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So far

Geometrical Optics

Reflection and refraction from planar and spherical interfaces

Imaging condition in the paraxial approximation Apertures & stops

Aberrations (violations of the imaging condition due to terms of

order higher than paraxial or due to dispersion)

Limits of validity of geometrical optics: features of interest are much

bigger than the wavelength

Problem: point objects/images aresmallerthan !!!

So light focusing at a single point is an artifact of ourapproximations

To understand light behavior at scales ~ we need to take intoaccount the wave nature of light.

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Step #1 towards wave optics: electro-dynamics

Electromagnetic fields (definitions and properties) in vacuo

Electromagnetic fields in matter

Maxwells equations Integral form

Differential form

Energy flux and the Poynting vector The electromagnetic wave equation

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Electric and magnetic forces

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+ +

r

dl

Fr

I

I

q q

F

free charges

Magnetic forceCoulomb force

2

04

1

r

qq =

F

r

II

l

2d

d0

=

F

(dielectric) permitivityof free space

(magnetic) permeabilityof free space

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Note the units

2

0 Distance

Charge1

force

Electric

=

2

04

1

r

qq =

F

r

II

l

2d

d0

=

F2

0

Time

Charge

force

Magnetic

=

( ) ( )SpeedTime

Distance2/1

00

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Electric and magnetic fields

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+

v

F

B

E

electric

charge

velocity

Lorentz

force

magnetic

induction

electric

field

( )BvEF += q

Observation Generation

+

Estatic charge:

electric field

q

v

++

B electric current(moving charges):

magnetic field

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Gauss Law: electric fields

E

++

E

da da

A

V

=A V

Vd1

d0

aEGauss theorem

0

= E

charge density

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Gauss Law: magnetic fields

B

A

V

there are nomagnetic

charges

da

=A

0daB

magnetic charge density

0= BGauss theorem

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Amperes Law: magnetic induction

dlB

C

A

I

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+=

C AAt

l aE

aJB ddd 00

current

dlB

capacitor

Stokes theorem

+= tEJB 00

Maxwells extension,

Displacement currentcurrent density

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Maxwells equations

(in vacuo)

=A V Vd1

d0

aE0= E Gauss/electric

=A

0daB 0= B Gauss/magnetic

t

=

BE =

C At

l aBE dd

dd Faraday

+=

C At

l aEJB dd 00

+=

tEJB 00

Ampere-Maxwell

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Electric fields in dielectric media

E

++++

++++

atom under electric field:

charge neutrality is preserved

spatial distribution of charges

becomes assymetric

atom in equilibrium

= pP++ = rrp qq

PolarizationDipole moment

Spatially variant polarization

induces localcharge imbalances(bound charges)

+

+

+

+

+

+

+

+

+

+

+

+ P=bound

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Electric displacement

( )boundfree0

total

0

11

+== EGauss Law:

( )P= free0

1

( ) free0 =+ PE

PED += 0 free= DElectric displacement field:

EP 0=

( ) EED += 10

Linear, isotropicpolarizability:

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General cases of polarization

EP 0=Linear, isotropicpolarizability: E

P

Linear, anisotropicpolarizability:

E

EP

=

333231

232221

131211

0

P

...)2(00 ++= EEEP Nonlinear, isotropicpolarizability:

etc.

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Constitutive relationships

E: electric fieldPED += 0

D: electric displacementpolarization

B: magnetic induction

MB

H =0H: magnetic field

magnetization

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Maxwells equations

(in matter)

=A V VddfreeaD Gauss/electricfree= D

=A

0daB 0= B Gauss/magnetic

t

=

BE =

C At

l aBE dd

dd Faraday

+=

C At

l aDJH dd free0

+=

tDJH free0

Ampere-Maxwell

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Maxwells equations wave equation

(in linear, anisotropic, non-magnetic matter, no free charges/currents)

( ) 0= E

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0= B

t

=

BE

( )t

=

EB

0

0= E

0= B

t

=

BE

t

=

EB 0

matter spatially and

temporally invariant

( ) ( ) 22

0ttt

=

=

= EBEBE

( ) ( ) EEE =

=0

02

2

0

2 =

t

EE

electromagnetic

wave equation

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Maxwells equations wave equation

(in linear, anisotropic, non-magnetic matter, no free charges/currents)

0= E

0= B

t

=

BE

t=E

B 0

02

2

0

2 =

t

EE

012

2

2

2 =

tc

EE201c

wave velocity

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Light velocity and refractive index

1

2

vacuum

00

c

cvacuum: speed of light

in vacuum

( ) 02

01 n+= n: index of refraction

22

vacuum

2

01

cc

n = ccvacuum/n: speed of lightin medium of refr. index n

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Simplified (1D, scalar) wave equation

01

2

2

22

2

=

t

E

cz

E Eis a scalar quantity (e.g. the componentEy of anelectric field E)

the geometry is symmetric inx,y

thex,yderivatives are zero

Solution:

++

= t

c

zgt

c

zftzE ),(

tcz =0z

zz +0

t

c

zf 0 t

z

t+t

z

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Special case: harmonic solution

t=0

z

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z

t+t

==

zatzf 2cos)0,(

ct2=

( )

= t

c

za

ctzatzf

2cos2cos),( =c

dispersionrelation

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Complex representation of waves

( )

( ) ( )

( ) ( )( ) ( ) ( )

