MIT 2.71/2.71010/17/05 wk7-a-1

Elementary waves: plane, spherical

MIT 2.71/2.71010/17/05 wk7-a-2

The EM vector wave equation

2

2

2

2

2

22

zyx ∂∂

+∂∂

+∂∂

=∇ zyxE ˆˆˆ zyx EEE ++=

01

01

01

2z

2

22z

2

2z

2

2z

2

2y

2

22y

2

2y

2

2y

2

2x

2

22x

2

2x

2

2x

2

=∂∂

−∂∂

+∂∂

+∂∂

=∂

∂−

∂

∂+

∂

∂+

∂

∂

=∂∂

−∂∂

+∂∂

+∂∂

tE

czE

yE

xE

tE

czE

yE

xE

tE

czE

yE

xE

012

2

22 =

∂∂

−∇tcEE

xE

zE

yE

E

MIT 2.71/2.71010/17/05 wk7-a-3

Harmonic solution in 3D: plane wave

zy

x

Φ=0Φ=2π

Φ=4πΦ=6π

E(z=0,t=0)

E(z=λ/2,t=0)

E(z=λ,t=0)

E(z=3λ/2,t=0)E(z=5λ/2,t=0)

E(z=2λ,t=0)E(z=3λ,t=0)

phase=constant

on the plane

t=0

MIT 2.71/2.71010/17/05 wk7-a-4

Plane wave propagating

zy

x

Φ=0Φ=2π

Φ=4πΦ=6π

E(z=0,t=∆t)

E(z=λ/2,t=∆t)

E(z=λ,t=∆t)

E(z=3λ/2,t=∆t)E(z=5λ/2,t=∆t)

E(z=2λ,t=∆t)E(z=3λ,t=∆t)

phase=constant

on the plane

t=∆tc∆t

propagationdirection

MIT 2.71/2.71010/17/05 wk7-a-5

Complex representation of 3D waves

( ) ( )

( ) ( )

( ) ( )

( )

( )

( ) const.,, surface : Wavefront""

,, wherephasor""or amplitudecomplex e

tionrepresenta complex e,,,ˆ

etc. ,cos2 cos,,,

coscoscos2cos,,,

0

,,

0

0

0

=

−++≡

=

=−−++=

⎟⎠⎞

⎜⎝⎛ −−++=

−

−−++

zyx

zkykxkzyxA

Atzyxf

ktzkykxkAtzyxf

tzyxAtzyxf

zyx

zyxi

tzkykxki

xzyx

zyx

φ

φφ

αλπφω

φωγβαλπ

φ

φω

MIT 2.71/2.71010/17/05 wk7-a-6

Plane wave

kkx

ky

kz

x

y

z

wave-vector

MIT 2.71/2.71010/17/05 wk7-a-7

Plane wave

kkx

ky

kz

x

y

z

wave-vector

( ) ( )

relation)n (dispersio

iffequation wavesolves

ˆˆˆ vector)coordinate (Cartesian

ˆˆˆe

0

c

kkk

zyxAa

zyx

ti

ω

ω

=

++=

++== −⋅

k

zyxk

zyxrr rk

MIT 2.71/2.71010/17/05 wk7-a-8

Plane wave

kkx

ky

kz

x

y

z

( ) ( )

plane a isfront - waveconst.

:condition phaseconstant

ˆˆˆ vector)coordinate (Cartesian

ˆˆˆe

0

⇒=−⋅

++=

++== −⋅

t

kkk

zyxAa

zyx

ti

ω

ω

rk

zyxk

zyxrr rk

const.by described surface

:wavefront""

=⋅rk

MIT 2.71/2.71010/17/05 wk7-a-9

Plane wave propagating

κ

Φ=0Φ=2π

Φ=4πΦ=6π

E(κ=0,t=0)

E(κ=λ/2,t=0)

E(κ=λ,t=0)

E(κ=3λ/2,t=0)E(κ=5λ/2,t=0)

E(κ=2λ,t=0)E(κ=3λ,t=0)

phase=constant

on the plane

t=0const. plane =⋅rk

wave-vectordirection

MIT 2.71/2.71010/17/05 wk7-a-10

Plane wave propagating

y

x

Φ=0Φ=2π

Φ=4πΦ=6π

E(κ=0,t=∆t)

E(κ=λ/2,t=∆t)

