1. The document contains details about a student named Mohit Sharma who is in the 4th semester of his BSC in computer science program at Raisahebbhanwar Singh College in Nasrullaganj. 2. It includes the Fresnel equations which describe the reflection and refraction of light when it passes from one medium to another at an interface. 3. The Fresnel equations are derived from the boundary conditions imposed by Maxwell's equations and take into account the polarization state of the light (TE or TM).

1. NAME : MOHIT SHARMA
CLASS : BSC IV SEM (C.S)
COLLEGE : RAISAHEBBHANWARSINGH
COLLEGENASRULLAGANJ
SUBMITTED TO : GYANRAO DHOTE SIR
4/9/2017 1
RAI SAHEB BHANWAR SINGH
COLLEGE NASRULLAGANJ

2. 4/9/2017 2
RAI SAHEB BHANWAR SINGH
COLLEGE NASRULLAGANJ

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

4. 1. This EM wave is TE
(transverse electric) polarized:
2. Snell’s Law only works for
TE polarized light.
3. Total internal reflection only occurs when light goes from
a high index material to a low index material.
Ei
Et
Er
Hi
Ht
Hr
•y

5. Refraction
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.
Water Waves E&M Waves
Waves refract at the top where the
water is shallower
ImagebyNOAAPhotolibrary
http://www.flickr.com/photos/noaap
hotolib/5179052751/onflickr

6. 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.

7. Moreover, this wheel analogy is mathematically equivalent to
the refraction phenomenon. One can recover Snell’s law from
it: n1sin1 = n2sin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.
Total Internal Reflection in Suburbia

8. Oblique Incidence (3D view)
Boundary
y
x
ki
z
kiz kix
θi
kix = ki sin(θi)
kiz = ki cos(θi)
Identical definitions for kr and kt

9. Oblique Incidence at Dielectric Interface
Transverse
Electric Field
Transverse
Magnetic Field
Ei
Et
Et
Ei
Er
Er
Hi
Hi
Ht
Ht
Hr
Hr
•y Why do we consider only these
two polarizations?

10. Partial TE Analysis
Ei
Et
Er
Hi
Ht
Hr
•y
Tangential E must be continuous at the boundary z = 0 for all x and for t.
z = 0
This is possible if and only if kix = krx = ktx and ωi = ωr = ωt. The
former condition is phase matching.

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

12. Snell’s Law
•
Snell’s Law

13. TE Analysis - Set Up
Ei
Et
Er
Hi
Ht
Hr
•y
z = 0
θi θt
θr
ki
kr
kt
Medium 1 Medium 2
To get H, use
Faraday’s Law

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

15. Solution Boundary
conditions are:
Ei
Et
Er
Hi
Ht
Hr
•y
z = 0
θi θt
θr
ki
kr
kt
Medium 1 Medium 2
Fresnel Equations

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

17. 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 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
The TM solution can be recovered from the TE solution. So,
consider only the TE solution in detail.

18. 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.

19. Fresnel Equations - Summary
TM Polarization
TE-polarization

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

21. 0 30 60 90
-1
-0.5
0
0.5
1
1.5
Fresnel Equations
Ei
Et
Et
Ei
Er
Er
Hi
Hi
Ht
Ht
Hr
Hr
•y
TE
TM
rTM
rTE
tTE
tTM
n1 = 1.0 n2 = 1.5
θB
Incidence Angle
ReflectionCoefficients

22. 0 30 60 90
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Fresnel Equations
rTM
rTE
θCθB
θC = sin-1(n2 / n1)
Total Internal Reflection
Ei
Et
Et
Ei
Er
Er
Hi
Hi
Ht
Ht
Hr
Hr
•y
TE
TM
n2 = 1.0n1 = 1.5
Incidence Angle
ReflectionCoefficients

23. Reflection of Light
(Optics Viewpoint … μ1 = μ2)
Incidence Angle
ReflectionCoefficients
TE:
TM:
Ei
Et
Et
Ei
Er
Er
Hi
Hi
Ht
Ht
Hr
Hr
•y
TE
TM

24. Sir David Brewster
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.
Brewster’s Angle
(Optics Viewpoint … μ1 = μ2)
Unpolarized
Incident Ray
TE Polarized
Reflected Ray
Slightly Polarized
Reflracted Ray
Snell’s Law
Requires
n1 = n2
Snell’s Law
Can be
Satisfied with
θB = arctan(n2/n1)
θB + θT = 90°
Image in the Public Domain

25. Cross-sectional Areas
Energy Transport
TE
TM

26. Transmitted Power Fraction:

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

28. 4/9/2017 28
RAI SAHEB BHANWAR SINGH
COLLEGE NASRULLAGANJ