Photonic Crystals – Introduction

Lecture 5: Photonic Crystals – Introduction

5 nm Photonic crystal: An Introduction

Photonic crystal: An Introduction

Photonic crystal:

Periodic arrangement of dielectric (metallic, polaritonic…) objects. Lattice constants comparable to the wavelength of light in the material. “ A worm ahead of its time”

“ A worm ahead of its time”

Sea Mouse and its hair

Normal incident light

Off-Normal incident light

20cm

http://www.physics.usyd.edu.au/~nicolae/seamouse.html Fast forward to 1987……

Fast forward to 1987……

E. Yablonovitch “Inhibited spontaneous emission in solid state physics and electronics” Physical Review Letters, vol. 58, pp. 2059, 1987

S. John “Strong localization of photons in certain disordered dielectric superlattices” Physical Review Letters, vol. 58, pp. 2486, 1987

Face-centered cubic lattice Complete photonic band gap Omni-directional reflector

Omni-directional reflector

Y. Fink, et al, Science, vol.282, p.1679 (1998) B. Temelkuran et al, Nature, vol.420, p.650 -3 (2002) Integrated photonic circuits and photonic crystal fibers

Integrated photonic circuits and photonic crystal fibers

J. D. Joannopoulos, et al, Nature, vol.386, p.143-9 (1997) R. F. Cregan, et al, Science, vol.285, p.1537-9 (1999) Three-dimensional photonic crystals

Three-dimensional photonic crystals

Y. A. Vaslov, Nature, vol.414, p.289-93 (2001) S. Lin et al, Nature, vol. 394, p. 251-3, (1998) The emphasis of recent breakthroughs

The emphasis of recent breakthroughs

•The use of strong index contrast, and the developments of nano- fabrication technologies, which leads to entirely new sets of phenomena.

Conventional silica fiber, δn~0.01, photonic crystal structure, δn ~ 1

•New conceptual framework in optics

Band structure concepts. Coupled mode theory approach for photon transport.

•Photonic crystal: semiconductors for light. Two-dimensional photonic crystal

Two-dimensional photonic crystal

λ ∼ 1.5 µm

High-index dielectric material, e.g. Si or GaAs Band structure of a two-dimensional

crystal

0.1

0.2 Band structure of a two-dimensional crystal TM

modes

0.3

dielectric band dielectric X 0.4

0.5 Displacement field parallel to the cylinder

0.6

0.7

air band air 0.8

X 0.8 M

! 0.6

0.4

0.2 Frequency (c/a)

0.0

Γ Wavevector (2π/a)

Wavevector determines the phase between nearest neighbor unit cells. X: (0.5*2π/a, 0): Thus, nearest neighbor unit cell along the x-direction is 180 degree out-of-phase M: (0.5*2π/a, 0.5*2π/a): nearest neighbor unit cell along the diagonal direction is 180 degree out-of-phase Maxwell’s equation in the steady state

Maxwell’s equation in the steady state

Time-dependent Maxwell’s equation in dielectric media:

&(%0E(r,t)) "• H(r,t) = 0 " # H(r,t) $%(r) = 0 &t

$(µ0H(r,t)) "•#E(r,t) = 0 " # E(r,t) + = 0 $t ! ! Time harmonic mode (i.e. steady state): ! "i#t ! H(r,t) = H(r)e E(r,t) = E(r)e"i#t Maxwell equation for the steady state: !

! H(r) + i((r)0E(r)) = 0

" # E(r) $ i%(µ0H(r)) = 0

! Master’s equation for steady state in dielectric

Master’s equation for steady state in dielectric

Expressing the equation in magnetic field only:

2 1 1 & % ) c = " # " # H(r) = ( + H(r) " µ $(r) ' c * 0 0

Thus, the Maxwe!ll ’s equation for the steady state !c an be expressed in terms of an eigenvalue problem, in direct analogy to quantum mechanics that governs the properties of electrons.

Quantum mechanics Electromagnetism Field "(r,t) = "(r)e j#t H(r,t) = H(r)ei"t $ 2 ' ˆ # Eigen-value problem H" (r) = E"(r) "H(r) = & 2 )H (r) % c ( ! " 2#2 ! 1 Operator Hˆ = h + V(r) " = # $ # $ 2m %(r) ! !

