Lecture13WaveOptics‐1Chapter22

PHYSICS270DennisPapadopoulos

March15,2011

Chapter22.WaveOpticsLight is an electromagnetic wave(!!!).The interference of light waves produces the colors reflected from a CD, the iridescence of bird feathers, and the technology underlying supermarket checkout scanners and optical computers. Chapter Goal: To understand and apply the wave model of light.

Topics:

•  LightandOptics•  TheInterferenceofLight•  TheDiffractionGrating•  Single‐SlitDiffraction•  Circular‐ApertureDiffraction•  Interferometers

WaveOptics

Whatisthenatureof

ModelsofLight• The wave model: under many circumstances, light

exhibits the same behavior as sound or water waves. The study of light as a wave is called wave optics.

• The ray model: The properties of prisms, mirrors, and lenses are best understood in terms of light rays. The ray model is the basis of ray optics.

• The photon model: In the quantum world, light behaves like neither a wave nor a particle. Instead, light consists of photons that have both wave-like and particle-like properties. This is the quantum theory of light.

The invention of radio? Hertzprovesthatlightisreallyanelectromagneticwave.

Wavescouldbegeneratedinonecircuit,andelectricpulseswiththesamefrequencycouldbeinducedinanantennasomedistanceaway.

Theseelectromagneticwavescouldbereflected,andrefracted,focused,polarized,andmadetointerfere—justlikelight!

“Okay…what you have to realize is…to first order…everything is a simple harmonic oscillator. Once you’ve got that, it’s all downhill from there.”

Anoscillationisatime‐varyingdisturbance.

oscillation

A

Phasor Representation

φ=ωt x

y

wave

Awaveisatime‐varyingdisturbancethatalsopropagatesinspace.

wavescaninterfere(addorcancel).

Consequencesarethat:

1+1notalways2,refraction(bendcorners),diffraction(spreadoutofhole),reflection..

wavesarenotlocalizedandcanbepolarized

particles1+1makes2,localized

TheMathematicsofInterference

€

AR2 = A1

2 + A22 + 2A1A2 cos(φ2 −φ1)

x

y

€

R = Acos(kx −ωt) + Acos(ky −ωt)

R = 2Acos[k(x − y)2

]cos[ωt +k(x + y)2

] =

= AR cos[ωt +k(x + y)2

]

AR = 2Acos[k(x − y)2

] = 2Acos[2π (x − y)2λ

]

I∝ AR2 = 4A2 cos2[π (x − y)

λ] = 4A2 cos2[π Δr

λ]

€

R = AR cos(ωt +φ1 + φ22

)

I∝ AR2 [ 1T

cos2(ωT +0

T

∫ φ1 + φ22

)] = AR2 /2

I∝ AR2 = 4A2 cos2(Δφ /2)

FOR UNEQUAL AMPLITUDES

wavescaninterfere(addorcancel)

Standing Waves

€

A = a1 cos(kx1 −ωt) + a2 cos(kx2 −ωt)

€

Vector addition of phasors

€

I∝ 4A2 cos2[π ΔLλ] = 4A2 cos2[π mλ

λ] = 4A2 cos2[mπ ]

€

I∝ 4A2 cos2[π ΔLλ] = 4A2 cos2[π (m +1/2)λ

λ] = 4A2 cos2[mπ + π /2] = 0

3/2 λ

InterferenceinWaves

€

Δφ = k(r2 − r1) = kΔr = kd sinθ = 2π dλsinθ

Maximum→Δφ = 2πm,m = 0,1,2,3

→ 2π dλsinθ = 2πm→ sinθm = m λ

dMinimum→Δφ = πm,m =1,2,3

→ 2π dλsinθ = πm→ sinθm = m 2λ

d

€

y = L tanθ ≈ Lθ

ym ≈ Lθm = m λLd

,m = 0,1,2,

Δy = ym+1 − ym = [(m +1) −m] λLd

=λLd

Independent of m - Same spacing

Positions of bright fringes

€

ymd ≈ (m + 12)

λLd,m = 0,1,2

Positions of dark fringes

Dark fringes exactly halfway between bright fringes

L>>d

MaximaMinima

Dark fringes exactly midway of bright ones How to measure the wavelength of light ?

AnalyzingDouble‐SlitInterferenceThe mth bright fringe emerging from the double slit is at an angle

where θm is in radians, and we have used the small-angle approximation. The y-position on the screen of the mth fringe is

while dark fringes are located at positions

€

I∝ AR2 = 4A2 cos2(Δφ /2)

Δφ = 2π Δrλ

= 2π d sinθλ

≈ 2π d tanθλ

Idouble = 4I1 cos2(πdλL

y)

Double slit intensity pattern

€

Δφ = k(r2 − r1) = kΔr = kd sinθ = 2π dλsinθ ≈ 2π d

λtanθ = 2π d

λLy

A = 2acos(Δφ2) = 2acos(πd

λLy)

I = cA2 = 4ca2 cos2(πdλL

y)

I2 = 4I1 cos2(πdλL

y)

I1 light intensity of single slit

Double slit intensity pattern If there was no interference intensity for two slits 2I1

TheDiffractionGratingSupposeweweretoreplacethedoubleslitwithanopaquescreenthathasNcloselyspacedslits.Whenilluminatedfromoneside,eachoftheseslitsbecomesthesourceofalightwavethatdiffracts,orspreadsout,behindtheslit.Suchamulti‐slitdeviceiscalledadiffractiongrating.Brightfringeswilloccuratanglesθm,suchthat

They‐positionsofthesefringeswilloccurat

Interference of N overlapped waves

€

ym = L tanθm

m is the order of diffraction

€

I∝ AR2 = 4A2 cos2(Δφ /2)

Δφ = 2π Δrλ

= 2π d sinθλ

≈ 2π d tanθλ

IN = N 2I1 cos2(πdλL

y)

I=4A2

I=0

I=2A2

ThePhysicsofTwoSlitInterference

λ mλ

mλ/2

mλ/4

Atotal=NA=10A

Itot=(NA)2=N2I1=100I1

KeydependenceN2