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1
REFLECTION
&
REFRACTION
SOLO HERMELIN
Updated: 4.11.06http://www.solohermelin.com
2
SOLO
TABLE OF CONTENT
REFLECTION & REFRACTION
History of Reflection and Refraction
Huygens Principle
Reflections Laws Development Using Huygens Principle
Refractions 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 Theorem
Stokes Treatment of Reflection and Refraction
References
3
REFLECTION & REFRACTIONSOLO
Introduction
4
REFLECTION & REFRACTIONSOLO
History of Reflection & Refraction
100-170A.D.
Claudius Ptolemey
Alexandria
“Optics” 130 A.D.
Tabulated angle of incidence and refraction for
several media.
c. 300 B.C.
Euclid
Greece
“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 & REFRACTIONSOLO
History of Reflection & Refraction
Willebrord van
Roijen Snell
1580-1626
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
=
θ
θ
correct
1
2
sin
sin
n
n
t
i
=
θ
θ
Descartes deduced
wrong
6
REFLECTION & REFRACTIONSOLO
History of Reflection & Refraction
Pierre de Fermat
1601-1665
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 Huygens
1629-1695
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 & REFRACTIONSOLO
Huygens Principle
Christiaan Huygens
1629-1695
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 & REFRACTIONSOLO
Reflection Laws Development Using Huygens Principle
Suppose a planar incident wave
AB is moving toward the boundary
AC between two media. The velocity
of 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 is
reaching the boundary at C, the
reflected ray at A will reach D and
the 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 & REFRACTIONSOLO
Refraction Laws Development Using Huygens Principle
Suppose a planar incident wave
AB is moving toward the boundary
AC between two media. The velocity
of 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 is
reaching the boundary at C, the
refracted ray at A will reach E and
the 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
==
θ
θ
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
21
21 =⋅







− rd
sd
rd
n
sd
rd
n
rayray 

In this case we have and21 nn =
( ) ( )
( ) 0ˆˆ
21
21 =⋅−=⋅







− 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
21
21 =⋅







− rd
sd
rd
n
sd
rd
n
rayray 

where is on the boundary between the two regions andrd
 ( ) ( )
sd
rd
s
sd
rd
s
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 − 21
ˆ −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&ˆˆ
0 =⋅== kkknkkk εµω








=⋅∇
=⋅∇
−=×∇
=×∇
0
0
H
E
HjE
EjH
ωµ
ωε











=⋅
=⋅
=×
−=×
0ˆ
0ˆ
ˆ
ˆ
0
0
00
00
Hk
Ek
HEk
EHk
ε
µ
µ
ε







=⋅−
=⋅−
−=×−
=×−
⇒
⋅−
⋅−
⋅−⋅−
⋅−⋅−
−=∇ ⋅−⋅−
0
0
0
0
00
00
rkj
rkj
rkjrkj
rkjrkj
ekje
eHkj
eEkj
eHjeEkj
eEjeHkj
rkjrkj










ωµ
ωε
( ) ∗∗
⋅==⋅==+= HHwEEwwcn
k
wwcn
k
S meme
22
&
2
ˆ
2
ˆ µε
Time 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
n
k
eEkH
eEE
iiii
rktj
iii
rktj
ii
ii
ii
1
00
11
0011
0
0
ω
εµ
εµ
εµωεµω
µ
ε ω
ω
===
×=
=
⋅−
⋅−




The monochromatic planar
reflected wave from the boundary is
( )
( )
1
1
1
1
0
0
&
n
c
v
vc
n
k
eEkH
eEE
r
rr
rktj
rrr
rktj
rr
rr
rr
===
×=
=
⋅−
⋅−
ω
ω
µ
ε ω
ω




The monochromatic planar
refracted wave from the boundary is
( )
( )
2
2
2
2
0
0
&
n
c
v
vc
n
k
eEkH
eEE
t
tt
rktj
ttt
rktj
tt
tt
tt
===
×=
=
⋅−
⋅−
ω
ω
µ
ε 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 points
on the plane at all times, implies
that the spatial and time
variations of
This implies that
Phase-Matching Conditions
( ) ( ) ( ) yxteEeEeE
z
rktj
t
z
rktj
r
z
rktj
i
ttrrii
,,,,
0
0
0
0
0
0 ∀
=
⋅−
=
⋅−
=
⋅−
  ωωω
( ) ( ) ( ) yxtrktrktrkt
z
tt
z
rr
z
ii ,,
000
∀⋅−=⋅−=⋅−
===

ωωω
ttri ∀=== ωωωω
( ) ( ) ( ) yxrkrkrk
z
t
z
r
z
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
( )
( )





−+=
++=
zyx
c
n
k
zyx
c
n
k
ttttttt
irirrrr
ˆcossinˆsinsinˆcos
ˆcossinˆsinsinˆcos
2
1
θαθααω
θαθααω


( )
( )








+=⋅
+=⋅
=⋅
=
=
=
yyx
c
n
rk
yx
c
n
rk
y
c
n
rk
ttt
z
t
irr
z
r
i
z
i
ˆsinsincos
sinsincos
sin
2
0
1
0
1
0
θααω
θααω
θω



( ) ( ) ( ) yxrkrkrk
z
t
z
r
z
i ,
000
∀⋅=⋅=⋅
===

2
π
αα == tr
ttri ∀=== ωωωω
x∀
y∀
Coplanar
Snell’s Law
( )




++=
−=
zzyyxxr
zy
c
n
k 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
εµ
εµ
θ
θ
===
v
v
n
n
t
iRefraction
Law
Reflection
Law
Phase-Matching Conditions
( )




++=
−=
zzyyxxr
zy
c
n
k iiii
ˆˆˆ
ˆcosˆsin1


θθω
( )
( )





−+=
++=
zyx
c
n
k
zyx
c
n
k
ttttttt
irirrrr
ˆcossinˆsinsinˆcos
ˆcossinˆsinsinˆcos
2
1
θαθααω
θαθααω


( ) ( )[ ]
( ) ( )[ ]





−+−=−×
−+−=−×
ynnyn
c
kkz
ynnyn
c
kkz
ittrti
irrrri
ˆsinsinsinˆcosˆ
ˆsinsinsinˆcosˆ
122
111
θθαα
ω
θθαα
ω


ttri ∀=== ωωωω
We can see that
( ) ( )








===
=−×=−×
ωωωω tri
tiri kkzkkz 0ˆˆ











===
==
==
ωωωω
θθθ
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
εµ
εµ
θ
θ
===
v
v
n
n
t
iRefraction
Law
Reflection
Law
Phase-Matching Conditions (Summary)
ttri ∀=== ωωωω
( ) ( )








===
=−×=−×
ωωωω tri
tiri kkzkkz 0ˆˆ











===
==
==
ωωωω
θθθ
παα
tri
tri
tr
nnn sinsinsin
2/
211
( ) ( ) ( ) yxrkrkrk
z
t
z
r
z
i ,
000
∀⋅=⋅=⋅
===

