April 03 LASERS 51

Interference of light• Interference of light waves similar to interference of

water waves– two different waves arrive at the observation point– the total influence is the sum of the two wave amplitudes at

each time and at each point in space

• High frequency of light has important consequences– Cannot follow the fast cycling of the field– Detectors measure the effect of many oscillations– Only interference that persists over many periods is

observable• Only two-beam interference is discussed in this module

– Multiple beam interference will be treated in next module

April 03 LASERS 51

What does an optical power meter measure?

Amplitude

=A Field wiggles electron causing it to escape from atom. Becomes electrical current

• Electron in atom

• Electron is released due to wiggling– Current from detector doesn’t go up and down with field– Responds to the amplitude not phase

• Fast detectors can measure changes in 100 psec (10-10sec)– This includes 60,000 cycles of the field– Unlike radio waves where individual cycles can be measured

• The irradiance (power per unit area) is proportional to A2, the square of the amplitude

April 03 LASERS 51

Power measurement with interfering waves

• Detector measures square of amplitude of resultant field (field obtained by adding waves at detector)– Two 1000 Watt beams impinging on detector will give a zero

power reading if out of phase and 4000 Watts if in phase!!!– Where does the power go? – somewhere else

• If the relative phase of the fields is not constant (e.g. incoherent light) then interference effects go away– This makes observation of interference difficult

Two point sources create interference pattern

• Along red lines– Crests of two sources always

coincide– Valleys of two sources always

coincide– Net disturbance has twice the

amplitude

•Along green lines–Crest of one wave always occurs with valley of the other

–Water is undisturbed along these lines

April 03 LASERS 51

Interference of two spherical waves• “Frozen” in time

– where two crests coincide, amplitude is double that of a single source

– where two troughs coincide, amplitude is negative and twice as deep as a single source

Along indicated arrows waves have twice amplitude of a single source

Constructive interference

source 1

source 2

crests of wave

crests intersect

troughsintersect

lines ofconstructiveinterference

April 03 LASERS 51

Interference of two spherical waves (cont)• “Frozen” in time

– If a crest of one wave coincides with a trough of another wave, there is no net disturbance

source 1

source 2

lines ofdestructiveinterference

crest wave 1intersectstrough wave 2

crest wave 2intersects troughwave 1

Along indicated arrows waves have zero amplitude!

Destructive interference

April 03 LASERS 51

Optical path difference (OPD)

observation point

d1 d2

•

source 2source 1

OPD = d1-d2

• Optical path length determines how long it takes light to travelfrom the source to the observation point– Phase at an instant of time depends on optical path length

• Optical path difference determines the phase difference at observation point between light from two sources

• Understanding interference given the OPD is easy• Finding the OPD in many cases is very complicated geometry

April 03 LASERS 51

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• Optical path from either source to observing point is the same

• Resulting wave 2x amplitude, same phase as either component

• Intensity is four times that of either source

source 1 source 2

Observationpoint, P

d1d2

d1=d2

•

Wave at P due to source 1

Wave at P due to source 2

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April 03 LASERS 51

OPD = λ/2, 180° out of phase

• Optical path difference (OPD)=λ/2– Phase difference=180°

• Resultant amplitude zero

d1=d2+λ/2

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ObservationPoint, P

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Conservation of energy works! Energy “missing” due to destructive interference is redistributed to regions of constructive interference

Wave at P due to source 1

Wave at P due to source 2

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April 03 LASERS 51

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Superposition of waves 90° out of phase

• Path length from source 1 and 2 not the same– Optical path difference

(OPD)=λ/4– Phase difference=90°

• Result: same frequency, amplitude 1.4x, phase different from either wave

• Intensity 2 times single source intensity

source 1 source 2

observation point

d1 d2d1=d2+λ/4

•

April 03 LASERS 51

More about superposition• To find interference pattern

you need to know the OPD– Often difficult to calculate.

