Physics 214

3: Interference, Diffraction and Polarization

• Young’s Double-Slit Experiment• Intensity Distribution of the Double-Slit

Interference Pattern• Interference in Thin Films• Single Slit Diffraction• Diffraction Grating• Diffraction by Crystals• Polarization of Light Waves

Double Slit Experiment

In order to observe interference in light rays, light must be:• Coherent• Monochromatic

Superposition Principle must apply

In phase Out of phase

r1

r2d

d

yq

L

x axis

P; (x=0)

path difference d=r2 - r1sin q =

dd

Þ d=dsin q=r2 - r1

We get constructive interference when d=dsinq =m l, m =0,±1,±2, K

We get destructive interference when d=dsin q= m +1

2æ è ç

ö ø ÷ l

for small q

m ld = sinq » tanq = y

L\ position of FRINGES

y bright = ml Ld

y dark = m + 12( )l Ld

Consider electric field intensity of the two interfering light waves at the point P

E1 = E0 sin(kx-wt)E 2 = E 0 sin

f only depends on path difference dpath difference of one wavelength l

cphase difference of 2 p radians

(kx-wt+f)

path difference of l2c

phase difference of p radians\d

l= f

2pÞ d

f= l

2p

\f= 2pld= 2p

ldsinq

i.e. f=f q( )

Electric field magnitude at point P, Ep

Ep =E1 +E2 =E0 sin +( )

=2E0cosf2

Amplitude1 2 4 3 4

sin

f=0,2 p, K Ûconstructive interferencef=p,3p, K Ûdestructive interferenceIntensity I of combined wave

I µEp max2

Amplitude squared

(kx-wt) (kx-wt+f)sin

(kx-wt+f/2)

Intensity of an electromagnetic wave is given by

I=Sav=EmaxBmax

2m0

=Emax2

2m0c=cBmax

2

2m0

=cuav

\Itot =4E0

2 cos 2f2

2m0c=4I0 cos 2f

2= Imax cos 2f

2

\Itot = Imaxcos 2 pdsinql

æ è ç

ö ø ÷

as sinq» yL we obtain

Itot = Imaxcos 2 pdlLy

æ è ç

ö ø ÷

Interference by Thin Films

airsoap

Get destructive and constructive interference depending on wavelength and position of observer: therefore see

colors at different positions.

white light

1800 phase change

no phase change

air

1

2

If ray 1 is 1800 out phase with ray 2 this

is equivalent to a path difference of l n2

wavelength of light in medium

whose refraction index is n is l n =ln

é

ë

ê ê ê

ù

û

ú ú ú

if 2t=l n2

rays will recombine in phase, in general

2t= m+ 12

æ è ç

ö ø ÷ l n Û 2nt= m+ 1

2æ è ç

ö ø ÷ l, m=0,1,2,3,K

constructive interference2nt=ml, m=1,2,3,K

destructive interference

Interference by Thin Films

airoilwater

white light

1800 phase change

1800 phase change

2nt = m l , m =1 ,2, 3, Kconstructive interference

2nt = m + 12

æ è

ö ø l , m = 0, 1 ,2, 3, K

destructive interference

t

Spreading out of light is called DIFFRACTION

This can occur when light passes

through small opening

or around object at sharp edges

• Fraunhofer Diffraction

• Light forms plane waves when

reaching screen

• long distance from source

• by converging lens• Fresnel Diffraction

• Wavefronts are not plane waves• short distance from source

qa/2a/2

P

Single Slit

In Fraunhofer Diffraction paths of waves are parallel wave 1 travels further than wave 3 by amount

= path difference = d =a2 sin q same for waves 2 & 4.

