ECE 5616Curtis
Nature of Light
• Boundary Conditions for dielectric• Scalar Wave Equation• Concept of polarization• Isotropic, uniaxial media• Concept of spatial frequency• Snell’s Law
• TIR, Brewster’s angle
ECE 5616Curtis
Maxwell’s Equations in Differential Form
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Constitutive RelationshipsInteractions with matter
Dispersive, Anisotropic, & Non linear
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Boundary Conditions
Any closed line integral of E must be 0, normal sides very small so
Similarly for B
Similarly for H
Simon Ramo, et al, “Fields and Waves in Communication Electronics”
ECE 5616Curtis
Boundary ConditionsDerived from Maxwell’s equation at sharp change in material
In absence of surface charge or current
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Monochromatic Fields
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Monochromatic Constitutive RelationshipsReason for monochromatic assumption
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Wave EquationEliminate all fields but E
3x108 m/s
Polarization of the field is direction of E
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Plane wave solution
Polarization is defined by direction of E vector
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Substitute in MEHkED o ×−=⋅= εωεω
EkHB ×== ωμω
HIkEo •×−=• )(εωε
Use k x I (antisymmetric matrix) to rewrite as dot products
EIkH •×= )(ωμSubstitute and solve for E
0))(( =•×+ EIkkoε
ECE 5616Curtis
Plane Wave Continued
See Chen, “Theory of Electromagnetic waves” page 184,185
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Special Case Isotropic
n = sqrt (ε/εo)
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Special Case uniaxial
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Uniaxial Surface and PolarizationsPositive uniaxial ne > no
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Biaxial Surface & Polarizations
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Spatial FrequenciesBasis of Fourier Optics
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Grating diffraction angle
Grating of 1000 line pairs per mmWavelength 500 nmAngle of incidence is normal
Find angle of diffraction into 1st order ?
d sin θ = λ(m) , where m is the diffraction order and d is period of grating
Sin θ = (5e-7) /(1e-6)
θ = 30 degrees
1/d=Fx= sin θ/λ = d sinθ = λ
F = 1000lp/mm => d= 1/1000 mm
Conservation of momentum
ECE 5616Curtis
Cartesian eigensolution at ∞ half space
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Booker Quartic
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Booker Quartic Continues
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Isotropic Refraction – Snell’s Law
Transverse components must be conserved
ninc sin (θinc) = ntrans sin (θ trans)
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Refraction at air/water interface
• Incident angle is 30 degrees• Air n = 1 into water n= 1.33• What are the reflected and refracted
angles?
Reflected = incident = 30 degrees
1 sin (30) = 1.33 sin θθ = 22 degrees
ECE 5616Curtis
k-space
LL
( )BAG kkKvvv
−±=
Akv
Bkv
Akv
Bkv
Real Space Momentum (k) Space (Eswald Sphere)
λπ in2
nnn Δ−→
λπ in2
or change λ
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Snell’s Law in k-space
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Total Internal Reflection (TIR)Evanescent waves …
n1>n2
ninc sin (θinc) = ntrans sin (θ trans)
ninc sin (θinc) = ntrans
sin (θinc) = ntrans/ ninc
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TIR – Fourier space
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Questions
• What is the velocity of light in water ?
• Critical angle for glass (1.5) and air interface ?
V = c/n = (3x108m/s)/1.33 = 2.25x108m/s
1.5 sinθ = 1, θ = 41.8 degrees
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Extraordinary RefractionFun with Crystals !!!!
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Extraordinary RefractionFun with Crystals !!!!
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Brewster’s Angle
n1<n2
ninc sin (θinc) = ntrans sin (θ trans)
ninc sin (θB) = ntrans sin (90-θB) = ntrans cos (θB)
θB = arctan (ntrans/ninc)
Can use a stack of thin plates tilted at Brewster’s angle to make polarizer
ECE 5616Curtis
Questions for the day
• Does Brewster’s angle (zero reflection) exist for a material with loss ?
• How much power gets through a linear polarizer if the beam is unpolarized ?
No
Unpolarizered light has polarization that has components in each axis (x,y) in equal amounts.Therefore 50% of the light would go through a perfect polarizer. In practice polarizers have their own losses which decreases this amount.
ECE 5616Curtis
Questions for the day
• Brewster’s angle for air and water (1.33) interface ?
θB = arctan(1.33/1) = 52.4 degrees
• Can you couple or detect evanescent fields ?
ECE 5616Curtis
Tuesday Reading
Chapter 1 in W. Smith – Modern Optical Engineering
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