Chapter 9Chapter 9

Light as a WaveLight as a Wave

24.1 Interference24.1 Interference

Light waves interfere with each other Light waves interfere with each other much like mechanical waves domuch like mechanical waves do

All interference associated with light All interference associated with light waves arises when the waves arises when the electromagnetic fields that constitute electromagnetic fields that constitute the individual waves combinethe individual waves combine

LINEAR SUPERPOSITION!LINEAR SUPERPOSITION!

Conditions for InterferenceConditions for Interference

For sustained interference between For sustained interference between two sources of light to be observed, two sources of light to be observed, there are two conditions which there are two conditions which must be metmust be met The sources must be The sources must be coherentcoherent

They must maintain a constant phase with They must maintain a constant phase with respect to each otherrespect to each other

The waves must have The waves must have identical identical wavelengthswavelengths

Producing Coherent Producing Coherent SourcesSources

Light from a monochromatic source is Light from a monochromatic source is allowed to pass through a narrow slitallowed to pass through a narrow slit

The light from the single slit is allowed The light from the single slit is allowed to fall on a screen containing two to fall on a screen containing two narrow slitsnarrow slits

The first slit is needed to insure the light The first slit is needed to insure the light comes from a tiny region of the source comes from a tiny region of the source which is coherentwhich is coherent

Old methodOld method

Producing Coherent Producing Coherent Sources, cont.Sources, cont.

CurrentlyCurrently, it is much more , it is much more common to use a common to use a laserlaser as a as a coherent sourcecoherent source

The laser produces an intense, The laser produces an intense, coherent, monochromatic coherent, monochromatic parallel beam over a width of parallel beam over a width of several millimetersseveral millimeters

24.2 Young’s Double 24.2 Young’s Double Slit ExperimentSlit Experiment

Thomas Young first demonstrated Thomas Young first demonstrated interference in light waves from two interference in light waves from two sources in 1801sources in 1801

Light is incident on a screen with a Light is incident on a screen with a narrow slit, Snarrow slit, Soo

The light waves emerging from this The light waves emerging from this slit arrive at a second screen that slit arrive at a second screen that contains two narrow, parallel slits, contains two narrow, parallel slits, SS11 and S and S22

Young’s Double Slit Young’s Double Slit Experiment, DiagramExperiment, Diagram

The narrow slits, SThe narrow slits, S11 and Sand S2 2 act as act as sources of wavessources of waves

The waves The waves emerging from the emerging from the slits originate from slits originate from the same wave the same wave front and therefore front and therefore are always in phaseare always in phase

Resulting Interference Resulting Interference PatternPattern

The light from the two slits form a The light from the two slits form a visible pattern on a screenvisible pattern on a screen

The pattern consists of a series of bright The pattern consists of a series of bright and dark parallel bands called and dark parallel bands called fringesfringes

Constructive interferenceConstructive interference occurs occurs where a bright fringe occurswhere a bright fringe occurs

Destructive interferenceDestructive interference results in a results in a dark fringedark fringe

Interference PatternsInterference Patterns

Constructive Constructive interference interference occurs at the occurs at the center pointcenter point

The two waves The two waves travel the same travel the same distancedistance Therefore, they Therefore, they

arrive in phasearrive in phase

Interference Patterns, 2Interference Patterns, 2 The upper wave has The upper wave has

to travel farther to travel farther than the lower wavethan the lower wave

The upper wave The upper wave travels one travels one wavelength fartherwavelength farther Therefore, the waves Therefore, the waves

arrive in phasearrive in phase A bright fringe A bright fringe

occursoccurs

Interference Patterns, 3Interference Patterns, 3 The upper wave The upper wave

travels one-half of a travels one-half of a wavelength farther wavelength farther than the lower wavethan the lower wave

The trough of the The trough of the bottom wave bottom wave overlaps the crest of overlaps the crest of the upper wave (180the upper wave (180 phase shift)phase shift)

This is destructive This is destructive interferenceinterference A dark fringe occursA dark fringe occurs

Interference EquationsInterference Equations The path difference, The path difference, δδ, is found from the tan triangle, is found from the tan triangle δδ = = rr22 – – rr11 = |PS = |PS22 – PS – PS11| | dd sin sin θθ

