chapter 25 wave optics lecture 19 - purdue university
TRANSCRIPT
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Chapter 25 Wave Optics – Lecture 19
25.1 Coherence and Conditions for Interference
25.2 The Michelson Interferometer
25.3 Thin-Film Interference
25.4 Light through a Single-Slit: Qualitative Behavior
25.5 Double-Slit Interference: Young’s Experiment
25.6 Single-Slit Diffraction: Interference of Light frm a Single
Slit
25.7 Diffraction Gratings
25.8 Optical Resolution and Rayleigh Criterion
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Wave Optics vs. Geometric Optics
• When discussing image characteristics over
distances much greater than the wavelength,
geometric optics is extremely accurate
• When dealing with sizes comparable to or smaller
than the wavelength, wave optics is required
• Examples include interference effects and propagation
through small openings
Introduction
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Interference
• One property unique to waves is interference
• Interference is a wave phenomenon
• Interference of sound waves can be produced by two speakers
Section 25.1
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Constructive Interference
• Require the phase difernece of 0,
2, 4 .... or a path difference x given by
+
=
0, 1, 2,...x m m m m
Destructive Interference
• Here the phase difference between the
two waves is radians, 180, or /2
1
0, 1, 2,...2
x m m m m
+
=
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Summary - Interference
• If light waves are traveling from some point, then the
phase difference can be related to the path difference between
the two waves
• The criterion for constructive interference is given by a path
difference x given by
• Destructive interference will take place if the path difference x
is a half wavelength plus an integer times the wavelength
• The inter m is called the order of the fringe
0, 1, 2,...x m m m m
1
0, 1, 2,...2
x m m m m
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Summary- Interference Conditions
• For constructive interference, Δx = m λ
• For destructive interference, Δx = (m + ½) λ
• m is an integer in both cases
• If the interference is constructive, the light
intensity at the detector is large
• Called a bright fringe
• If the interference is destructive, the light
intensity at the detector is zero
• Called a dark fringe
Section 25.2
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Interference
• Sunlight is composed of light containing a broad range
of frequencies and corresponding wavelengths
• We often see different colors
separated out of sunlight by
refraction in rainbows
• We also sometimes see various
colors from sunlight due to
constructive and destructive
interference phenomena on the
surface of DVD’s or CD’s
or in thin layers of oil or water
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Michelson Interferometer
• The Michelson
interferometer is based
on the interference of
reflected waves
• Two reflecting mirrors
are mounted at right
angles
• A third mirror is partially
reflecting
• Called a beam splitter
Section 25.2
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Michelson Interferometer, cont.
• The incident light hits the beam splitter and is divided
into two waves
• The waves reflect from the mirrors at the top and right
and recombine at the beam splitter
• After being reflected again from the
beam splitter, portions of the waves
combine at the detector
• The only difference between the two
waves is that they travel different
distances between their respective
mirrors and the beam splitter
• The path length difference is
ΔL = 2L2 – 2L1 Section 25.2
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Michelson Interferometer, final
• The path length
difference is related to the
wavelength of the light
• If N is an integer, the two
waves are in phase and
produce constructive
interference
• If N is a half-integer the
waves will produce
destructive interference
LN
D=
l
Section 25.2
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Measuring Length with a Michelson
Interferometer • Use the light from a laser and adjust the mirror to give
constructive interference
• This corresponds to one of the bright fringes
• The mirror is then moved, changing the path length
• The intensity changes from high to zero and back to high,
every time the path length changes by one wavelength
• If the mirror moves through N bright fringes, the distance
d traveled by the mirror is (for round trip) or N
d =2
l
Section 25.2
• The accuracy of the measurement depends on the
accuracy with which the wavelength is known
• Many laboratories use helium-neon lasers to make very
precise length measurements
d N2
He Ne nm632.99139822
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• When a light wave
passes from one
medium to another, the
waves must stay in
phase at the interface
• The frequency must be
the same on both sides
of the interface
Section 25.3
Light Traveling in an Optical Medium (1)
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Light Traveling in an Optical Medium (2)
• We have seen that the wavelength of light changes when traveling in an optical medium with index of refraction greater than one
• Taking with case 1 as a vacuum and case 2 as a medium with index of refraction n, we have found out that we can write
• Remembering that v = f we can write the frequency fn of light traveling in a medium as
• So the frequency does not change !
