Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Topics
 Transmission grating spectrometer:
• Measuring and calculating the angular dispersion.
• Understanding resolving power.
 Reflection grating spectrometer:
• Using a machinist scale to measure the laser wavelength.
 The far-field (Fraunhofer) diffraction pattern of randomly placed identical
particles:
• Measuring the particle size from the diffraction pattern.
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Transmission Grating – Normal Incidence
Qm
a
B
Qm
A
Path Difference
Intensity
Maxima
 AB  a sin Q m
: a sin Q m  m  ; m  0 ,  1,  2 ,  3 .....
What determines whether m=positive is above the dashed line
or below the dashed line?
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Transmission Grating – Oblique Incidence
Qm
D
C
a
Qi
Qi
Path Difference
Intensity
Maxima
B
Qm
A
 AB  CD  a sin Q m  a sin Q i
: a sin Q m  sin Q i   m  ; m  0 ,  1,  2 ,  3 .....
Our convention: ccw angles are positive; cw angles are negative.
In the example above: both Qi and Qm are positive.
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Angular Dispersion
Different colors (wavelengths) of light have their maxima at
slightly different transmitted angles given a particular
transmission grating and incident angle.
(White light gets “broken up” into rainbow colors at the
maxima for m  0. The m = 0 maximum remains white.)
Angular dispersion quantifies the change in the (transmitted)
angle at which the maxima occur per unit wavelength
change:
Angular
Dispersion

dQ m
d
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Calculating the Angular Dispersion
Intensity
Maxima
: a sin Q m  sin Q i   m  ; m  0 ,  1,  2 ,  3 .....
Hints :
Differenti
Think
d
d
ate both sides with respect to
of Q m as a function
a sin
 and use the chain rule.
of  .
Q m    sin Q i  
d
d
m  
The final result should be : Angular
Dispersion

dQ m
d

m
a cos Q m
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Resolving Power
Resolving
Using
power :  
m 


 Nm , where N  number
a sin Q m  sin Q i 


 



of illuminate
d slits
Na sin Q m  sin Q i 

Under which conditions can you resolve the sodium doublet?
Hint: You are given  and . You can make a (numerical)
statement involving N, a, Qm and Qi.
The final result should be : Angular
Dispersion

dQ m
d

m
a cos Q m
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
VIII.A Measurement of Angular Dispersion with White Light Source
Lens (138mm)
Diffraction
Grating
Lens (48mm)
Screen
White Light Source
138mm
48mm
 Look at “green” light to get some average wavelength.
 Measure Q from red to blue ( 400nm).
 Calculate the angular dispersion.
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
VIII.A Measurements with the Helium-Neon Laser
m=2
m=1
Laser
m=0
m=-1
m=-2
Screen
What happens as the grating is rotated?
How do the maxima move? Do they?
Laser
View from top
?
Screen
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
VIII.A Wavelength Measurements using Qi=0 and Qi=30
Know what you measure when doing the 30 incident angle measurement! An
example:
Qi
Laser
m=+1
Qm=+1
View from top
gm=+1
m=0
Qi
Qm=-1
gm=-1
m=-1
a sin Q m  sin Q i   m  ; m  0 ,  1,  2 , ...
For
m  1 :
Q m   1  Q i  g m   1 (note : g m   1 is positive
in this
example
- it is ccw.)
For
m  1 :
Q m   1  Q i  g m   1 (note : g m   1 is negative
in this
example
- it is cw.)
Screen
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Once you have figure out what Qm=-1 and Qm=-1 are, you can
calculate the wavelength as follows:
Remember
:
a sin Q m  sin Q i   m  ; m  0 ,  1,  2 , ...
For
m  1 :
a sin Q m   1  sin Q i    
For
m  1 :
a sin Q m   1  sin Q i    
(Subtract the first equation from the second on each side)
a sin Q m   1  sin Q i   a sin Q m   1  sin Q i        

 
a
2
sin
Q m   1  sin Q m   1 
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Normal Incidence
  a sin Q m
A
B
Laser
Qm
Convention:
ccw angles are positive;
cw angles are negative.
Qm
In this example:
Qm is negative  m negative
Path Difference
Intensity
Maxima
 AB  a sin Q m
: a sin Q m  m  ; m  0 ,  1,  2 ,  3 .....
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Normal Incidence
m=2
m=1
m=0
m=-1
m=-2
Laser
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Oblique Incidence
Qi
A
B
Convention:
ccw angles are positive;
cw angles are negative.
D
Qm
C
Path Difference
Intensity
Maxima
In this example:
Qi is positive
Qm is negative.
 AB  CD  a sin Q m  a sin Q i
: a sin Q m  sin Q i   m  ; m  0 ,  1,  2 ,  3 .....
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Oblique Incidence
Qi
Qm
m=0
Path Difference
for m  0 :
a sin Q m  a sin Q i  0
 Qm   Qi
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Oblique Incidence
m=2
m=1
m=0
m=-1
m=-2
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Let’s rotate the picture to see what we do in VIII.B
For grazing
incidence
2
2
d  xm  x0

 
2
2 m 
L
Screen
(small a ; x m  L ) :




xm
xo
a
Machinist Scale
xo
d
L
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Evaluation of Wavelength
For grazing
incidence
2
2
d  xm  x0

 
2
2 m 
L
(small a ; x m  L ) :
2

2L
2
2
  xm  x0 
m

d

2
xm
m
Get wavelength from slope
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Machinist Scale:
The “grid” spacing d depends on where the laser hits the scale!
Example:
d
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Another example:
d
An example of how not to do it:
d=?
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
VIII.C Measuring the Size and Shape of Randomly Distributed
Small Particles
Look at single slit pattern first. What effect does moving the slit
left or right have on the far-field diffraction pattern?
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
Imagine randomly placed slits of the same width:
Just as we found with the double slit pattern: We would see an
intensity variation in the far field diffraction pattern due to each
slit (a single slit pattern).
What about the diffraction pattern due to the interaction between
the slits (like the double slit pattern for a double slit, or the
diffraction grating pattern with regularly placed slits)?
Modern Optics Lab
Lab 8: Diffraction by Periodic and Non-Periodic Structures
VIII.C Measuring the Size and Shape of Randomly Distributed
Small Particles
For identical particles ,the diffraction pattern from each
individual object will look the same in the far field.
How about the pattern due to the interaction between the
objects?
What if the objects were regularly spaced in a pattern?