Huygen’s Principle• Any wave (including electromagnetic waves) is able to propagate

because the wave here affects nearby points there• In a sense, the wave is the source for more of the wave

• A wave here creates waves in all the forward directions• For a plane wave, the generated waves add up to make more plane

waves

• Mathematically, this works, but for plane waves, no one does it this way

Chs 37 and 38

Diffraction Through a Tiny Hole• The waves come out in all directions• It is only because the whole wave makes new waves that the

waves add up to only go forwards• What if we let the wave pass through a tiny hole?• Smaller than a wavelength

• Only one point acts as source• Waves spread out in all directions

0 sinE E kx t

0 sinE

E kr tr

r

sinE kr t

• What’s interesting is that oscillations depend on distance from slit

Diffraction is bending

CT -1 - Diffraction occurs when light passes a: A. pinhole. B. narrow slit. C. wide slit. D. sharp edge. E. all of the above

Ans E

Interference Through Two Slits• Now imagine we have two slits, equally sized• Each slit creates its own waves

• In some directions, crests add with crests to make bigger “brighter” crests

• In others, crests combine with troughs to make minimum areas

• In the end, what you get is a pattern of alternating light and dark bands

• We’re about to need an obscure math identity:

sin sin 2sin cos2 2

A B A BA B

Two Slit Interference• What do the EM waves look like far away?• Let the separation of the slits be d• Let’s find total E-field at point P

d

P

d sin

r1

r2

1 2E E E 1 2~ sin sinkr t kr t

1ave 2 122sin coskr t k r r

1ave 1 22r r r

2 1 sinr r d

1ave 22sin cos sinE kr t kd

2 2 2 1ave 2sin cos sinI E kr t kd

Two Slit Interference (2)

2 12cos sinI kd

2k

2max

sincos

dI I

• Where is it bright?

• Where is it dark?brightsin m d 0, 1, 2,m

1dark 2sin m d

CT -2- An interference pattern is formed on a screen by shining a planar wave on a double- slit arrangement (left). If we cover one slit with a glass plate (right), the phases of the two emerging waves will be different be-cause the wavelength is shorter in glass than in air. If the phase difference is 180°, how is the interference pattern, shown left, altered?

A. The pattern vanishes. B. The bright spots lie closer together. C. The bright spots are farther apart. D. There are no changes. E. Bright and dark spots are interchanged.

Ans E

Ex- (Serway 37-10) In a location where the speed of sound is 354 m/s, a 2000 Hz sound wave impinges on two slits 30.0 cm apart. (a) At what angle is the first maximum located? (b) If the sound wave is replaced by 3.00-cm microwaves, what slit separation gives the same angle for the first maximum? (c) If the slit separation is 1 M, what frequency light gives the same first maximum angle?

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Phases• When you combine two (or more) waves, you need to know the phase

shift between them:• The angle is the phase shift

• When the phase shift is zero, the waves add constructively• The result is bigger• Same thing for any even multiple of

• When the phase shift is , the waves add destructively• The result is smaller• Same thing for any odd multiple of

• To find maximum/minimum effects, set phase shift to even/odd multiples of

sinsinA x BE x

Phase Shift From Traveling• As a wave passes through any material, its phase shifts• For a distance d, we have: sinE kx t

kd 2 d

• Recall, wavelength changes inside a material

cf

n 0f c 0

n

0

2 dn

Light of wavelength 0.5 m takes two paths, both of length 1 m, one through air, the other through glass (n = 1.5). What is the difference in phase between the two waves in the end?A) 0 B) C) 2 D) 3 E) None of the above

1 2 1 1.5 0.5 6 2 2 1 1.0 0.5 4

2

1 m

Ex- (Serway 37-25) The intensity on the screen at a certain point in a double slit interference pattern is 64% of the maximum value. (a) What minimum phase difference (in radians) between the sources produces this result? (b) Express this phase difference in terms of a path difference for 486.1 nm light.

