electromagnetism can be described within the framework of the macroscopic Maxwell equations:
ρ=⋅∇ D
tBE
∂∂−=×∇
0=⋅∇ B
tDjH
∂∂+=×∇
withPED
+= 0ε
MHB
+= 0µ
: electric field : magnetic induction
: dielectric displacement : magnetic field
: polarization : magnetization
: free charge density : free current density
E
D
P
ρ
B
M
j
H
The material properties enter via the constitutive relations.
rdtdrtErrttrtP e ′′′′′′= ∫ ∫∞
∞−
∞
∞−
),(),,,(),( 0 χε
For low light intensities, one usually finds a linear relationship
between the polarization and the electric field as well as
between the magnetization and magnetic field:
rdtdrtHrrttrtM m ′′′′′′= ∫ ∫∞
∞−
∞
∞−
),(),,,(),( 0 χµ
Tensors!
Here, we consider only isotropic materials with a local response:
)(),(),,,( rrttrrtt ee ′−′=′′ δχχ
)(),(),,,( rrttrrtt mm ′−′=′′ δχχ
The response functions must be causal and do not explicitly depend on time (homogeneity in time):
)()(),( tttttt ee ′−Θ′−=′ χχ)()(),( tttttt mm ′−Θ′−=′ χχ
tdtEtttPt
e ′′′−= ∫∞−
)()()( 0
χε
tdtHtttMt
m ′′′−= ∫∞−
)()()( 0
χµ
⇒
)()()()()()( 0
F.T.
0 ωωχεωχε EPtdtEtttP e
t
e
=→′′′−= ∫
∞−
In the frequency domain, we get:
)()()()()()( 0
F.T.
0 ωωχεωχµ HMtdtHtttM m
t
m
=→′′′−= ∫
∞−
( ) )()()()(1)( 00 ωωεεωωχεω EED e
=+=
( ) )()()()(1)( 00 ωωµµωωχµω HHB m
=+=
This finally leads to
Magnetic permeability
Electric permittivity
see “Physik II“ and “THEORIE D“
tBE
∂∂−=×∇
tDH
∂∂=×∇
Ht
Et
2
2
0 ∂∂−=×∇
∂∂ µµ
Et
H
×∇∂∂=×∇×∇ εε 0
Electromagnetic waves in homogenous media without dispersion, free charges and free currents (ε = const, µ = const, ρ = 0, j = 0 ):
HB
0µµ=
t∂∂
ED
0εε=
×∇
0),( ),( 2
2
00 =∂∂−∆ rtHt
rtH ε µµε
With and we obtain the wave equation:( ) ∆−⋅∇∇=×∇×∇
0=⋅∇ B
see “Physik II“ and “THEORIE D“
tBE
∂∂−=×∇
tDH
∂∂=×∇
Ht
E
×∇∂∂−=×∇×∇ µµ 0
Et
Ht
2
2
0 ∂∂=×∇
∂∂ εε
Electromagnetic waves in homogenous media without dispersion, free charges and free currents (ε = const, µ = const, ρ = 0, j = 0 ):
HB
0µµ=
×∇
ED
0εε=
t∂∂
0),( ),( 2
2
00 =∂∂−∆ rtEt
rtE ε µµε
With and we obtain the wave equation:( ) ∆−⋅∇∇=×∇×∇
0=⋅∇ E
( )[ ] trkiE)r(tE ω−⋅= exp, 0
With the complex ansatz for E and H …
( )[ ] trkiH)r(tH ω−⋅= exp, 0
… we obtain from the wave equations:
200
2 ωε µµε=kCase 1: Plane waves
zyxiki ,,,0 ∈ℜ∈⇒>ε µ
Case 2: Evanescent modes zyxiki ,,,0 ∈ℑ∈⇒<ε µ
The physical electric and magnetic fields are obtained by taking the real parts of the complex quantities!
BiEkiEikEikEikEikEikEik
E
xyyx
xzzx
yzzy ω−=×=
−−−
=×∇!
( ) 0!
=⋅=++=⋅∇ BkiBkBkBkiB zzyyxx
( )[ ] trkiE)r(tE ω−⋅= exp, 0
Plane waves …
… are transversal:
( )[ ] trkiB)r(tB ω−⋅= exp, 0
Moreover, E and B are in phase.
