Transcript 27.4 Lloyd’s Mirror

27.4 Lloyd’s Mirror
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An arrangement for
producing an interference
pattern with a single light
source
Waves reach point P either
by a direct path or by
reflection
The reflected ray can be
treated as a ray from the
source S’ behind the mirror
Fig 27.6
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Interference Pattern
from the Lloyd’s Mirror
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This arrangement can be thought of as a
double slit source with the distance between
points S and S’ comparable to length d
An interference pattern is formed
The positions of the dark and bright fringes
are reversed relative to pattern of two real
sources
This is because there is a 180° phase change
produced by the reflection
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Phase Changes
Due To Reflection
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An electromagnetic wave
undergoes a phase
change of 180° upon
reflection from a medium
of higher index of
refraction than the one in
which it was traveling
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Analogous to a pulse on a
string reflected from a rigid
support
Fig 27.7
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Phase Changes
Due To Reflection, cont
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There is no phase
change when the
wave is reflected from
a boundary leading to
a medium of lower
index of refraction
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Analogous to a pulse
in a string reflecting
from a free support
Fig 27.7
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27.5 Interference in Thin Films
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Interference effects are commonly
observed in thin films
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Examples include soap bubbles and oil on
water
The varied colors observed when white
light is incident on such films result from
the interference of waves reflected from
the opposite surfaces of the film
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Interference in Thin Films, 2
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Facts to keep in mind
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An electromagnetic wave traveling from a medium
of index of refraction n1 toward a medium of index
of refraction n2 undergoes a 180° phase change
on reflection when n2 > n1
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There is no phase change in the reflected wave if n2 < n1
The wavelength of light λn in a medium with index
of refraction n is λn = λ/n where λ is the
wavelength of light in vacuum
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Interference in Thin Films, 3
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Assume the light rays are
traveling in air nearly
normal to the two surfaces
of the film
Ray 1 undergoes a phase
change of 180° with respect
to the incident ray
Ray 2, which is reflected
from the lower surface,
undergoes no phase
change with respect to the
incident wave
Fig 27.8
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Interference in Thin Films, 4
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Ray 2 also travels an additional distance of
2t before the waves recombine
For constructive interference
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2 n t = (m + ½ ) λ m = 0, 1, 2 …
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This takes into account both the difference in optical path
length for the two rays and the 180° phase change
For destructive interference
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2 n t = m λ m = 0, 1, 2 …
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Interference in Thin Films, 5
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Two factors influence interference
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Possible phase reversals on reflection
Differences in travel distance
The conditions are valid if the medium above
the top surface is the same as the medium
below the bottom surface
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If there are different media, these conditions are
valid as long as the index of refraction for both is
less than n
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Interference in Thin Films, 6
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If the thin film is between two different media,
one of lower index than the film and one of
higher index, the conditions for constructive
and destructive interference are reversed
With different materials on either side of the
film, you may have a situation in which there
is a 180o phase change at both surfaces or at
neither surface
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Be sure to check both the path length and the
phase change
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Interference in Thin Film,
Soap Bubble Example
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27.6 Diffraction
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Diffraction occurs when waves pass
through small openings, around
obstacles, or by sharp edges
Diffraction refers to the general behavior
of waves spreading out as they pass
through a slit
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A diffraction pattern is really the result of
interference
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Diffraction Pattern
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A single slit placed between a distant light
source and a screen produces a diffraction
pattern
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It will have a broad, intense central band
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The central band will be flanked by a series of
narrower, less intense secondary bands
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Called the central maximum
Called side maxima
The central band will also be flanked by a series of
dark bands
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Called minima
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Diffraction Pattern, Single Slit
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The central maximum
and the series of side
maxima and minima are
seen
The pattern is, in reality,
an interference pattern
Fig 27.12
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Diffraction Pattern, Penny
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The shadow of a penny
displays bright and dark
rings of a diffraction
pattern
The bright center spot is
called the Arago bright
spot
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Named for its discoverer,
Dominque Arago
Fig 27.13
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Diffraction Pattern,
Penny, cont
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The Arago bright spot is explained by the
