chapter 5. photonic crystals, plasmonics, and...
TRANSCRIPT
Chapter 5.
Photonic Crystals, Plasmonics,
and Metamaterials
Reading: Saleh and Teich Chapter 7
Novotny and Hecht Chapter 11 and 12
1. Photonic Crystals
http://optoelectronics.eecs.berkeley.edu/photonic_crystals.html https://alexandramjurgens.wordpress.com/tag/ens-cachan/
1D 2D 3D
Periodic photonic structures
Period 𝑎 ~ 𝜆
Natural Photonic Crystals
http://optoelectronics.eecs.berkeley.edu/photonic_crystals.html
Photonic Crystal Optical Fibers
Photonic Bandgap Fibers for Precision Surgery and Cancer Therapy
http://optoelectronics.eecs.berkeley.edu/photonic_crystals.html
Photonic Crystal Enhanced LED
Photonic Integrated Circuits
http://optoelectronics.eecs.berkeley.edu/photonic_crystals.html
Photonic Crystal Waveguide
http://spie.org/x104683.xml
A photonic crystal of dielectric media has a periodic lattice structure whose constituent media have distinctive dielectric constants:
𝜀𝑟 𝐫 = 𝜀𝑟 𝐫 + 𝐮
for all Bravais lattice vectors, 𝐮 = 𝑛1𝐚𝟏 + 𝑛2𝐚𝟐 + 𝑛3𝐚𝟑
The electromagnetic modes in the photonic crystal take the form,
𝐇 𝐫, 𝑡 = 𝐇 𝐫 𝑒−𝑖𝜔𝑡 , 𝐄 𝐫, 𝑡 =𝑖
𝜔𝜀0𝜀𝑟 𝐫𝛻 × 𝐇 𝐫, 𝑡
where the spatial mode function 𝐇 𝐫 is determined by the wave equation
𝛻 ×1
𝜀𝑟 𝐫𝛻 × 𝐇 𝐫 =
𝜔2
𝑐2𝐇 𝐫
Wave Equation in a Photonic Crystal
𝐇𝐤 𝐫 Eigenmodes
Eigenvalues 𝜔 𝐤
3D
𝑒𝑖𝑘𝑧
𝑟𝑇𝑒−𝑖𝑘𝑧
Bragg Reflection Incident light
𝑑
𝑟
The total reflection coefficient 𝑟 from a semi infinite structure:
𝑟𝑇 = 𝑟 + 𝑟𝑒2𝑖𝑘𝑑 + 𝑟𝑒4𝑖𝑘𝑑 + 𝑟𝑒6𝑖𝑘𝑑 + ⋯ =𝑟
1 − 𝑒2𝑖𝑘𝑑
Light cannot propagate in a crystal, when the frequency of the incident light satisfies the Bragg condition.
Origin of the photonic bandgap
Diverges if 𝑒2𝑖𝑘𝑑 = 1 → 𝑘 =𝜋
𝑑 Bragg condition
Constructive interference
e(x) = e(x+a) a
e1
Any 1d Periodic System has a Gap
w
0 π/a
sin
ax
cos
ax
x = 0
Treat it as “artificially” periodic
e(x) = e(x+a) a
e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 w
0 π/a
Add a small “real” periodicity 𝜀2 = 𝜀1 + Δ𝜀
sin
ax
cos
ax
x = 0
Any 1d Periodic System has a Gap
band gap
w
0 π/a
sin
ax
cos
ax
e(x) = e(x+a) a
e1 e2 e1 e2 e1 e2 e1 e2 e1 e2 e1 e2
x = 0
Splitting of degeneracy: state concentrated in higher index (𝜀2) has lower frequency
Any 1d Periodic System has a Gap
Add a small “real” periodicity 𝜀2 = 𝜀1 + Δ𝜀
1D Photonic Crystal
𝑛 𝑧 = 𝑛 𝑧 + Λ
𝐻 𝑧 = 𝐻𝑘 𝑧 𝑒𝑖𝑘𝑧
𝑔 =2𝜋
Λ Dispersion relation: cos 2𝜋
𝑘
𝑔= Re
1
𝑡 𝜔
cos 2𝜋𝑘
𝑔=
1
𝑡12𝑡21cos 𝜋
𝜔
𝜔𝐵− 𝑟12
2 cos 𝜋𝜁𝜔
𝜔𝐵
𝑡12𝑡21 =4𝑛1𝑛2
𝑛1 + 𝑛22 𝑟12
2 =𝑛2 − 𝑛1
2
𝑛1 + 𝑛22
𝜔𝐵 =𝑐𝜋
𝑛 Λ 𝑛 =
𝑛1𝑑1 + 𝑛2𝑑2
Λ
𝜁 =𝑛1𝑑1 − 𝑛2𝑑2
𝑛1𝑑1 + 𝑛2𝑑2
Bragg frequency
𝑛1 𝑛2
Λ
𝑑1 𝑑2
𝜔
2𝜔𝐵
𝜔𝐵
−𝑔
2
𝑔
2 𝑘
𝜔 ∝ 𝑘
photonic bangap
photonic bangap
cos 2𝜋𝑘
𝑔=
1
𝑡12𝑡21cos 𝜋
𝜔
𝜔𝐵− 𝑟12
2 cos 𝜋𝜁𝜔
𝜔𝐵 Band structure:
Y. Akahane et. al. Nature 425, 944 (2003)
Photonic Nanocavities
Photonic cavities strongly confine light.
