Lecture 13 Wave Optics‐1 Chapter 22

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Lecture 13 Wave Optics‐1 Chapter 22 PHYSICS 270 Dennis Papadopoulos March 15, 2011

Transcript of Lecture 13 Wave Optics‐1 Chapter 22

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