From Last time…

Diffraction grating

Two-source interference:

Diffraction = interference from many sources

Week3HW on Mastering Physicsdue Fri. Sep. 18

Week2HW due Fri. Sep. 11

New topic: Diffraction only one slit, but “wide”

• Interference-like pattern from a single slit.

€

θcentral width ~ 2λ

a

For a slit:

Long wavelength:wide pattern

Short wavelengthnarrow

pattern

Long wavelength:wide pattern

Short wavelengthnarrow

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

Angularseparation

• For a circular aperture (e.g. lens)

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θmin =1.22λ

D

36” Lick refractor at UC-Berkeley

Thursday, Sep. 10 Physics 208, Lecture 3 5

Large aperture -> good angular resolution

€

θmin =1.22λ

D

D

Diffraction from other objects

• General effect• Clearest w/single wavelength

Thursday, Sep. 10 Physics 208, Lecture 3 6

Light diffraction by pinhead

Thursday, Sep. 10 Physics 208, Lecture 3 7

Interference summary• Waves start in phase• Travel different distances (extra path length = )• No longer in phase when combined (Phase diff )

Shorter path

Longer pathHere, = λ/2Phase diff πCrest aligns with troughDestructive 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 interferenceBlack

Colors changing with thickness

Thursday, Sep. 10 Physics 208, Lecture 3 10

air: n=1

n>1t

/nExtra path length~2tair: 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

€

2π2t

λ /n

Contributions to the phase difference

Thin film

Thursday, Sep. 10 Physics 208, Lecture 3 11

€

π + 2π2t

λ /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 +1

2

⎛

⎝ ⎜

⎞

⎠ ⎟λ

n (m = 0,1,2K ) constructive interference

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2t = mλ

n (m = 0,1,2K ) 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!

Thursday, Sep. 4 Phy208 Lecture 2 13

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constructive = 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).

Interaction of light with matter

Absorption

And all occur simultaneouslyAnd 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

PlasticInter-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

€

v =c

n 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

invertedNo reflection/refraction if index of refraction is same.

Beaker of Wesson oil

Reflection• Angle of incidence

= angle of reflectionθi θr

Incidentray

Reflectedray

• 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 i= r

θi θr

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

n1Reflected 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 ExampleA 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.

Watern1=1.33

Airn2=1.00

θ1=45˚

€

n1 sinθ1 = n2 sinθ2

sinθ2 =n1

n2

sinθ1 = 0.94

θ2 = arcsinn1

n2

sinθ1

⎛

⎝ ⎜

⎞

⎠ ⎟= 70˚

θ2

Quick quizA 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

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° (sinθ = 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