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Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
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.
Topics
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Transmission Grating – Normal Incidence
m
m
a
maAB sin DifferencePath
AB
.....3,2,1,0 ; sin : MaximaIntensity mma m
What determines whether m=positive is above the dashed line or below the dashed line?
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Transmission Grating – Oblique Incidence
m
m
a
im aaCDAB sinsin DifferencePath
A
B
.....3,2,1,0 ; sinsin : MaximaIntensity mma im
Our convention: ccw angles are positive; cw angles are negative.
In the example above: both i and m are positive.
i
CD
i
Modern Optics LabLab 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:
d
d DispersionAngular m
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Calculating the Angular Dispersion
m
m
cosd
d DispersionAngular :be shouldresult final The
a
m
.....3,2,1,0 ; sinsin : MaximaIntensity mma im
sinsin
. offunction a as ofThink
rule.chain theuse and respect to with sidesboth ateDifferenti
:Hints
m
md
da
d
dim
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Resolving Power
m
m
cosd
d DispersionAngular :be shouldresult final The
a
m
slits dilluminate ofnumber N where, Nm :power Resolving
imim Naa
sinsinsinsin m Using
Under which conditions can you resolve the sodium doublet?Hint: You are given and. You can make a (numerical) statement involving N, a, m and i.
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
White Light Source
Lens (138mm)Screen
Diffraction Grating
VIII.A Measurement of Angular Dispersion with White Light Source
138mm
Lens (48mm)
48mm
Look at “green” light to get some average wavelength. Measure from red to blue (400nm). Calculate the angular dispersion.
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
VIII.A Measurements with the Helium-Neon Laser
Laser
View from topScreen
m=0
m=1
m=-1
m=2
m=-2
What happens as the grating is rotated?How do the maxima move? Do they?
Laser
Screen
?
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
VIII.A Wavelength Measurements using i=0 and i=30
View from top
Know what you measure when doing the 30 incident angle measurement! An example:
Laser
Screen
i
m=0
m=+1
m=-1
i
m=-1
m=+1
m=+1
...,2,1,0 ; sinsin mma im
ccw.) isit - example in this positive is :(note :1mFor 111 mmim
cw.) isit - example in this negative is :(note :1mFor 111 mmim
m=-1
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
...,2,1,0 ; sinsin :Remember mma im
Once you have figure out what m=-1 and m=-1 are, you can calculate the wavelength as follows:
sinsin :1mFor 1 ima
sinsin :1mFor 1 ima
(Subtract the first equation from the second on each side)
sinsinsinsin 11 imim aa
11 sinsin2 mm
a
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Normal Incidence
Laser
m
m
ma sin
maAB sin DifferencePath
.....3,2,1,0 ; sin : MaximaIntensity mma m
A
B
Convention: ccw angles are positive; cw angles are negative.
In this example:m is negative m negative
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Normal Incidence
Laserm=0
m=1
m=-1
m=2
m=-2
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Oblique Incidence
m
A
C
i
Laser
B
D
im aaCDAB sinsin DifferencePath
.....3,2,1,0 ; sinsin : MaximaIntensity mma im
Convention: ccw angles are positive; cw angles are negative.
In this example:i is positivem is negative.
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Oblique Incidence
Laser
m=0
i
m
imim aa 0sinsin :0mfor DifferencePath
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Reflection Grating Oblique Incidence
Laser
m=0
m=-1
m=1
m=-2
m=2
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Let’s rotate the picture to see what we do in VIII.B
m=0
m=1
m=2
Laser
Machinist Scale
d L
Screen
xo
xo
xm
2
20
2
m
2
:) x; (small incidence grazingFor
L
xx
m
d
L
m
Modern Optics LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Evaluation of Wavelength
md
Lxx
L
xx
m
d
L
mm
220
22
20
2
m
2
2
:) x; (small incidence grazingFor
2mx
m
Get wavelength from slope
Modern Optics LabLab 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 LabLab 8: Diffraction by Periodic and Non-Periodic Structures
Another example:
d
An example of how not to do it:
d=?
Modern Optics LabLab 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 LabLab 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 LabLab 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?