Applications • Coherent electron–photon
interactions • Ultra-small optical filters • Low-threshold lasers • Photonic chips • Nonlinear optics and
quantum information processing
PC Waveguide: High transmission through sharp bends
A. Mekis et al, PRL, 77, 3786 (1996)
Highly efficient transmission of light around sharp corners in photonic bandgap waveguides
Tunneling through localized resonant state
S. Fan et. al., PRL 80, 960 (1998).
Complete transfer can occur between the continuums by creating resonant states of different symmetry, and by forcing an accidental degeneracy between them.
2. Surface Plasmons
K. Yao and Y. Liu, Nanotech. Rev. 3, 177 (2014)
Recall the inhomogeneous wave equation:
Electromagnetic Waves in Conductors
𝜕2𝐄
𝜕𝑧2−
1
𝑐2
𝜕2𝐄
𝜕𝑡2= 𝜇0
𝜕2𝐏
𝜕𝑡2
The equation of motion based on the forced oscillator model is
Polarization 𝐏 when there are free electrons:
𝑑2𝑥 𝑡
𝑑𝑡2= 𝛾
𝑑𝑥 𝑡
𝑑𝑡+
𝑒𝐸0
𝑚𝑒 𝑒−𝑖𝜔𝑡
resistive force by scattering
force due to the incident light field
From this, we found the polarization:
𝑃 𝑡 = 𝑁𝑒𝑥 𝑡 = −𝑁𝑒2/𝑚𝑒
𝜔2 + 𝑖𝜔𝛾𝐸(𝑡)
We now plug this in for the polarization term in the wave equation.
Plasma Frequency
Polarization in conductor: 𝑃 𝑡 = −𝑁𝑒2/𝑚𝑒
𝜔2 + 𝑖𝜔𝛾𝐸(𝑡)
Define a new constant, the “plasma frequency” 𝜔𝑝:
𝜔𝑝2 =
𝑁𝑒2
𝜀0𝑚𝑒
Thus 𝑃 𝑡 = −𝜀0𝜔𝑝
2
𝜔2 + 𝑖𝜔𝛾𝐸(𝑡)
So this must be the (complex) refractive index for a metal.
Back to the wave equation
𝜕2𝐸
𝜕𝑧2−
1
𝑐2
𝜕2𝐸
𝜕𝑡2= 𝜇0
𝜕2𝑃
𝜕𝑡2= −𝜇0𝜀0
𝜔𝑝2
𝜔2 + 𝑖𝜔𝛾
𝜕2𝐸
𝜕𝑡2
𝜕2𝐸
𝜕𝑧2−
1
𝑐21 −
𝜔𝑝2
𝜔2 + 𝑖𝜔𝛾
𝜕2𝐸
𝜕𝑡2= 0
This is the wave equation for a wave propagating in a uniform medium, if we define the refractive index of the medium as:
𝑛2 𝜔 = 𝜀𝑟 𝜔 = 1 −𝜔𝑝
2
𝜔2 + 𝑖𝜔𝛾
𝜕2𝐸
𝜕𝑧2−
𝑛2
𝑐2
𝜕2𝐸
𝜕𝑡2= 0
Optical properties in the low-frequency limit, 𝝎 ≪ 𝜸
𝜀 𝜔 = 𝜀0 1 −𝜔𝑝
2
𝜔2 + 𝑖𝜔𝛾≈ 𝜀0 1 + 𝑖
𝜔𝑝2
𝜔𝛾 Dielectric function:
𝜀 𝜔 = 𝜀0 1 + 𝑖𝜎0
𝜀0𝜔
In the Drude model, 𝐽 = 𝜎0𝐸 𝜎0 =𝑁𝑒2𝜏
𝑚𝑒=
𝑁𝑒2
𝑚𝑒𝛾 where
From Drude theory, that 𝜏~10−14 sec, so 𝛾 = 1/𝜏 ~1014 Hz.
