THE PROPAGATION OF LIGHT

How to View LightHow to View Light

As a ParticleAs a Particle

As a RayAs a Ray As a WaveAs a Wave

Theories on nature of light:Light as a particle vs. Light as a wave

• Only corpuscular theory of light prevalent until 1660

• Francesco Maria Grimaldi (Bologna) described diffraction in 1660

Light as a particleSir Isaac Newton (1642-1727)• Embraces corpuscular theory of

light because of inability to explain rectilinear propagation in terms of waves

• Demonstrates that white light is mixture of a range of independent colors

• Different colors excite ether into characteristic vibrations---sensation of red corresponds to longer ether vibration

Light as a wave

Christiaan Huygens (1629-1695)Huygens’ principle (Traite de la

Lumière, 1678):Every point on a primary wavefront

serves as the source of secondary spherical wavelets, such that the primary wavefront at some later time is the envelope of these wavelets. Wavelets advance with speed and frequency of primary wave at each point in space

http://id.mind.net/~zona/mstm/physics/waves/propagation/huygens1.html

Light as a wave

Thomas Young (1773-1829)

1801-1803: double slit experiment, showing interference by light from a single source passing though two thin closely spaced slits projected on a screen far away from the slitshttp://vsg.quasihome.com/interfer.htm

Light as a waveAugustine Fresnel (1788-

1827)1818: Developed

mathematical wave theory combining concepts from Huygens’ wave propagation and wave interference to describe diffraction effects from slits and small apertures

Electromagnetic wave nature of light

• Michael Faraday (1791-1865)

• 1845: demonstrated electromagnetic nature of light by showing that you can change the polarization direction of light using a strong magnetic field

Electromagnetic theory

• James Clerk Maxwell (1831-1879)

• 1873: Theory for electromagnetic wave propagation

• Light is an electromagnetic disturbance in the form of waves propagated through the ether

Quantum mechanics• 1900: Max Planck postulates that

oscillating electric system imparts its energy to the EM field in quanta

• 1905: Einstein-photoelectric effect– Light consists of individual energy quanta, photons, that

interact with electrons like particle• 1900-1930 it becomes obvious that concepts of

wave and particle must merge in submicroscopic domain

• Photons, protons, electrons, neutrons have both particle and wave manifestations– Particle with momentum p has associated wavelength

given by p=h/l• QM treats the manner in which light is absorbed

and emitted by atoms

Max Planck

Niels Bohr

Louis de Broglie

Schrödinger

Heisenberg

Reflection and Refraction

vph = c/n1

vph = c/n2

qiqr

qt

normal

ri

1

2

sin

sin

nn

t

i

All of Geometrical optics boils down to…

Law of Reflection:

Snell’s Law:

The Snell’s Laws

Willebrordus Snellius

Fundamental Rules for Reflection and Refractionin the limit of Ray Optics

Fundamental Rules for Reflection and Refractionin the limit of Ray Optics

1. Huygens’s Principle2. Fermat’s Principle3. Electromagnetic Wave Boundary

Conditions

1. Huygens’s Principle2. Fermat’s Principle3. Electromagnetic Wave Boundary

Conditions

Huygen’s Principle

• Huygen assumed that light is a form of wave motion rather than a stream of particles

• Huygen’s Principle is a geometric construction for determining the position of a new wave at some point based on the knowledge of the wave front that preceded it

Christian Huygens (1629-1695)

Huygen’s Principle, cont.

• All points on a given wave front are taken as point sources for the production of spherical secondary waves, called wavelets, which propagate in the forward direction with speeds characteristic of waves in that medium– After some time has elapsed, the new position

of the wave front is the surface tangent to the wavelets

Huygen’s Construction for a Plane Wave

• At t = 0, the wave front is indicated by the plane AA’

• The points are representative sources for the wavelets

• After the wavelets have moved a distance cΔt, a new plane BB’ can be drawn, which is the tangent to the wavefronts

Huygen’s Construction for a Spherical Wave

• The points are representative sources for the wavelets

• The new wavefront is tangent at each point to the wavelet

Huygen’s Principle and the Law of Reflection

• The Law of Reflection can be derived from Huygen’s Principle

• AA’ is a wave front of incident light

• The reflected wave front is CD

Reflection According to Huygens

Reflection According to Huygens

Side-Side-SideDAA’C ADC1 = 1’

Side-Side-SideDAA’C ADC1 = 1’

Incoming ray Outgoing ray

Huygen’s Principle and the Law of Reflection, cont.