wave theory of light
Waves that diffract on the edges of the penny
all travel the same distance to the center
The center is a point of constructive
interference and therefore a bright spot
Geometric optics does not predict the
presence of the bright spot
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The penny should screen the center of the pattern
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Fraunhofer Diffraction Pattern
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Fraunhofer Diffraction
Pattern occurs when
the rays leave the
diffracting object in
parallel directions
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Screen very far from the
slit
Could be accomplished
by a converging lens
Fig 27.14
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Fraunhofer Diffraction
Pattern – Photo
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A bright fringe is
seen along the axis
(θ = 0)
Alternating bright
and dark fringes are
seen on each side
Fig 27.14
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Single Slit Diffraction
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The finite width of slits
is the basis for
understanding
Fraunhofer diffraction
According to Huygen’s
principle, each portion
of the slit acts as a
source of light waves
Therefore, light from
one portion of the slit
can interfere with light
from another portion
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Single Slit Diffraction, 2
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The resultant light intensity on a viewing
screen depends on the direction q
The diffraction pattern is actually an
interference pattern
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The different sources of light are different
portions of the single slit
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Single Slit Diffraction, Analysis
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All the waves that originate at the slit are in phase
Wave 1 travels farther than wave 3 by an amount
equal to the path difference
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(a/2) sin θ
If this path difference is exactly half of a
wavelength, the two waves cancel each other and
destructive interference results
In general, destructive interference occurs for a
single slit of width a when sin θdark = mλ / a
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m = ±1, ±2, ±3, …
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Single Slit Diffraction, Intensity
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The general features of the
intensity distribution are
shown
A broad central bright fringe is
flanked by much weaker
bright fringes alternating with
dark fringes
Each bright fringe peak lies
approximately halfway
between the dark fringes
The central bright maximum is
twice as wide as the
secondary maxima
Fig 27.15
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Resolution
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The ability of optical systems to distinguish
between closely spaced objects is limited
because of the wave nature of light
If two sources are far enough apart to keep
their central maxima from overlapping, their
images can be distinguished
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The images are said to be resolved
If the two sources are close together, the two
central maxima overlap and the images are
not resolved
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27.7 Resolved Images,
Example
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The images are far enough
apart to keep their central
maxima from overlapping
The angle subtended by the
sources at the slit is large
enough for the diffraction
patterns to be
distinguishable
The images are resolved
Fig 27.17
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Images Not Resolved,
Example
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The sources are so
close together that their
central maxima do
overlap
The angle subtended by
the sources is so small
that their diffraction
patterns overlap
The images are not
resolved
Fig 27.17
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Resolution,
Rayleigh’s Criterion
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When the central maximum of one
image falls on the first minimum of
another image, the images are said to
be just resolved
This limiting condition of resolution is
called Rayleigh’s criterion
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Resolution,
Rayleigh’s Criterion, Equation
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The angle of separation, qmin, is the angle
subtended by the sources for which the
images are just resolved
Since l << a in most situations, sin q is very
small and sin q q
Therefore, the limiting angle (in rad) of
resolution for a slit of width a is
qmin  l
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a
To be resolved, the angle subtended by the
two sources must be greater than qmin
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Circular Apertures
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The diffraction pattern of a circular aperture
consists of a central bright disk surrounded
by progressively fainter bright and dark rings
The limiting angle of resolution of the circular
aperture is
l
qmin  1.22
D
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D is the diameter of the aperture
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Circular Apertures,
Well Resolved
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The sources are far
apart
The images are well
resolved
The solid curves are the
individual diffraction
patterns
The dashed lines are the
resultant pattern
Fig 27.18
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Circular Apertures,
Just Resolved
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The sources are separated
by an angle that satisfies
Rayleigh’s criterion
The images are just resolved
The solid curves are the
individual diffraction patterns
The dashed lines are the
resultant pattern
Fig 27.18
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Circular Apertures,
Not Resolved
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The sources are close
together
The images are
unresolved
The solid curves are the
individual diffraction
patterns
The dashed lines are the
resultant pattern
Fig 27.18
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Resolution, Example
Fig 27.19
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Pluto and its moon, Charon
Left – Earth based telescope is blurred
Right – Hubble Space Telescope clearly
resolves the two objects
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27.8 Diffraction Grating
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The diffracting grating consists of a large