If d = l2

Ûphase shift of p( ) waves cancel through

destructive interference. This is true for any waves

that differ by a2 . \waves from upper half

that destructively interfere with waves from bottom half are at anglea2 sin q d =

l2

Û sin q d =la

The argument holds when dividing slit into 4 portionsa4 sin q d =

l2

Û sin q d =2la

Þ sin q d = mla ; m =±1,K

qd

By using the method of phasors one can find that the electric field at a point P

on the screen due to radiation from all points within the slit is given by

Eq = E 0

sinpal

sin q{ }p al

sin q

æ

è

ç ç ç

ö

ø

÷ ÷ ÷ = E 0 sinc pa

lsin q{ }

and thus the intensity of radiation by

Iq = I0 sinc2 pal

sin q{ }Þ minima occur at sinq = m

la

; m = ±1, KSinc

( )

( )

( ) ÷øö

çèæ

÷øö

çèæ=

÷øö

çèæ=

=÷øö

çèæ=

qlp

lqpq

qlpq

lqpq

sinsincsincos

sinsinc

4;sincos

22max

201

0max2

max2

adII

aII

IIdII

Fresnel / FraunhoferDiffraction from a Single Slit

Far from

the slit

zClose to the slit

Incident plane wave

Slit

Resolving between closely spaced sources

diffraction pattern for

two separate source points that

can be resolved

sources closer together

that can be justresolved

Sources so close that

they cannot be resolved

•Rayleighs Criterion•when central max. of one image falls on

first min. of other image, the images are said to be just resolved

first min in single slit occurs when

sinq = la

» q (as l < < a Þ q is small )

so qmin =la

q subtended by 2 sources must be ³qmin

in order to be resolvedFor circular apertures of diameter D

q min = 1 .22 lD

q

P

d

f

d = dsinqd = slit spacing

Diffraction Grating

d

If d=m l=d sin q, m=0,±1, Kwaves from all slits will be in phase at PÞ bright line at P; m is order # of diffraction patternmth order max. for each l occurs at some specific q

All l’s are seen at m = 0 Û q=0

m=1 Þsin ql =ld

m=2 Þsin ql =2ld

Resolving power of diffraction grating

R=lave

l2 - l1

=l aveDl

=Resolving power

l1,l2 two wavelengths that can be just resolvedl1 £l£l2; l1 »l2

gratings with high resolving power candistinguish small differences in l

R=Nm; N= # of lines of grating=resolving power of mth order diffraction

for m=0 all wavelengths are indistinguishable

for m=2 for grating with N=5000R=5000X2=10000

therefore min. wavelength difference that can be resolved for

waves with an average wavelength of 600 nm

is 6x10 -2 nm

Diffraction by Crystals

atomic spacings in crystals are approx. 10 -10 nm and therefore can act as 3D

diffraction grating

condition for constructive interference2dsin q =ml, m =1, Û Braggs Law

d q

Polarization

Electromagnetic Radiation is made of oscillating electric and magnetic fields, that are perpendicular to each other and to the direction of propagation of the radiation (Transverse Wave). These fields are proportional to each other in magnitude and are in phase.

EB

In general radiation is made up of a mixture of such fields, with each wave of

light having different orientation i.e

as the electric vectors are always perpendicular to the magnetic ones we

need only show the electric ones .

• Plane Polarized Light• Electric Field is in only one direction.• Light is Linearly Polarized

• E direction is constant in time• Light is Circularly Polarized

• E rotates • Ex = Ey at all times

• Light is Elliptically Polarized • E rotates • Ex Ey at all times

Producing Polarizationcan produce such light by passing through a polaroid sheet (Diochroic Material) this allows only one orientation of electric field through undiminished and completely absorbs the light with electric fields perpendicular to this direction. In general diminishes the intensity according to I = I0 cos2 q

Malus’s Law

polarized light is also produced by reflection

When light strikes a nonmetallic surface at any angle other than perpendicular, the reflected beam is polarized preferentially in the plane parallel to the surface. (light polarized in plane perpendicular to surface is preferentially absorbed or transmitted).

Why is the Sky Blue and daylight polarized?

• Higher frequencies are scattered more than lower ones (refracted more) by the oxygen and nitrogen molecules

• All the visible frequencies are scattered the same by larger objects e.g. water droplets in clouds.

• Scattered light is polarized.

Polarization by Scattering

Polarization by Double Refraction

•Materials that have two indices of refraction depending on the direction of incident rays are called Double Refracting or Birefringent

•These materials produce polarized light