Chapter 6.

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.htmlhttps://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 𝐫Eigenmodes

Eigenvalues 𝜔 𝐤

3D

𝑒𝑖𝑘𝑧

𝑟𝑇𝑒−𝑖𝑘𝑧

Bragg ReflectionIncident 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

𝑑Bragg 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 e2w

0 π/a

Add a small“real” periodicity𝜀2 = 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” periodicity𝜀2 = 𝜀1 + Δ𝜀

1D Photonic Crystal

𝑛 𝑧 = 𝑛 𝑧 + Λ

𝐻 𝑧 = 𝐻𝑘 𝑧 𝑒𝑖𝑘𝑧

𝑔 =2𝜋

ΛDispersion relation: cos 2𝜋

𝑘

𝑔= Re

1

𝑡 𝜔

cos 2𝜋𝑘

𝑔=

1

𝑡12𝑡21cos 𝜋

𝜔

𝜔𝐵− 𝑟12

2 cos 𝜋𝜁𝜔

𝜔𝐵

𝑡12𝑡21 =4𝑛1𝑛2𝑛1 + 𝑛2

2 𝑟122 =

𝑛2 − 𝑛12

𝑛1 + 𝑛22

𝜔𝐵 =𝑐𝜋

ത𝑛Λത𝑛 =

𝑛1𝑑1 + 𝑛2𝑑2Λ

𝜁 =𝑛1𝑑1 − 𝑛2𝑑2𝑛1𝑑1 + 𝑛2𝑑2

Bragg frequency

𝑛1 𝑛2

Λ

𝑑1 𝑑2

𝜔

2𝜔𝐵

𝜔𝐵

−𝑔

2

𝑔

2𝑘

𝜔 ∝ 𝑘

photonic bangap

photonic bangap

cos 2𝜋𝑘

𝑔=

1

𝑡12𝑡21cos 𝜋

𝜔

𝜔𝐵− 𝑟12

2 cos 𝜋𝜁𝜔

𝜔𝐵Band 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+𝑒𝐸0𝑚𝑒

𝑒𝑖𝜔𝑡

resistive force by scattering

force due to the incident light field

From this, we found the polarization:

𝑃 𝑡 = 𝑁𝑒𝑥 𝑡 = −𝑁𝑒2/𝑚𝑒𝜔2 + 𝑖𝜔𝛾

𝐸(𝑡)

We now plug this in for the polarization term in the wave equation.

Plasma Frequency

Polarization in conductor: 𝑃 𝑡 = −𝑁𝑒2/𝑚𝑒𝜔2 + 𝑖𝜔𝛾

𝐸(𝑡)

Define a new constant, the “plasma frequency” 𝜔𝑝:

𝜔𝑝2 =

𝑁𝑒2

𝜀0𝑚𝑒

Thus 𝑃 𝑡 = −𝜀0𝜔𝑝2

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 −𝜔𝑝2

𝜔2 + 𝑖𝜔𝛾

𝜕2𝐸

𝜕𝑧2−𝑛2

𝑐2𝜕2𝐸

𝜕𝑡2= 0

Optical properties in the low-frequency limit, 𝝎 ≪ 𝜸

𝜀 𝜔 = 𝜀0 1 −𝜔𝑝2

𝜔2 + 𝑖𝜔𝛾≈ 𝜀0 1 + 𝑖

𝜔𝑝2

𝜔𝛾Dielectric function:

𝜀 𝜔 = 𝜀0 1 + 𝑖𝜎0𝜀0𝜔

In the Drude model, 𝐽 = 𝜎0𝐸 𝜎0 =𝑁𝑒2𝜏

𝑚𝑒=𝑁𝑒2

𝑚𝑒𝛾where

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 −𝜔𝑝2

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/𝜀

0

Re 𝜀Im 𝜀

linear scale

𝜔𝑝 = 4000 cm-1

𝛾 = 40 cm-1

𝜔𝑝

10-3

10-1

101

103

105

𝜀𝜔/𝜀

0100 101 102 103 104

Frequency (cm-1)

log scale

𝜔𝑝

Re 𝜀

−Re 𝜀

Im 𝜀

10510-5

𝜀 𝜔 = 𝜀0 1 −𝜔𝑝2

𝜔2 + 𝑖𝜔𝛾Dielectric 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 −𝜔𝑝2

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

𝜅1𝜀1+𝜅2𝜀2

= 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 −𝜔𝑝2

𝜔2For

𝜔𝑠𝑝 =𝜔𝑝

1 + 𝜀2

Surface plasmonsare 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

Negativerefraction

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 = −𝜀𝜔𝐄𝜀 = 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 RingResonators(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

pointsource

imageInternalfocus

evanescent waves

𝑛 = 1 𝑛 = 1𝑛 = −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