number of equally spaced parallel slits
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A typical grating contains several thousand lines
per centimeter
The intensity of the pattern on the screen is
the result of the combined effects of
interference and diffraction
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Each slit produces diffraction, and the diffracted
beams interfere with one another to form the final
pattern
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Diffraction Grating, Types
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A transmission grating can be made by
cutting parallel grooves on a glass plate
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The spaces between the grooves are
transparent to the light and so act as separate
slits
A reflection grating can be made by cutting
parallel grooves on the surface of a
reflective material
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Diffraction Grating, cont
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The condition for maxima is
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d sin θbright = m λ
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m = 0, 1, 2, …
The integer m is the order
number of the diffraction
pattern
If the incident radiation
contains several
wavelengths, each
wavelength deviates through
a specific angle
Fig 27.20
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Diffraction Grating, Intensity
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All the wavelengths are
seen at m = 0
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This is called the zeroth
order maximum
The first order maximum
corresponds to m = 1
Note the sharpness of the
principle maxima and the
broad range of the dark
areas
Fig 27.21
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Diffraction Grating,
Intensity, cont
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Characteristics of the intensity pattern
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The sharp peaks are in contrast to the
broad, bright fringes characteristic of the
two-slit interference pattern
Because the principle maxima are so sharp,
they are much brighter than two-slit
interference patterns
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Diffraction Grating
Spectrometer
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The collimated beam is
incident on the grating
The diffracted light leaves
the gratings and the
telescope is used to view the
image
The wavelength can be
determined by measuring
the precise angles at which
the images of the slit appear
for the various orders
Fig 27.22
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Grating Light Valve
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A grating light valve consists of
a silicon microchip fitted with an
array of parallel silicon nitride
ribbons coated with a thin layer
of aluminum
When a voltage is applied
between a ribbon and the
electrode on the silicon
substrate, an electric force
pulls the ribbon down
The array of ribbons acts as a
diffraction grating
Fig 27.23
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54p. 918
Fig. 27-24,
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27.9 Diffraction of X-Rays
by Crystals
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X-rays are electromagnetic waves of
very short wavelength
Max von Laue suggested that the
regular array of atoms in a crystal could
act as a three-dimensional diffraction
grating for x-rays
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The spacing is on the order of 10-10 m
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Diffraction of X-Rays
by Crystals, Set-Up
Fig 27.25
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A collimated beam of monochromatic x-rays is incident on
a crystal
The diffracted beams are very intense in certain directions
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This corresponds to constructive interference from waves
reflected from layers of atoms in the crystal
The diffracted beams form an array of spots known as a
Laue pattern
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Laue Pattern for Beryl
Fig 27.26
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Laue Pattern for Rubisco
Fig 27.26
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61p. 920
Fig. 27-27,
62p. 920
Fig. 27-28,
27.10 Holography
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Holography is the production of threedimensional images of objects
The laser met the requirement of
coherent light needed for making
images
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Hologram of Circuit Board
Fig 27.29
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Hologram Production
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Light from the laser is
split into two parts by
the half-silvered mirror
at B
One part of the beam
reflects off the object
and strikes an
ordinary photographic
film
Fig 27.30
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Hologram Production, cont.
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The other half of the beam is diverged
by lens L2
It then reflects from mirrors M1 and M2
This beam then also strikes the film
The two beams overlap to form a
complicated interference pattern on the
film
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Hologram Production, final
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The interference pattern can be formed only if
the phase relationship of the two waves is
constant throughout the exposure of the film
This is accomplished by illuminating the
scene with light coming from a pinhole or
coherent laser radiation
The film records the intensity of the light as
well as the phase difference between the
scattered and reference beams
The phase difference results in the threedimensional perspective
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Viewing A Hologram
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A hologram is best viewed by allowing
coherent light to pass through the
developed film as you look back along
the direction from which the beam
comes
You see a virtual image, with light
coming from it exactly in the way the
light came from the original image
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Uses of Holograms
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Applications in display
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Example – Credit Cards
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Called a rainbow hologram
It is designed to be viewed in reflected
white light
Precision measurements
Can store visual information
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Exercises
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3, 9, 11, 18, 19, 22, 26, 33, 47, 49, 56,
58
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