reflection refraction
TRANSCRIPT
1
REFLECTION&
REFRACTION
SOLO HERMELIN
Updated: 4.11.06 http://www.solohermelin.com
2
SOLO
TABLE OF CONTENT
REFLECTION & REFRACTION
History of Reflection and Refraction
Huygens Principle
Reflections Laws Development Using Huygens PrincipleRefractions Laws Development Using Huygens Principle
Fermat’s Principle
Reflections Laws Development Using Fermat’s Principle
Refractions Laws Development Using Fermat’s Principle
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Monochromatic Wave Equations
Phase-Matching Conditions
Fresnel Conditions
Energy Reflected and Refracted for Normal Polarization
Malus-Dupin TheoremStokes Treatment of Reflection and Refraction
References
3
REFLECTION & REFRACTION SOLO
Introduction
4
REFLECTION & REFRACTION SOLO
History of Reflection & Refraction
100-170 A.D.Claudius Ptolemey
Alexandria
“Optics” 130 A.D.Tabulated angle of incidence and refraction for several media.
c. 300 B.C.EuclidGreece
“Optica” 280 B.C.Rectilinear propagation of Light. Law of Reflection. Light originate in the eye, illuminates the object seen,and then returns to the eye.
5
REFLECTION & REFRACTION SOLO
History of Reflection & Refraction
Willebrord van Roijen Snell1580-1626
Professor at Leyden, experimentally discovered the law of refraction in 1621
René Descartes 1596-1650
Was the first to publish the law of refraction in
terms of sinuses in “La Dioptrique” in 1637.
Descartes assumed that the component of velocity of light parallel to the interface was unaffected, obtaining
ti vv θθ sinsin 21 =
from which
1
2
sin
sin
v
v
t
i =θθ
correct1
2
sin
sin
n
n
t
i =θθ
Descartes deduced
wrong
6
REFLECTION & REFRACTION SOLO
History of Reflection & Refraction
Pierre de Fermat1601-1665
Principle of Fermat (1657) of the extremality of (usually a minimum) optical path enables the derivation of reflection and refraction laws.
∫2
1
P
P
dsn
Christiaan Huygens1629-1695
In a communication to the Académie des Science in 1678 reported his wave theory (published in his “Traité de Lumière” in 1690). He considered that light is transmitted through an all-pervading aether that is made up of small elastic particles, each of each can act as a secondary source of wavelets. On this basis Huygens explained many of the known properties of light, including the double refraction in calcite.
Augustin Jean Fresnel
1788-1827
Presented the laws which enable the calculation of the intensity and polarization of reflected and refracted light in 1823.
7
REFLECTION & REFRACTION SOLO
Huygens Principle
Christiaan Huygens1629-1695
Every point on a primary wavefront serves the source of spherical secondary wavelets such that the primary wavefront at some later time is the envelope o these wavelets. Moreover, the wavelets advance with a speed and frequency equal to that of the primary wave at each point in space.
“We have still to consider, in studying the spreading of these waves, that each particle of matter in which a wave proceeds not only communicates its motion to the next particle to it, which is on the straight line drawn from the luminous point, but it also necessarily gives a motion to all the other which touch it and which oppose its motion. The result is that around each particle there arises a wave of which this particle is a center.”
Huygens visualized the propagtion of light in terms of mechanical vibration of an elastic medium (ether).
8
REFLECTION & REFRACTION SOLO
Reflection Laws Development Using Huygens Principle
Suppose a planar incident waveAB is moving toward the boundary AC between two media. The velocityof light in the first media is v1.
The incident rays are reflected at the boundary AB. At the time the incident ray passing through B isreaching the boundary at C, thereflected ray at A will reach D andthe ray passing through F will be reflected at G and reaches H.
According to Huygens Principle a reflected wavefront CHD, normal to the reflected rays AD, GH is formed and CBGHFGAD =+=
ADCABC ∆=∆
DCABAC ri ∠=∠= θθ &From the geometry
DCABAC ∠=∠
ri θθ =
9
REFLECTION & REFRACTION SOLO
Refraction Laws Development Using Huygens Principle
Suppose a planar incident waveAB is moving toward the boundary AC between two media. The velocityof light in the first media is v1.
The incident rays are refracted at the boundary AB. At the time the incident ray passing through B isreaching the boundary at C, therefracted ray at A will reach E andthe ray passing through F will be refracted at G and reaches H.
According to Huygens Principle a reflected wavefront CH’E, normal to the refracted rays AD, GH’ is formed and tvCBtvAE 12 ===
ECABAC ti ∠=∠= θθ &From the geometry
( ) ( ) ACAEECAACBCBAC /sin&/sin =∠=∠ 2
1
sin
sin
v
v
EA
BC
t
i ==θθ
10
SOLO
Fermat’s Principle (1657)
The Principle of Fermat (principle of the shortest optical path) asserts that the optical length
of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certai neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length).
∫2
1
P
P
dsn
An other form of the Fermat’s Principle is:
Principle of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum).
REFLECTION & REFRACTION
11
SOLO
1. The optical path is reflected at the boundary between two regions
( ) ( )0
2121 =⋅
− rd
sd
rdn
sd
rdn rayray
In this case we have and21 nn =( ) ( ) ( ) 0ˆˆ
2121 =⋅−=⋅
− rdssrd
sd
rd
sd
rd rayray
We can write the previous equation as:
i.e. is normal to , i.e. to the boundary where the reflection occurs.
21 ˆˆ ss − rd
( ) 0ˆˆˆ 2121 =−×− ssn
REFLECTION & REFRACTION
Reflection Laws Development Using Fermat Principle
ri θθ = Incident ray and Reflected ray are in the same plane normal to the boundary.
This is equivalent with:
&
12
SOLO
2. The optical path passes between two regions with different refractive indexes n1 to n2. (continue – 1)
( ) ( )0
2121 =⋅
− rd
sd
rdn
sd
rdn rayray
where is on the boundary between the two regions andrd ( ) ( )
sd
rds
sd
rds rayray 2
:ˆ,1
:ˆ 21
==
Therefore is normal to .
2211 ˆˆ snsn − rd
Since can be in any direction on the boundary between the two regions is parallel to the unit vector normal to the boundary surface, and we have
rd
2211 ˆˆ snsn −21ˆ −n
( ) 0ˆˆˆ 221121 =−×− snsnn
We recovered the Snell’s Law from Geometrical Optics
REFLECTION & REFRACTION
Refraction Laws Development Using Fermat Principle
ti nn θθ sinsin 21 = Incident ray and Refracted ray are in the same plane normal to the boundary.