( ) ( ) ( ) yxtrktrktrkt
z
tt
z
rr
z
ii ,,
000
∀⋅−=⋅−=⋅−
===

ωωω
Vector
Notation
Scalar
Notation
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 a
source-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
2
2
1
1
1
21
µ
ε
µ
ε
Reflections and Refractions Laws Development Using the Electromagnetic Approach
Fresnel Equations
Boundary
conditions
20
SOLO REFLECTION & REFRACTION
( ) 0ˆ 00021

=−+×− tri EEEn
0
111
ˆ 0
2
0
1
0
1
21

=





×−×+××− ttrrii EkEkEkn
µµµ
( ) 0ˆ 02010121 =−+⋅− tri EEEn

εεε
( ) 0ˆ 00021 =×−×+×⋅− ttrrii EkEkEkn

Using
,ˆ,ˆ,ˆˆ
221111
1
ttrriii kkkkkk
c
n
k 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 0
2
2
00
1
1
=−− ttirii EEE θ
µ
ε
θθ
µ
ε
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
θθ
θεµθθεµ4
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
( ) 0cos
1
cos
1
0
00
22
2
00
00
11
1
21
=−− tt
n
iri
n
EEE θ
εµ
εµ
µ
θ
εµ
εµ
µ

1
0000 =−+ tri EEE2
From and
ti
ti
i
r
nn
nn
E
E
r
θ
µ
θ
µ
θ
µ
θ
µ
coscos
coscos
2
2
1
1
2
2
1
1
0
0
+
−
=





=
⊥
⊥
ti
i
i
t
nn
n
E
E
t
θ
µ
θ
µ
θ
µ
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
=
θ
θ
( )
( )ti
ti
i
r
E
E
r
θθ
θθ
+
−
−=





=
⊥
⊥
sin
sin
0
0
( )ti
it
i
t
E
E
t
θθ
θθ
+
=





=
⊥
⊥
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
E
r
θ
µ
θ
µ
θ
µ
θ
µ
coscos
coscos
2
2
1
1
2
2
1
1
0
0
+
−
=





=
⊥
⊥
ti
i
i
t
nn
n
E
E
t
θ
µ
θ
µ
θ
µ
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
=
θ
θ
( )
( )ti
ti
i
r
E
E
r
θθ
θθµµ
+
−
−=





=
=
⊥
⊥
sin
sin21
0
0
( )ti
it
i
t
E
E
t
θθ
θθµµ
+
=





=
=
⊥
⊥
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
( ) ( ) 0
11
0ˆ 0
00
22
2
00
00
11
1
0
2
2
00
1
1
21
=−+=





−+ t
n
ri
n
tri EEEoryEEE


µε
µε
µµε
µε
µµ
ε
µ
ε
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
11
2211
0
2
2
00
1
1
=−+
=⇒=
=





−+
ttiri
t
i
ti
tri
EEE
yEEE
θεθε
θε
θε
µε
µε
θεµθεµ
µ
ε
µ
ε 
1
Identical to 3
We have two independent equations
( ) 0coscos 000 =−− ttiri EEE θθ2
( ) 00
2
2
00
1
1
=−+ tri E
n
EE
n
µµ
1 ti
ti
i
r
nn
nn
E
E
r
θ
µ
θ
µ
θ
µ
θ
µ
coscos
coscos
1
1
2
2
1
1
2
2
||0
0
||
+
−
=





=
ti
i
i
t
nn
n
E
E
t
θ
µ
θ
µ
θ
µ
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
E
r
θ
µ
θ
µ
θ
µ
θ
µ
coscos
coscos
1
1
2
2
1
1
2
2
||0
0
||
+
−
=





=
ti
i
i
t
nn
n
E
E
t
θ
µ
θ
µ
θ
µ
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
=
θ
θ
( )
( )ti
ti
i
r
E
E
r
θθ
θθµµ
+
−
=





=
=
tan
tan21
||0
0
||
( ) ( )titi
it
i
t
E
E
t
θθθθ
θθµµ
−+
=





=
=
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
E
r
θ
µ
θ
µ
θ
µ
θ
µ
coscos
coscos
1
1
2
2
1
1
2
2
||0
0
||
+
−
=







=
ti
i
i
t
nn
n
E
E
t
θ
µ
θ
µ
θ
µ
coscos
cos2
1
1
2
2
1
1
||0
0
||
+
=







=
ti
ti
i
r
nn
nn
E
E
r
θ
µ
θ
µ
θ
µ
θ
µ
coscos
coscos
2
2
1
1
2
2
1
1
0
0
+
−
=







=
⊥
⊥
ti
i
i
t
nn
n
E
E
t
θ
µ
θ
µ
θ
µ
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 Lorentz
1853-1928
30
SOLO REFLECTION & REFRACTION
Discussion of Fresnel Equations
it
n
ni
θθ
θ
2
1
0→
=
0→iθ
[ ] [ ]
ti
ti
nn
nn
rr ii
θθ
θθ
θθ
+
−
=
+
−
=−= =⊥=
21
12
00||
[ ] [ ]
ti
t
nn
n
tt ii
θθ
θ
θθ
+
=
+
== =⊥=
22
21
1
00||
1
2
sin
sin
n
n
t
i
=
θ
θ
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
E
t
θθ
θθ
θθ
θµµ
+
=
+
=







=
=
⊥
⊥
sin
cossin2
coscos
cos2
21
1
0
0
21
( ) ( )titi
it
ti
i
i
t
nn
n
E
E
t
θθθθ
θθ
θθ
θµµ
−+
=
+
=







=
=
cossin
cossin2
coscos
cos2
12
1
||0
0
||
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
−=
+
−
=







=
=
⊥
⊥
sin
sin
coscos
coscos
21
21
0
0
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
=
+
−
=







=
=
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
=
θ
θ
Snell’s law
Reflections and Refractions Laws Development Using the Electromagnetic Approach
David Brewster
1781-1868
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/2
The 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
nnn
it
θθθ
θθ
cossinsin 2
90
2
'
1
−=
==

pi θθ =
→
1
2
tan
n
n
p
=θ
( )ti
it
ti
i
i
t
nn
n
E
E
t
θθ
θθ
θθ
θµµ
+
=
+
=







=
=
⊥
⊥
sin
cossin2
coscos
cos2
21
1
0
0
21
( ) ( )titi
it
ti
i
i
t
nn
n
E
E
t
θθθθ
θθ
θθ
θµµ
−+
=
+
=







=
=
cossin
cossin2
coscos
cos2
12
1
||0
0
||
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
−=
+
−
=







=
=
⊥
⊥
sin
sin
coscos
coscos
21
21
0
0
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
=
+
−
=







=
=
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
=
θ
θ
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
21
sin
n
n
icθCritical Angle
Brewster Angle
( )
( )
1
2/sin
2/sin21
2/
0
0
=
+
−
−=







=
=
=
⊥
⊥
πθ
πθµµ
πθ
i
i
i
r
t
E
E
r
( )
( )
1
2/tan
2/tan21
2/
||0
0
||
=
+
−
=







=
=
=
πθ
πθµµ
πθ
i
i
i
r
t
E
E
r
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.
1sinsin
21
2
1
nn
it
ici
n
n >
>
>=
θθ
θθFrom Snell’s law
therefore is no solution for θt , but we can use for
1sinsin1cos 2
2
2
12
−