• Triangle ADC is congruent to triangle AA’C

• θ1 = θ1’• This is the Snell’s Law

of Reflection

Huygen’s Principle and the Law of Refraction

• Every point on a wave front can be considered to be a source of secondary waves. The figure explains the refraction at an interface between media with different optical densities.

Air

Huygens’s Principle and the Law of Refraction, cont.

• Ray 1 strikes the surface and at a time interval ∆t later, ray 2 strikes the surface– During this time interval, the wave

at A sends out a wavelet, centered at A, toward D

– The wave at B sends out a wavelet, centered at B, toward C

Huygens’s Principle and the Law of Refraction, cont.

• The two wavelets travel in different media, therefore their radii are different

• From triangles ABC and ADC, we find

Huygens’s Principle and the Law of Refraction, final

• The preceding equation can be simplified to

• This is Snell’s law of refraction

26

Pierre de Fermat’s principle

• 1657 – Fermat (1601-1665) proposed a Principle of Least Time encompassing both reflection and refraction

• “The actual path between two points taken by a beam of light is the one that is traversed in the least time”

Fermat’s Principle

The path a beam of light takes between two points is the one which is traversed in the least time.

A B

Isotropic medium: constant velocity.

Minimum time = minimum path length.

28

Optical path length

n1

n4

n2

n5

nm

n3

S

P

Optical Path Length (OPL)

When n constant, OPL = n geometric length.

nvac vac

LL

n > 1n = 1

For n = 1.5, OPL is 50% larger than L

For n = 1.5, OPL is 50% larger than L

P

SdxxnOPL )(

S P

30

Fermat’s principle

• t = OPL/c• Light, in going from

point S to P, traverses the route having the smallest optical path length

c

OPLt

31

Optical path length

• Transit time from S to P

m

iiisn

ct

1

1

m

iiisnOPL

1

P

SdssnOPL )(

P

S

dsv

cOPL

Same for all rays

32

n1

n2

Fermat’s principle

n1 < n2

A

O

B

θi

θr

x

a

h

bWhat geometry gives the shortest time between the points A and B?

n1

n2

qi

qt

normalA

B

O

Method 1

a

b

c

ti vOB

vAO

t

x

ti v

xcbv

xat

2222

2222 xcbv

xc

xav

xdxdt

ti

0

sinsin t

t

i

i

vvdxdt

ttii nn sinsin

Method 2

Minimizing the time (optical path length) between points Q and Q’ yields Snell’s Law:

2

2/122

1

2/122

21

)'())((

'

v

xh

v

xpht

V

AQ

v

QAt

'sin'sin

''

)'('

])([

,

02)'(

2/')22(

)([

2/

:

)'('])([

:

)'('

))((

'

2/1222/122

2/1222/12

2/1222/122

2/122

2/122

nn

andd

xn

d

xpn

or

xh

xn

xph

xpn

thus

xxh

nxp

xph

n

dx

d

atingdifferenti

xhnxphn

ngsubstituti

xhd

xphd

dnndOPL

Fermat’s Principle and ReflectionFermat’s Principle and Reflection

A light ray traveling from one fixed point to another will followa path such that the time required is an extreme point – either amaximum or a minimum.

Electromagnetic Waves

Maxwell’s Equations for time varying electric and magnetic fields in free space

0

E

t

BE

0 B

t

EIB

000

(where r is the charge density)

Simple interpretation

Divergence of electric field is a functionof charge density

A closed loop of E field lines will exist whenthe magnetic field varies with time

Divergence of magnetic field =0(closed loops)

A closed loop of B field lines will exist inThe presence of a current and/or time varying electric field

Description of Light

Wave Equation (derived from Maxwell’s equations)Any function that satisfies this eqn is a waveIt describes light propagation in free space and in time

operatorLaplacian

fieldinductionmagnetic

fieldelectricE

lightofspeedc

wheretc

t

E

cE

2

2

2

22

2

2

22

,

1

1

B

BB

(see calculus review handout)

Its general solutions (plane wave) :

trkie

B

E

B

E

0

0

TE TM

Electromagnetic Wave Boundary Conditions

(E fields)

Light at a Plane Dielectric Interface

TE TM

trkjjoiinc

ii eeE

E

tωrkjjφorref

rrr eeE

E tωrkjjφ

ottransttt eeE

E

ki kr

kt

ki kr

kt

n

Assume:

A plane wave is incident:

A plane wave is reflected:

A plane wave is transmitted:

What are the relative amplitudes, wave numbers, frequencies, and phases?