&
13
ELECTROMAGNETICSSOLO
To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations
we haveUsing: 1ˆˆ&ˆˆ0 =⋅== kkknkkk εµω
=⋅∇=⋅∇
−=×∇=×∇
0
0
H
E
HjE
EjH
ωµωε
=⋅
=⋅
=×
−=×
0ˆ
0ˆ
ˆ
ˆ
0
0
00
00
Hk
Ek
HEk
EHk
εµ
µε
=⋅−
=⋅−
−=×−
=×−
⇒
⋅−
⋅−
⋅−⋅−
⋅−⋅−
−=∇ ⋅−⋅−
0
0
0
0
00
00
rkj
rkj
rkjrkj
rkjrkj
ekje
eHkj
eEkj
eHjeEkj
eEjeHkjrkjrkj
ωµ
ωε
( ) ∗∗ ⋅==⋅==+= HHwEEwwcnk
wwcnk
S meme 22&
2
ˆ
2
ˆ µεTime Average Poynting Vector of the Planar Wave
Reflections and Refractions Laws Development Using the Electromagnetic Approach
14
SOLO REFLECTION & REFRACTION
Consider an incident monochromatic planar wave
( )
( )
c
nk
eEkH
eEE
iiii
rktjiii
rktjii
ii
ii
1
00
110011
0
0
ωεµεµεµωεµω
µε ω
ω
===
×=
=
⋅−
⋅−
The monochromatic planar reflected wave from the boundary is
( )
( )
11
1
1
0
0
&n
cv
vc
nk
eEkH
eEE
rrr
rktjrrr
rktjrr
rr
rr
===
×=
=
⋅−
⋅−
ωω
µε ω
ω
The monochromatic planar refracted wave from the boundary is
( )
( )
22
2
2
0
0
&n
cv
vc
nk
eEkH
eEE
ttt
rktjttt
rktjtt
tt
tt
===
×=
=
⋅−
⋅−
ωω
µε ω
ω
Reflections and Refractions Laws Development Using the Electromagnetic Approach
15
SOLO REFLECTION & REFRACTION
The Boundary Conditions at z=0 must be satisfied at all pointson the plane at all times, impliesthat the spatial and time variations of
This implies that
Phase-Matching Conditions
( ) ( ) ( ) yxteEeEeEz
rktjt
z
rktjr
z
rktji
ttrrii ,,,,0
00
00
0 ∀=
⋅−
=
⋅−
=
⋅− ωωω
( ) ( ) ( ) yxtrktrktrktz
ttz
rrz
ii ,,000
∀⋅−=⋅−=⋅−===
ωωω
ttri ∀=== ωωωω
( ) ( ) ( ) yxrkrkrkz
tz
rz
i ,000
∀⋅=⋅=⋅===
must be the same
Reflections and Refractions Laws Development Using the Electromagnetic Approach
16
SOLO REFLECTION & REFRACTION
tri nnn θθθ sinsinsin 211 ==
Phase-Matching Conditions
( )
( )
−+=
++=
zyxc
nk
zyxc
nk
ttttttt
irirrrr
ˆcossinˆsinsinˆcos
ˆcossinˆsinsinˆcos
2
1
θαθααω
θαθααω
( )
( )
+=⋅
+=⋅
=⋅
=
=
=
yyxc
nrk
yxc
nrk
yc
nrk
tttz
t
irrz
r
iz
i
ˆsinsincos
sinsincos
sin
2
0
1
0
1
0
θααω
θααω
θω
( ) ( ) ( ) yxrkrkrkz
tz
rz
i ,000
∀⋅=⋅=⋅===
2
παα == tr
ttri ∀=== ωωωω
x∀
y∀
Coplanar
Snell’s Law
( )
++=
−=
zzyyxxr
zyc
nk iiii
ˆˆˆ
ˆcosˆsin1
θθω
Given:
Let find:
Reflections and Refractions Laws Development Using the Electromagnetic Approach
17
SOLO REFLECTION & REFRACTION
Second way of writing phase-matching equations
ri θθ =11
22
2
1
1
2
sin
sin
εµεµ
θθ ===
v
v
n
n
t
iRefraction Law
Reflection Law
Phase-Matching Conditions
( )
++=
−=
zzyyxxr
zyc
nk iiii
ˆˆˆ
ˆcosˆsin1
θθω
( )
( )
−+=
++=
zyxc
nk
zyxc
nk
ttttttt
irirrrr
ˆcossinˆsinsinˆcos
ˆcossinˆsinsinˆcos
2
1
θαθααω
θαθααω
( ) ( )[ ]
( ) ( )[ ]
−+−=−×
−+−=−×
ynnync
kkz
ynnync
kkz
ittrti
irrrri
ˆsinsinsinˆcosˆ
ˆsinsinsinˆcosˆ
122
111
θθααω
θθααω
ttri ∀=== ωωωω
We can see that ( ) ( )
====−×=−×
ωωωω tri
tiri kkzkkz 0ˆˆ
=====
==
ωωωωθθθ
παα
tri
tri
tr
nnn sinsinsin
2/
211
Reflections and Refractions Laws Development Using the Electromagnetic Approach
18
SOLO REFLECTION & REFRACTION
ri θθ =11
22
2
1
1
2
sin
sin
εµεµ
θθ ===
v
v
n
n
t
iRefraction Law
Reflection Law
Phase-Matching Conditions (Summary)
ttri ∀=== ωωωω
( ) ( )
====−×=−×
ωωωω tri
tiri kkzkkz 0ˆˆ
=====
==
ωωωωθθθ
παα
tri
tri
tr
nnn sinsinsin
2/
211
( ) ( ) ( ) yxrkrkrkz
tz
rz
i ,000
∀⋅=⋅=⋅===
( ) ( ) ( ) yxtrktrktrktz
ttz
rrz
ii ,,000
∀⋅−=⋅−=⋅−===
ωωω
Vector Notation
ScalarNotation
Reflections and Refractions Laws Development Using the Electromagnetic Approach
19
SOLO REFLECTION & REFRACTION
( ) 0ˆ 2121
=−×− EEn
( ) 0ˆ 2121
=−×− HHn
( ) 0ˆ 2121 =−⋅− DDn
( ) 0ˆ 2121 =−⋅− BBn
Boundary conditions for asource-less boundary
( ) 0ˆ 00021
=−+×− tri EEEn
[ ] 0ˆ/ˆ/ˆ/ˆ 02201101121
=×−×+××− ttrrii EkEkEkn µεµεµε
( ) 0ˆ 02010121 =−+⋅− tri EEEn
εεε
( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn
εµεµεµ
In our case ( ) ttrrii
tri
EkHEkEkH
EEEEE
×=×+×=
=+=
ˆ&ˆˆ
&
2
22
1
11
21
µε
µε
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Fresnel EquationsBoundary conditions
20
SOLO REFLECTION & REFRACTION