( ) ( ){ }( ) ( ) ( )phasor""oramplitudecomplexe

tionrepresentacomplexe,aka,,

Re,i.e.

sincos,cos,

2,2cos,

22

cos,

i

tkzi

A

Atzftzftzftzf

tkziAtkzAtzf

tkzAtzf

ktkzAtzf

tzAtzf

=

=

+==

===

=

wave-number

angular frequency

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Time reversal

( )uf

z

( )tkzf

( )uf

z

( )tkzf +

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Superposition

( )uf ( )uf

z

( )tkzf ( )tkzf +

( ) ( )tkzftkzf ++More generally,

is also a solution

( ) ( )tkzgtkzf ++ is a solution

Even more generally, LINEARITY

SUPERPOSITION

( ) ( )

( ) ( )

+++++

++

tkzgtkzg

tkzftkzf

21

21is a solution

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What is the solution to the wave

equation? In general: the solution is an (arbitrary) superposition of propagating

waves

Usually, we have to impose

initial conditions (as in any differential equation)

boundary condition (as in most partial differential equations)

Example: initial value problem

( ) ( )0,wave0 zfuf =

z

z=0

z

z=0 z=t/k

( )tzf ,wave

( ) ( ) ( ) ( )tkzftzfufzf = 0wave0wave ,0,

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What is the solution to the wave

equation? In general: the solution is an (arbitrary) superposition of propagating

waves

Usually, we have to impose

initial conditions (as in any differential equation)

boundary condition (as in most partial differential equations)

Boundary conditions: we will not deal much with them in this class,

but it is worth noting that physically they explain interesting

phenomena such as waveguiding from the wave point of view (we sawalready one explanation as TIR).

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Elementary waves:

plane, spherical

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The EM vector wave equation

2

2

2

2

2

22

zyx

+

+

= zyxE zyx EEE ++=

01

01

01

2

z

2

22

z

2

2

z

2

2

z

2

2

y2

22

y2

2

y2

2

y2

2

x

2

22

x

2

2

x

2

2

x

2

=

+

+

=

+

+

=

+

+

t

E

cz

E

y

E

x

E

t

E

cz

E

y

E

x

E

t

E

cz

E

y

E

x

E

01

2

2

2

2 =

tc

EE

xE

zE

yE

E

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Harmonic solution in 3D: plane wave

zy

x

E(z=0,t=0)

E(z=/2,t=0)

E(z=,t=0)

E(z=3/2,t=0)E(z=5/2,t=0)

E(z=2,t=0)

E(z=3,t=0)

phase

=con

stant

onthe

plan

e

t=0

=0=2

=4

=6MIT 2.71/2.710

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Plane wave propagating

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zy

x

=0

E(z=0,t=t)

E(z=/2,t=t)

E(z=,t=t)

E(z=3/2,t=t) E(z=5/2,t=t)

E(z=2,t=t)E(z=3,t=t)

phase

=con

stant

ontheplan

ect

propagation

direction t=t

=2

=4

=6

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Complex representation of 3D waves

( ) ( )

( ) ( )

( ) ( )( )

( )

( ) const.,,surface:Wavefront""

,,where

phasor""oramplitudecomplexe

tionrepresentacomplexe,,,

etc.,cos2

cos,,,

coscoscos2

cos,,,

0

,,

0

0

0

=

++

=

=++=

++=

++

zyx

zkykxkzyx

A

Atzyxf

ktzkykxkAtzyxf

tzyxAtzyxf

zyx

zyxi

tzkykxki

xzyx

zyx

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Plane wave

x

wave-vector

kx k

kz

z

ky

y

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Plane wave

ky

y

( ) ( )

relation)n(dispersio

iffequationwavesolves

vector)coordinate(Cartesian

e 0

c

kkk

zyx

Aa

zyx

ti

=

++=

++=

=

k

zyxk

zyxr

rrk

x

wave-vector

kx k

kz

z

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E(=0,t=0)

E(=/2,t=0)

E(=,t=0)

E(=3/2,t=0)E(=5/2,t=0)

E(=2,t=0)

E(=3,t=0)

phase

=con

stant

onthep

lane

t=0const.plane =rk

wave-vector

direction

=0=2

=4

=6MIT 2.71/2.710

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y

x

=0

E(=0,t=t)

E(=/2,t=t)

E(=,t=t)

E(

=3/2,t=t) E(=5/2,t=t)

E(=2,t=t)

E(=3,t=t)

t=t

ctpropagation

direction

phase

=con

stant

onthep

lane

const.plane =rk

wave-vector

direction

=2=4

=6

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Spherical waveequation of wavefront

Outgoing

rays

Outgoing

wavefronts

constant= tkR

R

( )R

tkRAtzyxa

2/cos),,,(

+=

( ){ }iR

tkRiAtzyxa

exp),,,(

=

+

+=

z

yxi

zi

iR

Azyxa

22

2exp),,(

exponential

notationpointsource

paraxial

approximation

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Spherical wave

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point

source

=2

=4

=6

spherical wavefronts

pointsource

=2

=4

=6

parabolic wavefronts

paraxial approximation/paraxial approximation/

/Gaussian beams

exactexact

/Gaussian beams

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The role of lenses

plane wave

point

source

spherical wave

(divergent)

plane wave

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point

image

spherical wave

(convergent)

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The role of lenses

plane wave

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point

image

spherical wave

(divergent)

point

source

plane wave spherical wave

(convergent)

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