Calculation not needed tounderstand interference.

– Changing OPD by λ, 2λ, 3λ, etc.doesn’t change interference

– Constructive interference: OPD=0, λ, 2λ, 3λ, etc. (integral number of wavelengths)

– Destructive interference: OPD=λ/2, 3λ/2, 5λ/2, etc. (half-integral number of wavelengths)

• Interference between waves with unequal amplitudes– OPD=half-integral number of waves

• incomplete cancellation• Dark fringes are not completely dark

– When OPD=integral no. of waves• intensity less than four times that of a single wave

laser beam

Diverging lens

Glass plate

Interference observed on card

April 03 LASERS 51

Coherence

• No interference observed– On a femtosecond timescale

interference still occurs but it is not observable

• If phase jumps in each source are the sameinterference returns

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Wave due to source 1 has random phase jumps

Wave due to source 1 has differentrandom phase jumps

Superposition has some regions of constructive other regions of destructive interference

April 03 LASERS 51

Young’s double-slit experiment• First demonstration of wave nature of light

– Thomas Young, 1802– not accepted until Fresnel’s work 12 years later

• Division of wavefront– single wavefront from point source strikes both

slits simultaneously– source is nearly monochromatic– point source must be nearly same distance

from each pinhole

opaque screen with two slits

point source

April 03 LASERS 51

Young’s experiment—fringe spacing

• Assume L is very large compared to D or d

• By (approximately) similar triangles

slits screen

d D

Lδ=OPD

dLD δ=

Essentially two sources, Huygen’s principle

Photo of pattern on screen

OPD calculation easy if:

Observation point

• Constructive interference (bright fringes) if δ is 0 or an integral number of wavelengths

L,2,1,0 where == ndLnD λ

Spacing between bright fringes is λL/d

April 03 LASERS 51

Two slit interference pattern• Interference of two slits

gives a sinusoidal variation in intensity– This pattern is charcteristic

of all two beam interference

• This pattern can be modulated by an overall intensity pattern– Due to diffraction as in

next module, or uniformity of illumination in other cases

Position on screen

Inte

nsity

If intensity at low points don’t go to zero (unequal illumination of slits or partial coherence) fringes are indistinct or fuzzed out (low visibility)

April 03 LASERS 51

Interference using nonmonochromatic sources• Each wavelength produces

interference fringes – spacings are different

• At center n=0, all wavelengths have a bright fringe

• For larger n, fringes become colored, red on outside blue on inside

• Finally for larger n fringes become completely washed out

screen

slits

Fringes in red light

Fringes in blue light

White-light interference can only be observed for OPD << λ

April 03 LASERS 51

Young’s fringes - large source• For a large source each

point on the source produces a set of fringes

• Fringes are shifted relative to each other

• Waves from each point interfere add “incoherently”– Interfere with themselves

only• A light field incident on

the set of slits is said to have spatial coherence if an interference pattern is produced on the screen

screen

slits Fringes from point 1

point 2

point 1

Fringes from point 2

April 03 LASERS 51

Why do fringes from two different points wash out?

• Atoms excited by some energy source

• Excited atoms decay at random times– average lifetime (~10-8 sec)– each emission results in very

short burst (wavepacket)– no phase relationship

between different packets

• Emission between different points, or even same point at different times are incoherent

Atoms in gas

Wave packets emitted by each atom at random times in random directions

April 03 LASERS 51

Interference between incoherent sources• Shown is superposition of

a randomly restarted wave with a perfect sine wave

• sudden jumps represent termination of light from one atom and start of light from another

• Superposition sometimes in phase sometimes out of phase– average over many cycles

shows no enhancement by interference

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April 03 LASERS 51

Coherence length of a laser• Because of feedback,

laser light has longer “memory” of its phase

• Nevertheless, the phase drifts and eventually goes out of phase

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April 03 LASERS 51

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Superposition of waves with different frequencies