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Four Slit Interference• What if we have more than two slits?• Four slits, each spaced distance d apart

• Treat it as two double slits

P

r1,2

r3,4

1 2 3 4E E E E E 1,2 3,4E E

11,2 2

13,4 2

2sin cos sin

2sin cos sin

kr t kd

kr t kd

ave4sin cos sin

cos 2 sin

kr t d

d

2I E

• For four slits, every third band is bright

More Slits and Diffraction gratings• This process can be continued for more slits• For N slits, every N – 1’th band is bright• For large N, bands become very narrow

N = 8N = 16N = 32

• A device called a diffraction grating is just transparent with closely spaced regular lines on it• You already used it in lab

brightsin m d

0, 1, 2,m

• Diffraction gratings are another way to divide light into different colors• More accurate way of measuring wavelength than a prism• Commonly used by scientists

CT -3 - A diffraction grating is illuminated with yellow light at normal incidence. The pattern seen on a screen behind the grating consists of three yellow spots, one at zero degrees (straight through) and one each at ±45°. You now add red light of equal intensity, coming in the same direction as the yellow light. The new pattern consists of

A. red spots at 0° and ±45°. B. yellow spots at 0° and ±45°. C. orange spots at 0° and ±45°. D. an orange spot at 0°, yellow spots at ±45°, E. and red spots slightly farther out. F. an orange spot at 0°, yellow spots at ±45°,and red spots slightly closer in.

Ans D

Resolution of Diffraction Gratings• Note that the angle depends on the wavelength• With a finite number of slits, nearby wavelengths

may overlap

• The width of the peaks is about

• The difference between peaks is

• We can distinguish two peaks if:

1 1

2 2

sin

sin

m d

m d

sin dN N = 8

1.1

sin m d

• This quantity (mN) is called the resolving power

• Even if N is very large, effectively N is how many slits the light beam actually falls on

m d Nd

mN

Diffraction through a single slit• What if our slit is NOT small compared to a wavelength?• Treat it as a large number of closely spaced sources, by Huygen’s

principle

rave

P

r

a x

• Let the slit size be a, and rave the distance to the center• Let x be the distance of some point from the center• The distance r will be slightly different from here to P

ave sinr r x ~ sinE kr t

avesin sinkr kx t

12

12

avesin sina

aE kr kx t dx

/2

ave /2

1cos sin

sin

a

akr kx t

k

Diffraction through a single slit (2)

1ave 2

1ave 2

cos sin1

sin cos sin

kr ka tE

k kr ka t

12

ave12

sin sinsin

sin

kakr t

k

2

12

max 12

sin sin

sin

kaI I

ka

2

max

sin sin

sin

aI I

a

darksin m a

1, 2,m

• Very similar to equation for multi-slit diffraction, but . . .

• a is the size of the slit• This equation is for dark, not light• Note m= 0 is missing• Central peak twice as wide

Ex - (Serway 38-7) A screen is placed 50 cm from a single slit, which is illuminated with 690 light. If the distance between the first and third minima in the diffraction pattern is 3 mm, what is the width of the slit?

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If you used a little wider slit, the pattern wouldA) Get wider and dimmerB) Get wider and brighterC) Get narrower and dimmerD) Get narrower and brighter

Screens and Small Angles• Usually your slit size/separation is large compared to the wavelength• Multi-slit: Diffraction:

• When you project them onto a screen, you need to calculate locations of these bright/dark lines

• For small angles, sin and tan are the same

brightsin m d darksin m a

L

x

tanx

L sin

bright

Lx m

d

dark

Lx m

a

Diffraction and Interference Together

d

a

a

• Now go through two finite sized slits• Result is simply sum of each slit• Resulting amplitude looks like:

112

12 2

sinsin sin

sin sin

kr tkaE

k kr t

2

2max

sin sin sinsin

sin

a dI I

a

a = d/5 • Resulting pattern has two kinds of variations:

• Fast fluctuations from separation d• Slow fluctuations from slit size a

Diffraction Limit:• When light goes through a “small” slit, its

direction gets changed• Can’t determine direction better than this

a

a• If we put light through rectangular (square) hole,

we get diffraction in both dimensions• A circular hole of diameter D is a trifle smaller,

which causes a bit more spread in the outgoing wave• For homework, use this formula; for tests, the

approximate formula is good enough

minsin a

min a

D

min 1.22 D min D

CT -4 - For a given lens diameter, which light gives the best resolution in a microscope?

A. red B. yellow C. green D. blue E. All give the same resolution.

Ans D

Diffraction Limit (2)

• A degree is 1/360 of a circle, an arc-minute is 1/60 of a degree, an arc-second is 1/60 of an arc minute

• Telescopes require large apertures to see small angles

min 1.22 D

If the pupil of your eye in good light is 2 mm in diameter, what’s the smallest angle you can see using 500 nm visible light?

