Download Diffraction Intensity, resolving power, Xray diffraction

Colorful interference (herrings and butterfly wings)
200 microns
scale photo
Herring (“nishin” in
Japanese)
Thin film interference
homework problem on
herring.
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The color of butterfly wings is not from
pigmentation (dull brown) but from
interference in small structures on the wings
Last time: Interference pattern of several slits
• The figure below shows the interference pattern for 2, 8, and 16 equally
spaced narrow slits.
If there are N slits, how many minima in the
diffraction pattern between the maxima ?
Ans: N-1
The height of each maxima is proportional to N2.
So what does this mean for the width ?
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Width ~1/N
to conserve
energy
Today
• Diffraction grating
• Diffraction limit and Rayleigh criterion.
• Crystal Diffraction.
• Next time: Holography, review of diffraction (Chap
36)
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The diffraction grating
•
A diffraction grating is an array of a large number of
slits having the same width and equal spacing. The
intensity maxima occur at
d sin   m
Looks similar to equations, be careful
Some points to note:
1)d is the spacing between slits
1)The angle θ is the angle
between the center of the slit
array to the mth bright region on
the screen.
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Diagram of a grating spectrograph
• A diagram of a diffraction-grating spectrograph for use in
astronomy.
Chromatic
resolving
power
l
R=
Dl
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Grating spectrographs
• A diffraction grating can be used to disperse light into a spectrum.
• The greater the number of slits, the better the resolution.
l
R=
= Nm
Dl
• Figure 36.18(a) below shows our sun in visible light, and in (b) dispersed into a
spectrum by a diffraction grating. See description of Eschelle spectrograph:
http://www.vikdhillon.staff.shef.ac.uk/teaching/phy217/instruments/phy217_inst_echelle.html
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Interesting diffraction grating example
The intensity maxima for a diffaction grating occur at
d sin   m
b = 0.05 Þ v = 0.05c
•An astronomer examining a distant galaxy observes a
line in the hydrogen spectrum that has a wavelength of
656.3 nm (Hα line) in first order. Using a transmission
diffraction grating 5758 lines/cm she finds that the
bright fringe for the Hα occur at ±23.410 from the
central spot. How fast is the galaxy moving ?
d=
1 cm
= 1.74 ´10-4 cm=1.74 x 10 -6 m
5758 lines
l = 690nm
l = d sinq / m = 1.74 ´10-6 sin(23.410 )
Why is the line shifted ?
Ans: Doppler effect
1+ b
(690 / 656.3) =
» (1+ b )(1- b ) » 1- 2b
1- b
2
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Example of a diffraction grating
The intensity maxima for a diffaction grating occur at
d sin   m
•Example 36.4: The wavelengths of the visible spectrum
are approximately 380 nm (violet) to 750 nm (red). (a)
Find the angular limits of the first-order visible
spectrum produced by a plane grating with 600 slits per
millimeter when white light falls normally on the
grating. (b) Do the first order and second order spectra
overlap? What about the 2nd and 3rd orders?
•(a) distance between slits is
1 mm
d

 1.67
106 m
st
Violet light for 1 600
order
occurs
at
slits
Red light
for 1storder
 arcsin
/ d  occurs
arcsinat 3.8 107 /1.67 10 6   13.2
•(b) recalculate for m = 2and
m = 3./ d   arcsin  7.5 107 /1.67 106   26.7
arcsin
The 2nd-order spectrum extends from 27.1-63.9° while the 3rd order is from 43-90.
Yes there is overlap
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Airy disk (circular aperture)
What is different about this
diffraction pattern ?
Ans: It is produced by a circular
aperture and has rings
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Circular apertures
• An aperture of any shape forms a diffraction pattern.
• The figures below illustrate diffraction by a circular aperture. The Airy
disk is the central bright spot.
• The first dark ring occurs at an angle given by sinθ1 = 1.22 λ/D.
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Diffraction and image formation
• Diffraction limits the resolution of
optical equipment, such as
telescopes.
• The larger the aperture, the better
the resolution. The figure on the
right illustrates this effect.
However, in practice the
atmosphere provides a more
stringent limit than diffraction
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Summary of Diffraction Limit (Rayleigh’s criterion)
Angular radius of the first dark ring in the circular diffraction pattern
sinq1 = 1.22l / D
Rayleigh criterion for the diffraction limit
Lord Rayleigh 1904
Nobel Prize in Physics
Idea: center of one diffraction pattern coincides with the
first minimum of the other. (Note that D is the diameter of
the aperture for a lens or telescope.)
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Bigger telescope, better resolution
• Because of diffraction, large-diameter telescopes, such as
the VLA (Very Large Array) radio telescope in New
Mexico below, give sharper images than small ones.
• 27 telescopes, 25m in diameter effective aperture 36 km !
Rayleigh
criterion gives
(λ=1.5cm,
D=36 km)
5 x 10-7 rad
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Sites for the SKA (Square Kilometer Array)
The effective area will be 3000 km
when completed (sites on every
continent except South America)
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Crystal Diffraction Basics
• Can we observe crystal diffraction with visible or ultraviolet
light ? (Hint: what is the typical spacing of the components
of a crystal.)
Ans: No, for example the wavelength of
visible light is 400-700 nm, but the plane
spacing in NaCl is 0.28nm.
UV is 100nm to 380nm so that won’t work
either.
How can we observe crystal diffraction and understand the
atomic structure of a crystal (or a protein) ?
Ans: use coherent x-rays
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X-ray diffraction
• When x rays pass through a crystal, the crystal behaves like
a diffraction grating, causing x-ray diffraction. The figure
below illustrates this phenomenon.
A version of this experiment
is done at UH in PHYS481L.
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A simple model of x-ray diffraction
The Bragg condition for constructive
interference is 2d sinθ = mλ
The path difference is the due to the dimensions of the crystals
(reflected x-rays have slightly different path lengths)
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A simple model of x-ray diffraction
The Bragg condition for constructive
interference is 2d sinθ = mλ
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An example of holography
• Photographs of a holographic image from two different
angles, showing the changing perspective.
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What is holography?
• By using a beam splitter and mirrors, coherent laser light
illuminates an object from different perspectives.
Interference effects provide the depth that makes a threedimensional image from two-dimensional views.
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How does holography work?
• Follow the analysis using the Figure below.
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