( ) 0ˆ 00021
=−+×− tri EEEn
0111
ˆ 02
01
01
21
=
×−×+××− ttrrii EkEkEkn
µµµ
( ) 0ˆ 02010121 =−+⋅− tri EEEn
εεε
( ) 0ˆ 00021 =×−×+×⋅− ttrrii EkEkEkn
Using ,ˆ,ˆ,ˆˆ221111
1ttrriii kkkkkk
c
nk εµωεµωεµωω ====
( ) 0ˆ 00021
=−+×− tri EEEn
[ ] 0ˆ/ˆ/ˆ/ˆ 02201101121
=×−×+××− ttrrii EkEkEkn µεµεµε
( ) 0ˆ 02010121 =−+⋅− tri EEEn
εεε
( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn
εµεµεµ
Boundary Conditions
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Fresnel Equations
21
SOLO REFLECTION & REFRACTION
( ) ( ) ( )( ) ( ) ( ) ( ) xEknEEknxEknEEkn
xEknEEnkEkn
tttttirrrrri
iiiiiiii
ir
i
ˆcosˆˆˆˆ&ˆcosˆˆˆˆ
ˆcosˆˆˆˆˆˆ
0
cos
2100210
cos
210021
0
cos
210
0
021021
θθ
θ
θθ
θ
=⋅−=××−=⋅−=××
=⋅−⋅=××
−−
−
−−
−−−
( ) ( ) ( ) ( )( ) ( ) tttttt
rrrrrriiiiii
EEzzEkn
EEzzEknEEzzEkn
θθ
θθθθ
sinsinˆˆˆˆ
sinsinˆˆˆˆ&sinsinˆˆˆˆ
00021
0002100021
=−⋅−=×⋅
=−⋅−=×⋅=−⋅−=×⋅
−
−−
zn ˆˆ 21 −=−
( ) 0ˆ 00021
=−+×− tri EEEn
[ ] 0ˆ/ˆ/ˆ/ˆ 02201101121
=×−×+××− ttrrii EkEkEkn µεµεµε
( ) 0ˆ 02010121 =−+⋅− tri EEEn
εεε
( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn
εµεµεµ
1
2
3
4
zykzykzyk tttiiriii ˆcosˆsin&ˆcosˆsin&ˆcosˆsin θθθθθθ −=+=−=
Assume is normal o plan of incidence(normal polarization)E
xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥
Boundary Conditions
Reflections and Refractions Laws Development Using the Electromagnetic Approach
22
SOLO REFLECTION & REFRACTION
( ) 0coscoscos 02
200
1
1 =−− ttirii EEE θµεθθ
µε
1
0000
=−+ tri EEE2
( ) 0ˆ 00021
=−+×− tri EEEn
[ ] 0ˆ/ˆ/ˆ/ˆ 02201101121
=×−×+××− ttrrii EkEkEkn µεµεµε
( ) 0ˆ 02010121 =−+⋅− tri EEEn
εεε
( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn
εµεµεµ
1
2
3
4
( )0
sinsin
sinsin
0sinsinsin
000
2211
0220011
=−+⇒
=
==−+
tri
ti
ri
ttrrii
EEE
EEE
θεµθεµ
θθθεµθθεµ4
Identical to 2
3 00 =
Assume is normal o plan of incidence(normal polarization)E
xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥
Boundary Conditions
Reflections and Refractions Laws Development Using the Electromagnetic Approach
23
SOLO REFLECTION & REFRACTION
( ) 0cos1
cos1
000
22
200
00
11
1
21
=−− tt
n
iri
n
EEE θεµεµ
µθ
εµεµ
µ
1
0000 =−+ tri EEE2
From and
ti
ti
i
r
nn
nn
E
Er
θµ
θµ
θµ
θµ
coscos
coscos
2
2
1
1
2
2
1
1
0
0
+
−=
=
⊥⊥
ti
i
i
t
nn
n
E
Et
θµ
θµ
θµ
coscos
cos2
2
2
1
1
1
1
0
0
+=
=
⊥⊥
For most of media μ1= μ2 , and using refraction law:
1
2
sin
sin
n
n
t
i =θθ
( )( )ti
ti
i
r
E
Er
θθθθ
+−−=
=
⊥⊥ sin
sin
0
0
( )ti
it
i
t
E
Et
θθθθ
+=
=
⊥⊥ sin
cossin2
0
0
1 2
Assume is normal o plan of incidence(normal polarization)E
xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥
Reflections and Refractions Laws Development Using the Electromagnetic Approach
24
SOLO REFLECTION & REFRACTION
ti
ti
i
r
nn
nn
E
Er
θµ
θµ
θµ
θµ
coscos
coscos
2
2
1
1
2
2
1
1
0
0
+
−=
=
⊥⊥
ti
i
i
t
nn
n
E
Et
θµ
θµ
θµ
coscos
cos2
2
2
1
1
1
1
0
0
+=
=
⊥⊥
For most of media μ1= μ2 ,
and using refraction law: 1
2
sin
sin
n
n
t
i =θθ
( )( )ti
ti
i
r
E
Er
θθθθµµ
+−−=
=
=
⊥⊥ sin
sin21
0
0
( )ti
it
i
t
E
Et
θθθθµµ
+=
=
=
⊥⊥ sin
cossin221
0
0
Assume is normal o plan of incidence(normal polarization)E
xEExEExEE ttrrii ˆ&ˆ&ˆ 000000 −=−=−= ⊥⊥⊥
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Fresnel Equations
25
SOLO REFLECTION & REFRACTION
Assume is parallel to plan of incidence(parallel polarization)E
( )( )
( )zyEE
zyEE
zyEE
tttt
rrrr
iiii
ˆsinˆcos
ˆsinˆcos
ˆsinˆcos
0||0
0||0
0||0
θθ
θθ
θθ
+=
+−=
+=
( ) ( ) ( )( ) ( ) yEEknyEEkn
yEknEEnkEkn
ttirri
iiiiiii
ii
ˆˆˆ&ˆˆˆ
ˆˆˆˆˆˆˆ
00210021
0
cos
210
sin
021021
−=××−=××
−=⋅−⋅=××
−−
−
−
−−
θθ
( ) ( ) ( ) 0ˆˆˆˆˆˆ 021021021 =×⋅=×⋅=×⋅ −−− ttrrii EknEknEkn
zn ˆˆ 21 −=−
zykzykzyk tttiiriii ˆcosˆsin&ˆcosˆsin&ˆcosˆsin θθθθθθ −=+=−=
xEEnxEEnxEEn tttirriii ˆcosˆ&ˆcosˆ&ˆcosˆ 002100210021 θθθ =×−=×=× −−−
tttirriii EEnEEnEEn θθθ sinˆ&sinˆ&sinˆ 002100210021 −=⋅−=⋅−=⋅ −−−
Reflections and Refractions Laws Development Using the Electromagnetic Approach
26
SOLO REFLECTION & REFRACTION
