Effect of diffraction

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Effect of diffraction
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Fraunhofer diffraction
The math is simplified if the rays are parallel.
This is called Fraunhofer diffraction. The
text achieves this condition by making the source
and the screen far away from the
slit. Converging lenses can be used to achieve
the condition in a practical way.
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The general case, illustrated here, where the
rays are not parallel, is called Fresnel
diffraction. We do not deal with this case.
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To calculate the intensity at the screen,
contributions from each small part (?x in the
figure) of the slit are added. Then the finite
sum is converted to an integral by letting ?x go
to zero.
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Relative intensity in single-slit diffraction for
three different values of a/?. The narrower the
slit the wider the central diffraction maximum.
For the same slit width, the relative intensity
for two different wavelengths. Since the
central maximum for B is wider, we can see that
?B gt ?A,.
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Homework Problem

• For Fraunhofer diffraction, the intensity of the
  center of the central maximum decreases if the
  slit size is reduced. This is clear from the
  figure. If the slit is narrower, fewer rays
  (representing the paths of Huygens wavelets)
  reach P0.
• The first minimum occurs for a sin ? ?, so
  when a is reduced ? is increased.
• Thus, the answer is A.
• The intensity equation may be misinterpreted to
  imply that intensity at ?0 is independent of a,
  so C may seem correct. It is not.

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Diffraction imposes a fundamental limit on
optical devices, since the light from two
separate objects cannot be distinguished (or
resolved) if the central diffraction maxima of
the light from two sources overlap.
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Light enters the human eye through the pupil,
which is a circular aperture of diameter around 2
mm. Diffraction therefore limits the ability to
resolve distance objects. Applying Rayleighs
criterion, two dots (as shown) cannot be
resolved if ? (?D/L) is less than ?R 1.22 ?/d,
where d is the diameter of the pupil. Note the
index of refraction inside the eye is similar to
that of water (n 1.33), so the wavelength
inside the eye is less than outside. It is the
wavelength inside that matters since the
diffraction takes place inside the eye. Taking ?
550 nm, n 1.33, and d 2 mm as typical
values, ?R is about 0.025 rad. So, two objects 1
m away from you cannot be resolved by the eye if
they are less than 0.25 mm apart. This is not
the result of defective eyesight. It is the
result of diffraction.
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Example The aperture of a telescope has
diameter 6.0 cm. For white light, take the
wavelength to be 550 nm. For this telescope, what
is the minimum angular separation between objects
that can be resolved according to Rayleighs
criterion? What minimum distance does this
imply between resolvable objects on the
Moon? Note the Earths atmosphere limits the
angular resolution of a telescope to no better
than 1 arc second (i.e., 2? rad/(360x60x60) 5 x
10?6 rad).
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Active and Adaptive Optics
Material properties limit the size of precision
mirrors for telescopes (the largest is 200
inches, 5.1 m). Active optics overcomes this
by making a larger mirror out of smaller mirrors
that can be independently aimed, achieving the
overall effect of a bigger mirror. As a result,
8 m mirrors are in use. Adaptive optics
addresses the limit from atmospheric turbulence.
A wavefront sensor detects the distortions, a
computer calculates corrections, and the mirror
segments are rapidly adjusted (in milliseconds)
to correct for the distortions.
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Intensity from double-slit interference if
diffraction is ignored (corresponding to the
case of vanishingly narrow slits, altlt?).
Single-slit diffraction.
Double-slit interference including diffraction.
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Double-slit Intensity
I(?) for three different slit widths (shown) for
slit separation d 50?.
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Diffraction grating (showing five slits)
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The lines formed by a diffraction grating become
narrower if the number of rulings is increased.
Image on screen for monochromatic source
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Grating spectrometer
diffraction grating
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Atoms emit light when excited. The light is
emitted at specific wavelengths that are
characteristic of the particular atom. Thus, a
grating spectrometer can be used to identify
atoms from their emission spectrum. Hydrogen
emits light at four visible wavelengths.
Hydrogen lines from a grating spectrometer (m3
is not shown, for clarity, since the lines
overlap with m2 and m4)
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X-ray diffraction
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