from last time… new topic: diffraction only one slit, but ... · (reflection, refraction,...
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From Last time…
Diffraction grating
Two-source interference:
Diffraction = interference from many sources
Week3HW on Mastering Physics due Fri. Sep. 18
Week2HW due Fri. Sep. 11
New topic: Diffraction only one slit, but “wide”
• Interference-like pattern from a single slit.
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θcentral width ~ 2 λa
For a slit:
Long wavelength: wide pattern
Short wavelength narrow pattern
Thursday, Sep. 10 Physics 208, Lecture 3 3
Huygen’s principle
• Huygen’s principle: each portion of the slit acts as a source of waves
• These sources interfere according to path-length difference.
Overlapping diffraction patterns • Two sources ->two
diffraction patterns.
• Width central max determined by aperture.
• Larger aperture gives better angular resolution
θ
Angular separation
• For a circular aperture (e.g. lens)
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θmin =1.22 λD
36” Lick refractor at UC-Berkeley
Thursday, Sep. 10 Physics 208, Lecture 3 5
Large aperture -> good angular resolution
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θmin =1.22 λD
D
Diffraction from other objects
• General effect • Clearest w/single wavelength
Thursday, Sep. 10 Physics 208, Lecture 3 6
Light diffraction by pinhead
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Thursday, Sep. 10 Physics 208, Lecture 3 7
Interference summary • Waves start in phase • Travel different distances (extra path length = δ) • No longer in phase when combined (Phase diff φ)
Shorter path
Longer path Here, δ = λ/2 Phase diff π Crest aligns with trough Destructive interference
Another source of phase difference • In some cases reflection gives phase shift
n1 n2>n1
n1 n2<n1
π phase shift
no phase shift
Thursday, Sep. 10 Physics 208, Lecture 3 9
Thin film interference Black
Colors changing with thickness
Thursday, Sep. 10 Physics 208, Lecture 3 10
air: n=1
n>1 t
λ
λ /n Extra path length~2t
air: n=1
• Phase difference from reflection – Top reflection has π phase shift, bottom not
• Phase difference from path length difference – Path length difference = 2t
– Gives phase difference
180˚ (π radians) phase shift from reflection
no phase shift from reflection
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2π 2tλ /n
Contributions to the phase difference
Thin film
Thursday, Sep. 10 Physics 208, Lecture 3 11
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π + 2π 2tλ /n( )
=Phase difference = Reflection phase shift # wavelengths in
extra path length
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2mπ constructive
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2m +1( )π destructive
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2t = m +12
λn
(m = 0,1,2…) constructive interference
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2t = m λn
(m = 0,1,2…) destructive interference
Convert to phase
What happens when: t << all λ in light? Constructive int. condition for some λ?
Thursday, Sep. 10 Physics 208, Lecture 3 12
Biological iridescence
• Some organisms seem to reflect incredibly vivid colors. Not by pigment, but interference!
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Thursday, Sep. 4 Phy208 Lecture 2 13 €
λconstructive = 446nm
Waves and geometry
• Interference and diffraction demonstrate that light is a wave.
• Doesn’t always appear as a straight ‘ray’ of light
… but sometimes it almost does!
Geometric optics: Tracing the path of light rays
What is a light ray?
• Light ray is a line in the direction along which light energy is flowing.
Ray enters eye -> you can see the light source
Wavefronts (crests of waves)
What does a light ray do?
• Light rays travel forever in straight line unless they interact with matter (reflection, refraction, absorption)
What about diffraction? • Light really behaves as a wave • The concept of a light ray is an approximation
i.e. a lie
Wavelength << aperture size, rays are good approximation
Light rays from point source
• Light rays are not always parallel. – E.g. light bulb visible from all directions – Rays must be traveling in all directions
Light ray perpendicular to local wavefront (crest of wave).