! ! Electromagnetism as an eigenvalue problem

Electromagnetism as an eigenvalue problem

The master equations define an operator:

1 "H(r) # $ % $ % H(r) &(r)

Importantly, the Θ operator is a Hermitian operator. If we define the inner product !o f two vector fields F(r) and G(r) as:

(F,G) = # drF* (r) "G(r) then * " 1 # (F,$G)= drF %&'( &'G ) ! , * ! + * " 1 # = dr (&' F) %( &'G ) , * ! + * " 1 # = dr ( &' F) %(&'G) , * ! + * " 1 # = dr (&' &' F) %G = ($F,G) , * ! + General property of the harmonic modes

General property of the harmonic modes

Having the Θ operator to be Hermitian leads to a number of nice properties about the harmonic modes

2 $ # ' Assuming that H(r) is an eigen-mode, i.e. "H(r) = & ) H(r) % c (

2 2 2 ω is real. $ # ' $ #* ' (H,"H) = & ) (H,H) = ("H,H) = & ) (H,H) % c ( ! % c (

ω2 is positive. 2 ! # " & 1 2 % ( (H,H) = (H,)H) = dr , - H $ c ' * +(r)

Two modes H1(r) and H2(r) at different frequencies ω1 and ω2 are orthogonal, i.e. (H1, H2) = 0 ! 2 2 # "1 & # "2 & % ( (H 2 ,H1) = (H 2 ,)H1) = ()H 2 ,H1) = % ( (H 2 ,H1 ) $ c ' $ c '

Thus if ω1 and ω2 are different, then (H1, H2) = 0. ! Consequence of the orthogonality

Consequence of the orthogonality

For two real one-dimensional function f(x) and g(x) to be orthogonal, i.e.

0 = ( f ,g )= ! dx f (x)g (x)

Thus, the product fg must be negative as much as it is positive over the interval of interest, so that the net integral vanishes.

1 Since the operator "H(r) # $ % $ % H(r) &(r) contains derivative with respect to the field, higher-frequency mode tends to have more spatial variation! in their field patterns.

By orthogonality, higher-frequency mode tends to have more nodal plane in the field pattern. Scale Invariance

Scale Invariance

The solution at one scale determines the solution at all other length scales.

Suppose, for example, we have an electromagnetic steady state H(r) in a dielectric configuration ε(r)

2 1 & % ) " # " # H(r) = ( + H(r) $(r) ' c *

Then, in a configuration of dielectric ε’(r’) that is just a compressed or expanded version of ε: ε’(r’) = ε(r’/s), Using r’ = sr, H(r’/s) = H’(r’), and ! ! ' = ! / s 2 1 # ! $ After changing the length scale by & '' & '' H%(r')= ( ) H%(r') "%(r') * cs + s, we just scale the old mode and its frequency by the same factor λ λ equivalent

a a Normalized units

Normalized units

The lattice constant: a

The units of the following physical quantities become: Frequency: c/a

Angular frequency: 2πc/a a

Wavevector: 2π/a

0 Wavelength: a

0.1

0.2

modes

0.3

dielectric band dielectric X 0.4

0.5

A0.6 simple example for reading the band diagram

0.7

air band air

0.8 X

M 0.8 Gap extends from 0.2837 c/a to 0.4183 c/a

! The mid gap frequency is at 0.3510 c/a 0.6

0.4 To design a crystal such that 1.55 micron light falls at the center of the gap, we have M 0.2 Frequency (c/a) Γ X c/(1.55micron) = 0.3510 c/a, hence 0.0 a = 0.3510 * 1.55 micron = 0.5440 micron Wavevector (2π/a) Electromagnetic energy and the variational principle

Electromagnetic energy and the variational principle

From the Master equations: First two bands, at M-point Displacement field 2 1 & % ) !H(r) = " # " # H(r) = ( + H(r)

$ (r) ' c *

0.1

0.2

TM modes

Integral form: 0.3

dielectric band dielectric X 0.4

0.5

1 2 0.6

0.7

2 dr * + H(r) band air 0.8

" ! % ( ) (r) X

M 0.8 $ ' = 2

# c & dr H(r! ) ( 0.6

0.4

Concentration of the displacement field D M 0.2 in the high dielectric constant region Frequency (c/a) Γ X minimizes the frequency. 0.0 Γ X M Γ Wavevector (2π/a) A simple example of the band- structure: vacuum (1d)

A simple example of the band-structure: vacuum (1d)

Vacuum: ε=1, µ=1, plane-wave solution to the Maxwell’s equation:

k Hei(k"r#$t ) with a transversality constraints:k •H = 0

! ! A band structure, or dispersion relation defines the relation between the frequency ω, and the wavevector k.