• May be from two different lasers or two different atoms with different Doppler shifts

• Resultant varies periodically in amplitude as the two waves go in and out of phase

• Hetrodyne detection of fm signal

“A photon can only interfere with itself.”-P. Dirac

April 03 LASERS 51

Fringe visibility• Perfectly coherent sources of

equal intensity give maximaof intensity 4x intensity of onesource and zero intensity minima

• Coherent source of unequalintensities give fringes withmaxima of I1+I2+2 √(I1I2) andminima of I1+I2-2 √(I1I2)

• Partially coherent sources givefringes with lower maxima and higher minima depending on the degree of coherence

• Completely incoherent sources give no fringes (intensity everywhere = 2x intensity of one source)

Visibility ≡Imax − Imin

Imax + Imin

High visibility

Low visibility

April 03 LASERS 51

Lloyd’s mirror

Interference fringesin overlap region

point source

mirrorimage ofpoint source

• Light reflected from mirror interferes with light directly from point source

• Considered as interference between the point source and its image, this is almost identical to Young’s fringes– one significant difference is that center fringe is dark

due to phase change on reflection

April 03 LASERS 51

Division of amplitudeMichelson interferometer

• Source may also be a non-laser point source or even an extended source– compensator plate required in

this case

• Tilting one mirror produces straight-line fringes

• Often used for optical testing

observationscreen

mirror 1

mirror 2

beamsplitterexpandedlaser beam

April 03 LASERS 51

input wave

wave reflectedfrom second surface

wave reflectedfrom first surface

d

x

α

Interference from wedged plates• Waves from two surfaces

interfere at observation point– OPD determined only by path

between reflecting surfaces –other parts of path are common

• Interference from other surfaces occurs with laser sources– With incoherent sources OPD

for other surfaces is greater than coherence length

– Small gap, nearly monochromatic source needed to observe interference with incoherent sources

– Spatial coherence not needed

2/2 λ+= dOPD

phase change on reflection

Bright fringes when OPD=integral number of waves

angles smallfor xd α=2/2 λα += xOPD

etcx ,49,

45,

4 αλ

αλ

αλ

=

April 03 LASERS 51

Newton’s rings

• Common optical testing technique– optical flat may be replaced by a curved test plate

• Optical path difference varies with shape of tested part– radius of part, as well as size of defects can be measured– dark fringe in center due to phase change on reflection

monochromatic,extended light source

High qualityoptical flat

part under test

eye

OPD, opticalpath difference

April 03 LASERS 51

Antireflection (AR) coatingsAir, index na

Coating, index nc

Reflection coefficient of uncoated glass

Glass, index ng 2

2

)()(

ag

ag

nnnn

R+

−=

• Reflection from air/coating interface interferes with reflection from coating/glass interface– If nc=√(ng*na) and the coating is λ/(4nc) thick the two reflections

cancel completely for one wavelength– For ng=1.5 and na=1.0 this requires nc=1.225 (unknownium)– a 1/4-wave coating of a relatively low index material MgF(n=1.38)

can be applied to obtain a reflectance of 1.4% (at one wavelength)– to get lower reflectance or more than one wavelength multiple

layers must be used

April 03 LASERS 51

Shear plate interferometer• For a plane input beam,

fringes are straight lines perpendicular to tilt direction– Spacing of fringes depends

on tilt angle and wavelength

• Working out the OPD for any other given wavefront is complicated

• Shear can be measured using shadow of object in the input beam

Back surfacereflection

Front surfacereflection

Tilt

shear

Top view

Side view

input beam

April 03 LASERS 51

Shear-plate interferometer fringe patterns

• Horizontal lines indicates a plane wave– Infinite radius of curvature

• Tilted straight lines indicates a spherical wavefront– Tilts one direction for converging wave,

opposite direction for diverging wave

• Deviation from straight lines means wavefront is not plane or spherical– i.e. aberrations