Transcript Example

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 > 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 Huygen’s 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 Rayleigh’s 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 Rayleigh’s criterion? What minimum distance does this imply
between resolvable objects on the Moon?
Note: the Earth’s 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, a<<).
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
(m=3 is not shown, for clarity, since the
lines overlap with m=2 and m=4)
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X-ray diffraction
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