For a typical metal, 𝜔𝑝 is 100 or even 1000 times larger.
(corresponding to the frequency of infrared light)
(corresponding to the frequency of ultraviolet light)
In the high-frequency limit, 𝝎 ≫ 𝜸 𝜀 𝜔 ≈ 𝜀0 1 −𝜔𝑝
2
𝜔2
A plot of Re 𝜀 and Im 𝜀 for illustrative values:
• Imaginary part gets very small for high frequencies • Real part has a zero crossing at the plasma frequency
0 2000 4000 6000 8000 10000
0
-1
-2
-3
-4
-5
1
2
3
4
5
Frequency (cm-1)
𝜀𝜔
/𝜀0
Re 𝜀 Im 𝜀
linear scale
𝜔𝑝 = 4000 cm-1
𝛾 = 40 cm-1
𝜔𝑝
10-3
10-1
101
103
105
𝜀𝜔
/𝜀0
100 101 102 103 104
Frequency (cm-1)
log scale
𝜔𝑝
Re 𝜀
−Re 𝜀
Im 𝜀
105
10-5
𝜀 𝜔 = 𝜀0 1 −𝜔𝑝
2
𝜔2 + 𝑖𝜔𝛾 Dielectric function of metals:
Drude theory at optical frequencies
large and negative (below 𝜔𝑝)
small and positive
Re{𝜀}
𝜀0
Im{𝜀}
𝜀0
𝜀 𝜔
𝜀0= 1 −
𝜔𝑝2
𝜔2 + 𝑖𝜔𝛾
≅ 1 −𝜔𝑝
2
𝜔2+ 𝑖
𝛾
𝜔
𝜔𝑝2
𝜔2
𝜔𝑝 > 𝜔 ≫ 𝛾
IR visible
In the regime where 𝜔 > 𝛾,
• For frequencies below the plasma frequency, 𝑛 is complex, so the wave is attenuated and does not propagate very far into the metal.
• For high frequencies above the plasma frequency, 𝑛 is real. The metal becomes transparent! It behaves like a non-absorbing dielectric medium.
Reflectivity drops abruptly at the plasma frequency.
This is why x-rays can pass through metal objects.
𝑛 𝜔 =𝜀 𝜔
𝜀0= 1 −
𝜔𝑝2
𝜔2
High frequency optical properties
The uppermost part of the atmosphere, where many of the atoms are ionized. There are a lot of free electrons floating around here.
For 𝑁~1012 m−3, the plasma frequency is:
Radiation above 9 MHz is transmitted, while radiation at lower frequencies is reflected back to earth. That’s why AM radio broadcasts can be heard very far away.
Radio Waves in Ionosphere
𝜔𝑝 =𝑁𝑒2
𝜀0𝑚𝑒= 2𝜋 × 9 MHz
𝜔𝑝 = 15000 cm−1
𝛾 = 40 cm−1
For real metals, there is a very broad range of frequencies for which Im 𝜀 ~0 and Re 𝜀 < 0.
0 4000 8000 12000 16000 20000
0
-4 -8
-12
-16
-20
4
8
12
16
20
Frequency (cm-1)
e(w
)/e 0
Re 𝜀
Im 𝜀
linear
scale
𝜔𝑝
This has interesting implications!!!
Dielectric Function of Real Metals
Consider a wave at the interface between two semi-infinite non-magnetic media (𝜇1 = 𝜇2 = 𝜇0).
Is there a solution to Maxwell’s equations describing a wave that propagates along the surface?
𝜺𝟏 𝜺𝟐
𝑧
𝑥
𝑧 = 0
This propagates along the interface, and decays exponentially into both media.
(Note: this is not a transverse wave, but that’s OK.)
Waves trapped at an interface
We can guess a solution of the form for media 1 and 2:
𝐄𝑚 = 𝐸1𝑥 , 0, 𝐸1𝑧 𝑒−𝜅𝑚 𝑧 𝑒𝑖 𝑘𝑥−𝜔𝑡
𝐁𝑚 = 0, 𝐵1𝑦 , 0 𝑒−𝜅𝑚 𝑧 𝑒𝑖 𝑘𝑥−𝜔𝑡 𝑚 = 1,2
In order to exist, the wave must satisfy Maxwell’s equations:
Since 𝜅1 and 𝜅2 are always positive, this shows that interface waves only exist if 𝜀1 and 𝜀2 have opposite signs.
In a metal, 𝜀 < 0 for frequencies less than 𝜔𝑝.
