lecture 13 wave optics‐1 chapter 22
TRANSCRIPT
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
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.”
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
€
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
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)