Lecture 27-1
Thin-Film Interference-Cont’d
(Assume near-normal incidence.)
Path length difference:
l  2t
destructive
m

 
 ( m  1 / 2 )  constructive
where
 
0
n
• ray-one got a phase change of 180o due to reflection from air to
glass.
• the phase difference due to path length is: f ' 
•then total phase difference: f = f’+180.
l

 2 
l

'
n
 2
Lecture 27-2
Two (narrow) slit Interference
Young’s double-slit experiment
• According to Huygens’s principle,
each slit acts like a wavelet. The
the secondary wave fronts are
cylindrical surfaces.
• Upon reaching the screen C, the
two wave interact to produce an
interference pattern consisting of
alternating bright and dark bands
(or fringes), depending on their
phase difference.
Constructive vs. destructive
interference
Lecture 27-3
Interference Fringes
For D >> d, the difference in path lengths between
the two waves is  L  d sin 
• A bright fringe is produced if the path lengths
differ by an integer number of wavelengths,
d sin   m  , m  0,  1,
y ~ D*tan(θ)~ D*mλ/d
• A dark fringe is produced if the path lengths
differ by an odd multiple of half a wavelength,
d sin   ( m  1 / 2 )  , m  0,  1,
y ~ D*tan(θ)~ D*(m+1/2)λ/d
Lecture 27-4
Intensity of Interference Fringes
Let the electric field components of the two
coherent electromagnetic waves be
E 1  E 0 sin  t
E 2  E 0 sin(  t  f )
The resulting electric field component point P
is then
E  E E
1
2
 E 0  sin  t  sin(  t  f ) 
f
f

 2 E 0 cos sin   t  
2
2

Intensity is proportional to E2
I
I0
2

Em
E
2
0
 I  4 I 0 cos
2
f
2
I=0 when f = (2m+1) , i.e. half cycle + any number of cycle.
Lecture 27-5
Dark and Bright Fringes of Single-Slit Diffraction
Lecture 27-6
Phasor Diagram
f2
f1
Lecture 27-7
Phasor Diagram for Single-Slit Diffraction
The superposition of wavelets can be illustrated by a phasor diagram.
If the slit is divided into N zones, the phase difference between
adjacent wavelets is
a

sin 
 N
  2  





 a sin 


( N / 2)


total phase difference:
f  N  
f 
2  a sin 

Am ax
r
A  2 r sin
f
2
I  A

2 Am ax
f
2
I ( )
I m ax

A
A
2
2
m ax
 I ( )  I m a x
sin
f
2

f
s
in


2

f


2






2
Lecture 27-8
Intensity Distribution 1
I ( )  I m ax

f
sin


2

f


2






2
w here f 
2  a sin 

maxima:
f  0
central maximum because
sin x
or
s in
 1 as x  0
x
f
  1 or a sin   ( m 
2
1
)
2
minima:
sin
f
 0 or a sin   m 
2
m   1,  2,  3, ...
m 0
Lecture 27-9
Intensity Distribution 2
for small 
• Fringe widths are proportional to /a.
• y ~ D*θ
•Bright fringe: D*(m+1/2)λ/a
•Dark fringe: D*mλ/a
•Width: D*λ/a except central maximum
• Width of central maximum is twice any
other maximum.
•Width = D*λ/a – D*(-1)λ/a = 2D*λ/a
• Intensity at first side maxima is (2/3)2
that of the central maximum.
y
Lecture 27-10
Young’s Double-Slit Experiment Revisited
• Intensity pattern for an ideal double-slit experiment with narrow slits (a<<)
Light leaving each slit has a unique phase. So there is no
superimposed single-slit diffraction pattern but only the
phase difference between rays leaving the two slits matter.
D
slit separation
d
2   d sin  
I  4 I 0 cos 




d
a

where I0 is the intensity
if one slit were blocked
• If each slit has a finite width a (not much smaller than ), single-slit
diffraction effects must be taken into account!
Lecture 27-11
Intensity Distribution from Realistic Double-Slit Diffraction
I  4 I 0 cos 
2
double-slit intensity
 sin  
replace by I m 

  
2
 sin  
2
I ( )  I m (cos  ) 




2
 
d

sin 
single-slit intensity envelope
a
 
sin 

Lecture 27-12
Diffraction by a Circular Aperture
• The diffraction pattern consists of a bright circular
region and concentric rings of bright and dark fringes.
• The first minimum for the
diffraction pattern of a circular
aperture of diameter d is located by
sin   1.22

d
geometric factor
• Resolution of images from a lens
is limited by diffraction.
• Resolvability requires an angular
separation of two point sources to
be no less than R where central
maximum of one falls on top of
the first minimum of the other:
  R
  R
  R
Rayleigh’s criterion
 R  sin
1
1.22 
d

1.22 
d
Lecture 27-13
Diffraction Gratings
• Devices that have a great number of slits or rulings to produce an
interference pattern with narrow fringes.
• One of the most useful optical tools. Used to analyze wavelengths.
D
Types of gratings:
• transmission gratings
• reflection gratings
up to thousands
per mm of rulings
Maxima are produced when every pair of
adjacent wavelets interfere constructively, i.e.,
D
d
d sin   m  , m  0,  1,
mth order maximum
Lecture 27-14
Spectral Lines and Spectrometer
• Due to the large number of rulings,
the bright fringes can be very narrow
and are thus called lines.
• For a given order, the location of a
line depends on wavelengths, so light
waves of different colors are
spread out, forming a spectrum.
Spectrometers are devices that can
be used to obtain a spectrum, e.g.,
prisms, gratings, …
Lecture 27-15
X Ray Diffraction
• X rays are EM radiation of the wavelength on the order of 1
Å, comparable to atomic separations in crystals.
• X rays are produced, e.g., when core electrons in atoms are
inelastically excited. They are also produced when electrons
are decelerated or accelerated.
• Vacuum tubes, synchrotrons, …
 Standard gratings cannot be used as X ray spectrometers.
(Slit separation must be comparable to the wavelength!)
 Von Laue discovered the use of crystals as 3-dimensional
diffraction gratings.
Nobel 1914