Assume is parallel to plan of incidence(parallel polarization)E ( )
( )( )zyEE
zyEE
zyEE
tttt
rrrr
iiii
ˆsinˆcos
ˆsinˆcos
ˆsinˆcos
0||0
0||0
0||0
θθ
θθ
θθ
+=
+−=
+=
( ) 0ˆ 00021
=−+×− tri EEEn
[ ] 0ˆ/ˆ/ˆ/ˆ 02201101121
=×−×+××− ttrrii EkEkEkn µεµεµε
( ) 0ˆ 02010121 =−+⋅− tri EEEn
εεε
( ) 0ˆˆˆˆ 02201101121 =×−×+×⋅− ttrrii EkEkEkn
εµεµεµ
1
2
3
4
( ) 0sinsin 02001 =−+ ttiri EEE θεθε3
( )[ ] 0ˆcoscos 000 =−− xEEE ttiri θθ2
( ) ( ) 011
0ˆ 000
22
200
00
11
10
2
200
1
1
21
=−+=
−+ t
n
ri
n
tri EEEoryEEE
µεµε
µµεµε
µµε
µε
1
4 00 =
Boundary Conditions
Reflections and Refractions Laws Development Using the Electromagnetic Approach
27
SOLO REFLECTION & REFRACTION
Assume is parallel to plan of incidence(parallel polarization)E
( )( )
( )zyEE
zyEE
zyEE
tttt
rrrr
iiii
ˆsinˆcos
ˆsinˆcos
ˆsinˆcos
0||0
0||0
0||0
θθ
θθ
θθ
+=
+−=
+=
( )( ) 0sinsin
sin
sin
/
/sinsin
0ˆ
02001
2
1
22
112211
02
200
1
1
=−+
=⇒=
=
−+
ttiri
t
iti
tri
EEE
yEEE
θεθε
θεθε
µεµε
θεµθεµ
µε
µε
1
Identical to 3
We have two independent equations
( ) 0coscos 000 =−− ttiri EEE θθ2
( ) 002
200
1
1 =−+ tri En
EEn
µµ1 ti
ti
i
r
nn
nn
E
Er
θµ
θµ
θµ
θµ
coscos
coscos
1
1
2
2
1
1
2
2
||0
0||
+
−=
=
ti
i
i
t
nn
n
E
Et
θµ
θµ
θµ
coscos
cos2
1
1
2
2
1
1
||0
0||
+=
=
Reflections and Refractions Laws Development Using the Electromagnetic Approach
28
SOLO REFLECTION & REFRACTION
Assume is parallel to plan of incidence(parallel polarization)E
( )( )
( )zyEE
zyEE
zyEE
tttt
rrrr
iiii
ˆsinˆcos
ˆsinˆcos
ˆsinˆcos
0||0
0||0
0||0
θθ
θθ
θθ
+=
+−=
+=
ti
ti
i
r
nn
nn
E
Er
θµ
θµ
θµ
θµ
coscos
coscos
1
1
2
2
1
1
2
2
||0
0||
+
−=
=
ti
i
i
t
nn
n
E
Et
θµ
θµ
θµ
coscos
cos2
1
1
2
2
1
1
||0
0||
+=
=
For most of media μ1= μ2 ,
and using refraction law: 1
2
sin
sin
n
n
t
i =θθ
( )( )ti
ti
i
r
E
Er
θθθθµµ
+−=
=
=
tan
tan21
||0
0|| ( ) ( )titi
it
i
t
E
Et
θθθθθθµµ
−+=
=
=
cossin
cossin221
||0
0||
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Fresnel Equations
29
SOLO REFLECTION & REFRACTION
ti
ti
i
r
nn
nn
E
Er
θµ
θµ
θµ
θµ
coscos
coscos
1
1
2
2
1
1
2
2
||0
0||
+
−=
=
ti
i
i
t
nn
n
E
Et
θµ
θµ
θµ
coscos
cos2
1
1
2
2
1
1
||0
0||
+=
=
ti
ti
i
r
nn
nn
E
Er
θµ
θµ
θµ
θµ
coscos
coscos
2
2
1
1
2
2
1
1
0
0
+
−=
=
⊥
⊥
ti
i
i
t
nn
n
E
Et
θµ
θµ
θµ
coscos
cos2
2
2
1
1
1
1
0
0
+=
=
⊥
⊥
The equations of reflection and refraction ratio are called Fresnel Equations, that first developed them in a slightly less general form in 1823, using the elastic theory of light.
Augustin Jean Fresnel
1788-1827
The use of electromagnetic approach to prove those relations, as described above, is due to H.A. Lorentz (1875)
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Hendrik Antoon Lorentz1853-1928
30
SOLO REFLECTION & REFRACTION
Discussion of Fresnel Equations
it n
ni
θθθ
2
10→
=
0→iθ
[ ] [ ]ti
ti
nn
nnrr
ii θθθθ
θθ +−
=+−=−= =⊥=
21
1200||
[ ] [ ]ti
t
nn
ntt
ii θθθ
θθ +=
+== =⊥=
22
21
100||
1
2
sin
sin
n
n
t
i =θθ
Snell’s law
Reflections and Refractions Laws Development Using the Electromagnetic Approach
n2 / n1 =1.5 n2 / n1 =1/1.5
0cos90 =→→ ii θθ
[ ] [ ] 19090|| −=−= =⊥= ii
rr θθ
[ ] [ ] 09090|| == =⊥= ii
tt θθ
( )ti
it
ti
i
i
t
nn
n
E
Et
θθθθ
θθθµµ
+=
+=
=
=
⊥
⊥ sin
cossin2
coscos
cos2
21
1
0
021
( ) ( )titi
it
ti
i
i
t
nn
n
E
Et
θθθθθθ
θθθµµ
−+=
+=
=
=
cossin
cossin2
coscos
cos2
12
1
||0
0||
21
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
−=+−
=
=
=
⊥
⊥ sin
sin
coscos
coscos
21
21
0
021
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
=+−
=
=
=
tan
tan
coscos
coscos
12
12
||0
0||
21
31
SOLO REFLECTION & REFRACTION
Discussion of Fresnel Equations
1
2
sin
sin
n
n
t
i =θθ
Snell’s law
Reflections and Refractions Laws Development Using the Electromagnetic Approach
David Brewster1781-1868
David Brewster , “On the laws which regulate the polarization of light by reflection from transparent bodies”, Philos. Trans. Roy. Soc., London 105, 125-130, 158-159 1815).
