Chapter 352011).pdf¥ Intensity distribution, 6 ¥ Interference pattern consists of equally spaced...
Transcript of Chapter 352011).pdf¥ Intensity distribution, 6 ¥ Interference pattern consists of equally spaced...
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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