Transcript Math 5900 – Summer 2010 Lecture 1: Simple Harmonic

Math 5900 – Summer 2011
Lecture 4: Interference of Light
Gernot Laicher
University of Utah - Department of Physics & Astronomy
Light = travelling electromagnetic wave (em wave)
Different colors = different frequencies of em waves.
Mechanical waves need medium to travel (e.g., air molecules
for sound waves).
Light can travel through vacuum (needs no medium)
Speed of light in vacuum approx. 3x108m/s (different in other
media, e.g. 2x108m/s in glass).
Wave propagation:
Wave crests move with the speed of light.
Period T (of electric field vector oscillation) :
Time required for one crest to move forward by a distance
equal to its wavelength.
Frequency f:
f=1/T
In short:
Huygen’s Principle:
Every point on a wave-front may be considered a source of
secondary spherical wavelets which spread out in the forward
direction at the speed of light. The new wave-front is the
tangential surface to all of these secondary wavelets.
Diffraction: Occurs when wave encounters obstacle.
Example +Demo: Plane wave encounters a single slit.
Intensity distribution of diffraction pattern is mathematically quite
involved. Location of the so-called minima fairly easy to understand.
a<<

y

a

d


a
y

d
Dark region on screen: Sum of all the waves coming from the
opening interfere destructively (amplitudes add up to zero).
E.g, when 


a
y

d
Use trigonometry to relate path length difference  to
position y on the screen:
Destructive interference occurs for
Relate min to distance y on the screen:
For small angles
Between these two minima is “central maximum” for y=0 (=0).
Between these two minima : “Central maximum” for y=0
(=0).
Determine wavelength of laser by measuring distance
between the two minima (2 ymin) on both sides of the central
maximum and d and a.
Additional minima from single slit diffraction:
Shape of intensity distribution on screen:
Diffraction pattern of a circular aperture
Without derivation: For circular aperture (a hole) of diameter
c the diffraction pattern is an “airy disk”.
Angle of first minimum (dark circle) (measured from axis that goes
through center of the central bright disc):
Small angle approximation ( in radians)
For small angles
ymin = radius of first dark ring
c = diameter of hole
d = distance from hole to screen
Babinet’s Principle
The diffraction patterns from an opaque body is identical to
that from a hole of the same size and shape except for the
overall forward beam intensity.
Example:
Diffraction pattern produced by circular disc = same from
circular hole of the same size.
We will use this principle to determine the thickness of a hair and the
size of small particles from their respective diffraction patterns.
Double Slit Diffraction
Wave fronts of light
Dark
Screen
Double slit
Wave going through this slit travels a bit
further to get to this particular place on
the screen.
Waves from the two slits are out of
phase by half a wavelength in that direction
Waves annihilate each other in that direction
(“destructive interference”).
Darkness on that place on the screen.
Dark
Bright
The waves going through both slits travel
the same distance to the screen.
Waves from the two slits in phase.
Waves add together to twice the
amplitude (“constructive interference”).
 Bright spot in center.
The light exits the slits in all directions
simultaneously.
A pattern of bright and dark
regions appears.

y

d



d

Condition for intensity maximum:
y
For small angles 
Intensity distribution of diffraction pattern of double slit depends on
each single slit width.
Single slit pattern superimposed on double slit pattern!
Basically, “pure” double slit intensity distribution is multiplied by the
single slit intensity distribution.
Diffraction grating acts in many ways like a double slit.
However, bright spots are much more narrow/less broad.
Makes it easier to separate peak location of two separate
but close wavelengths.
d sin n (= difference in path length)
n
d
Diffraction
grating
Whenever d sin n = n: All waves are in phase
 (constructive interference); n = 0,1,2….
Otherwise they cancel each other (destructive interference).
Different  means
constructive interference
for different n !