−=−= itt
n
n
i θθθ
( )
( )2
21
2
2
21
2
21
21
0
0
/sincos
/sincos
coscos
coscos21
nni
nni
nn
nn
E
E
r
ii
ii
ti
ti
i
r
−−
−+
=
+
−
=







=
=
⊥
⊥
θθ
θθ
θθ
θθµµ
( ) ( )
( ) ( )2
21
22
21
2
21
22
21
12
12
||0
0
||
/sincos/
/sincos/
coscos
coscos21
nninn
nninn
nn
nn
E
E
r
ii
ii
ti
ti
i
r
−−
−+
=
+
−
=







=
=
θθ
θθ
θθ
θθµµ
We can see that
( )⊥⊥
= ϕjr exp
( )|||| exp ϕjr =
( )
i
i nn
θ
θϕ
cos
/sin
2
tan
2
21
2
−
=




 ⊥
( )
( ) i
i
nn
nn
θ
θϕ
cos/
/sin
2
tan 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
E
t
θθ
θθ
θθ
θµµ
+
=
+
=







=
=
⊥
⊥
sin
cossin2
coscos
cos2
21
1
0
0
21
( ) ( )titi
it
ti
i
i
t
nn
n
E
E
t
θθθθ
θθ
θθ
θµµ
−+
=
+
=







=
=
cossin
cossin2
coscos
cos2
12
1
||0
0
||
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
−=
+
−
=







=
=
⊥
⊥
sin
sin
coscos
coscos
21
21
0
0
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
=
+
−
=







=
=
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
||i
E

⊥i
E

⊥rE

ik

rk

tk

Boundary
21ˆ −n
z
x y
iθ rθ
tθ
iθ rθ
tθ
tk

rk

ik

21ˆ −n
Boundary
Plan of
incidence
iE

||i
E

⊥i
E

||rE

⊥r
E

pi nn <
pi θθ =
p
n
i
n
p
n
in
||iE

⊥i
E

⊥rE

ik

rk

tk

Boundary
21ˆ −n
z
x y
iθ rθ
tθ
iθ rθ
tθ
tk

rk

ik

21
ˆ −n
Boundary
Plan of
incidence
iE

||iE

⊥i
E

||rE

⊥r
E

pi nn <
pi
θθ >
pn
in
p
n
i
n
||r
E

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
E
t
θθ
θθ
θθ
θµµ
+
=
+
=







=
=
⊥
⊥
sin
cossin2
coscos
cos2
21
1
0
0
21
( ) ( )titi
it
ti
i
i
t
nn
n
E
E
t
θθθθ
θθ
θθ
θµµ
−+
=
+
=







=
=
cossin
cossin2
coscos
cos2
12
1
||0
0
||
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
−=
+
−
=







=
=
⊥
⊥
sin
sin
coscos
coscos
21
21
0
0
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
=
+
−
=







=
=
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
E
t
θθ
θθ
θθ
θµµ
+
=
+
=







=
=
⊥
⊥
sin
cossin2
coscos
cos2
21
1
0
0
21
( ) ( )titi
it
ti
i
i
t
nn
n
E
E
t
θθθθ
θθ
θθ
θµµ
−+
=
+
=







=
=
cossin
cossin2
coscos
cos2
12
1
||0
0
||
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
−=
+
−
=







=
=
⊥
⊥
sin
sin
coscos
coscos
21
21
0
0
21
( )
( )ti
ti
ti
ti
i
r
nn
nn
E
E
r
θθ
θθ
θθ
θθµµ
+
−
=
+
−
=







=
=
tan
tan
coscos
coscos
12
12
||0
0
||
21
38
SOLO REFLECTION & REFRACTION
Phase Shifts
||i
E

⊥iE

⊥rE

ik

rk

tk

Boundary
21
ˆ −n
z
x y
iθ rθ
tθ
iθ rθ
tθ
tk

rk

ik

21
ˆ −n
Boundary
Plan of
incidence
i
E

||i
E

⊥iE

||r
E

⊥rE

pi nn <
pi θθ =
pn
in
pn
i
n
||i
E

⊥iE

⊥rE

ik

rk

tk

Boundary
21
ˆ −n
z
x y
iθ rθ
tθ
iθ rθ
tθ
tk

rk

ik

21
ˆ −n
Boundary
Plan of
incidence
iE

||iE

⊥iE

||r
E

⊥rE

pi
nn <
pi
θθ >
p
n
in
pn
i
n
||r
E

rE

n1 > n2
n1 < n2
39
SOLO REFLECTION & REFRACTION
Energy Reflected and Refracted for Normal Polarization
2
01
12
ˆ
⊥
⊥
= i
i
i
E
n
ck
S ε

Time Average Poynting Vectors (Irradiances) of the Planar waves are
2
01
12
ˆ
⊥
⊥
= r
r
r
E
n
ck
S ε
 2
02
22
ˆ
⊥
⊥
= t
t
t
E
n
ck
S ε

2
2
2
1
1
2
2
2
1
1
2
0
2
0
coscos
coscos
ˆ
ˆ






+






−
==
⋅
⋅
=
⊥
⊥
⊥
⊥
⊥
ti
ti
i
r
i
r
nn
nn
E
E
zS
zS
R
θ
µ
θ
µ
θ
µ
θ
µ


2
2
2
1
1
21
12
2
1
1
2
021
2
012
coscos
cos
cos
cos2
cos
cos
ˆ
ˆ






+






==
⋅
⋅
=
⊥
⊥
⊥
⊥
⊥
ti
i
t
i
ii
tt
i
t
nn
n
nn
En
En
zS
zS
T
θ
µ
θ
µ
θε
θε
θ
µ
θε
θε




titi
cn
cn
i
t
i
nn
n
nn
n
nn
θθ
µµ
θθ
µεµ
εµ
µθε
θε
θ
µ
coscos4coscos
1
4
cos
cos
cos2
21
21
2211
122
1
2
1
21
12
2
1
1
22
1
22
2
==





2
2
2
1
1
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
zS
T
θ
µ
θ
µ
θθ
µµ
θε
θε


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
2
0
2
0
coscos
coscos
ˆ
ˆ






+






−
==
⋅
⋅
=
⊥
⊥
⊥
⊥
⊥
ti
ti
i
r
i
r
nn
nn
E
E
zS
zS
R
θ
µ
θ
µ
θ
µ
θ
µ


2
2
2
1
1
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
zS
T
θ
µ
θ
µ
θθ
µµ
θε
θε


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
ˆ
i
i
i
E
n
ck
S ε=

Time Average Poynting vector of the Planar waves are
2
||01
1
|| 2
ˆ
r
r
r
E
n
ck
S ε=
 2
||02
2
|| 2
ˆ
t
t
t
E
n
ck
S 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
zS
R
θ
µ
θ
µ
θ
µ
θ
µ