Interference Equations, 2Interference Equations, 2

This assumes the paths are parallelThis assumes the paths are parallel Not exactly, but a very good approximation Not exactly, but a very good approximation ((LL>>>>dd))

Interference Equations, 3Interference Equations, 3

For a bright fringe, produced by For a bright fringe, produced by constructive interferenceconstructive interference, the , the path path difference must be either zero or some difference must be either zero or some integral multiple of of the wavelengthintegral multiple of of the wavelength

δδ = = dd sin sin θθbrightbright = = m λm λ mm = 0, ±1, ±2, … = 0, ±1, ±2, … mm is called the is called the order numberorder number

When When mm = 0, it is the zeroth order maximum = 0, it is the zeroth order maximum When When mm = ±1, it is called the first order = ±1, it is called the first order

maximummaximum

Interference Equations, 4Interference Equations, 4

When When destructive interferencedestructive interference occurs, a dark fringe is observedoccurs, a dark fringe is observed

This needs a path difference of an This needs a path difference of an odd half wavelengthodd half wavelength

δδ = = dd sin sin θθdarkdark = ( = (n -n - ½) ½) λλ nn = ±1, ±2, … = ±1, ±2, …

Interference Equations, 5Interference Equations, 5 The positions of the The positions of the

fringes can be fringes can be measured vertically measured vertically from the zeroth from the zeroth order maximumorder maximum

xx = = LL tan tan θθ LL sin sin θθ AssumptionsAssumptions

LL>>>>dd dd>>>>λλ tan tan θθ sin sin θθ θθ is small and therefore the is small and therefore the

approximation approximation tan tan θθ sin sin θθ can be usedcan be used

Interference Equations, Interference Equations, finalfinal

For For bright fringesbright fringes (use sin (use sinθθbrightbright==m m λλ//dd))

For For dark fringesdark fringes (use sin (use sinθθdarkdark==λλ ( (n -n - ½)/½)/dd))

21,0,m(bright) mmd

Lx

21,2

1n(dark)

nn

d

Lx

Interference Equations, Interference Equations, finalfinal For For distance between adjacent fringesdistance between adjacent fringes

n=1n=1

This can be rearranged to the formThis can be rearranged to the form

d

Lx

L

xd

Uses for Young’s Double Uses for Young’s Double Slit ExperimentSlit Experiment

Young’s Double Slit Experiment Young’s Double Slit Experiment provides a method for measuring provides a method for measuring wavelength of the lightwavelength of the light

This experiment gave the wave This experiment gave the wave model of light a great deal of model of light a great deal of credibilitycredibility It is inconceivable that particles of It is inconceivable that particles of

light could cancel each otherlight could cancel each other

24.4 Interference in 24.4 Interference in Thin FilmsThin Films

Interference effects are Interference effects are commonly observed in thin filmscommonly observed in thin films Examples are soap bubbles and oil Examples are soap bubbles and oil

on wateron water Assume the light rays are Assume the light rays are

traveling in air nearly normal to traveling in air nearly normal to the two surfaces of the filmthe two surfaces of the film

Interference in Thin Films, Interference in Thin Films, 22

Rules to rememberRules to remember An electromagnetic wave traveling from a An electromagnetic wave traveling from a

medium of index of refraction medium of index of refraction nn11 toward a toward a medium of index of refraction medium of index of refraction nn22 undergoes a undergoes a 180° phase change180° phase change on reflection when on reflection when nn22 > > nn11

There is There is no phase changeno phase change in the reflected in the reflected wave if wave if nn22 < < nn11

The wavelength of light The wavelength of light λλnn in a medium with in a medium with index of refraction index of refraction nn is is λλnn = = λλ//nn, where , where λλ is the is the wavelength of light in vacuumwavelength of light in vacuum

Interference in Thin Films, Interference in Thin Films, 33

Ray 1 undergoes a Ray 1 undergoes a phase change of phase change of 180° with respect 180° with respect to the incident rayto the incident ray

Ray 2, which is Ray 2, which is reflected from the reflected from the lower surface, lower surface, undergoes no undergoes no phase change with phase change with respect to the respect to the incident waveincident wave

180 phase change

iiii

Interference in Thin Films, Interference in Thin Films, 44

Ray 2 also travels an additional Ray 2 also travels an additional distance of 2distance of 2tt before the waves before the waves recombinerecombine

