Lecture 21 Wave Optics-2 Chapter 22 PHYSICS 270 Dennis Papadopoulos March 31, 2010.
Lecture 13 Wave Optics‐1 Chapter 22
Transcript of Lecture 13 Wave Optics‐1 Chapter 22
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
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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
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ym = L tanθm
m is the order of diffraction
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I∝ AR2 = 4A2 cos2(Δφ /2)
Δφ = 2π Δrλ
= 2π d sinθλ
≈ 2π d tanθλ
IN = N 2I1 cos2(πdλL
y)