2
1
1
2
2
21
12
2
1
1
2
||021
2
||012
||
||
||
coscos
cos
cos
cos2
cos
cos
ˆ
ˆ






+






==
⋅
⋅
=
ti
i
t
i
ii
tt
i
t
nn
n
nn
En
En
zS
zS
T
θ
µ
θ
µ
θε
θε
θ
µ
θε
θε




titi
cn
cn
i
t
i
nn
n
nn
n
nn
θθ
µµ
θθ
µεµ
εµ
µθε
θε
θ
µ
coscos4coscos
1
4
cos
cos
cos2
21
21
2211
122
1
2
1
21
12
2
1
1
22
1
22
2
==





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
zS
T
θ
µ
θ
µ
θθ
µµ
θε
θε


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
zS
R
θ
µ
θ
µ
θ
µ
θ
µ


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
zS
T
θ
µ
θ
µ
θθ
µµ
θε
θε


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
zS
R
θθ
θθµµ
+
−
=
⋅
⋅
=
=
2
2
||
||
||
tan
tan
ˆ
ˆ 21


( ) ( )titi
ti
i
t
zS
zS
T
θθθθ
θθµµ
−+
=
⋅
⋅
=
=
22
||
||
||
cossin
2sin2sin
ˆ
ˆ 21

 1|||| =+TR
Reflections and Refractions Laws Development Using the Electromagnetic Approach
( )
( )ti
ti
i
r
zS
zS
R
θθ
θθµµ
+
−
=
⋅
⋅
=
=
⊥
⊥
⊥ 2
2
sin
sin
ˆ
ˆ 21


( )ti
ti
i
t
zS
zS
T
θθ
θθµµ
+
=
⋅
⋅
=
=
⊥
⊥
⊥ 2
sin
2sin2sin
ˆ
ˆ 21


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 Malus
1775-1812
A surface passing through the end points of rays which have traveled equal optical path
lengths 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 is
the wavefront), they are said to form a normal congruence.
The Malus-Dupin Theorem (introduced in 1808 by Malus
and modified in 1812 by Dupin) states that:
“The set of rays that are orthogonal to a wavefront remain
normal to a wavefront after any number of refraction or reflections.”
Charles Dupin
1784-1873
Using Fermat principle
[ ] [ ]'' BQBAVApathoptical ==
[ ] [ ] ( )2
'' εOAVAAQA +=
VQ=ε is a small quantity
[ ] [ ] ( )2
'' εOBQBAQA +=
Since ray BQ is normal to wave W at B
[ ] [ ] ( )2
εOBQAQ +=
[ ] [ ] ( )2
'' εOQBQA += ray BQB’ is normal to wave W’ at B’
Proof for Refraction:
n
'n
P
Q
VA
P’
A'
B B'
Wavefront
from P Wavefront
to P'
46
Stokes Treatment of Reflection and RefractionSOLO
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 RefractionSOLO
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'
θθ
θθθθ
rr
rrtt
−=
=+
Stokes relations
θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
48
Stokes Treatment of Reflection and RefractionSOLO
( ) ( ) ( ) ( )
( ) ( )12
1112
'
1'
θθ
θθθθ
rr
rrtt
−=
=+
Stokes relations
Let check that Fresnel Equation do satisfy Stokes relations
( )
2211
11
2
coscos
cos221
θθ
θ
θ
µµ
nn
n
t
+
=
=
⊥
2112
11
||
coscos
cos221
θθ
θµµ
nn
n
t
+
=
=
( )
2211
2211
1
coscos
coscos21
θθ
θθ
θ
µµ
nn
nn
r
+
−
=
=
⊥
( )
2112
2112
1||
coscos
coscos21
θθ
θθ
θ
µµ
nn
nn
r
+
−
=
=
( )
2211
22
1
coscos
cos2
'
21
θθ
θ
θ
µµ
nn
n
t
+
=
=
⊥
( )
2211
1122
2
coscos
coscos
'
21
θθ
θθ
θ
µµ
nn
nn
r
+
−
=
=
⊥
1
( ) ( ) ( ) ( )
( ) ( )12
1112
'
1'
θθ
θθθθ
⊥⊥
⊥⊥⊥⊥
−=
=+
rr
rrtt
We can see that:
2
2112
22
||
coscos
cos2
'
21
θθ
θµµ
nn
n
t
+
=
=
( )
2112
1221
1||
coscos
coscos
'
21
θθ
θθ
θ
µµ
nn
nn
r
+
−
=
=
( ) ( ) ( ) ( )
( ) ( )1||2||
1||1||1||2||
'
1'
θθ
θθθθ
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
( ) ( ) ldtHtHhldtHldtHldH
h
C
2211
0
2211
ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅
→→
∫

where are unit vectors along C in region (1) and (2), respectively, and21
ˆ,ˆ tt
2121
ˆˆˆˆ −×=−= nbtt
- a unit vector normal to the boundary between region (1) and (2)21
ˆ −n
- a unit vector on the boundary and normal to the plane of curve Cbˆ
Using we obtainbaccba

⋅×≡×⋅
( ) ( ) ( )[ ] ldbkldbHHnldnbHHldtHH e
ˆˆˆˆˆˆ 21212121121 ⋅=⋅−×=×⋅−=⋅− −−

Since this must be true for any vector that lies on the boundary between
regions (1) and (2) we must have:
bˆ
( ) ekHHn

=−×− 2121
ˆ
∫∫∫ ⋅







∂
∂
+=⋅
→
S
e
C
Sd
t
D
JdlH



( ) dlbkbdlh
t
D
JSd
t
D
J e
h
e
S
e
ˆˆ
0
⋅=⋅







∂
∂
+=⋅







∂
∂
+
→
∫∫





AMPÈRE’S LAW
[ ]1
0
lim: −
→
⋅







∂
∂
+= mAh
t
D
Jk e
h
e


55
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 1)
( ) ( ) ldtEtEhldtEldtEldE
h
C
2211
0
2211
ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅
→→
∫

where are unit vectors along C in region (1) and (2), respectively, and21
ˆ,ˆ tt
2121
ˆˆˆˆ −×=−= nbtt
- a unit vector normal to the boundary between region (1) and (2)21
ˆ −n
- a unit vector on the boundary and normal to the plane of curve Cbˆ
Using we obtainbaccba

⋅×≡×⋅
( ) ( ) ( )[ ] ldbkldbEEnldnbEEldtEE m
ˆˆˆˆˆˆ 21212121121 ⋅−=⋅−×=×⋅−=⋅− −−

Since this must be true for any vector that lies on the boundary between
regions (1) and (2) we must have:
bˆ
( ) mkEEn

−=−×− 2121
ˆ
∫∫∫ ⋅







∂
∂
+−=⋅
→
S
m
C
Sd
t
B
JdlE



( ) dlbkbdlh
t
B
JSd
t
B
J m
h
m
S
m
ˆˆ
0
⋅=⋅







∂
∂
+=⋅







∂
∂
+
→
∫∫





FARADAY’S LAW
[ ]1
0
lim: −
→
⋅







∂
∂
+= mVh
t
B
Jk m
h
m


56
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 2)
( ) ( ) SdnDnDhSdnDSdnDSdD
h
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)21
ˆ −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=−⋅− 2121
ˆ

( ) dSdShdv e
h
e
V
e σρρ
0→
==∫∫∫
GAUSS’ LAW - ELECTRIC
[ ]1
0
lim: −
→
⋅⋅= msAhe
h
e ρσ
∫∫∫∫∫ =•
V
e
S
dvSdD ρ

