Polarization and Diffraction
A Presentation by:
Dan Stark
Isaac Childres
Tim Wofford
Patrick Zabawa
Experimental Setup for Diffraction
Light Sensor
Light sensor is
mounted on rotary
motion sensor
Laser
and Slit
Rotary motion sensor mounted
on linear motion accessory which
is mounted at other end of track
Rotary
Motion Sensor
Laser and slit mounted 3 cm
apart at one end of track
Track
Linear
Motion
Accessory
Procedure:
• Assemble track, placing neutral density filter(s) between laser and slit in
order to prevent saturation of light sensor
• Connected sensors to the computer using a breakout box
• Took data using Data Studio and passing the light sensor across the linear
motion accessory
Single Slit Diffraction Theory
dE 
L
R
sin( t  kr) dx
dE Electric field at a point due
L
R
to a slit segment dx
Amplitude of incident wave
Fraunhofer Approximation: r  R  x sin 
E
L
R

d / 2
d / 2
sin[ t  k ( R  x sin  )] dx
 L d sin 
E
sin( t  kR)
R 
Where
I ( )  E 2
  (k d / 2) sin   ( d /  ) sin 

1   Ld 


2 R 
2
 sin 

 
 sin 
 I (0)
 



2



2
Full-Width at Half Max (FWHM)
 
sin 
1
Solve 

2
2
numerically.
Full width at half max of the envelope is
β  2.783114.
So as d increases, the peak sharpens.
1
I(0)
0.8
0.6
FWHM – 2.78 β
0.4
0.2
β
-6
-4
-2
2
4
6
Results: Single Slit Diffraction Patterns (Plastic)
Normalized Light Intensity (%)
100
90
0.02 mm
80
0.04 mm
0.08 mm
0.16 mm
70
60
50
40
30
20
10
0
0.03
0.05
0.07
0.09
Position (m)
0.11
0.13
0.15
Results: Single Slit Diffraction Patterns (Razor)
100
90
0.2 mm
0.1 mm
Light Intensity (%)
Normalized intensity
80
70
60
50
40
30
20
10
0
0.065
0.07
0.075
0.08
0.085
position
Position (m)
0.09
0.095
0.1
0.105
A study of the slight interference pattern
observed at the peak of single slit diffraction
The Problem:
When sending light through a medium in order to accomplish
single slit diffraction, we find a slight aberration at the peak
Plastic medium (Pasco instrument)
No medium (razor blades)
100
Normalized Light Intensity (%)
Normalized Light Intensity (%)
100
95
90
85
80
75
70
0.095
0.096
0.097
0.098
Position (m)
0.099
0.1
95
90
85
80
75
70
0.085
0.086
0.087
Position (m)
0.088
0.089
What’s going on?
Given that this only happens when the light travels through a medium,
we can only assume the aberration has something to do with the
reflection that occurs when light moves from one medium to another.
Light comes in at an angle
slightly different than zero
Note: Rays after the slit
represent diffraction patterns,
which will have some
interference with each other
As it passes through the medium,
some transmits and some reflects
The reflected light will reflect again
and eventually come out at some different
point, resembling a double slit experiment
Exploratory Experiment
We performed similar single slit experiments with a
piece of glass 2.33mm thick and two pieces of black tape
We find “a” to be 0.03m,
a slit separation comparable
to our other experiments
We can find the slit separation, a, based on this equation.
If m=1, then y=0.25m, wavelength=670nm and D=1.024m
Experimental Results
At some minute angle, say 1 degree, given a width of 2.33mm, which must be traveled
twice by the reflected light, we would have an effective double slit separation of 0.08mm,
which is on the same order of magnitude as our double slit experiments.
Normalized Light Intensity (%)
100
90
80
70
60
50
40
30
20
10
0
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
0.12
Position (m)
Given these conditions, a dip can be observed
0.13
0.14
Experimental Results, cont.
Given a larger angle, say 15 degrees, the effective double slit separation would increase
to 1.25mm and the dip would disappear due to the overly large separation.
100
Normalized Light Intensity (%)
90
80
70
60
50
40
30
20
10
0
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Position (m)
Given these conditions, less of a dip can be observed
0.12
0.13
Multiple Slit Diffraction Theory
Consider A system with N slits of
width d separated by a distance a,
E can be written as:
b / 2
a b / 2
b / 2
a b / 2
E  C
F ( x) dx  C 
 C
( N 1) a  b / 2
( N 1) a b / 2
F ( x) dx
F ( x) dx
Where F ( x)  sin[ t  k (r  x sin  )]
1
2
 sin    sin N 
 
