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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

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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”

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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

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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

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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

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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

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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.

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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 ?

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Tuesday Reading

Chapter 1 in W. Smith – Modern Optical Engineering