Fresnel Equations and EM Power Flow

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Fresnel Equations and EM Power Flow

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

TRUE / FALSE

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

Today

’ s Culture Moment

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:

H Gauss:

Faraday:

Ampere: k∙E = 0 k∙H = 0 k х E = ωμH k х H = - ωεE

E → H

H → - E

ε → μ

μ → ε

E Gauss:

H Gauss:

Faraday:

Ampere: k∙E = 0 k∙H = 0 k х E = ωμH 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

Fresnel Equations

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

Fresnel Equations

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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