7

3

1.22 5 10 m

2 10 m

43.05 10 rad

1 arc-min

phase shift

Reflection and Phase Shift• When you reflect off of a mirror, the reflected wave must cancel the

incoming wave• It has a phase shift

• When you go from a low index of refraction medium to a high one, some of the wave is reflected• It also has a phase shift

phase shift• When you go from a high index

of refraction medium to a low one, some of the wave is reflected• This has a 0 phase shift 0 phase shift

Concept QuestionSuppose we are in a glass medium, and we have a wave that goes from glass to air to glass. If the layer of air is much smaller than one wavelength, then the two reflected waves will addA) Constructively B) Destructively C) Insufficient Info

• First transition: high to low• no phase shift

• Second transition: low to high• phase shift

• Compared to each other, the two waves are out of phase with each other

• They will have a tendency to cancel• Very little effect from layer if thinner than a wavelength

Interference from Thin Films• Suppose we go through a thin soap film• Index goes up then down

Front surface:• Phase shift of from reflection (low-high)Back surface:• Phase shift of 2t/ from traveling• Phase shift of 0 from reflection• Phase shift of 2t/ from travelingTotal phase shift between two reflected waves:• Weak reflection when odd times :• Strong reflection when even• Same results for index down then up• Opposite for:• Index up, then up• Index down, then down

t

4 t

weak42 1

tm

weak2t m

1strong 22t m

Applications of Thin Film Interference• What if the light isn’t monochromatic?

d

weak2t m 1strong 22t m

• Some wavelengths are enhanced, others are not• Soap bubbles• Oil on water

• Newton’s rings: convex lens on flat glass plate• Air gap changes thickness in circular pattern• Alternating light/dark regions

narrow air gap

Ex- (Serway 37-34) An oil film (n = 1.45) floating on water is illuminated by white light at normal incidence. The film is 280 nm thick. Find (a) The dominant observed color in the reflected light and (b) the dominant color in the transmitted light.

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Michelson Interferometer

Laser

Detector

Mirrors

• Interference easy to measure• Can see much smaller than one wavelength

• LIGO, state of the art, can see 10-15 m!

Hanford, Washington

Crystal Scattering of X-rays• Mysterious rays were discovered by Röntgen in

1895• Suspected to be short-wavelength EM waves• Order 1-0.1 nm wavelength

• Scattered very weakly off of atoms• Bragg, 1912, measured wavelength accurately

ddcos

dcos

• Scattering strong only if waves are in phase

• Must be integer multiple of wavelength

2 cosd m

Polarization• Recall that light waves have electric and magnetic fields

perpendicular to the direction of motion• But there are two independent ways of arranging this• Called polarization

• Our eyes can’t tell these two polarizations apart• But some instruments can measure or take advantage of

polarization• We describe polarization by telling which direction the

electric field points, e.g. vertically or horizontally• A polarizer polarizes light along its transmission axis.• Malus’s Law

E0

B0

E0

B0

20I = I cos (θ)

n1

n2

Methods of Producing Polarization (1)Direct production• Antennas produce waves that are automatically polarizedScattering• Light waves of all orientations hit small targets• Target has vibrating charges, like an antennaReflection and Brewster’s Angle:• When light hits a substance, some of it reflects and some refracts• Fraction of each depends on polarization• There’s a special angle – Brewster’s angle – where reflected is

completely polarized

– – – – – –

++++++

E0

2 1tan P n n

P

Methods of Producing Polarization (2)Birefringent Crystals• Index of refraction has to do with electric fields from the wave pushing

atoms around• In some crystals, it is easier to push them one way than another• Index of refraction depends on polarization• You can use such birefringent crystals to sort light based on

polarizationSelective absorption• Similarly, some materials absorb one polarization better than another

E0

E0

E0

E0E0

Sugar water

Some uses for PolarizationPolarized Sun Glasses• “Glare” comes mostly from light scattered in the atmosphere and

reflected from water• Mostly polarized• Sun glasses use selective absorption to eliminate it

Optical Activity• Some materials are capable of rotating the plane of polarization• These materials are not mirror-symmetric• Enantiomers, especially biological molecules

• Studying rotation of polarized light detects presence of these molecules• Someday use these to detect life on other planets?

E0 E0

CT -5 - When a ray of light is incident on two polarizers with their polarization axes perpendicular, no light is transmitted. If a third polarizer is inserted between these two with its polarization axis at 45° to that of the other two, does any light get through to point P?

A. yes B. no

Ans A

Ex - Serway 38-43. Plane-polarized light is incident on a single polarizing disk with the direction of E0 parallel to the direction of the transmission axis. Through what angle should the disk be rotated so that the intensity in the transmitted beam is reduced by a factor of (a) 3 (b) 5, (c) 10?

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