I ( )  I 0 

    sin  
2
I(0)
0.8
• Envelope
0.6
• High
Frequency
• Wave group
0.4
Where   (a /  ) sin  and   (d /  ) sin  .
Note: N = 2, α = β
0.2
β
-6
-4
-2
2
4
6
Results: Double Slit Diffraction Patterns
0.04 mm slit size – 0.25 and 0.5 slit separations
Double Slit 0.04mm d with 0.5mm Separation, First Run
100
Light Intensity (%)
NormalizedIntensity
90
0.04 mm/0.5 mm
0.04 mm/0.25 mm
80
70
60
50
40
30
20
10
0
0.07
0.08
0.09
0.1
Distance(m)
(in m)
Position
0.11
0.12
0.13
Results: Double Slit Diffraction Patterns
0.04 mm slit size – 0.25 mm slit separation
0.08 mm slit size – 0.25 mm slit separation
100
90
Normalized Light Intensity (%)
Normalized Light Intensity (%)
80
70
60
50
40
30
20
10
0
0.07
0.09
0.11
Position (m)
0.13
Position (m)
Results: Multiple Slit Diffraction Patterns
0.04 mm slit size – 0.125 slit separation
100
Normalized Light Intensity (%)
90
5 slits
3 slits
80
70
60
50
40
30
20
10
0
0.06
0.07
0.08
0.09
0.1
Position (m)
0.11
0.12
0.13
0.14
Experimental Setup for Polarization
Aperture Disk
Light
Sensor
Polarizers
Laser
Rotary Motion
Sensor
Optics Bench
Procedure:
• Assemble track, placing
neutral density filter(s)
between laser and 1st
polarizer in order to
prevent saturation of
light sensor
• Connected sensors to the
computer using a
breakout box
• Took data using Data
Studio and rotating the
2nd polarizer connected
to the rotary motion
sensor
Polarization Theory
 Ax 
J 
 Ay 
Jones vector with

I  



Ax A y  2
2
2
Where   impedance of the medium.

T  T 11
T 21
T
T


12
12
Jones matrix, characterizes
polarization device
1 0
T

0 0 
Transmits only xcomponent of wave, a
linear polarizer
 cos 2  
sin   cos 
T

2






sin

cos

sin



Jones matrix for linear polarizer with
a transmission axis making an angle
θ with x-axis
T  R  TR 
Where R(θ) is the
rotation matrix
1  cos 2   
J  T   

0




sin

cos

  

2
I
Jx  Jy
2
2
cos 2  

2
Experimental Results of Malus’ Law
This experiment uses two polarizers, one that polarizes the light
initially and one that varies to show angular dependence.
100
Normalized Light Intensity (%)
90
80
70
60
50
40
30
20
10
0
0
50
100
150
200
250
Angular Position (Degrees )
300
350
400
Polarization of the Laser
This data was taken with one polarizer to show that the laser light is already
almost linearly polarized. (Basically we did not need the first polarizer)
100
Normalized Light Intensity (%)
90
80
70
60
50
40
30
20
10
0
0
60
120
180
240
Angular Position (Degrees )
300
360
Resources
http://www.pa.msu.edu/courses/1997spring/PHY232/lectures/interference/oneslit.html
http://rhs.rocklin.k12.ca.us/gclarion/physics2/chapter19/singleslit.htm
http://www.mu.edu/courses/phys/matthysd/Lab0122.htm
http://bednorzmuller87.phys.cmu.edu/demonstrations/optics/interference/demo321.html
http://www.colorado.edu/physics/phys2020/phys2020_f98/lab_manual/Lab5/lab5.html
http://rhs.rocklin.k12.ca.us/gclarion/apphysics/chapter27/diffractioninterference.htm
http://www.ccmr.cornell.edu/~muchomas/P214/Notes/Interference/node19.html
http://www3.ltu.edu/~s_schneider/physlets/main/doubleslitintensity.shtml
http://electron9.phys.utk.edu/optics421/modules/m5/Diffraction.htm
http://hyperphysics.phy-astr.gsu.edu/hbase/phyopt/mulslid.html
Saleh, B. E. A. and Teich, M. C. Fundamentals of Photonics. John Wiley and Sons, Inc.:
New York: New York. 1991.