In contrast r|| changes sign (for both n2> n1
and n2<n1) when tan(θi+θt)=∞ → θi + θt=π/2The incident angle, θi, when this occurs is denoted θp and is referred as polarization or Brewster angle (after David Brewster who found it in 1815).
n2 / n1 =1.5
n2 / n1 =1/1.5
For n2>n1 we have from Snell’s law θi > θt ,therefore r┴ is negative for all values of θi.
it
LawsSnell
i nnnit
θθθθθ
cossinsin 2
90
2
'
1
−=
==
pi θθ =
→1
2tann
np =θ
( )ti
it
ti
i
i
t
nn
n
E
Et
θθθθ
θθθµµ
+=
+=
=
=
⊥
⊥ sin
cossin2
coscos
cos2
21
1
0
021
( ) ( )titi
it
ti
i
i
t
nn
n
E
Et
θθθθθθ
θθθµµ
−+=
+=
=
=
cossin
cossin2
coscos
cos2
12
1
||0
0||
21
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
−=+−
=
=
=
⊥
⊥ sin
sin
coscos
coscos
21
21
0
021
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
=+−
=
=
=
tan
tan
coscos
coscos
12
12
||0
0||
21
32
SOLO REFLECTION & REFRACTION
Discussion of Fresnel Equations (continue)
n2 / n1 =1.5
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Brewster Angle
n1 < n2
33
SOLO REFLECTION & REFRACTION
Discussion of Fresnel Equations (continue)
n2 / n1 =1/1.5
Reflections and Refractions Laws Development Using the Electromagnetic Approach
1
2
sin
sin
n
n
t
i =θθ
Snell’s law
For n2<n1 we have from Snell’s law θi < θt therefore when θi increases,θt increases until it reaches 90°(no refraction and total reflection ).The incident angle when this occurs is denoted θic and is referred as the critical angle.
= −
1
21sinn
nicθCritical Angle
Brewster Angle
( )( ) 1
2/sin
2/sin21
2/0
0 =+−
−=
=
=
=⊥
⊥ πθπθµµ
πθi
i
i
r
tE
Er
( )( ) 1
2/tan
2/tan21
2/||0
0|| =
+−=
=
=
= πθπθµµ
πθi
i
i
r
tE
Er
n1 > n2
34
SOLO REFLECTION & REFRACTION
n1 > n2 Total Reflection
Reflections and Refractions Laws Development Using the Electromagnetic Approach
For n1>n2 and θi > θic we have total reflection.
1sinsin21
2
1nn
iticin
n >
>>=
θθθθFrom Snell’s law
therefore is no solution for θt , but we can use for
1sinsin1cos 2
2
2
12 −
−=−= itt n
ni θθθ
( )( ) 221
2
2
21
2
21
21
0
0
/sincos
/sincos
coscos
coscos21
nni
nni
nn
nn
E
Er
ii
ii
ti
ti
i
r
−−
−+=
+−
=
=
=
⊥
⊥
θθ
θθθθθθµµ
( ) ( )( ) ( ) 221
22
21
2
21
22
21
12
12
||0
0||
/sincos/
/sincos/
coscos
coscos21
nninn
nninn
nn
nn
E
Er
ii
ii
ti
ti
i
r
−−
−+=
+−
=
=
=
θθ
θθθθθθµµ
We can see that ( )⊥⊥ = ϕjr exp
( )|||| exp ϕjr =
( )i
i nn
θθϕcos
/sin
2tan
2
21
2 −=
⊥
( )( ) i
i
nn
nn
θθϕ
cos/
/sin
2tan
2
21
2
21
2
|| −=
n2 / n1 =1/1.5
35
SOLO REFLECTION & REFRACTION
Phase Shifts
( )ti
it
ti
i
i
t
nn
n
E
Et
θθθθ
θθθµµ
+=
+=
=
=
⊥
⊥ sin
cossin2
coscos
cos2
21
1
0
021
( ) ( )titi
it
ti
i
i
t
nn
n
E
Et
θθθθθθ
θθθµµ
−+=
+=
=
=
cossin
cossin2
coscos
cos2
12
1
||0
0||
21
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
−=+−
=
=
=
⊥
⊥ sin
sin
coscos
coscos
21
21
0
021
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
=+−
=
=
=
tan
tan
coscos
coscos
12
12
||0
0||
21
n2 / n1 =1/1.5
n2 / n1 =1.5
On can see that and are always positive, therefore is no phase difference between incidence and refracted waves.
⊥t||t
On the other hand and can be either positive or negative depending of the sign of (θi-θt).
⊥r||r
The phase change of the reflected wave, in the cases where refraction is possible, can be either π or 0, depending on whether the index n1 of the medium in which the wave originates is less or greater than n2 of the medium in which it travels.