57
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (continue – 3)
( ) ( ) SdnBnBhSdnBSdnBSdB
h
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)21
ˆ −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=−⋅− 2121
ˆ

( ) dSdShdv m
h
m
V
m σρρ
0→
==∫∫∫
GAUSS’ LAW – MAGNETIC
[ ]1
0
lim: −
→
⋅⋅= msVhm
h
m ρσ
∫∫∫∫∫ =•
V
m
S
dvSdB ρ

58
SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS
Boundary Conditions (summary)
( ) mkEEn

−=−×− 2121
ˆ FARADAY’S LAW
( ) ekHHn

=−×− 2121
ˆ AMPÈRE’S LAW [ ]1
0
lim: −
→
⋅







∂
∂
+= mAh
t
D
Jk e
h
e


[ ]1
0
lim: −
→
⋅







∂
∂
+= mVh
t
B
Jk m
h
m


( ) eDDn σ=−⋅− 2121
ˆ
 GAUSS’ LAW
ELECTRIC
[ ]1
0
lim: −
→
⋅⋅= msAhe
h
e ρσ
( ) mBBn σ=−⋅− 2121
ˆ
 GAUSS’ LAW
MAGNETIC
[ ]1
0
lim: −
→
⋅⋅= msVhm
h
m ρσ
Fresnel Equations
January 4, 2015 59
SOLO
Technion
Israeli Institute of Technology
1964 – 1968 BSc EE
1968 – 1971 MSc EE
Israeli Air Force
1970 – 1974
RAFAEL
Israeli Armament Development Authority
1974 – 2013
Stanford University
1983 – 1986 PhD AA

