Resource Lesson
Physical Optics - Interference and Diffraction Patterns
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Background Information
The central region of
constructive interference
is known as the
central maximum
, or A
_{o}
. On either side of the central maximum are the
first order nodes
, N
_{1}
. These are regions of
destructive interference
. On either side of N
_{1}
are the next antinodes, A
_{1}
. This alternating pattern of
nodes and antinodes
continues throughout the construction.
Utilize this ripple tank wave
physlet
to assist you in understanding the following sets of interference properties:
When two in-phase point sources are
moved closer together
, there is
less interference
produced, as evidenced by fewer nodes. When the amount of interference decreases, the width of any given antinode increases.
When two in-phase point sources are
moved further apart
, there is a
greater amount of interference
produced, as evidenced by a larger number of nodes. When the amount of interference increases, the width of any given antinode decreases.
Another way to change the amount of interference produced by two in-phase point sources is to change their frequency but leave their separation distance unchanged.
When the
frequency is decreased, less interference
is produced since the wavelengths will increase, generating fewer wavefronts between the two sources (the equivalent to moving the point sources closer together).
When the
frequency is increased, more interference
is produced since the wavelengths will decrease, generating more wavefronts between the two sources (the equivalent to moving the point sources further apart).
Young's Double Slit Experiment
Historically, why was
Young’s discovery of the interference of light
important? Prior to his experiment, the controversy regarding the nature of light was centered in two camps:
Newton’s corpuscular theory
Huygen’s wavelet theory
Both groups had developed theories that explained the reflection and refraction of light. Since Newton was the most eminent scientist of his day, his corpuscular theory [light rays were composed of tiny bullets of finite mass that traveled at extremely high speeds] received wider acceptance, even though his dynamics erroneously demanded that light would travel faster in denser media. Huygen’s wave theory supported the phenomena of interference and diffraction of light - it had just never been observed.
Young’s interference experiment, along with the diffraction effects seen in Fresnel's prediction and the subsequent
demonstration
by Dominique Arago of
Poisson's Spot
, showed that light DID have a wave nature. The reason these effects had not be seen previously was a result of light’s extremely small wavelengths.
Newton’s corpuscular theory was completely abandoned after
Fizeau
and
Foucault
showed that light SLOWED down while it traveled through water. Einstein’s explanation of the photoelectric effect re-instituted the particle nature of light - now called a photon which represents, not a particle with mass, but a bundle of radiant energy (E = hf) that interacts with matter, principally electrons.
Young began his
experiment by sending waves of coherent light towards a barrier with two slits
. The two slits are separated by a distance
d
. On a screen, a perpendicular distance
L
from the slits, a series of interference fringes were viewed. The formulas we will now develop will allow us to determine if a point
P
in the interference pattern a distance
y
from its center will fall into a bright (maximum, constructive interference) zone or into a dark (minimum, destructive interference) zone.
EPD
represents
e
xtreme
p
ath
d
ifference, or the effective
difference in the distances
the light must travel to reach a given position on the screen from each of the slits. For
bright fringes
, this path difference must be a multiple of a wavelength to insure constructive interference. When this overall path difference equals an odd-multiple of a half wavelength, then destructive interference is insured and a
dark fringe
is formed on the screen.
Thomas Young's equation for double slit interference
is
where
For
bright fringes
(constructive interference)
where
For
dark fringes
(destructive interference)
where
How is an interference pattern affected when the distance between the slits is decreased but the wavelength remains unchanged?
How is an interference pattern affected when the distance between the slits is increased but the wavelength remains unchanged?
How is an interference pattern affected when the distance between the slits remains constant but the wavelength of the impinging light decreases?
Diffraction Gratings
Many times these interference properties are investigated through the use of
diffraction gratings
. A grating is a huge collection of tightly spaced slits - usually upwards of 5 x 10
^{5}
slits per meter. The distance between the slits in a grating is calculated as the reciprocal of its grating spacing. That is
d = 1/[number of slits/meter]
The principal difference between an interference pattern caused by two slits and that caused by a grating is that a grating has more intense bright fringes that are more widely-spaced. These two effects should make sense.
The brightness would increase since more light can reach the screen (there are more openings, or slits, in the barrier).
The greater spacing can be accounted for by the fact that since
d
is getting increasingly smaller,
y
would get increasing larger.
A common commercial grating has 13400 lines per inch. What is the distance between each slit in meters?
Light of wavelength 500
nm
is shown through this grating onto a screen located on a wall 3 meters away. How wide is the central maximum?
What is the maximum number of interference fringes that could be viewed on the screen?
Diffraction
Diffraction
is the bending of a wave around an obstacle or through an opening. Diffraction fringes are caused by the interference between the component rays as they pass through the same slit.
The formula used to calculate diffraction fringes for a single slit of width w is very similar to that used for double-slit interference.
Notice how similar this is to the formula for interference. The distance between the slits,
d
, is replaced with the width of a single slit,
w
. Since diffraction represents the interference between component rays WITHIN a single slit,
dark fringes
occur when the
. This results from the fact that the slit can be divided into a even number of half-wavelength regions which cancel out each other. The location of bright fringes is not calculated.
In the "zoomed-in" diagram below, point
P
would be located in D
_{1}
since the EPD between the top ray and the bottom ray emanating from the slit equals 1l. The slit can be effectively divided into two regions where all of the component rays reaching the screen from the top half of the slit have counterparts in the bottom half that are exactly a ½λ out-of-phase with each other. This results in complete destructive interference - or a dark fringe on the screen.
The width of the central maximum can be calculated by letting m = 1, finding the value for y and then doubling it since the diffraction pattern is symmetric.
Remember in this formula that y represents the linear deviation along the screen from the center of the central maximum to the center of a specified dark fringe.
If the EPD equals 1½ λ, then are three ½λ-regions and P would be located in B
_{2}
where the intensity would be 1/3
^{rd}
that of the central maximum. Remember that only light from one of the three sections reaches the screen.
If the EPD equals 2λ then there are four ½λ-regions and P would be located in D
_{2}
.
If the EPD equals 2½ λ, then are five ½λ-regions and P would be in B
_{3}
where the intensity would be 1/5
^{th}
that of the central maximum.
How does the number of fringes change when the width of the slit is decreased?
What happens when the width of the slit remains unchanged, but the wavelength is decreased?
Take a moment and investigate these relationships with this
physlet
on
single slit diffraction
.
Further differences between interference and diffraction patterns.
Note that the intensity of the bright fringes in an interference is equally bright, equally spaced. In a diffraction pattern, the central maximum has the greatest brightness, with each successive bright fringe getting narrower and dimmer.
Given below is a sketch of an intensity pattern that shows the combined effects of both diffraction and interference when light passes through multiple slits.
Notice that the diffraction pattern forms "an envelope" around the interference pattern that defines the intensity of the interference fringes. Also notice that since the interference fringes REMAIN equally spaced, some of the bright fringes are "squelched" by diffraction minima.
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