E(κ=λ,t=∆t)

E(κ=3λ/2,t=∆t)E(κ=5λ/2,t=∆t)

E(κ=2λ,t=∆t)E(κ=3λ,t=∆t)

t=∆tc∆t

propagationdirection

phase=constant

on the plane

const. plane =⋅rk

wave-vectordirection

κ

MIT 2.71/2.71010/17/05 wk7-a-11

Spherical wave

“point”source

Outgoingrays

Outgoingwavefronts

equation of wavefront

constant=− tkR ω

R( )

RtkRAtzyxa 2/cos),,,( πω +−

=

( ){ }iR

tkRiAtzyxa exp),,,( ω−=

⎭⎬⎫

⎩⎨⎧ +

+=zyxizi

iRAzyxa

λπ

λπ

22

2 exp),,(

exponentialnotation

paraxialapproximation

MIT 2.71/2.71010/17/05 wk7-a-12

Spherical wave

“point”source

Φ=2π

Φ=4π

Φ=6π

spherical wavefronts

“point”source

Φ=2π

Φ=4π

Φ=6π

parabolic wavefronts

exactexact paraxial approximation/paraxial approximation//Gaussian beams/Gaussian beams

MIT 2.71/2.71010/17/05 wk7-a-13

The role of lenses

“point”image

“point”source

spherical wave(divergent)

plane wave

plane wave spherical wave(convergent)

MIT 2.71/2.71010/17/05 wk7-a-14

The role of lenses

“point”image

“point”source

plane wave

plane wave

spherical wave(divergent)

spherical wave(convergent)

MIT 2.71/2.71010/17/05 wk7-a-15

Polarization

MIT 2.71/2.71010/17/05 wk7-a-16

Propagation and polarization

x

y

z

kwave-vector

Eelectric field vector

const.efrontplanar wav

=⋅rk

) toparallelnot have could

one crystals, e.g.media, canisotropi

in :(reminder0

generally, More

i.e.0

etc.) glass, amorphousspace, free (e.g.

media isotropicIn

DE

Dk

EkEk

=⋅

⊥=⋅

MIT 2.71/2.71010/17/05 wk7-a-17

Linear polarization (frozen time)

zy

x

Φ=0

Φ=2π

E(z=0,t=0)

E(z=λ/2,t=0)

E(z=λ,t=0)

phase=constant

on the plane

t=0

MIT 2.71/2.71010/17/05 wk7-a-18

Linear polarization (fixed space)

ty

x

Φ=0

Φ=2π

E(z=0,t=0)

E(z=0,t=π/ω)

E(z=0,t=2π/ω)

phase=constant

on the plane

z=0

MIT 2.71/2.71010/17/05 wk7-a-19

Circular polarization (frozen time)

zy

x

Φ=0

Φ=2π

E(z=0,t=0)E(z=λ/2,t=0)

E(z=λ,t=0)

phase=constant

on the plane

t=0

MIT 2.71/2.71010/17/05 wk7-a-20

Circular polarization: linear components

+

Ex

Ey

z

z

MIT 2.71/2.71010/17/05 wk7-a-21

Circular polarization (fixed space)

z y

x

E(z=0,t=0)

E(z=0,t=π/ω)

E(z=0,t=π/2ω) E(z=0,t=3π/2ω)

rotationdirection

MIT 2.71/2.71010/17/05 wk7-a-22

z

λ/4 plate

birefringentλ/4 plate

LinearLinearpolarizationpolarization

CircularCircularpolarizationpolarization

MIT 2.71/2.71010/17/05 wk7-a-23

z

λ/2 plate

birefringentλ/2 plate

LinearLinearpolarizationpolarization

Linear (90Linear (90oo--rotated)rotated)polarizationpolarization

MIT 2.71/2.71010/17/05 wk7-a-24

Think about that

birefringentλ/4 plate

LinearLinearpolarizationpolarization

z

mirror

IncomingIncoming

????????

OutgoingOutgoing

MIT 2.71/2.71010/17/05 wk7-a-25

Relationship between E and B

EB

k

( )

EkB

k

xEBE rk

×=⇒

−≡∂∂

×≡∇×⇒

=∂∂

−=×∇ −⋅

ω

ω

ω

1

and

eˆ where 0

it

i

Et

ti

Vectors k, E, B form aright-handed triad.

Note: free space or isotropic media only