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Reflection refraction

  • 2. 2 SOLO TABLE OF CONTENT REFLECTION & REFRACTION History of Reflection and Refraction Huygens Principle Reflections Laws Development Using Huygens Principle Refractions 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 Theorem Stokes Treatment of Reflection and Refraction References
  • 4. 4 REFLECTION & REFRACTIONSOLO History of Reflection & Refraction 100-170A.D. Claudius Ptolemey Alexandria “Optics” 130 A.D. Tabulated angle of incidence and refraction for several media. c. 300 B.C. Euclid Greece “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. 5 REFLECTION & REFRACTIONSOLO History of Reflection & Refraction Willebrord van Roijen Snell 1580-1626 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 = θ θ correct 1 2 sin sin n n t i = θ θ Descartes deduced wrong
  • 6. 6 REFLECTION & REFRACTIONSOLO History of Reflection & Refraction Pierre de Fermat 1601-1665 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 Huygens 1629-1695 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. 7 REFLECTION & REFRACTIONSOLO Huygens Principle Christiaan Huygens 1629-1695 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. 8 REFLECTION & REFRACTIONSOLO Reflection Laws Development Using Huygens Principle Suppose a planar incident wave AB is moving toward the boundary AC between two media. The velocity of 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 is reaching the boundary at C, the reflected ray at A will reach D and the 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. 9 REFLECTION & REFRACTIONSOLO Refraction Laws Development Using Huygens Principle Suppose a planar incident wave AB is moving toward the boundary AC between two media. The velocity of 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 is reaching the boundary at C, the refracted ray at A will reach E and the 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 == θ θ
  • 10. 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. 11 SOLO 1. The optical path is reflected at the boundary between two regions ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   In this case we have and21 nn = ( ) ( ) ( ) 0ˆˆ 21 21 =⋅−=⋅        − 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. 12 SOLO 2. The optical path passes between two regions with different refractive indexes n1 to n2. (continue – 1) ( ) ( ) 0 21 21 =⋅        − rd sd rd n sd rd n rayray   where is on the boundary between the two regions andrd  ( ) ( ) sd rd s sd rd s 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 − 21 ˆ −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. 13 ELECTROMAGNETICSSOLO To satisfy the Maxwell equations for a source free media we must have: Monochromatic Planar Wave Equations we haveUsing: 1ˆˆ&ˆˆ 0 =⋅== kkknkkk εµω         =⋅∇ =⋅∇ −=×∇ =×∇ 0 0 H E HjE EjH ωµ ωε            =⋅ =⋅ =× −=× 0ˆ 0ˆ ˆ ˆ 0 0 00 00 Hk Ek HEk EHk ε µ µ ε        =⋅− =⋅− −=×− =×− ⇒ ⋅− ⋅− ⋅−⋅− ⋅−⋅− −=∇ ⋅−⋅− 0 0 0 0 00 00 rkj rkj rkjrkj rkjrkj ekje eHkj eEkj eHjeEkj eEjeHkj rkjrkj           ωµ ωε ( ) ∗∗ ⋅==⋅==+= HHwEEwwcn k wwcn k S meme 22 & 2 ˆ 2 ˆ µε Time Average Poynting Vector of the Planar Wave Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 14. 14 SOLO REFLECTION & REFRACTION Consider an incident monochromatic planar wave ( ) ( ) c n k eEkH eEE iiii rktj iii rktj ii ii ii 1 00 11 0011 0 0 ω εµ εµ εµωεµω µ ε ω ω === ×= = ⋅− ⋅−     The monochromatic planar reflected wave from the boundary is ( ) ( ) 1 1 1 1 0 0 & n c v vc n k eEkH eEE r rr rktj rrr rktj rr rr rr === ×= = ⋅− ⋅− ω ω µ ε ω ω     The monochromatic planar refracted wave from the boundary is ( ) ( ) 2 2 2 2 0 0 & n c v vc n k eEkH eEE t tt rktj ttt rktj tt tt tt === ×= = ⋅− ⋅− ω ω µ ε ω ω     Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 15. 15 SOLO REFLECTION & REFRACTION The Boundary Conditions at z=0 must be satisfied at all points on the plane at all times, implies that the spatial and time variations of This implies that Phase-Matching Conditions ( ) ( ) ( ) yxteEeEeE z rktj t z rktj r z rktj i ttrrii ,,,, 0 0 0 0 0 0 ∀ = ⋅− = ⋅− = ⋅−   ωωω ( ) ( ) ( ) yxtrktrktrkt z tt z rr z ii ,, 000 ∀⋅−=⋅−=⋅− ===  ωωω ttri ∀=== ωωωω ( ) ( ) ( ) yxrkrkrk z t z r z i , 000 ∀⋅=⋅=⋅ ===  must be the same Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 16. 16 SOLO REFLECTION & REFRACTION tri nnn θθθ sinsinsin 211 == Phase-Matching Conditions ( ) ( )      −+= ++= zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1 θαθααω θαθααω   ( ) ( )         +=⋅ +=⋅ =⋅ = = = yyx c n rk yx c n rk y c n rk ttt z t irr z r i z i ˆsinsincos sinsincos sin 2 0 1 0 1 0 θααω θααω θω    ( ) ( ) ( ) yxrkrkrk z t z r z i , 000 ∀⋅=⋅=⋅ ===  2 π αα == tr ttri ∀=== ωωωω x∀ y∀ Coplanar Snell’s Law ( )     ++= −= zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1   θθω Given: Let find: Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 17. 17 SOLO REFLECTION & REFRACTION Second way of writing phase-matching equations ri θθ = 11 22 2 1 1 2 sin sin εµ εµ θ θ === v v n n t iRefraction Law Reflection Law Phase-Matching Conditions ( )     ++= −= zzyyxxr zy c n k iiii ˆˆˆ ˆcosˆsin1   θθω ( ) ( )      −+= ++= zyx c n k zyx c n k ttttttt irirrrr ˆcossinˆsinsinˆcos ˆcossinˆsinsinˆcos 2 1 θαθααω θαθααω   ( ) ( )[ ] ( ) ( )[ ]      −+−=−× −+−=−× ynnyn c kkz ynnyn c kkz ittrti irrrri ˆsinsinsinˆcosˆ ˆsinsinsinˆcosˆ 122 111 θθαα ω θθαα ω   ttri ∀=== ωωωω We can see that ( ) ( )         === =−×=−× ωωωω tri tiri kkzkkz 0ˆˆ            === == == ωωωω θθθ παα tri tri tr nnn sinsinsin 2/ 211 Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 18. 18 SOLO REFLECTION & REFRACTION ri θθ = 11 22 2 1 1 2 sin sin εµ εµ θ θ === v v n n t iRefraction Law Reflection Law Phase-Matching Conditions (Summary) ttri ∀=== ωωωω ( ) ( )         === =−×=−× ωωωω tri tiri kkzkkz 0ˆˆ            === == == ωωωω θθθ παα tri tri tr nnn sinsinsin 2/ 211 ( ) ( ) ( ) yxrkrkrk z t z r z i , 000 ∀⋅=⋅=⋅ ===  ( ) ( ) ( ) yxtrktrktrkt z tt z rr z ii ,, 000 ∀⋅−=⋅−=⋅− ===  ωωω Vector Notation Scalar Notation Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 19. 19 SOLO REFLECTION & REFRACTION ( ) 0ˆ 2121  =−×− EEn ( ) 0ˆ 2121  =−×− HHn ( ) 0ˆ 2121 =−⋅− DDn  ( ) 0ˆ 2121 =−⋅− BBn  Boundary conditions for a source-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 2 2 1 1 1 21 µ ε µ ε Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations Boundary conditions