36
SOLO REFLECTION & REFRACTION
Phase Shifts
||iE
⊥iE
⊥rE
ik
rk
tk
Boundary
21̂−n
z
x yiθ rθ
tθ
iθ rθ
tθ
tk
rk
ik
21̂−n
Boundary
Plan ofincidence
iE
| |iE
⊥iE
||rE
⊥rE
pi nn <
pi θθ =
pn
in
pn
in
||iE
⊥iE
⊥rE
ik
rk
tk
Boundary
21̂−n
z
x yiθ rθ
tθ
iθ rθ
tθ
tk
rk
ik
21̂−n
Boundary
Plan ofincidence
iE
| |iE
⊥iE
||rE
⊥rE
pi nn <
pi θθ >
pn
in
pn
in
||rE
rE
n2 / n1 =1.5
n1 < n2
The phase change of the reflected wave is:- π for ,- 0 for 0 ≤ θi ≤ θp, and π for θi > θp for
⊥r
||r
( )ti
it
ti
i
i
t
nn
n
E
Et
θθθθ
θθθµµ
+=
+=
=
=
⊥
⊥ sin
cossin2
coscos
cos2
21
1
0
021
( ) ( )titi
it
ti
i
i
t
nn
n
E
Et
θθθθθθ
θθθµµ
−+=
+=
=
=
cossin
cossin2
coscos
cos2
12
1
||0
0||
21
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
−=+−
=
=
=
⊥
⊥ sin
sin
coscos
coscos
21
21
0
021
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
=+−
=
=
=
tan
tan
coscos
coscos
12
12
||0
0||
21
37
SOLO REFLECTION & REFRACTION
Phase Shifts
n2 / n1 =1/1.5
n1 > n2
The phase change of the reflected wave is:-0 for 0 ≤ θi ≤ θc and changes
from 0 to π for θi > θc for- π for 0 ≤ θi ≤ θp, π for θp>θi > θc for
⊥r
||r
( )ti
it
ti
i
i
t
nn
n
E
Et
θθθθ
θθθµµ
+=
+=
=
=
⊥
⊥ sin
cossin2
coscos
cos2
21
1
0
021
( ) ( )titi
it
ti
i
i
t
nn
n
E
Et
θθθθθθ
θθθµµ
−+=
+=
=
=
cossin
cossin2
coscos
cos2
12
1
||0
0||
21
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
−=+−
=
=
=
⊥
⊥ sin
sin
coscos
coscos
21
21
0
021
( )( )ti
ti
ti
ti
i
r
nn
nn
E
Er
θθθθ
θθθθµµ
+−
=+−
=
=
=
tan
tan
coscos
coscos
12
12
||0
0||
21
38
SOLO REFLECTION & REFRACTION
Phase Shifts
||iE
⊥iE
⊥rE
ik
rk
tk
Boundary
21̂−n
z
x yiθ rθ
tθ
iθ rθ
tθ
tk
rk
ik
21̂−n
Boundary
Plan ofincidence
iE
| |iE
⊥iE
||rE
⊥rE
pi nn <
pi θθ =
pn
in
pn
in
||iE
⊥iE
⊥rE
ik
rk
tk
Boundary
21̂−n
z
x yiθ rθ
tθ
iθ rθ
tθ
tk
rk
ik
21̂−n
Boundary
Plan ofincidence
iE
| |iE
⊥iE
||rE
⊥rE
pi nn <
pi θθ >
pn
in
pn
in
||rE
rE
n1 > n2
n1 < n2
39
SOLO REFLECTION & REFRACTION
Energy Reflected and Refracted for Normal Polarization
201
12
ˆ⊥⊥
= ii
iE
n
ckS ε
Time Average Poynting Vectors (Irradiances) of the Planar waves are
201
12
ˆ⊥⊥
= rr
rE
n
ckS ε
202
22
ˆ⊥⊥
= tt
tE
n
ckS ε
2
2
2
1
1
2
2
2
1
1
20
20
coscos
coscos
ˆ
ˆ
+
−
==⋅
⋅=
⊥
⊥
⊥
⊥⊥
ti
ti
i
r
i
r
nn
nn
E
E
zS
zSR
θµ
θµ
θµ
θµ
2
2
2
1
1
21
12
2
1
1
2021
2012
coscos
coscos
cos2
cos
cos
ˆ
ˆ
+
==⋅
⋅=
⊥
⊥
⊥
⊥⊥
ti
i
ti
ii
tt
i
t
nn
nnn
En
En
zS
zST
θµ
θµ
θεθεθ
µθεθε
titi
cn
cn
i
ti
nn
n
nn
n
nn θθµµ
θθµεµ
εµµθε
θεθµ
coscos4coscos1
4cos
coscos2
21
21
2211
122
1
21
21
12
2
1
1
221
222
==
2
2
2
1
1
2
2
1
1
2021
2012
coscos
coscos4
cos
cos
ˆ
ˆ
+
==⋅
⋅=
⊥
⊥
⊥
⊥⊥
ti
ti
ii
tt
i
t
nn
nn
En
En
zS
zST
θµ
θµ
θθµµ
θεθε
Reflectance Transmittance
Reflections and Refractions Laws Development Using the Electromagnetic Approach
40
SOLO REFLECTION & REFRACTION
2
2
2
1
1
2
2
2
1
1
20
20
coscos
coscos
ˆ
ˆ
+
−
==⋅
⋅=
⊥
⊥
⊥
⊥⊥
ti
ti
i
r
i
r
nn
nn
E
E
zS
zSR
θµ
θµ
θµ
θµ
2
2
2
1
1
2
2
1
1
2021
2012
coscos
coscos4
cos
cos
ˆ
ˆ
+
==⋅
⋅=
⊥
⊥
⊥
⊥⊥
ti
ti
ii
tt
i
t
nn
nn
En
En
zS
zST
θµ
θµ
θθµµ
θεθε
Reflectance
Transmittance
We can see that
1=+ ⊥⊥ TR
Energy Reflected and Refracted for Normal Polarization
Reflections and Refractions Laws Development Using the Electromagnetic Approach
41
SOLO REFLECTION & REFRACTION
Energy Reflected and Refracted for Parallel Polarization
2||01
1|| 2
ˆi
i
iE
n
ckS ε=
Time Average Poynting vector of the Planar waves are
2||01
1|| 2
ˆr
r
rE
n
ckS ε=
2||02
2|| 2
ˆt
t
tE
n
ckS ε=
2
1
1
2
2
2
1
1
2
2
2||0
2||0
||
||
||
coscos
coscos
ˆ
ˆ
+
−
==⋅
⋅=
ti
ti
i
r
i
r
nn
nn
E
E
zS
zSR
θµ
θµ
θµ
θµ
2
1
1
2
2
21
12
2
1
1
2||021
2||012
||
||
||
coscos
coscos
cos2
cos
cos
ˆ
ˆ
+
==⋅
⋅=
ti
i
ti
ii
tt
i
t
nn
nnn
En
En
zS
zST
θµ
θµ
θεθεθ
µθεθε
titi
cn
cn
i
ti
nn
n
nn
n
nn θθµµ
θθµεµ
εµµθε
θεθµ
coscos4coscos1
4cos
coscos2
21
21
2211
122
1
2
1
21
12
2
1
1
221
222
==
2
1
1
2
2
2
2
1
1
2||021
2||012
||
||
||
coscos
coscos4
cos
cos
ˆ
ˆ
+
==⋅
⋅=
ti
ti
ii
tt
i
t
nn
nn
En
En
zS
zST
θµ
θµ
θθµµ
θεθε
Reflectance Transmittance
Reflections and Refractions Laws Development Using the Electromagnetic Approach
42
SOLO REFLECTION & REFRACTION
2
1
1
2
2
2
1
1
2
2
2||0
2||0
||
||
||
coscos
coscos
ˆ
ˆ
+
−
==⋅
⋅=
ti
ti
i
r
i
r
nn
nn
E
E
zS
zSR
θµ
θµ
θµ
θµ
Reflectance
Transmittance
2
1
1
2
2
2
2
1
1
2||021
2||012
||
||
||
coscos
coscos4
cos
cos
ˆ
ˆ
+
==⋅
⋅=
ti
ti
ii
tt
i
t
nn
nn
En
En
zS
zST
θµ
θµ
θθµµ
θεθε
We can see that
1|||| =+TR