2
20
20
002
22 1
ncc
kc ====
ε µµε µεω
The planes of constant phase propagate with the phase velocity :
εεµµ
0
0=Z Ω≈= 7.3760
00 ε
µZ
… and the impedance
c
see Metamaterials
ε µε µ ±=⇒= nn 2
The material properties enter via the refractive index …
Dielectric materials: ε µ=n( )0, >µε
( )∗∗ ⋅+⋅ℜ= 000041 BHDEw
The energy density w of an electromagnetic wave in a nondispersiv medium is given by
( )∗×ℜ= 0021 HES
The corresponding time averaged energy flux density is given by the Poynting vector
This formula is valid only for nondispersive media!
kS
||0 ⇒>µ
These quantities are spatially constant for plane waves.
The magnitude of denotes the intensity of the electromagnetic field:
S
2
0021 EcSI
εε==
Plane waves …
constrk =⋅
Wavelength λ
E and B are in phase!
[ ] [ ] tirekE)r(tE k ω−⋅−= exp||exp, 0
Evanescent modes …
[ ] [ ] tirekB)r(tB k ω−⋅−= exp||exp, 0
… exhibit exponentially decaying field strengths (E and B).
Evanescent modes …
… do not transport energy since the time-averaged normal component of the Poynting vector vanishes :
[ ]( ) [ ]( ) 0)(2
121
000
00 =××⋅ℜ=×⋅ℜ= ∗∗ EkEeHEeS kkek
µω µ
Purely imaginary!
Therefore, evanescent modes do only have a noticeable field strength at interfaces.
Classification of electromagnetic modes
n2 = ε·µ > 0
⇒ propagating waves
n2 = ε·µ < 0
⇒ evanescent waves
n2 = ε·µ > 0
⇒ propagating waves
n2 = ε·µ < 0
⇒ evanescent waves
Electromagnetic fields at interfaces
• Use 3rd Maxwell equation
• Use Gauss-Theorem
( )12)(
030 BBnfdBfBr
V VSx
−⋅ →⋅=⋅∇= ∫ ∫
∆ ∆→∆d d
n f
d
f
d
x∆
⇒ Normal component of B must be continous
Electromagnetic fields at interfaces
n
tFF
∆=∆
t x∆
1l∆
2l∆
( )tjlDft
jfHf FF
xFF
⋅∆ →⋅
∂∂+⋅=×∇⋅∫ ∫∫
∆→∆
∆∆0 d d d f
( ) ( )120 drot d HHntlHsHfF
xF
−⋅×∆ →⋅=⋅∫ ∫
∆→∆
∆∂
• Use 4th Maxwell equation
• Use Stokes-Theorem
⇒ (no surface current) Tangential component of H must be continous
( ) ( )tj
HHnt
F
⋅=
−⋅× 12
Electromagnetic fields at interfaces
With a similar derivation for E and D follows:
•D normal•E tangential•B normal•H tangential
have to be continous across charge- and current-free interfaces.
We obtain for the other field components:
nn HH 12
12 µ
µ=tt BB 11
22 µ
µ=nn EE 12
12 ε
ε=tt DD 11
22 ε
ε=
Refraction at an interface – Fresnel formulas
ri Θ=Θ ( ) ( )ttii nn Θ=Θ sinsin
( ) ( )( ) ( ) ( ) ( )itttii
iii
pi
tp nn
nEEt
Θ+ΘΘ=
=
cos/cos/cos/2
µµµ
( ) ( ) ( ) ( )( ) ( ) ( ) ( )itttii
tiiitt
pi
rp nn
nnEEr
Θ+ΘΘ−Θ=
=
cos/cos/cos/cos/
µµµµ
p-polarization
Refraction at an interface – Fresnel formulas
s-polarization
ri Θ=Θ ( ) ( )ttii nn Θ=Θ sinsin
( ) ( )( ) ( ) ( ) ( )tttiii
iii
si
ts nn
nEEt
Θ+ΘΘ=
=
cos/cos/cos/2
µµµ
( ) ( ) ( ) ( )( ) ( ) ( ) ( )tttiii
tttiii
si
rs nn
nnEEr
Θ+ΘΘ−Θ=
=
cos/cos/cos/cos/
µµµµ
Refraction at an interface – Fresnel formulas
Parameters: ε1=1.0, µ1=1.0, ε2=2.25, µ2=1.0
Brewster’s angle
A slab of matter: Fabry-Perot modes
0E
tE0
20
δiettE ′
δierttE ′′0
232
0δierttE ′′
254
0δierttE ′′
276
0δierttE ′′
δ230
ierttE ′′
δ350
ierttE ′′
δ470
ierttE ′′
rE0
slabn
d)cos(20 tslabdnk ϑδ =
Phase difference due to propagation (single round trip):
20
δiertE ′
tϑ
A slab of matter: Fabry-Perot modes
The total transmitted electric field is given by the superposition of all partially transmitted electric fields:
...27
25
23
2 60
40
200 +′′+′′+′′+′= δδδδ iiii
t erttEerttEerttEettEE
After a little bit of algebra we obtain the intensity of the total transmitted electromagnetic field (for lossless media)