  • 20. 20 SOLO REFLECTION & REFRACTION ( ) 0ˆ 00021  =−+×− tri EEEn 0 111 ˆ 0 2 0 1 0 1 21  =      ×−×+××− ttrrii EkEkEkn µµµ ( ) 0ˆ 02010121 =−+⋅− tri EEEn  εεε ( ) 0ˆ 00021 =×−×+×⋅− ttrrii EkEkEkn  Using ,ˆ,ˆ,ˆˆ 221111 1 ttrriii kkkkkk c n k 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. 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. 22 SOLO REFLECTION & REFRACTION ( ) 0coscoscos 0 2 2 00 1 1 =−− ttirii EEE θ µ ε θθ µ ε 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 θθ θεµθθεµ4 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. 23 SOLO REFLECTION & REFRACTION ( ) 0cos 1 cos 1 0 00 22 2 00 00 11 1 21 =−− tt n iri n EEE θ εµ εµ µ θ εµ εµ µ  1 0000 =−+ tri EEE2 From and ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 2 2 1 1 2 2 1 1 0 0 + − =      = ⊥ ⊥ ti i i t nn n E E t θ µ θ µ θ µ 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 = θ θ ( ) ( )ti ti i r E E r θθ θθ + − −=      = ⊥ ⊥ sin sin 0 0 ( )ti it i t E E t θθ θθ + =      = ⊥ ⊥ 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. 24 SOLO REFLECTION & REFRACTION ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 2 2 1 1 2 2 1 1 0 0 + − =      = ⊥ ⊥ ti i i t nn n E E t θ µ θ µ θ µ 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 = θ θ ( ) ( )ti ti i r E E r θθ θθµµ + − −=      = = ⊥ ⊥ sin sin21 0 0 ( )ti it i t E E t θθ θθµµ + =      = = ⊥ ⊥ 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. 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. 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 ( ) ( ) 0 11 0ˆ 0 00 22 2 00 00 11 1 0 2 2 00 1 1 21 =−+=      −+ t n ri n tri EEEoryEEE   µε µε µµε µε µµ ε µ ε 1 4 00 = Boundary Conditions Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 27. 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 11 2211 0 2 2 00 1 1 =−+ =⇒= =      −+ ttiri t i ti tri EEE yEEE θεθε θε θε µε µε θεµθεµ µ ε µ ε  1 Identical to 3 We have two independent equations ( ) 0coscos 000 =−− ttiri EEE θθ2 ( ) 00 2 2 00 1 1 =−+ tri E n EE n µµ 1 ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 1 1 2 2 1 1 2 2 ||0 0 || + − =      = ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 1 1 2 2 1 1 ||0 0 || + =      = Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 28. 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 E r θ µ θ µ θ µ θ µ coscos coscos 1 1 2 2 1 1 2 2 ||0 0 || + − =      = ti i i t nn n E E t θ µ θ µ θ µ 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 = θ θ ( ) ( )ti ti i r E E r θθ θθµµ + − =      = = tan tan21 ||0 0 || ( ) ( )titi it i t E E t θθθθ θθµµ −+ =      = = cossin cossin221 ||0 0 || Reflections and Refractions Laws Development Using the Electromagnetic Approach Fresnel Equations
  • 29. 29 SOLO REFLECTION & REFRACTION ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 1 1 2 2 1 1 2 2 ||0 0 || + − =        = ti i i t nn n E E t θ µ θ µ θ µ coscos cos2 1 1 2 2 1 1 ||0 0 || + =        = ti ti i r nn nn E E r θ µ θ µ θ µ θ µ coscos coscos 2 2 1 1 2 2 1 1 0 0 + − =        = ⊥ ⊥ ti i i t nn n E E t θ µ θ µ θ µ 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 Lorentz 1853-1928
  • 30. 30 SOLO REFLECTION & REFRACTION Discussion of Fresnel Equations it n ni θθ θ 2 1 0→ = 0→iθ [ ] [ ] ti ti nn nn rr ii θθ θθ θθ + − = + − =−= =⊥= 21 12 00|| [ ] [ ] ti t nn n tt ii θθ θ θθ + = + == =⊥= 22 21 1 00|| 1 2 sin sin n n t i = θ θ 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 E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
  • 31. 31 SOLO REFLECTION & REFRACTION Discussion of Fresnel Equations 1 2 sin sin n n t i = θ θ Snell’s law Reflections and Refractions Laws Development Using the Electromagnetic Approach David Brewster 1781-1868 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/2 The 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 nnn it θθθ θθ cossinsin 2 90 2 ' 1 −= ==  pi θθ = → 1 2 tan n n p =θ ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
  • 32. 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. 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 = θ θ 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 21 sin n n icθCritical Angle Brewster Angle ( ) ( ) 1 2/sin 2/sin21 2/ 0 0 = + − −=        = = = ⊥ ⊥ πθ πθµµ πθ i i i r t E E r ( ) ( ) 1 2/tan 2/tan21 2/ ||0 0 || = + − =        = = = πθ πθµµ πθ i i i r t E E r n1 > n2
  • 34. 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. 1sinsin 21 2 1 nn it ici n n > > >= θθ θθFrom Snell’s law therefore is no solution for θt , but we can use for 1sinsin1cos 2 2 2 12 −      −=−= itt n n i θθθ ( ) ( )2 21 2 2 21 2 21 21 0 0 /sincos /sincos coscos coscos21 nni nni nn nn E E r ii ii ti ti i r −− −+ = + − =        = = ⊥ ⊥ θθ θθ θθ θθµµ ( ) ( ) ( ) ( )2 21 22 21 2 21 22 21 12 12 ||0 0 || /sincos/ /sincos/ coscos coscos21 nninn nninn nn nn E E r ii ii ti ti i r −− −+ = + − =        = = θθ θθ θθ θθµµ We can see that ( )⊥⊥ = ϕjr exp ( )|||| exp ϕjr = ( ) i i nn θ θϕ cos /sin 2 tan 2 21 2 − =      ⊥ ( ) ( ) i i nn nn θ θϕ cos/ /sin 2 tan 2 21 2 21 2 || − =      n2 / n1 =1/1.5
  • 35. 35 SOLO REFLECTION & REFRACTION Phase Shifts ( )ti it ti i i t nn n E E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = 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. 36 SOLO REFLECTION & REFRACTION Phase Shifts ||i E  ⊥i E  ⊥rE  ik  rk  tk  Boundary 21ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21ˆ −n Boundary Plan of incidence iE  ||i E  ⊥i E  ||rE  ⊥r E  pi nn < pi θθ = p n i n p n in ||iE  ⊥i E  ⊥rE  ik  rk  tk  Boundary 21ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21 ˆ −n Boundary Plan of incidence iE  ||iE  ⊥i E  ||rE  ⊥r E  pi nn < pi θθ > pn in p n i n ||r E  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 E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
  • 37. 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 E t θθ θθ θθ θµµ + = + =        = = ⊥ ⊥ sin cossin2 coscos cos2 21 1 0 0 21 ( ) ( )titi it ti i i t nn n E E t θθθθ θθ θθ θµµ −+ = + =        = = cossin cossin2 coscos cos2 12 1 ||0 0 || 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − −= + − =        = = ⊥ ⊥ sin sin coscos coscos 21 21 0 0 21 ( ) ( )ti ti ti ti i r nn nn E E r θθ θθ θθ θθµµ + − = + − =        = = tan tan coscos coscos 12 12 ||0 0 || 21
  • 38. 38 SOLO REFLECTION & REFRACTION Phase Shifts ||i E  ⊥iE  ⊥rE  ik  rk  tk  Boundary 21 ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21 ˆ −n Boundary Plan of incidence i E  ||i E  ⊥iE  ||r E  ⊥rE  pi nn < pi θθ = pn in pn i n ||i E  ⊥iE  ⊥rE  ik  rk  tk  Boundary 21 ˆ −n z x y iθ rθ tθ iθ rθ tθ tk  rk  ik  21 ˆ −n Boundary Plan of incidence iE  ||iE  ⊥iE  ||r E  ⊥rE  pi nn < pi θθ > p n in pn i n ||r E  rE  n1 > n2 n1 < n2
  • 39. 39 SOLO REFLECTION & REFRACTION Energy Reflected and Refracted for Normal Polarization 2 01 12 ˆ ⊥ ⊥ = i i i E n ck S ε  Time Average Poynting Vectors (Irradiances) of the Planar waves are 2 01 12 ˆ ⊥ ⊥ = r r r E n ck S ε  2 02 22 ˆ ⊥ ⊥ = t t t E n ck S ε  2 2 2 1 1 2 2 2 1 1 2 0 2 0 coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   2 2 2 1 1 21 12 2 1 1 2 021 2 012 coscos cos cos cos2 cos cos ˆ ˆ       +       == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti i t i ii tt i t nn n nn En En zS zS T θ µ θ µ θε θε θ µ θε θε     titi cn cn i t i nn n nn n nn θθ µµ θθ µεµ εµ µθε θε θ µ coscos4coscos 1 4 cos cos cos2 21 21 2211 122 1 2 1 21 12 2 1 1 22 1 22 2 ==      2 2 2 1 1 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 zS T θ µ θ µ θθ µµ θε θε   Reflectance Transmittance Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 40. 40 SOLO REFLECTION & REFRACTION 2 2 2 1 1 2 2 2 1 1 2 0 2 0 coscos coscos ˆ ˆ       +       − == ⋅ ⋅ = ⊥ ⊥ ⊥ ⊥ ⊥ ti ti i r i r nn nn E E zS zS R θ µ θ µ θ µ θ µ   2 2 2 1 1 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 zS T θ µ θ µ θθ µµ θε θε   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. 41 SOLO REFLECTION & REFRACTION Energy Reflected and Refracted for Parallel Polarization 2 ||01 1 || 2 ˆ i i i E n ck S ε=  Time Average Poynting vector of the Planar waves are 2 ||01 1 || 2 ˆ r r r E n ck S ε=  2 ||02 2 || 2 ˆ t t t E n ck S 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 zS R θ µ θ µ θ µ θ µ   2 1 1 2 2 21 12 2 1 1 2 ||021 2 ||012 || || || coscos cos cos cos2 cos cos ˆ ˆ       +       == ⋅ ⋅ = ti i t i ii tt i t nn n nn En En zS zS T θ µ θ µ θε θε θ µ θε θε     titi cn cn i t i nn n nn n nn θθ µµ θθ µεµ εµ µθε θε θ µ coscos4coscos 1 4 cos cos cos2 21 21 2211 122 1 2 1 21 12 2 1 1 22 1 22 2 ==      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 zS T θ µ θ µ θθ µµ θε θε   Reflectance Transmittance Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 42. 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 zS R θ µ θ µ θ µ θ µ   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 zS T θ µ θ µ θθ µµ θε θε   We can see that 1|||| =+TR Average Poynting vector of the Planar waves are Reflections and Refractions Laws Development Using the Electromagnetic Approach