Average Poynting vector of the Planar waves are
Reflections and Refractions Laws Development Using the Electromagnetic Approach
43
SOLO REFLECTION & REFRACTION
( )( )ti
ti
i
r
zS
zSR
θθθθµµ
+−
=⋅
⋅=
=
2
2
||
||
|| tan
tan
ˆ
ˆ21
( ) ( )titi
ti
i
t
zS
zST
θθθθθθµµ
−+=
⋅
⋅=
=
22
||
||
|| cossin
2sin2sin
ˆ
ˆ21
1|||| =+TR
Reflections and Refractions Laws Development Using the Electromagnetic Approach
( )( )ti
ti
i
r
zS
zSR
θθθθµµ
+−
=⋅
⋅=
=
⊥
⊥⊥ 2
2
sin
sin
ˆ
ˆ21
( )ti
ti
i
t
zS
zST
θθθθµµ
+=
⋅
⋅=
=
⊥
⊥⊥ 2sin
2sin2sin
ˆ
ˆ21
1=+ ⊥⊥ TR
Summary
44
SOLO REFLECTION & REFRACTION
Reflections and Refractions Laws Development Using the Electromagnetic Approach
||R⊥R
||R⊥R
45
Malus-Dupin TheoremSOLO
Étienne Louis Malus1775-1812
A surface passing through the end points of rays which have traveled equal optical pathlengths from a point object is called an optical wavefront.
1808 1812
If a group of ray is such that we can find a surface that is orthogonal to each and every one of them (this surface isthe wavefront), they are said to form a normal congruence.
The Malus-Dupin Theorem (introduced in 1808 by Malusand modified in 1812 by Dupin) states that:“The set of rays that are orthogonal to a wavefront remainnormal to a wavefront after any number of refraction or reflections.”
Charles Dupin1784-1873
Using Fermat principle[ ] [ ]'' BQBAVApathoptical ==
[ ] [ ] ( )2'' εOAVAAQA +=
VQ=ε is a small quantity [ ] [ ] ( )2'' εOBQBAQA +=
Since ray BQ is normal to wave W at B [ ] [ ] ( )2εOBQAQ +=[ ] [ ] ( )2'' εOQBQA += ray BQB’ is normal to wave W’ at B’
Proof for Refraction:
n 'n
P
Q
VAP’
A'
B B'
Wavefrontfrom P Wavefront
to P'
46
Stokes Treatment of Reflection and Refraction SOLO
An other treatment of reflection and refraction was given by Sir George Stokes.
Suppose we have an incident wave of amplitude E0i
reaching the boundary of two media (where n1 = ni and n2 = nt) at an angle θ1. The amplitudes of the reflected and transmitted (refracted) waves are, E0i·r and E0i·t, respectively (see Fig. a). Here r (θ1) and t (θ2) are the reflection and transmission coefficients.
According to Fermat’s Principle the situation where the rays direction is reversed (see Fig. b) is also permissible. Therefore we have two incident rays E0i·r in media with refraction index n1 and E0i·t in media with refraction index n2.E0i·r is reflected, in media with refraction index n1, to obtain a wave with amplitude (E0i·r )·t and refracted, in media with refraction index n2, to obtain a wave with amplitude (E0i·r )·r (see Fig. c).
E0i·t is reflected, in media with refraction index n2, to obtain a wave with amplitude (E0i·t )·r’ and refracted, in media with refraction index n1, to obtain a wave with amplitude (E0i·t )·t’ (see Fig. c).
θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
47
Stokes Treatment of Reflection and Refraction SOLO
To have Fig. c identical to Fig. b the following conditions must be satisfied:
( ) ( ) ( ) ( ) iii ErrEttE 0110120 ' =+ θθθθ( ) ( ) ( ) ( ) 0' 220210 =+ θθθθ rtEtrE ii
Hence:
( ) ( ) ( ) ( )( ) ( )12
1112
'
1'
θθθθθθ
rr
rrtt
−==+
Stokes relations
θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
48
Stokes Treatment of Reflection and Refraction SOLO
( ) ( ) ( ) ( )( ) ( )12
1112
'
1'
θθθθθθ
rr
rrtt
−==+
Stokes relations
Let check that Fresnel Equation do satisfy Stokes relations
( )2211
112 coscos
cos221
θθθθ
µµ
nn
nt
+==
⊥
2112
11|| coscos
cos221
θθθµµ
nn
nt
+==
( )2211
22111 coscos
coscos21
θθθθθ
µµ
nn
nnr
+−=
=
⊥
( )2112
21121|| coscos
coscos21
θθθθθ
µµ
nn
nnr
+−=
=
( )2211
221 coscos
cos2'
21
θθθθ
µµ
nn
nt
+==
⊥( )2211
11222 coscos
coscos'
21
θθθθθ
µµ
nn
nnr
+−=
=
⊥
1
( ) ( ) ( ) ( )( ) ( )12
1112
'
1'
θθθθθθ
⊥⊥
⊥⊥⊥⊥
−==+
rr
rrtt We can see that:
2
2112
22|| coscos
cos2'
21
θθθµµ
nn
nt
+==
( )2112
12211|| coscos
coscos'
21
θθθθθ
µµ
nn
nnr
+−=
=
( ) ( ) ( ) ( )( ) ( )1||2||
1||1||1||2||
'
1'
θθθθθθ
rr
rrtt
−==+
We can see that:
θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
49
SOLO REFLECTION & REFRACTION
Thinks to complete
Photons and Laws of Reflection & Refraction (Hecht & Zajac pp. 93)
http://physics.nad.ru/Physics/English/index.htm
50
ELECTROMAGNETICSSOLO
References
J.D. Jackson, “Classical Electrodynamics”, 3rd Ed., John Wiley & Sons, 1999
R. S. Elliott, “Electromagnetics”, McGraw-Hill, 1966
J.A. Stratton, “Electromagnetic Theory”, McGraw-Hill, 1941
