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Chapter 35

Interference of Light Waves

Prof. Raymond Lee,revised 4-20-2012

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• Wave optics

• Wave optics involves phenomena difficult or

impossible to explain via geometric (ray) optics

• Phenomena include:

• interference

• diffraction

• polarization

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• Interference

• In constructive interference,

resultant wave’s amplitude >

either individual wave

• In destructive interference,

resultant wave’s amplitude <

either individual wave

• All light-wave interference

comes from combining EM

fields of individual waves

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• Conditions for interference

• To see light-wave interference, must have:1) Coherent sources (i.e., fixed phase relationship)

2) Monochromatic (i.e., single-!) light

• Monochromatic light illuminates 2 narrow slits

• Light emerging from slits is coherent since 1source produces original light beam(commonly used method)

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• Diffraction

• From Huygens’ principle,

waves spread out from slits

• Call this divergence of light

from its initial path diffraction

(compare Fig. 35-8, p. 964)

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• Young’s double-slit experiment

• Thomas Young first demonstrateslight-wave interference from twosources in 1801

• Narrow slits S1 & S2 act as wavesources

• Waves emerging from slitsoriginate from same wave front &so are always in phase

(compare Fig.35-8, p. 964)

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• Resulting interference pattern

• Light from 2 slits forms patternvisible on screen

• Pattern is a series of bright &dark || bands called fringes

• Constructive interference "bright fringes

• Destructive interference "dark fringes

(compare Fig. 35-9, p. 965)

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• Interference patterns

• Constructive interference

occurs at point P, where the

2 waves travel same

distance & so arrive in phase

• As a result, constructive

interference occurs here,

yielding a bright fringe

(compare Fig.35-10, p. 965)

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• Interference patterns, 2

• Upper wave must travelfarther than lower wave toreach point Q

• Since upper wave travels1 ! farther, waves arrive inphase & produce a brightfringe at Q

(compare Fig.35-10, p. 965)

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• Interference patterns, 3

• To reach point R, upper

wave must travel !/2farther than lower wave

• Trough of bottom waveoverlaps crest of upperwave, resulting indestructive interference &a dark fringe (compare Fig.

35-10, p. 965)

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• Geometry of Young’s double slit

• Path difference # (see tan

triangle) is ! = r2–r1 = d sin($)

• This assumes paths

are ||, good approx-

imation if L » d

• Also, assume that

slit width a « ! (see

text, p. 1003)

(Eq. 35-12,p. 966)

(SJ 2008 Fig. 37.5, p. 1054)

a

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• Interference equations

• For constructive interference’s bright fringes,# must be zero or some integer multiple of !:

! = d sin($bright) = m" (Eq. 35-14, p. 966)

• order number m = 0, ±1, ±2, … (m = 0 is zeroth-

order maximum; m = ±1 is first-order maximum)

• For destructive interference’s dark fringes, #must be an odd half-!:

! = d sin($dark) = (m + 1/2)" (Eq. 35-16, p. 966)

• m = 0, ±1, ±2, …

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• Interference equations, 2

• Measure fringe positions from zeroth-ordermaximum

• If L » d, d » ", & $ is small ($ < 4°) so that tan($) ~

sin($), then y = L tan($) ! L sin($) & sin($) ! y/L(SJ 2008 Eq. 37.4, p. 1054)

• For bright fringes, ybright = "Lm/d for m = 0, ±1, ±2,… (SJ 2008 Eq. 37.7, p. 1055)

• For dark fringes, ydark = "L(m+1/2)/d for m = 0, ±1,±2, … (SJ 2004 Eq. 37.6, p. 1181)

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• Double-slit implications

• Experiment " way to measure !light & gave

wave model credibility

• In particular, light particles couldn’t cancel

each other in a way that explains dark fringes

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• Intensity distribution

• Interference pattern’s bright fringes lack sharp

edges

• So far, equations only locate centers of bright

& dark fringes

• Now calculate distribution of light intensity

throughout double-slit interference pattern

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• Intensity distribution, 2

• Assumptions:

• 2 slits represent coherent sources of sinusoidal

waves

• waves from slits have same angular frequency %

• waves have fixed initial phase difference &

• |E| at any point on screen is superposition of

the 2 waves

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• Intensity distribution, 3

• For constant peak amplitude E0,

wave magnitudes at point P are:

• E1 = E0*sin(%t)

• E2 = E0*sin(%t + &)

(SJ 2008 Fig. 37.6, p. 1057)

(Eqs. 35-20 & 21; p. 969)

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• Intensity distribution, 4

• Phase difference between waves at P dependson path difference ! = r2–r1 = d sin($)

• Path difference = 1" corresponds to phase

difference & = 2# radians

• Path difference of ! is same fraction of 1" as

phase difference & is fraction of 2#

• Thus & = 2#!/" = 2# d sin($)/" (Eq. 35-23, p. 970)

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• Intensity distribution, 5

• Magnitude of resultant E-field (= ER) comes from

superposition principle, ER = E1+ E2 =E0{sin(%t) + sin (%t + &)} (Eq. 35-21, p. 969) or,

ER = 2E0 cos(&/2) sin(%t + &/2) (SJ 2008 Eq. 37.11, p. 1057)

• ER has same frequency as light illuminating slits,

but |ER| differs from |E0| by factor = 2cos(&/2)

• Wave intensity I ' |E|2 at any point, so

I = Imaxcos2(# d sin($)/") ! Imaxcos2(# d y/("L))

{for $ < 4°} (SJ 2008 Eqs. 37.13-14, p. 1058)

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• Intensity distribution, 6

• Interference pattern consists of equally

spaced fringes of equal Imax

• This result is valid only if L»d & $ is small

(compare Fig.35-12, p. 970)