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Interaction of light with matter
Absorption
And all occur simultaneously
Reflection
Refraction
Reflection/refraction occur at interfaces between different materials
Reflection and Refraction • Direction of light can be changed by
– Reflection (lets you see an object) – Refraction (transmits light thru object)
… at an interface between different materials
• Ray is the incident ray • Ray is the reflected ray • Ray is refracted into the
lucite • Ray is reflected inside the
lucite • Ray is refracted as it enters
the air from the lucite
Air
Plastic Inter- face
When are materials different?
• For reflection/refraction – materials are different if they have
different index of refraction – Light propagates at different speed
in different materials. – Due to interaction of
electromagnetic wave with atoms in material.
Material Index of refraction
Vacuum 1.00 exactly
Air (actual) 1.0003
Air (accepted) 1.00
Ice 1.31
Water 1.33
Ethyl Alcohol 1.36
Oil 1.46
Pyrex glass 1.46
Crown glass 1.52
Polystyrene plastic 1.59
Flint glass 1.66
Diamond 2.41
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v =cn c=speed of light in vacuum
What do you think? Pyrex stirring rod (n=1.46) dipped into beaker of
Wesson oil (n=1.46). What happens to the rod?
Pyrex stirring rod
A. Appears dark
B. Appears bright
C. Appears invisible
D. Appears curved
E. Appears inverted
No reflection/refraction if index of refraction is same.
Beaker of Wesson oil
Reflection • Angle of incidence
= angle of reflection θi θr
Incident ray
Reflected ray
• Multiple reflections
• Apply θi=θr at each surface – trace ray
Why θi=θr? • Christian Huygens modeled this in 1690
– Said that each point on wavefront acts as source of spherical wavelets
– Superposition of wavelets gives reflected plane wave such that θi=θr
θi θr
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What about refraction? • Refraction occurs when light moves into
medium with different index of refraction. • Result: light direction bends according to
Snell’s law
θi,1 θr
θ2
Angle of refraction
n1
n2 €
n1 sinθ1 = n2 sinθ2
Why Snell? • Can analyze in exactly the same way • Light moves at different speed in different media
θi θr
n1
n2>n1
θ2 v2<v1
Refraction angle
n2 < n1
v2>v1
Reflected ray
n1
n2 >n1
v2<v1
n1 Reflected ray
n2>n1 Refracted ray bent toward normal
n2<n1 Refracted ray bent away from normal
slower in medium 2 faster in medium 2
Quick quiz Which of these fluids has the
smallest index of refraction (highest light speed)?
A
B
C
A. Fluid A
B. Fluid B
C. Fluid C
D. All equal
Numerical Example A beam of light is traveling underwater, aimed up at the
surface at 45˚ away from the surface normal. Part of it is reflected back into the water, and part is transmitted into the air.
Water n1=1.33
Air n2=1.00
θ1=45˚
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n1 sinθ1 = n2 sinθ2
sinθ2 =n1n2sinθ1 = 0.94
θ2 = arcsin n1n2sinθ1
= 70˚
θ2
Quick quiz A trout looks up through
the surface at the setting sun, and at the moon directly overhead. He sees
A. Moon directly overhead, sun ~ parallel to water surface
B. Moon directly overhead, sun ~ 40˚ above water surface
C. Moon ~ 40˚ from vertical, sun ~ parallel to water surface
D. Moon and sun aligned at 40˚ from vertical.
n2=1.0
n1=1.33
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Total Internal Reflection • Is possible when light is directed from n1 > n2
⇒ refracted rays bend away from the normal
• Critical angle: angle of incidence that will result in an angle of refraction of 90° (sinθ2 = 1)
For water:
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sinθc =1
1.333= 0.75⇒θc = 48.75˚
Optical Fibers The cladding has a lower n than the core
• Plastic or glass light pipes • Applications:
– Medicine: endoscope (light can be directed even if bent and the surgeon can view areas in the body using a camera.)
– Telecommunications