Reading - Shen and Kong – Ch. 4
Outline
• Review of Oblique Incidence
• Review of Snell ’ s Law
• Fresnel Equations
• Evanescence and TIR
• Brewster ’ s Angle
• EM Power Flow

H r
1. This EM wave is TE
(transverse electric) polarized: E r
E i
H i
2. Snell ’ s Law only works for
TE polarized light.
E t
H t y •
3. Total internal reflection only occurs when light goes from a high index material to a low index material.

Water Waves
Refraction
E&M Waves
Waves refract at the top where the water is shallower
Refraction involves a change in the direction of wave propagation due to a change in propagation speed. It involves the oblique incidence of waves on media boundaries, and hence wave propagation in at least two dimensions.

Refraction in Suburbia
Think of refraction as a pair of wheels on an axle going from a sidewalk onto grass. The wheel in the grass moves slower, so the direction of the wheel pair changes.
Note that the wheels move faster (longer space) on the sidewalk, and slower
(shorter space) in the grass.

Total Internal Reflection in
Suburbia
Moreover, this wheel analogy is mathematically equivalent to the refraction phenomenon. One can recover Snell ’ s law from it: n
1 sin

1
= n
2 sin

2
.
The upper wheel hits the sidewalk and starts to go faster, which turns the axle until the upper wheel re-enters the grass and wheel pair goes straight again.

Oblique Incidence (3D view) k iz k i k ix
θ i z y x
Boundary k ix
= k i sin( θ i
) k iz
= k i cos( θ i
)
Identical definitions for k r and k t

E i
E r
E r
E i
Oblique Incidence at Dielectric Interface
H r
E t
Transverse
Electric Field
H t
H r
H i
E t
H t
Transverse
Magnetic Field
H i y •
Why do we consider only these two polarizations?

Partial TE Analysis
H r z = 0
E t E r
E i
H t
H i y •
Tangential E must be continuous at the boundary z = 0 for all x and for t.
This is possible if and only if k ix
= k rx
= k
The former condition is phase matching.
tx and ω i
= ω r
= ω t
.

Snell
s Law Diagram
Tangential E field is continuous …
Refraction Total Internal Reflection

Snell ’ s Law
•
Snell
’ s Law

TE Analysis - Set Up z = 0 k r
E r
θ r
E i
θ i
H r
θ t
E t
H t k i
H i
Medium 1 Medium 2 y • k t
To get H, use
Faraday ’ s
Law

TE Analysis – Boundary Conditions
Incident Wavenumber:
Phase Matching:
Tangential E:
Normal μH:
Tangential H:

Solution Boundary conditions are: z = 0 k r
E r
θ r
E i
θ i
H r
θ t
E t
H t k i
H i
Medium 1 Medium 2 y • k t
Fresnel Equations

• Scottish scientist
Sir David Brewster
• Studied at University of Edinburgh at age 12
• Independently discovered Fresnel lens
• Editor of Edinburgh Encyclopedia and contributor to Encyclopedia Britannica (7 th and 8 th editions)
• Inventor of the Kaleidoscope
• Nominated (1849) to the National Institute of
France.
1781 –1868
Fresnel Lens
Kaleidoscope
All Images are in the public domain

TM Case is the dual of TE
E Gauss: k∙E = 0
H Gauss: k∙H = 0
Faraday: k х E = ωμH
Ampere: k х H = - ωεE
E → H
H → - E
ε → μ
μ → ε
E Gauss: k∙E = 0
H Gauss: k∙H = 0
Faraday: k х E = ωμH
Ampere: k х H = - ωεE
The TM solution can be recovered from the TE solution.
So, consider only the TE solution in detail.

TE & TM Analysis – Solution
TE solution comes directly from the boundary condition analysis
TM solution comes from ε ↔ μ
Note that the TM solution provides the reflection and transmission coefficients for H, since TM is the dual of TE.

Fresnel Equations - Summary
TE-polarization
TM Polarization

Fresnel Equations - Summary
From Shen and Kong … just another way of writing the same results
TE Polarization TM Polarization

E i
E r n
1
= 1.0
H r
E r
E i
H r
H i
TE n
E t
E t
2
= 1.5
H t
H i
TM
H t y •
1 . 5
1
0 . 5
0
- 0 . 5
- 1
0 t
TE t
TM r
TM r
TE
θ
B
3 0
Incidence Angle
6 0 9 0

E i
E r n
1
= 1.5
H r
E r
E i
H r
H i
TE n
2
= 1.0
E t
E t
H t
H i
TM
H t y •
1
0 . 8
0 . 6
0 . 4
0 . 2
0
- 0 . 2
- 0 . 4
0
Total Internal Reflection r
TE
θ
B
θ
C r
TM
θ
C
= sin -1 (n
2
/ n
1
)
3 0
Incidence Angle
6 0 9 0

Reflection of Light
(Optics Viewpoint … μ
1
= μ
2
)
E i
E r
E r
E i
H r
E t
TE H t
H r
H i
E t
H i
TM
H t y •
TE:
TM:
Incidence Angle

Image in the Public Domain
Sir David Brewster
Brewster ’ s Angle
(Optics Viewpoint … μ
1
= μ
2
)
Snell ’ s Law
Requires n
1
= n
2
Unpolarized
Incident Ray
TE Polarized
Reflected Ray
Snell ’ s Law
Can be
Satisfied with
θ
B
θ
B
= arctan(n
2
/n
1
)
+ θ
T
= 90 °
Slightly Polarized
Reflracted Ray
Light incident on the surface is absorbed, and then reradiated by oscillating electric dipoles (Lorentz oscillators) at the interface between the two media. The dipoles that produce the transmitted (refracted) light oscillate in the polarization direction of that light. These same oscillating dipoles also generate the reflected light. However, dipoles do not radiate any energy in the direction along which they oscillate.
Consequently, if the direction of the refracted light is perpendicular to the direction in which the light is predicted to be specularly reflected, the dipoles will not create any TM-polarized reflected light.

TE
TM
Energy Transport
Cross-sectional Areas

Transmitted Power Fraction:

Reflected Power Fraction:
… and from ENERGY CONSERVATION we know:

Summary CASE I:
E-field is polarized perpendicular to the plane of incidence
… then E-field is parallel (tangential) to the surface, and continuity of tangential fields requires that:

Summary CASE II:
H-field is polarized perpendicular to the plane of incidence
… then H-field is parallel (tangential) to the surface, and continuity of tangential fields requires that:

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6.007 Electromagnetic Energy: From Motors to Lasers
Spring 2011
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