𝑖𝜅1𝐵1𝑦 = 𝜀1𝜔
𝑐2𝐸1𝑥
𝑖𝜅2𝐵2𝑦 = −𝜀2𝜔
𝑐2𝐸2𝑥
and also the continuity boundary conditions at 𝑧 = 0:
𝐵1𝑦 𝑧 = 0 = 𝐵2𝑦 𝑧 = 0 𝐸1𝑥 𝑧 = 0 = 𝐸2𝑥 𝑧 = 0
It is easy to show that these conditions can only be satisfied if: 𝜀1𝜅1
+𝜀2𝜅2
= 0
Interface Waves
𝛻 × 𝐁 = 𝜇𝜀𝜕𝐄
𝜕𝑡
Surface Plasmon Polariton (SPP) - a surface wave moving along the interface between a metal and a dielectric (e.g., air)
The electrons in the metal oscillate in conjunction with the surface wave, at the same frequency. In fact, an SPP is both an electromagnetic wave and a collective oscillation of the electrons.
Surface Plasmon Polaritons
SPP Dispersion Relation
𝜅1
𝜀1+
𝜅2
𝜀2= 0
𝐸 = 𝐸0𝑒−𝜅 𝑧 𝑒𝑖 𝑘𝑥−𝜔𝑡
𝑘2 + 𝜅𝑖2 = 𝜀𝑖
𝜔
𝑐
2
SPP Electric field:
and 𝑖 = 1,2
Dispersion relation:
𝑘 =𝜔
𝑐
𝜀1𝜀2𝜀1 + 𝜀2
1/2
𝜀1 𝜔 = 1 −𝜔𝑝
2
𝜔2 For
𝜔𝑠𝑝 =𝜔𝑝
1 + 𝜀2
Surface plasmons are very sensitive to molecules on the metal surface.
Surface plasmon sensors
Instead of considering a semi-infinite piece of metal, what if the metal object is small?
e.g., a metal nanosphere
We can still excite a plasmon, but in this case it does not propagate! The electrons just collectively slosh back and forth.
excess negative charge
excess positive charge
There is a restoring force on the electron cloud! Once again, we encounter something like a mass on a spring, with a resonance…
Surface plasmons on small objects
2.8 nm copper nanoparticles
Pedersen et al., J Phys Chem C (2007)
The sloshing electrons interact with light most strongly at the resonant frequency of their oscillation.
gold nanoparticles give rise to the red colors in stained glass
windows
Surface plasmon resonance
Controlling the surface plasmon resonance
gold nano-shells
The frequency of the plasmon resonance can be tuned by changing the geometry of the metal nano-object.
Halas group, Rice U.
3. Metamaterials
L. Billings, Nature 500, 138 (2013)
Negative refractive index
K. Yao and Y. Liu, Nanotech. Rev. 3, 177 (2014)
Metadevices
N. I. Zheludev and Y. S. Kivshar, Nature Mat. 11, 917 (2012)
Is an invisibility cloak magic or reality?
Invisibility Skin Cloak for Visible Light
Ni et. al. Science 349, 1310 (2015).
AFM Cloak on Cloak off
𝜆 ≫ 𝑎, 𝑏
a b
Electromagnetic Metamaterial
Physics Today Jun 2004 Physics Today Feb 2007
Negative refraction
Invisibility
Meta-atom
Y. Liu and X. Zhang, Chem. Soc. Rev. 40, 2494 (2011)
Material Parameter Space by 𝜺 and 𝝁
𝑛2 = 𝜀𝑟𝜇𝑟
𝑛 = ± 𝜀𝑟 𝜇𝑟
𝐤 × 𝐄 = 𝜇𝜔𝐇
𝐤 × 𝐇 = −𝜀𝜔𝐄 𝜀 = 𝜀0𝜀𝑟, 𝜇 = 𝜇0𝜇𝑟 𝐒 = 𝐄 × 𝐇
𝐒 ∥ 𝐤
𝐒 ∥ −𝐤
Y. Liu and X. Zhang, Chem. Soc. Rev. 40, 2494 (2011)
Basic metamaterial structures to implement artificial electric and magnetic Responses
Periodic Wires
Split Ring Resonators (SRR)
Smith et. al., PRL14, 234 (2000)
First Negative Index Material
Negative Refraction and Perfect Focusing
RHM LHM
𝜃 𝜃
𝜃𝑟 > 0
𝜃𝑟 < 0
𝑛 = 1 𝑛 > 1 𝑛 = 1 𝑛 < 1
sin 𝜃 = 𝑛 sin 𝜃𝑟
point source
image Internal focus
evanescent waves
𝑛 = 1 𝑛 = 1 𝑛 = −1
Fang et. al. Science 308, 534 (2005)
FIB 40nm
AFM with superlens
AFM w/o superlens
Shelby et. al. Science 292, 77 (2001)
Y. Liu and X. Zhang, Chem. Soc. Rev. 40, 2494 (2011)
Invisibility Cloak