For constructive interferenceFor constructive interference 22ntnt = ( = (mm + ½ ) + ½ ) λλ mm = 0, 1, 2 … = 0, 1, 2 …

This takes into account both the difference in This takes into account both the difference in optical path length for the two rays and the optical path length for the two rays and the 180° phase change180° phase change

For destruction interferenceFor destruction interference 2 2 n tn t = = m m λλ mm = 0, 1, 2 … = 0, 1, 2 …

Interference in Thin Films, 5Interference in Thin Films, 5

Two factors influence interferenceTwo factors influence interference Possible phase reversals on reflectionPossible phase reversals on reflection Differences in travel distanceDifferences in travel distance

The conditions are valid if the medium The conditions are valid if the medium above the top surface is the same as the above the top surface is the same as the medium below the bottom surfacemedium below the bottom surface

If the thin film is between two different If the thin film is between two different media, one of lower index than the film and media, one of lower index than the film and one of higher index, the conditions for one of higher index, the conditions for constructive and destructive interference constructive and destructive interference are are reversedreversed

Interference in Thin Films, Interference in Thin Films, finalfinal To form a nonreflecting coating a thin-To form a nonreflecting coating a thin-

film on glass with a minimum film film on glass with a minimum film thickness of thickness of λλ /(4/(4nn11) is required.) is required.

Destructive Destructive interference interference when: 2when: 2tt==λλ/(2/(2nn11))

Minimum Minimum thickness for thickness for nonreflecting nonreflecting surfaces: surfaces: tt==λλ/(4/(4nn11))

Newton’s RingsNewton’s Rings Another method for viewing interference is to Another method for viewing interference is to

place a planoconvex lens on top of a flat glass place a planoconvex lens on top of a flat glass surfacesurface

The air film between the glass surfaces varies The air film between the glass surfaces varies in thickness from zero at the point of contact to in thickness from zero at the point of contact to some thickness some thickness tt

A pattern of light and dark rings is observedA pattern of light and dark rings is observed This rings are called This rings are called Newton’s RingsNewton’s Rings The particle model of light could not explain the The particle model of light could not explain the

origin of the ringsorigin of the rings Newton’s Rings can be used to test optical Newton’s Rings can be used to test optical

lenseslenses

Newton’s rings, cont. Newton’s rings, cont.

Ray 1 undergoes Ray 1 undergoes a phase change of a phase change of 180180 on reflection, on reflection, whereas ray 2 whereas ray 2 undergoes no undergoes no phase changephase change

Problem Solving Strategy Problem Solving Strategy with Thin Films, 1with Thin Films, 1

Identify the thin film causing the Identify the thin film causing the interferenceinterference

The type of interference – The type of interference – constructive or destructive – that constructive or destructive – that occurs is determined by the phase occurs is determined by the phase relationship between the upper relationship between the upper and lower surfacesand lower surfaces

Problem Solving with Thin Problem Solving with Thin Films, 2Films, 2

Phase differences have two causesPhase differences have two causes differences in the distances traveleddifferences in the distances traveled phase changes occurring on reflectionphase changes occurring on reflection BothBoth must be considered when determining must be considered when determining

constructive or destructive interferenceconstructive or destructive interference The interference is The interference is constructiveconstructive if the if the path path

difference is an integral multiple of difference is an integral multiple of λλ and and destructive if the path difference is an odd destructive if the path difference is an odd half multiple of half multiple of λλ The conditions are reversed if one of the waves The conditions are reversed if one of the waves

undergoes a phase change on reflectionundergoes a phase change on reflection

24.5 CD’s and DVD’s24.5 CD’s and DVD’s

Data is stored digitallyData is stored digitally A series of ones and zeros read by laser light A series of ones and zeros read by laser light

reflected from the diskreflected from the disk Strong reflectionsStrong reflections correspond to correspond to

constructive interferenceconstructive interference These reflections are chosen to These reflections are chosen to represent represent

zeroszeros Weak reflectionsWeak reflections correspond to correspond to

destructive interferencedestructive interference These reflections are chosen to These reflections are chosen to represent onesrepresent ones

CD’s and Thin Film CD’s and Thin Film InterferenceInterference

A CD has multiple tracks A CD has multiple tracks The tracks consist of a sequence of The tracks consist of a sequence of

pits of varying length formed in a pits of varying length formed in a reflecting information layerreflecting information layer