To remain constant at a certain place:

tri

rkrkrk tri

To remain constant at a certain time:

ki, kr, kt are all co-planar

incident, reflected, and refracted all at same frequency.

tωrktωrktωrk ttrrii

Relationship between fields at the interface should not depend on position or time:

vk ''kv

tkx sin

A boundary at one point in space for all time:

The left side shakes the right at frequency w, which creates a wave with a different velocity (different medium) and therefore different wavelength.

ki krn

O

r

(could be in any direction)ki dot r makes this happen… kr dot r makes this happen…

0 rkk ri

ki - kr

ki-kr dot r makes a plane, but it must be the surface since the boundary condition is for r at the surface.

But this can only be true if kr is also in the plane of incidence!

kikr

kt

r

rrii rkrk 22 coscos

Same medium, same velocity, same wavelength, same wavenumber, so:

ttii rkrk 22 coscos

qi qr

ri

ttii nn sinsin

qt

to

ti

o

i nn

sin2

sin2

Law of Reflection

Snell’s Law

SCad

tld

BE

SCad

ttld

PEB 000

0tangentialinside

tangentialoutside lElE

tangentialinside

tangentialoutside EE

0tangentialinside

tangentialoutside lBlB

tangentialinside

tangentialoutside BB

Tangential components of both E and B are continuous at the boundary.

Therefore, for all points on the boundary at all times:

tωrkjjφot

tωrkjjφor

tωrkjjφoi

tttrrriii eeEeeEeeE

Now for the relative amplitudes: reflection and transmission

TETM

tri EEE

ttrrii BBB coscoscos

tri BBB

ttrrii EEE coscoscos

BvBE nc

BiBr

E E

Bt

TE

TM

tri EEE

tttiriiii EnEnEn coscoscos

ttriii EnEnEn

ttirii EEE coscoscos

For reflection: eliminate Et, separate Ei and Er, and get ratio:

TE

tnn

i

tnn

i

i

r

i

t

i

t

EE

r

coscos

coscos

TM

tinn

tinn

i

r

i

t

i

t

EE

r

coscos

coscos

Get all in terms of E, and recall that qi = qr:

Apply Snell’s law (let n = nt/ni)

TE

ii

ii

n

nr

22

22

sincos

sincos

TM

ii

ii

nn

nnr

222

222

sincos

sincos

Coefficient of transmission: t

ii

i

i

t

nE

Et

22 sincos

cos2

TE

ii

i

i

t

nn

n

E

Et

222 sincos

cos2

TM

internal reflection: n = 0.667

-0.2

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80

r

Angle of incidence

TM

TE

-1

-0.5

0

0.5

1

0 20 40 60 80

r

Angle of incidence

TM

TE

external reflection: n = 1.5

TE/TM wave optical reflection• TE (transverse electric) polarization

– Electric field parallel to substrate surface

• TM (transverse magnetic) polarization– Magnetic field parallel to substrate surface

low index high index high index low index

TETM

TETM

RS

the critical angle for total reflection

If i cri, then it is total reflection and no power can be transmitted, these fields are referred as evanescent waves.

1 2critical

1

( ) sini

Brewster’s angle for total transmission

For lossless, non-magnetic media, we have

Total transmission for TM polarization

2 2 21 2 2 1

2 2 2 22 1 1 2

( )sini BA

1

1

2

1sin

1BA

r

r

RS

Ex1 A 2 GHz TE wave is incident at 30 angle of incidence from air on to a thick slab of nonmagnetic, lossless

dielectric with r = 16. Find TE and TE.

RS

Ex2 A uniform plane wave is incident from air onto glass at an angle from the normal of 30. Determine the fraction of the incident power that is reflected and transmitted for a) and b). Glass has refractive index n2 =

1.45.

a) TM polarization

b) TE polarization

Photons and The Laws of Reflection and Refraction