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• Lloyd’s mirror

• " interference pattern

using a single light source

• Waves reach point P either

directly or by reflection

• Treat reflected ray as if its

S' is behind mirror

(SJ 2008 Fig.37.9, p. 1059)

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• Lloyd’s mirror, 2

• Think of this arrangement as 2-slit source withdistance between points S & S' = length d

• An interference pattern is formed, but positionsof its dark & bright fringes are reversed relativeto pattern from 2 real sources

• Occurs because reflection from glass " 180°phase change

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• Phase changes on reflection

• EM wave undergoes"& = 180° on reflection

from a medium with

nincident < nreflected

• Analogous to a pulse on

string reflected by a rigid

(i.e., more massive) support (compare Fig.35-16b, p. 974)

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• Phase changes on reflection, 2

• But "& = 0° on reflectionfrom a medium with nincident

> nreflected

• Analogous to a pulse onstring reflected by a free(i.e., less massive) support

(compare Fig.35-16a, p. 974)

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• Thin-film interference

• Commonly see interference in thin films such

as soap bubbles & oil on water

• Colors occur if such films are lit by white light

& result from interference of waves reflected

from film’s 2 surfaces

• EM wave traveling from nincident < nreflected

undergoes "& = 180° on reflection

• "n in medium with index of refraction n is

"n = "/n, where " = wavelength in vacuum(Eq. 35-35, p. 975)

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• Thin-film interference, 2

• Assume light rays travel inair ! ( film’s 2 surfaces

• Ray 1 undergoes "& = 180°w.r.t incident wave

• Ray 2, which is reflected bylower surface, undergoes "&= 0° w.r.t. incident wave

(compare Fig. 35-17, p. 975)

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• Thin-film interference, 3

• Ray 2 also travels additional distance = 2t before

waves recombine

• Constructive interference: 2nt = (m + 1/2)"

(for m = 0, 1, 2 …) (Eq. 35-36, p. 975)

• Takes into account both difference in optical

pathlength = nt for 2 rays & the 180° phase change

• Destructive interference: 2nt = m" (m = 0, 1, 2 …)(Eq. 35-37, p. 976)

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• Thin-film interference, 4

• 2 factors influence interference

• Possible phase reversals on reflection

• Differences in n & in travel distance

• Constructive/destructive interference equations

hold if same medium surrounds film

• If 2 different media surround film, equations are

still valid if for both media nsurround < nfilm

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• Thin-film interference, 5

• If thin film is between 2 differentmedia, 1 with n < nfilm & 1 withn > nfilm, then reverse equationsfor constructive & destructiveinterference

• If different materials surroundfilm, may have "& = 180° at 0,1, or 2 surface(s)

• Check both pathlength nt &phase change "& whencalculating thin-film interference (SJ 2008, p. 1062)

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• Newton’s rings

• View interference by placing plano-convex lenson top of glass flat

• Air film between glass surfaces varies inthickness from 0 at contact point to somethickness t

• Pattern of light & dark rings results calledNewton’s rings, an ironic name choice since hisparticle model of light couldn’t explain them

• Can use Newton’s rings to test lens symmetry

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• Newton’s rings, 2

(SJ 2008 Fig. 37.12, p. 1061)

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• Solving thin-film problems

• Identify thin film causing interference

• Constructive or destructive interference depends on& relationship between 2 surfaces

• "& has 2 causes: (1) differences in rays’ opticalpaths & (2) phase changes on reflection

• Consider both causes when determiningconstructive or destructive interference

• Interference is destructive if optical path difference2nt is integer multiple of " & constructive if 2nt isodd integer multiple of "/2

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• Michelson interferometer

• Interferometer was invented by USNA’s ownAlbert A. Michelson (class of 1873)

• It splits light beam into 2 parts & thenrecombines them to form an interference pattern

• Can use device to measure ! or other lengthswith great precision

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• Michelson interferometer, 2

• Split incident light ray into 2 rayswith beam-splitter mirror M0 rotated45° w.r.t. incident direction

• Beam splitters typically transmit50% of incident light & reflect 50%

• Reflected ray " mirror M1, &transmitted ray " mirror M2

• Rays travel separate paths L1 & L2

• After reflecting from M1 & M2, raysrecombine at M0, producing aninterference pattern

(SJ 2008Fig. 37.14,p. 1064)

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• Michelson interferometer, 3

• Pathlength difference for 2 rays determineswhether they " constructive or destructiveinterference

• If you move M1, fringe pattern collapses orexpands, depending on direction & distancethat M1 is moved

• Fringe pattern shifts by 1/2 fringe each time

M1 is moved a distance = "/4

• Then measure ! light by counting # of fringeshifts for given M1 displacement

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• Michelson interferometer, 4

• Michelson used interferometer to disprove idea

that earth moves through a luminiferous ether

supposedly required for light’s propagation

• Modern applications include

• Fourier Transform Infrared Spectroscopy (FTIR)

• Laser Interferometer Gravitational-Wave

Observatory (LIGO)

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• Fourier Transform Infrared Spectroscopy

• Used to create a high-resolution spectrum veryquickly, a big advantage

• Results in a complex dataset giving light intensityas a function of mirror position, an interferogram

• A Fourier analysis of interferogram extracts all itsspectral intensity (i.e., I(!)) information

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• Laser Interferometer Gravitational-

Wave Observatory (LIGO)

• General relativity predicts gravitational waves

• In Einstein’s theory, gravity is equivalent to adistortion of space, through which these wavescan propagate

• LIGO detects gravitational waves that pass nearearth by using laser beams with effectivepathlengths of several km

• At each end of interferometer, a mirror is mountedon a massive pendulum

• When a gravitational wave passes, pendulummoves, & laser-beam interference patternchanges for the 2 arms

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• LIGO in Richland, WA