The pits appear as bumps to the The pits appear as bumps to the laser beamlaser beam The laser beam shines on the metallic The laser beam shines on the metallic

layer through a clear plastic coatinglayer through a clear plastic coating

Reading a CDReading a CD As the disk rotates, the As the disk rotates, the

laser reflects off the laser reflects off the sequence of bumps and sequence of bumps and lower areas into a lower areas into a photodectorphotodector The photodector converts The photodector converts

the fluctuating reflected the fluctuating reflected light intensity into an light intensity into an electrical string of zeros electrical string of zeros and onesand ones

The pit depth is made The pit depth is made equal to one-quarter of equal to one-quarter of the wavelength of the the wavelength of the lightlight

Reading a CD, contReading a CD, cont

When the laser beam hits a rising or When the laser beam hits a rising or falling bump edge, part of the beam falling bump edge, part of the beam reflects from the top of the bump and reflects from the top of the bump and part from the lower adjacent areapart from the lower adjacent area This ensures destructive interference and This ensures destructive interference and

very low intensity when the reflected beams very low intensity when the reflected beams combine at the detectorcombine at the detector

The bump edges are read as onesThe bump edges are read as ones The flat bump tops and intervening flat The flat bump tops and intervening flat

plains are read as zerosplains are read as zeros

DVD’sDVD’s

DVD’s use shorter wavelength DVD’s use shorter wavelength laserslasers The track separation, pit depth and The track separation, pit depth and

minimum pit length are all smallerminimum pit length are all smaller Therefore, the DVD can store about 30 Therefore, the DVD can store about 30

times more information than a CDtimes more information than a CD Therefore the industry is very much Therefore the industry is very much

interested in blue (semiconductor) interested in blue (semiconductor) lasers lasers

Geometrical Optics Geometrical Optics Against Wave OpticsAgainst Wave Optics

Geometrical opticsGeometrical optics   Particles are “running” along a line forming the ray Particles are “running” along a line forming the ray  However, geometrical optics cannot explain However, geometrical optics cannot explain

deflection (shadows, twilight) deflection (shadows, twilight)  WAVE OPTICS CAN!WAVE OPTICS CAN!

Deflection Diffraction

Linear Superposition

Interference

Example:Example:

Red light (Red light (=664 nm) =664 nm) is used in Young’s is used in Young’s experiment according experiment according to the drawing. Find to the drawing. Find the distance the distance yy on the on the screen between the screen between the central bright and the central bright and the third-order bright third-order bright fringe. fringe.

Solution:Solution:

0.95m101.20

m)103(664sinsin

4

91

md

y=Ltan=(2.75 m)(tan0.95)=0.046 m

24.3 Lloyd’s Mirror24.3 Lloyd’s Mirror An arrangement for An arrangement for

producing an producing an interference pattern interference pattern with a single light with a single light sourcesource

Wave reach point P Wave reach point P either by a direct either by a direct path or by reflectionpath or by reflection

The reflected ray can The reflected ray can be treated as a ray be treated as a ray from the source S’ from the source S’ behind the mirrorbehind the mirror

Interference Pattern from Interference Pattern from Lloyd’s MirrorLloyd’s Mirror

An interference pattern is formed An interference pattern is formed The positions of the dark and The positions of the dark and

bright fringes are bright fringes are reversedreversed relative to pattern of two real relative to pattern of two real sourcessources

This is because there is a 180° This is because there is a 180° phase change produced by the phase change produced by the reflectionreflection

Phase Changes Due To Phase Changes Due To ReflectionReflection

An electromagnetic An electromagnetic wave undergoes a wave undergoes a phase change of phase change of 180° upon reflection 180° upon reflection from a medium of from a medium of higher index of higher index of refractionrefraction than the than the one in which it was one in which it was travelingtraveling Analogous to a Analogous to a

reflected pulse on a reflected pulse on a stringstring

Phase Changes Due To Phase Changes Due To Reflection, cont.Reflection, cont.

There is no phase There is no phase change when the change when the wave is reflected wave is reflected from a boundary from a boundary leading to a leading to a medium of lower medium of lower index of refractionindex of refraction Analogous to a Analogous to a

pulse in a string pulse in a string reflecting from a reflecting from a free supportfree support