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Physics 116
Session 27
Diffraction gratings
Postulates of Relativity
Nov 14, 2011
R. J. Wilkes
Email: [email protected]
Announcements
•! HW 5 due today!
Lecture Schedule
(up to exam 3)
Today
3
Diffraction for a circular aperture: resolution
•! Pinholes also show diffraction fringes
Last time:
–! Similar to single slit pattern, but with circular symmetry
–! Mathematical form is called the Airy function
–! Can just resolve 2 pinholes if their
1st minima overlap:
(a): One pinhole
(b): Two, just separable
(c): Two, not separable!
–! Angle to first dark fringe for a pinhole is
•! Rayleigh Criterion: resolution for aperture of diameter D
Telescope, camera, binoculars and human eye = circular apertures !
Rayleigh criterion lets us estimate resolution limits for optical devices
Examples
•!
Light with wavelength 676 nm strikes a single slit of width 7.64 microns, and
goes to a screen 185 cm away. What is distance in cm of 1st bright fringe above
the central fringe?
First bright fringe is about halfway between dark fringes for m=1 and m=2
We can find its angle by using “m=1.5” in the dark-fringe formula:
(same as finding location of
dark fringes 1 and 2, and
then finding their midpoint)
y
L
Tangent of angle = y/L:
•!
A spy satellite orbits at 160 km altitude. What camera lens diameter is needed
to resolve objects of size 30 cm (meaning: distinguish two objects a foot apart)?
Assuming light wavelength 550 nm:
(~ center of human vision range)
5
2-slits revisited
•! Now that we know about diffraction, we can understand details of 2-slit
interference patterns
•! Each slit’s diffraction pattern modulates the 2-slit interference pattern
a
w
diffraction pattern due to each slit
!
w
Interference pattern for 2 slits
w=0.25mm, a=0.9mm, "=632nm
Result:
From 2 to N slits: diffraction gratings
•!
For N>>2 slits, uniformly spaced, we get an interference pattern that has
–! Many sharply defined bright fringes, equally spaced (principal maxima)
•! Approximately same intensity for all
•! Increasing N sharpens the principal maxima
–! In between, many very dim secondary fringes (secondary maxima)
•!
For very large N, slit mask = “diffraction grating”
–! for m=0 all wavelengths have max at q=0
–! for m>0, maxima at
! ! " : can use grating as a spectroscope
For w=0.25mm, a=0.5mm, "=632nm:
N=4
N=8
Diffraction gratings
•! N-slit interference produces pattern with fringe spacings
dependent on wavelength
•! Diffraction grating = thousands of closely spaced slits
–! Very sharp fringes build up for each color, with contributions from
all slits
–! Better than a prism (no light absorption in glass)
–! Can use mirror surface with fine-line pattern also: reflection grating
Diffraction grating with red + blue wavelengths incident:
colors are separated due to different angles for their maxima
Rainbow effect
looking at white
light reflected in a
CD
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Examples
typos corrected:
•!
A laser emits 2 wavelengths, 420nm and 630nm. At what angle and for what order
will maxima for both wavelengths coincide, for a grating with N=450 lines/mm?
We want same angle to be order number m for one line, and n for the other; both
must be integers:
d sin ! m = m"1
d sin ! n = n"2
spacing d (in mm) = 1 / N lines / mm
(
! m = sin #1
m"1
= sin #1 m"1 N
d
(
)(
(
)
)
450 lines / mm = 450 $ 10#6 lines / nm
)
(
)
= sin #1 %& m 420 nm 450 $ 10#6 nm #1 '( = sin #1 %& m 0.189 '(
! n = sin #1 %& n 630 nm 450 $ 10#6 nm #1 '( = sin #1 %& n 0.2835 '(
* 0.2835 so we want m 0.189 = n 0.2835 ) m = n ,
= 1.5n
+ 0.189 /.
Try n=1, 2… and find first value of n that gives integer m
n= 2 gives m=3: so find angle in degrees by putting in m=3 above
(
(
m = -3
n = -2
-2
-1
-1
)(
)
(
)
(
)
)
0
0
+1
+2
+1
+3
+2
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“Modern” physics
•! Next set of topics: a brief introduction to relativity and quantum
theory, atomic and nuclear physics
–! Modern physics (term coined in mid-20th century!)
–! As opposed to: “classical physics” (Newton and Maxwell), where
•! Time ticks on, independent of physical objects or their motions
•! EM radiation consists of waves, not particles
•! With fully detailed info on its physical state now, motion of a
body can be predicted precisely, into the future
•! In 1895, physics was thought to be “almost finished”
–! Just a few little problems remained to be settled… (sound familiar?)
1.! How to tweak Maxwell’s equations to make them obey Galilean Relativity?
2.! Why do atoms emit light only at specific wavelengths (“line spectra”)
3.! Explanations for a few peculiar experimental results:
“Blackbody radiation” (thermal emission of EM waves) spectrum
“Photoelectric effect” (emission of charge when light hits metal surfaces)
First, item 1 above: relativity
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Review: Galileo’s “common sense” relativity (c. 1600)
•! Example: Bill and Phil are (at first) both standing still
–! Bill fires a gun: bullet’s speed is 1000 m/s relative to gun
•! Both agree: bullet speed is 1000 m/s
–! Next, Phil rides on train with speed 100 m/s, shoots gun forward
•! Phil says v=1000 m/s, but Bill says it is 1100 m/s
•! Both are right: describing motion in their reference frames
–! Bill agrees he would say 1000 m/s if he were on train
–! Phil agrees he would say 1100 m/s if he were on ground
–! Both agree bullet “really” moves 1100 m/s (Earth reference)
First: Both are at rest
Next: Phil rides on 100 m/s train moving past Bill
Q: what if Phil fired backward ?
P
B
P
B
1000 m/s
1000 m/s
Earth
Train
100 m/s
Earth
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Oops: A little problem with Maxwell’s equations
•! By 1880s, Maxwell’s work was in everyday technology
–! Every generator, motor, telegraph, telephone proved him right
•! Just like quantum theory today…computer chips, lasers
•! Problem: Maxwell equations don’t obey Galilean relativity !
–! Simple example: imagine Phil holds an electric charge
•! Both standing still: both see only electric field of charge
•! Phil is on moving train, Bill remains at rest:
–! Bill sees moving charge = current ! magnetic field B
–! Phil still sees only electric field E of static charge
First: Both are at rest
Both see E field and
no B field
B
P
+
Next: Phil rides on 100 m/s train moving past Bill
Phil sees only E field, but Bill sees a B field
P
B
+
Static charge
Train
Earth
Moving charge = current I
100 m/s
Earth
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