n
v
c n
/ ( / )n
n
v cf v v c f
1 2
1 2
tv v
nf f
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Light Traveling in an Optical Medium (3)
• So the frequency of light traveling in an optical medium with n >
1 is the same as the frequency of that light traveling in vacuum
• We perceive color by frequency rather than wavelength
• Thus placing an object under water does not change our
perception of the color of the object
• Easy to demonstrate: take a colored object and put it in a jar of
water. Water has index of refraction n = 1.33. The object
appears to have the same color under water as in air
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Phase Change and Reflection
• When a light wave reflects from a surface it may be
inverted
• Inversion corresponds to a phase change of 180°
• There is a phase change whenever the index of
refraction on the incident side is less than the index
of refraction of the opposite side
• If the index of refraction is larger on the incident side
the reflected ray is not inverted and there is no
phase change
Section 25.3
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Phase Change and Reflection, Diagram
Section 25.3
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Phase Changes in a Thin Film
• The total phase change in a thin film must be
accounted for
• The phase difference due to the extra distance
traveled by the ray
• Any phase change due to reflection
• For a soap film on glass, nair < nfilm < nglass
• There are phase changes for both reflections at the
soap-film interfaces
• The reflections at both the top and bottom surfaces
undergo a 180° phase change no phase change !
• The nature of the interference is determine only by
the extra path length
Section 25.3
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Thin-Film Interference (1)
• Assume a thin soap film rests on a flat glass
surface (Fig. A and Fig. B)
• The upper surface of the soap film is similar to
the beam splitter in the interferometer
• It reflects part of the incoming light and allows the
rest to be transmitted into the soap layer after
refraction at the air-soap interface Section 25.3
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Thin-Film Interference (2)
• The transmitted ray is partially reflected at the bottom
surface
• The two outgoing rays meet the conditions for interference
• They travel through different regions
• One travels the extra distance through the soap film
• They recombine when they leave the film
• They are coherent because they originated from the same
source and initial ray Section 25.3
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Thin-Film Interference (3), nair < nfilm > nair
• Assume the soap bubble is
surrounded by air
• There is a phase change at the top
of the bubble
• There is no phase change at the
bottom of the bubble
• Since only one wave undergoes a
phase change, the interference
conditions are
film
film
m
d constructive interferencen
md destructive interference
n
æ ö+ç ÷
è ø=
=
1
22
2
l
l
Section 25.3
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Thin-Film Interference (4), White Light
• Each color can interfere
constructively, but at
different angles
• Blue will interfere
constructively at a
different angle than
red
• When you look at the
soap film the white light
illuminates the film over
a range of angles
Section 25.3
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Thin Film Interference (5) • When light travels from an optical medium with an index of refraction n1 into
a second optical medium with index of refraction n2,
(1) The light can be transmitted through the boundary
no change of the phase of light
(2) The light can be reflected
• If n1 < n2, the phase of the reflected wave will be changed by or
• If n1 > n2 then there will be no phase change
/ 2
Oil floating on water: The color seen corresponds to the
wavelength of light that is interfering constructively
n2
n1
n1
n2
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• Due to the phase change of 1800, the criterion for
constructive interference is given by
• The minimum thickness tmin that will produce contructive
interference corresponds to (for m=0)
• Note that this result applies only to the case where we
have a material with index of refraction n and air on both
sides, like a soap bubble
min4
airtn
1
2 0, 1, 2,...2
x m t m m m
min min
12 0 4 =
2 4 4
airx m t m t tn
Thin Film Interference (6), nair < nfilm > nair
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• The criterion for destructive interference is given by
• The minimum thickness tmin that will produce destructive
interference corresponds to (for m=1)
