Interference and DiffractionHuygen’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

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

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

Interference Through Two Slits (2)•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

Interference Through Two Slits (3)

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

Interference Through Four Slits•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 1 1

1,2 3,42 22sin cos sin 2sin cos sinkr t kd kr t kd

ave4sin cos sin cos 2 sinkr 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

brightsinm

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

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:

11

22

sin

sin

m

dm

d

sindN

N = 8 1.1

sin

m

d

•This quantity (mN) is called theresolving power•Even if N is very large, effectively N is how many slits the light beam actually falls on

m N

d d

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

1ave 2

1ave 2

cos sin1

sin cos sin

kr ka tE

k kr ka t

12

ave12

sin sinsin

sin

kakr t

k

Diffraction Through a Single Slit (2)

212

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

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

brightsinm

d

darksinm

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:

12

1 212

sin sinsin sin

sin

kaE kr t kr t

k

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

The 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

minsina

min a

D

min

1.22

D

min D

Sample problem

•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

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

Spherical Mirrors: Finding the Image•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

Reflection and Phase Shift

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 shift0 phase shift

Interference From Thin Films•Suppose we go through a thin soap film

•Index goes up then downFront 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 Films 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

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

E0

B0

E0

B0

Methods of Producing Polarization

n1

n2

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

1

tan 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

Some Uses for Polarization

Sugar water

Polarized 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