W.K.H. Panofsky, M. Phillips, “Classical Electricity and Magnetism”, Addison-Wesley, 1962
F.T. Ulaby, R.K. More, A.K. Fung, “Microwave Remote Sensors Active and Passive”, Addson-Wesley, 1981
A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”,John Wiley & Sons, 1988
51
SOLO
References
Foundation of Geometrical Optics
[1] M. Born, E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation, Interference and Diffraction of Light”, 6th Ed., Pergamon Press, 1980, Ch. 3 and App. 1
[2] C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996, pp. 692-694
52
SOLO
References
Foundation of Geometrical Optics
[3] E.Hecht, A. Zajac, “Optics ”, 3th Ed., Addison Wesley Publishing Company, 1997,
[4] M.V. Klein, T.E. Furtak, “Optics ”, 2nd Ed., John Wiley & Sons, 1986
53
ELECTROMAGNETICSSOLO
References
1. W.K.H. Panofsky & M. Phillips, “Classical Electricity and Magnetism”,
2. J.D. Jackson, “Classical Electrodynamics”,
3. R.S. Elliott, “Electromagnetics”,
4. A.L. Maffett, “Topics for a Statistical Description of Radar Cross Section”,
54
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions
( ) ( ) ldtHtHhldtHldtHldHh
C
2211
0
2211ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅
→→
∫
where are unit vectors along C in region (1) and (2), respectively, and 21ˆ,ˆ tt
2121 ˆˆˆˆ−×=−= nbtt
- a unit vector normal to the boundary between region (1) and (2)21ˆ −n- a unit vector on the boundary and normal to the plane of curve Cb̂
Using we obtainbaccba ⋅×≡×⋅
( ) ( ) ( )[ ] ldbkldbHHnldnbHHldtHH eˆˆˆˆˆˆ
21212121121 ⋅=⋅−×=×⋅−=⋅− −−
Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have:
b̂
( ) ekHHn
=−×− 2121ˆ
∫∫∫ ⋅
∂∂+=⋅
→
S
e
C
Sdt
DJdlH
( ) dlbkbdlht
DJSd
t
DJ e
h
e
S
eˆˆ
0
⋅=⋅
∂∂+=⋅
∂∂+
→
∫∫
AMPÈRE’S LAW
[ ]1
0lim: −
→⋅
∂∂+= mAht
DJk e
he
55
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 1)
( ) ( ) ldtEtEhldtEldtEldEh
C
2211
0
2211ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅
→→
∫
where are unit vectors along C in region (1) and (2), respectively, and 21ˆ,ˆ tt
2121 ˆˆˆˆ−×=−= nbtt
- a unit vector normal to the boundary between region (1) and (2)21ˆ −n- a unit vector on the boundary and normal to the plane of curve Cb̂
Using we obtainbaccba ⋅×≡×⋅
( ) ( ) ( )[ ] ldbkldbEEnldnbEEldtEE mˆˆˆˆˆˆ
21212121121 ⋅−=⋅−×=×⋅−=⋅− −−
Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have:
b̂
( ) mkEEn
−=−×− 2121ˆ
∫∫∫ ⋅
∂∂+−=⋅
→
S
m
C
Sdt
BJdlE
( ) dlbkbdlht
BJSd
t
BJ m
h
m
S
mˆˆ
0
⋅=⋅
∂∂+=⋅
∂∂+
→
∫∫
FARADAY’S LAW
[ ]1
0lim: −
→⋅
∂∂+= mVht
BJk m
hm
56
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 2)
( ) ( ) SdnDnDhSdnDSdnDSdDh
S
2211
0
2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅→
∫∫
where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and
21 ˆ,ˆ nn
2121 ˆˆˆ −=−= nnn
- a unit vector normal to the boundary between region (1) and (2)21ˆ −n
( ) ( ) SdSdnDDSdnDD eσ=⋅−=⋅− −2121121 ˆˆ
Since this must be true for any dS on the boundary between regions (1) and (2) we must have:
( ) eDDn σ=−⋅− 2121ˆ
( ) dSdShdv e
h
e
V
e σρρ0→
==∫∫∫
GAUSS’ LAW - ELECTRIC
[ ]1
0lim: −
→⋅⋅= msAhe
he ρσ
∫∫∫∫∫ =•V
e
S
dvSdD ρ
57
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 3)
( ) ( ) SdnBnBhSdnBSdnBSdBh
S
2211
0
2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅→
∫∫
where are unit vectors normal to boundary pointing in region (1) and (2), respectively, and
21 ˆ,ˆ nn
2121 ˆˆˆ −=−= nnn
- a unit vector normal to the boundary between region (1) and (2)21ˆ −n
( ) ( ) SdSdnBBSdnBB mσ=⋅−=⋅− −2121121 ˆˆ
Since this must be true for any dS on the boundary between regions (1) and (2) we must have:
( ) mBBn σ=−⋅− 2121ˆ
( ) dSdShdv m
h
m
V
m σρρ0→
==∫∫∫
GAUSS’ LAW – MAGNETIC
[ ]1
0lim: −
→⋅⋅= msVhm
hm ρσ
∫∫∫∫∫ =•V
m
S
dvSdB ρ
58
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (summary)
( ) mkEEn
−=−×− 2121ˆ FARADAY’S LAW
( ) ekHHn
=−×− 2121ˆ AMPÈRE’S LAW [ ]1
0lim: −
→⋅
∂∂+= mAht
DJk e
he
[ ]1
0lim: −
→⋅
∂∂+= mVht
BJk m
hm
( ) eDDn σ=−⋅− 2121ˆ GAUSS’ LAW
ELECTRIC [ ]1
0lim: −
→⋅⋅= msAhe
he ρσ
( ) mBBn σ=−⋅− 2121ˆ GAUSS’ LAW
MAGNETIC [ ]1
0lim: −
→⋅⋅= msVhm
hm ρσ
Fresnel Equations
January 4, 2015 59
SOLO
TechnionIsraeli Institute of Technology
1964 – 1968 BSc EE1968 – 1971 MSc EE
Israeli Air Force1970 – 1974
RAFAELIsraeli Armament Development Authority
1974 – 2013
Stanford University1983 – 1986 PhD AA