introduction to condensed matter physics · the optical properties of the material do not depend on...

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M.P. Vaughan

Introduction to Condensed Matter Physics

Diffraction I – Basic Physics

Coláiste na hOllscoile Corcaigh, Éire University College Cork, Ireland

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Department of Physics

PY3105 Introduction to

Condensed Matter Physics

Diffraction

• Electromagnetic waves• Geometric wavefront• The Principle of Linear Superposition• Diffraction regimes• Single slit diffraction• Multiple slit diffraction• Multiple slits as a 1D lattice

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Electromagnetic waves


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Maxwell’s Equations

,0=⋅∇ B

,fρ=⋅∇ D

,t∂

∂−=×∇B

E

,t

f ∂∂

+=×∇D

jH

(Gauss’ Law)

(no magnetic monopoles)

(Faraday’s Law of Induction)

(Ampere’s Law & Continuity)

where ρf and jf are the free charge density and current respectively,

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Wave solutions of Maxwell’s Equations

Using Maxwell’s Equations, a wave equation may be found of the form

,1

2

2

2

2

tv ∂∂

=∇E

E

where, in free space, the wave speed is

( ) .2/1

00

−== µεcv

Similar forms may be found in materials provided that the material is...


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linearThe electrical polarisation does not introduce nonlinear terms into the wave equation

homogenous

isotropic

The optical properties of the material are the same everywhere.

The optical properties of the material do not depend on the direction of the optical polarisation.


,n

cv =

If these conditions are met, then

where n is the refractive index.

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Plane wave solutions may be found of the form

( ) ( ),, 0

tiet ω−⋅= rkErE

where k is the wave vector and w is the angular frequency. From the wave equation, we find that

.k

vω

=

This speed is often called the phase velocity, for reasons we shall illuminate.


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Phase velocity

The condition

0φω =−⋅ trk

where φ0 is a constant must be satisfied for all points in space having the same phase.

Differentiating with respect to time, we obtain

ωα ==⋅ cosvkvk

where v is the wave speed (or phase velocity) and α is the angle between k and v.

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Phase velocity

Hence, putting v = |v| and k = |k|,

Note that the phase velocity is not the same as the group velocity.

.cosαω

kv =

If v and k are parallel, then

.k

vω

=


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Wave propagation in free space

Taking the gradient of

,0φω =−⋅ trk

where φ0 is a constant, will yield a vector normal to the surface of constant optical phase. Thus

.krk =⋅∇=∇φ

Therefore, for a linear, isotropic and homogeneous material, k is parallel to the phase velocity v.

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Spherical waves

rkek =

In this case, it is convenient to use spherical polar coordinates. The wave vector k has constant magnitude and always points away from the centre of radiation. Thus we put

and

rrer =

so that the equation for surfaces of constant phase becomes

.0φω =− tkr


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Spherical waves

Since the intensity of an EM wave is proportional to the squared modulus of the amplitude, by the conservation of energy, the amplitude must vary as 1/r.

Moreover, the requirement that the amplitude be finite at r = 0 means that the spherical wave must be of the form

( ) .sin, krer

EtrE tir ω−=

Note that such an equation cannot exactly model an EM

wave.

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Spherical waves

A 2D section of the amplitude of a spherical wave.

Geometric wavefront

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Geometric wavefront

A geometric wavefront is the surface in

space containing all points in an optical

field that have the same phase.

A ray is a path through space that is

everywhere perpendicular to the

wavefront.


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Geometric wavefront - spherical

Wavefronts –contours of

constant phase

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Geometric wavefront - spherical


constant phaseRays –

everywhere perpendicular to wavefronts


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Geometric wavefront - plane


constant phase

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Geometric wavefront - plane


constant phaseRays –

everywhere perpendicular to wavefronts


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Huygens’ Principle

Each point on a wavefront acts as a source

of secondary, spherical wavelets.

At a later time, t, a new wavefront is

constructed from the sum of these

wavelets.

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Huygens’ Principle – rectilinear propagation

Consider the wavefront of a

plane wave at z = 0

z0


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All points on the wavefront act as sources of

spherical wavelets

z0

constant phase over surface of sphere

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Every point on the line contributes a similar sphere

z0


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Since all points on the spheres must have the same

phase, the tangent to the leading edge of all the spheres must also be at a constant phase.

z0

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This leading edge will be a new

wavefront. Since it is parallel to the original wavefront, the light must be propagating in a straight line.

z0


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Huygens’ Principle – spherical waves

Application of Huygens’ Principle to spherical waves.