  • 43. 43 SOLO REFLECTION & REFRACTION ( ) ( )ti ti i r zS zS R θθ θθµµ + − = ⋅ ⋅ = = 2 2 || || || tan tan ˆ ˆ 21   ( ) ( )titi ti i t zS zS T θθθθ θθµµ −+ = ⋅ ⋅ = = 22 || || || cossin 2sin2sin ˆ ˆ 21   1|||| =+TR Reflections and Refractions Laws Development Using the Electromagnetic Approach ( ) ( )ti ti i r zS zS R θθ θθµµ + − = ⋅ ⋅ = = ⊥ ⊥ ⊥ 2 2 sin sin ˆ ˆ 21   ( )ti ti i t zS zS T θθ θθµµ + = ⋅ ⋅ = = ⊥ ⊥ ⊥ 2 sin 2sin2sin ˆ ˆ 21   1=+ ⊥⊥ TR Summary
  • 44. 44 SOLO REFLECTION & REFRACTION Reflections and Refractions Laws Development Using the Electromagnetic Approach || R ⊥ R || R ⊥ R
  • 45. 45 Malus-Dupin TheoremSOLO Étienne Louis Malus 1775-1812 A surface passing through the end points of rays which have traveled equal optical path lengths 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 is the wavefront), they are said to form a normal congruence. The Malus-Dupin Theorem (introduced in 1808 by Malus and modified in 1812 by Dupin) states that: “The set of rays that are orthogonal to a wavefront remain normal to a wavefront after any number of refraction or reflections.” Charles Dupin 1784-1873 Using Fermat principle [ ] [ ]'' BQBAVApathoptical == [ ] [ ] ( )2 '' εOAVAAQA += VQ=ε is a small quantity [ ] [ ] ( )2 '' εOBQBAQA += Since ray BQ is normal to wave W at B [ ] [ ] ( )2 εOBQAQ += [ ] [ ] ( )2 '' εOQBQA += ray BQB’ is normal to wave W’ at B’ Proof for Refraction: n 'n P Q VA P’ A' B B' Wavefront from P Wavefront to P'
  • 46. 46 Stokes Treatment of Reflection and RefractionSOLO 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. 47 Stokes Treatment of Reflection and RefractionSOLO 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' θθ θθθθ rr rrtt −= =+ Stokes relations θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
  • 48. 48 Stokes Treatment of Reflection and RefractionSOLO ( ) ( ) ( ) ( ) ( ) ( )12 1112 ' 1' θθ θθθθ rr rrtt −= =+ Stokes relations Let check that Fresnel Equation do satisfy Stokes relations ( ) 2211 11 2 coscos cos221 θθ θ θ µµ nn n t + = = ⊥ 2112 11 || coscos cos221 θθ θµµ nn n t + = = ( ) 2211 2211 1 coscos coscos21 θθ θθ θ µµ nn nn r + − = = ⊥ ( ) 2112 2112 1|| coscos coscos21 θθ θθ θ µµ nn nn r + − = = ( ) 2211 22 1 coscos cos2 ' 21 θθ θ θ µµ nn n t + = = ⊥ ( ) 2211 1122 2 coscos coscos ' 21 θθ θθ θ µµ nn nn r + − = = ⊥ 1 ( ) ( ) ( ) ( ) ( ) ( )12 1112 ' 1' θθ θθθθ ⊥⊥ ⊥⊥⊥⊥ −= =+ rr rrtt We can see that: 2 2112 22 || coscos cos2 ' 21 θθ θµµ nn n t + = = ( ) 2112 1221 1|| coscos coscos ' 21 θθ θθ θ µµ nn nn r + − = = ( ) ( ) ( ) ( ) ( ) ( )1||2|| 1||1||1||2|| ' 1' θθ θθθθ rr rrtt −= =+ We can see that: θ1 and θ2 are related by Snell’s Law: 2211 sinsin θθ nn =
  • 49. 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. 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. 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. 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. 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. 54 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions ( ) ( ) ldtHtHhldtHldtHldH h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbHHnldnbHHldtHH e ˆˆˆˆˆˆ 21212121121 ⋅=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) ekHHn  =−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +=⋅ → S e C Sd t D JdlH    ( ) dlbkbdlh t D JSd t D J e h e S e ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e  
  • 55. 55 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 1) ( ) ( ) ldtEtEhldtEldtEldE h C 2211 0 2211 ˆˆˆˆ ⋅+⋅=Θ+⋅+⋅=⋅ →→ ∫  where are unit vectors along C in region (1) and (2), respectively, and21 ˆ,ˆ tt 2121 ˆˆˆˆ −×=−= nbtt - a unit vector normal to the boundary between region (1) and (2)21 ˆ −n - a unit vector on the boundary and normal to the plane of curve Cbˆ Using we obtainbaccba  ⋅×≡×⋅ ( ) ( ) ( )[ ] ldbkldbEEnldnbEEldtEE m ˆˆˆˆˆˆ 21212121121 ⋅−=⋅−×=×⋅−=⋅− −−  Since this must be true for any vector that lies on the boundary between regions (1) and (2) we must have: bˆ ( ) mkEEn  −=−×− 2121 ˆ ∫∫∫ ⋅        ∂ ∂ +−=⋅ → S m C Sd t B JdlE    ( ) dlbkbdlh t B JSd t B J m h m S m ˆˆ 0 ⋅=⋅        ∂ ∂ +=⋅        ∂ ∂ + → ∫∫      FARADAY’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m  
  • 56. 56 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 2) ( ) ( ) SdnDnDhSdnDSdnDSdD h 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)21 ˆ −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=−⋅− 2121 ˆ  ( ) dSdShdv e h e V e σρρ 0→ ==∫∫∫ GAUSS’ LAW - ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ∫∫∫∫∫ =• V e S dvSdD ρ 
  • 57. 57 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (continue – 3) ( ) ( ) SdnBnBhSdnBSdnBSdB h 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)21 ˆ −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=−⋅− 2121 ˆ  ( ) dSdShdv m h m V m σρρ 0→ ==∫∫∫ GAUSS’ LAW – MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ ∫∫∫∫∫ =• V m S dvSdB ρ 
  • 58. 58 SOLO ELECTROMAGNETICS BOUNDARY CONDITIONS Boundary Conditions (summary) ( ) mkEEn  −=−×− 2121 ˆ FARADAY’S LAW ( ) ekHHn  =−×− 2121 ˆ AMPÈRE’S LAW [ ]1 0 lim: − → ⋅        ∂ ∂ += mAh t D Jk e h e   [ ]1 0 lim: − → ⋅        ∂ ∂ += mVh t B Jk m h m   ( ) eDDn σ=−⋅− 2121 ˆ  GAUSS’ LAW ELECTRIC [ ]1 0 lim: − → ⋅⋅= msAhe h e ρσ ( ) mBBn σ=−⋅− 2121 ˆ  GAUSS’ LAW MAGNETIC [ ]1 0 lim: − → ⋅⋅= msVhm h m ρσ Fresnel Equations
  • 59. January 4, 2015 59 SOLO Technion Israeli Institute of Technology 1964 – 1968 BSc EE 1968 – 1971 MSc EE Israeli Air Force 1970 – 1974 RAFAEL Israeli Armament Development Authority 1974 – 2013 Stanford University 1983 – 1986 PhD AA

Editor's Notes

  1. J.P.Mathieu, “Optics”, Pergamon Press, 1975, pp.52-57 E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  2. E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  3. E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  4. E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  5. E. Hechtm “Optics”, Addison Wesley, 2002, pp. 116-119
  6. Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.105-106 V.N. Mahajan, “Optical Imaging and Aberrations” Part I, Ray Geometrical Optics, SPIE, 1998, pp.11,12 M. Born, E. Wolf, “Principles of Optics”, Pergamon Press, 5th Ed., 1975, pp.130-132
  7. Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.136-137 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 4th Ed., 1979, pp.91-93
  8. Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.136-137 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 4th Ed., 1979, pp.91-93
  9. Hecht, “Optics”, 4th Ed, .Addison Wesley, 2002, pp.136-137 E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 4th Ed., 1979, pp.91-93