• Note that this result applies only to the case where we
have a material with index of refraction n and air on both
sides, like a soap bubble
min2
airtn
2 , 1 0, 1, 2,...x m t m m m m
1 1 1 2
2 2 2 1
/
/
v c n n
v c n n
2 2, 1
air
air air
n
n n
1 2
1 2
tv v
min min2 1 2 = 2 2
airx m t m t tn
Thin Film Interference (7), nair < nfilm > nair
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Thin-Film Interference, nair < nfilm > nair
• Equations are
• These equations apply whenever
nair < nfilm > n(substance below the film)
film
film
md constructive interference
n
m
d destructive interferencen
2
1
22
Section 25.3
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Thin Film Interference (8) - Demo
• The reflected wave undergoes a phase shift
of half a wavelength when it is reflected
because nair < n
• The light that is transmitted has no phase
shift and continues to the back surface of the
film
• At the back surface, again part of the
wave is transmitted and part of the wave is
reflected
• The reflected light has no phase shift because n >
nair and travels back to the front surface of the film
• The transmitted light has traveled a longer distance
than the originally reflected light and has a phase
shift given by the path length difference that is
twice the film thickness t
soap bubble
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Phase Changes in a Thin Film, nair < nfilm < nglass
• For a soap film on glass, nair < nfilm < nglass
• There are phase changes for both reflections at the
soap-film interfaces
• The reflections at both the top and bottom surfaces
undergo a 180° phase change no phase change
!
• For nair < nfilm < n(substance below the film)
Section 25.3
film
film
md constructive interference
n
m
d destructive interferencen
2
1
22
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Antireflection Coatings (1)
• Nearly any flat piece of
glass may act like a
partially reflecting mirror
• To avoid reductions in
intensity due to this
reflection, antireflective
coatings may be used
• The coating makes a
lens appear slightly dark
in color when viewed in
reflected light
Section 25.3
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Antireflective Coatings (2)
• Many coatings are
made from MgF2
• nMgF2 = 1.38
• There is a 180° phase
change at both
interfaces
• Destructive interference
occurs when
Section 25.3
MgF MgF
m
d dn n
2 2
1
22
4
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• The light transmitted through the
coating has no phase change: When it
is reflected from the lens, it will
undergo a phase change
• the criterion for destructive interference
is
n=1
n=1.38 n=1.51
1
2 0, 1, 2,...2
air
coating
m t m m mn
min
4
air
coating
tn
• Light reflected at the surface of
the coating will undergo a phase
change of half a wavelength
because nair < ncoating
Antireflective Coatings (3) – more details
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• The criterion for destructive interference is given by
• The minimum thickness tmin that will produce destructive
interference corresponds to (for m=0)
• Note that this result applies to the case where we have a
material with index of refraction n coated on a lens and
air on one side, like a coated lens.
min4
airtn
1
2 0, 1, 2,...2
x m t m m m
min min
12 0 4 =
2 4 4
airx m t m t tn
Antireflective Coatings (4) – more details
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Summary - Thin-Film Interference
• For nair < nfilm < n(substance below the film)
• For nair < nfilm > n(substance below the film)
Section 25.3
film
film
md destructive interference
n
m
d costructive interferencen
2
1
22
film
film
md constructive interference
n
m
d destructive interferencen
2
1
22
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Light Through a Single Slit
• Light passes through a slit or opening and then
illuminates a screen
• As the width of the slit becomes closer to the
wavelength of the light, the intensity pattern on the
screen and additional maxima become noticeable
Section 25.4
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Single-Slit Diffraction
• Water wave example of
single-slit diffraction
• All types of waves
undergo single-slit
diffraction
• Water waves have a
wavelength easily
visible
• Diffraction is the
bending or spreading of
a wave when it passes
through an opening
Section 25.4
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Huygen’s Principle
• It is useful to draw the
wave fronts and rays for
the incident and
diffracting waves
• Huygen’s Principle
can be stated as all
points on a wave front
can be thought of as
new sources of
spherical waves
Section 25.4