" = c k ω

For a one-dimensional system, the band ω=ck structure c!a n be simply depicted as:

k Visualization of the vacuum band structure (2d)

Visualization of the vacuum band structure (2d)

For a two-dimensional system: ω

" = c k 2 + k 2 x y Light cone This function depicts a cone: light cone.

kx k !A few ways to visualize this band structure : y

k y ω ω ω Continuum of plane wave Light line kx ω=ckx

kx k 0 (0,0) (k,0) (k,k) (0,0) Constant frequency contour Projected band diagram Band diagram along several “special” directions Electromagnetic energy and the variational principle

Electromagnetic energy and the variational principle

From the Master equations: First two bands, at M-point Displacement field 2 1 & % ) !H(r) = " # " # H(r) = ( + H(r)

$ (r) ' c *

0.1

0.2

TM modes

Integral form: 0.3

dielectric band dielectric X 0.4

0.5

1 2 0.6

0.7

2 dr * + H(r) band air 0.8

" ! % ( ) (r) X

M 0.8 $ ' = 2

# c & dr H(r! ) ( 0.6

0.4

Concentration of the displacement field D M 0.2 in the high dielectric constant region Frequency (c/a) Γ X minimizes the frequency. 0.0 Γ X M Γ Wavevector (2π/a) Visualization of the vacuum band structure (2d)

Visualization of the vacuum band structure (2d)

For a two-dimensional system: ω

" = c k 2 + k 2 x y Light cone This function depicts a cone: light cone.

kx k !A few ways to visualize this band structure : y

k y ω ω ω Continuum of plane wave Light line kx ω=ckx

kx k 0 (0,0) (k,0) (k,k) (0,0) Constant frequency contour Projected band diagram Band diagram along several “special” directions Bloch theorem for electromagnetism

Bloch theorem for electromagnetism

In a periodic dielectric media, i.e. ε(r+a)=ε(r), the solution H(r) to the Master’s equation:

2 1 % $ ( ! " ! " H(r) = ' * H(r) a1 # (r) & c ) a2 has to satisfy the following relations:

i(k!r) H(r) = e uk (r)

where uk(r) = uk(r+a) is a periodic function. Bloch wave functions

Bloch wave functions

eikx

u(x) ! eikxu(x) !

! A simple proof of Bloch theorem

A simple proof of Bloch theorem (Solid state Phys. Kittel, p179-180) Proof in 1 dimension x • Consider N identical lattice points on a ring of length Na o a

• The dielectric function is periodic in a, with ε(x)=ε(x+sa), where s is an integer

• Translational symmetry Expect solutions of the wave equation

H(x+a) = C H(x) n-times • Going once around the ring: H(x+Na) = H(x) = CN H(x)

C is one of the N roots of unity: C = exp(i2πs/N); s = 0, 1, 2, …, N-1

• Bloch function H(x) = uk(x) exp(i2πsx/(Na)) satisfies H(x+a) = C H(x) H(x+Na) = H(x) Where uk(x+a) = uk(x) Bragg scattering

Bragg scattering

Incident light a eikx

re-ikx

Regardless of how small the reflectivity r is from an individual scatter, the total reflection R from a semi infinite structure: 1 R = re!ikx + re !2ikae!ikx + re !4ikae!ikx + ... = re!ikx 1! e!2ika Diverges if ! e2ika = 1 k = Bragg condition a

Light can not propagate in a crystal, when the Origin of the photonic frequency of the incident light is such that the band gap Bragg condition is satisfied A brief review of the reciprocal lattice

A brief review of the reciprocal lattice See Kittel, pp. 32-33

•The reciprocal lattice vector G is defined by: eiG"a = 0 where a is any lattice vector of the crystal.

! • For a given set of lattice vectors, a1, a2 and a3, the set of basis vectors for the reciprocal lattice is:

a 2 "a 3 a 3 " a1 a1 "a 2 b1 = # 2$ b2 = # 2$ b3 = # 2$ a1 •(a 2 "a 3 ) a1 •(a 2 "a 3 ) a1 •(a 2 "a 3 )

The reciprocal lattice vector G are: G = n 1 b 1 + n 2 b 2 + n 3 b 3 , where n1, n2, n3 are arbitrary integers.

! b2 a1

! ! ! Summary

Summary

•Photonic crystals are artificial media with a periodic index

contrast.

•Electromagnetic wave in a ! 0

0.1

0.2

photonic crystal is described by a TM

modes

0.3

dielectric band dielectric X

band structure, which relates the 0.4 0.5

frequency of modes to the 0.6

0.7

air band air 0.8

wavevectors. X

M 0.8 ! •Fundamental properties of 0.6 modes: scale invariance, 0.4 orthogonality. M 0.2 Frequency (c/a) Γ X 0.0 Γ X M Γ Wavevector (2π/a)