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We have used Huygens’ Principle to prove the Law of Rectilinear Propagation for a plane wave.

Note: we did not invoke the Principle of Superposition to prove this result.

When the Principle of Superposition is explicitly added to Huygens’ Principle, it is becomes the Huygens-Fresnel Principle.

Linear Superposition

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Linear superposition

Consider the differential equation

Suppose E1 and E2 are both solutions of this. Then we may construct the linear superposition

.01

2

2

22

2

=∂∂

−∂∂

t

E

vx

E

.21 bEaEE +=


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We then have

In other words, the linear superposition of E1 and E2 is also a solution of the differential equation.

.0

1

11

2

2

2

22

2

2

2

1

2

22

1

2

2

2

22

2

=

∂∂

−∂∂

+

∂∂

−∂∂

=∂∂

−∂∂

t

E

vx

Eb

t

E

vx

Ea

t

E

vx

E

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The differential equation

is an example of a linear differential equation. On the other hand, consider the equation

.01

2

2

22

2

=∂∂

−∂∂

t

E

vx

E

02

22

=∂

∂t

E

and suppose E1 and E2 are both solutions of this.


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If we substitute in the linear superposition as before, we find

( )( )

.02

2

2

21

2

2

2

2

21

2

1

2

2

2

2

22

≠∂

∂=

++∂∂

=∂

∂

t

EEab

EbEabEEatt

E

Thus, in general, we cannot find new solutions by linearly adding known solutions. This type of differential equation is called nonlinear.

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In free space or a linear medium,

the electric field is linearly

additive.

Hence, we may apply the Principle

of Linear Superposition to the

electric field vector.


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Interference effects

Linear superposition of the electric

field vector will lead to interference

effects.

Diffraction is an interference

effect.

Physically interference and diffraction

are one and the same phenomenon.

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The Huygens-Fresnel Principle

For light of a given frequency, every

point on a wavefront acts as a

secondary source of spherical wavelets

with the same frequency and the same

initial phase.

The wavefront at a later time and

position is then the linear superposition

of all of these wavelets.

Diffraction regimes: near field and far field

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Light passing through a narrow aperture

Maximum possible path difference

.max DABBPAP ==−=∆


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Limiting cases: λ >> D

∆max always less than λ – wavelets add constructively in all directions.

Emergent field looks like point

source.

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Limiting cases: λ << D

Both constructive and destructive interference outside shaded region

Wavelets add constructively in this region


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Fresnel and Fraunhofer diffraction

Near field (Fresnel diffraction)

Far field (Fraunhofer diffraction)

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Fresnel and Fraunhofer diffraction

• Fresnel diffraction• Diffraction pattern varies with increasing distance from aperture

• Fraunhofer diffraction• Diffraction pattern settles down to a constant profile

• Applies for when the radial distance from the aperture satisfies the Fraunhofer condition (see later).

Single slit diffraction

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EL is the field strength per unit

length

EP is the total field a the point P


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Field at x

.dxEdE L=

Contribution to field EP due to dE

( )( )[ ] .sin dxxkrt

xr

EdE L

P −= ω

Total field EP

( )( )[ ] .sin

2/

2/∫− −=D

D

LP dxxkrt

xr

EE ω

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r(x) is given by the cosine rule

( ) ( )θπ −−+=2

222 cos2RxxRxr

x

or

( ) .sin2

1

2/1

2

2

−+= θR

x

R

xRxr

To find a closed form solution, we must approximate this expression.


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Taylor series expansion of r(x)

The Taylor series expansion for a function (1 + ξ)1/2 is

( ) K+−+=+82

112

2/1 ξξξ

Hence,

( )

++−= Kθθ 2

2

2

cos2

sin1R

x

R

xRxr

and

( ) .cos2

sin 22

K++−= θθR

kxkxkRxkr

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The Fraunhofer condition

The third term in the expression for kr(x) takes its maximum when x ± D/2 and θ = 0. That is

.48

cos2 2

2

2

22

2

R

D

R

kD

R

kx

λπ

θ =→

The condition that this term makes a negligible contribution to the phase is

.4 2

2

πλπ

<<R

D


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The Fraunhofer condition

Neglecting the factor of 4 in the denominator of the condition just found, it may be re-written as

.DR

D λ<<

This is the Fraunhofer condition for far field diffraction.

