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