)2/(sin11
20 δFII t +
=
with the finesse factor2
212
−=
rrF
See, e.g., “Optik“, E. Hecht, Addison-Wesley
A slab of matter: Fabry-Perot modes
F=0.1F=1F=10F=100
The transmittance is maximal for
...)1( 6420
2 +′+′+′+′= rrrettEE it
δ
since all partial waves are in phase:
,...2,1,0,2 ∈= mmπδ
A slab of matter: Fabry-Perot modes
The total reflected electric field is given by the superposition of all partially reflected electric fields:
...350
23000 +′′+′′+′′+= δδδ iii
r erttEerttEerttErEE
After a little bit of algebra we obtain the intensity of the total reflected electromagnetic field (for lossless media)
)2/(sin1)2/(sin
2
2
0 δδ
FFII r +
=
with the finesse factor2
212
−=
rrF
See, e.g., “Optik“, E. Hecht, Addison-Wesley
A slab of matter: Fabry-Perot modes
F=0.1F=1F=10F=100
The reflectance is maximal for ,...2,1,0,)12( ∈+= mm πδ
? ?
Classification of optical materials
? ?
Optical frequencies:
µ = 1
Classification of optical materials
Lorentz Oscillator modell
Lorentz dielectric function:0
2200
2 11)(ω γωωε
ωεim
Nq−−
+=
Equation of motion: tieqExmdt
xdmdt
xdm ωωγ −−=++ 02002
2
Lorentz Oscillator modell – without losses
Lorentz Oscillator modell – with losses
0. Introduction
1. Reminder:E-Dynamics in homogenous media and at interfaces
2. Photonic Crystals2.1 Introduction2.2 1D Photonic Crystals2.3 2D and 3D Photonic Crystals2.4 Numerical Methods2.5 Fabrication2.6 Non-linear optics and Photonic Crystals2.7 Quantumoptics2.8 Chiral Photonic Crystals2.9 Quasicrystals2.10 Photonic Crystal Fibers – „Holey“ Fibers
3. Metamaterials and Plasmonics3.1 Introduction3.2 Background3.2 Fabrication3.3 Experiments
Silicon, a semiconductor crystal
Is there such a thing as a “semiconductor for light“ ?
S. John, Phys. Rev. Lett. 58, 2586 (1987)E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987)
Semiconductors
Periodic potential for electrons Band structure for electrons
⇒
Photonic Crystals
• Dielectric or metallic materials with a dielectric function that is periodically modulated along at least one spatial direction:
( )ω,rε
1D 2D 3D
Photonic Crystals
Band structure for photons
⇒
Periodic “potential” for photons
1D Photonic Crystals in nature
• Mother-of-pearl
taken from: http://www.biosbcc.net/ocean/marinesci/06future/abrepro.htmhttp://www.solids.bnl.gov/~dimasi/bones/abalone/
Aragonite [CaCO3] / protein layers
Taken from: A.R. Parker et al., Nature 409, 36 (2001)
20 cm
300 nm
Sea-mouse
2D Photonic Crystals in nature
• Pachyrhynchus argus
3D Photonic Crystals in nature
Taken from: •http://www.shgresources.com/nv/symbols/gemstonep•A. R. Parker et al., Nature 426, 786 (2003)
Morpho Rhetenor und Parides Sesostris
Overview: P. Vukusic and J.R. Sambles, Nature 424, 852 (2003)
1.2µm
3D Photonic Crystals in nature
Opals: 3D Photonic Crystals
Taken from: eBay.com
A closer look at an Opal
Taken from: J.B. Pendry, Current Science 76, 1311 (1999)
400nm
Visions for Photonic Crystals
• Custom designed electromagnetic vacuum
• Control of spontaneous emission
• Zero threshold lasers
• Ultrasmall optical components
• Ultrafast all-optical switching
• Integration of components on many layers
‘Photonic Micropolis’J. Joannopoulos Research Group (MIT) http://ab-initio.mit.edu/
‘Optical Microchip’S. John Research Group (Toronto)http://www.physics.utoronto.ca/~john/
Visions for photonic crystals