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Far field approximations

Assuming that the Fraunhofer condition is valid, the third term in the expression for kr(x) may be neglected and we have

( ) .sinθkxkRxkr −≈

The 1/r(x) factor appearing in the integral for EP is less sensitive to changes in r(x) than the phase and we may simply put

( ).11

Rxr≈


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Integrating over x

Using these approximations, the expression for the total field EP becomes

To perform this integral, we note that

[ ] ( ){ }.Imsinsin sinθωθω kxkRtiekxkRt +−=+−

[ ] .sinsin2/

2/∫− +−=D

D

LP dxkxkRt

R

EE θω

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The total field EP

Integrating over the x-dependent part

where

.sin2

θβkD

=

,sin

sin

2/

2/

sin2/

2/

sin

ββ

θ

θθ D

ik

edxe

D

D

ikxD

D

ikx =

=

−−∫

Hence, the total field EP is

( ).sinsin

kRtR

DEE LP −= ω

ββ


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Intensity profile for a single slit

Averaging EP over time gives

The squared modulus of this will be proportional to the intensity, i.e.

.sin

2 ββ

R

DEE LP =

( ) ( ) .sin

0

2

ββ

θ II =

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The zeros of the peaks occur at values of

where m is an integer. Hence, the first zeros around the central peak are given by

,sin2

πθβ mkD

==

.sinD

λθ =

Note that this result is only valid for λ < D. In other cases, there are no zeros from –π to π.

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Multiple slit diffraction


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It will be useful to define

,sin2

θαka

=

in analogy to the previously defined

.sin2

θβkD

=


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For N slits, the total contribution of the field EP is

[ ] .sinsin1

0

2/

2/∑∫

−

=

+

−+−=

N

n

na

Dna

LP dxkxkRt

R

EE θω

Again, we make use of the earlier approximations for far field diffraction.

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∑∑−

=

−

=

+

−

=

1

0

21

0

2/

2/

sin

.sin

sin

N

n

niN

n

na

Dna

ikx

Deik

e

ββ

θα

θ

Focussing on the x-dependent part of the integral and factorising as before, we find

The new factor is a geometric progression with common factor ei2α

∑−

=

=1

0

2 .N

n

ni

N eS α


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Multiplying SN by ei2α

so

,1

22 ∑=

=N

n

nii

N eeS αα

( ) ,11 22 αα Nii

N eeS −=−

which gives

( ) .sin

sin1

ααα N

eS Ni

N

−=

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Intensity profile for multiple slits

The phase factor may be dropped from this expression. Note also that since

it is useful to incorporate a normalising factor 1/N into this ratio. Hence, the intensity takes the form

,sin

sinlim

0N

N=

→ αα

α

( ) ( ) .sin

sin

sin0

22

=ββ

αα

θN

NII


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Intensity profile for multiple slits

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Diffraction condition

Note that the condition for constructive interference is

.sin λθ na =


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We can re-write this as

.sin2

nka

πθ =

But this is just,nπα =

( ) ( ) .sin

sin

sin0

22

=ββ

αα

θN

NII

which gives the condition for the local maxima of the intensity

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Multiple slits as a 1D lattice

We can begin to see the connection between the diffraction pattern and crystal structure by imagining the array of slits to be a 1D lattice with lattice vector

.xa a=

The associated reciprocal lattice vector would then be

.ˆ2xb

a

π=


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Incident and diffracted wavevectors:

,zk k=

( ).cosˆsinˆ' θθ zxk += k

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We take the wavevector of the incident wave to be

,zk k=

We then have

and that of the diffracted wave to be

( ).cosˆsinˆ' θθ zxk += k

( )[ ].1cosˆsinˆ' −+=−=∆ θθ zxkkk k


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Considering just the x component,

this becomes

Asserting the diffraction condition

.sinθkkx =∆

,sin2

nka

πθ =

.2

a

nkx

π=∆

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But this is just an integral multiple of the x component of the reciprocal lattice vector

Writing this as a G vector, assertion of the diffraction condition then gives

.2

xx nba

nk ==∆

π

.xx Gk =∆

introduction to condensed matter physics · the optical properties of the material do not depend on...

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