Reflection refraction

$Page 1: Reflection refraction$
1

REFLECTION&

REFRACTION

SOLO HERMELIN

Updated: 4.11.06 http://www.solohermelin.com

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


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



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

a, 08/14/2005

Descarte work is described inR.S.Elliot,"Electromagnetics",pg.6M.V.Klein, T.E.Furtak,"Optics", pp. 7-9

6



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


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 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


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).


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 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


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


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


15


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


( ) ( ) ( ) 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


16


tri nnn θθθ sinsinsin 211 ==


( )

( )

−+=

++=

zyxc

nk

zyxc

nk

ttttttt

irirrrr

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


tz

rz

i ,000

∀⋅=⋅=⋅===

2

παα == tr


x∀

y∀

Coplanar

Snell’s Law

( )

++=

−=

zzyyxxr

zyc

nk iiii

ˆˆˆ

ˆcosˆsin1

θθω

Given:

Let find:


17


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


( )

++=

−=

zzyyxxr

zyc

nk iiii

ˆˆˆ

ˆcosˆsin1

θθω

( )

( )

−+=

++=

zyxc

nk

zyxc

nk

ttttttt

irirrrr



2

1

θαθααω

θαθααω

( ) ( )[ ]

( ) ( )[ ]

−+−=−×

−+−=−×

ynnync

kkz

ynnync

kkz

ittrti

irrrri

ˆsinsinsinˆcosˆ

ˆsinsinsinˆcosˆ

122

111

θθααω

θθααω


We can see that ( ) ( )

====−×=−×

ωωωω tri

tiri kkzkkz 0ˆˆ

=====

==

ωωωωθθθ

παα

tri

tri

tr

nnn sinsinsin

2/

211


18


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)


( ) ( )

====−×=−×

ωωωω tri

tiri kkzkkz 0ˆˆ

=====

==

ωωωωθθθ

παα

tri

tri

tr

nnn sinsinsin

2/

211


tz

rz

i ,000

∀⋅=⋅=⋅===

( ) ( ) ( ) yxtrktrktrktz

ttz

rrz

ii ,,000

∀⋅−=⋅−=⋅−===

ωωω

Vector Notation

ScalarNotation


19


( ) 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


Fresnel EquationsBoundary conditions

20


( ) 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


( ) 0ˆ 02010121 =−+⋅− tri EEEn

εεε


εµεµεµ

Boundary Conditions


Fresnel Equations

21


( ) ( ) ( )( ) ( ) ( ) ( ) 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


( ) 0ˆ 02010121 =−+⋅− tri EEEn

εεε


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


22


( ) 0coscoscos 02

200

1

1 =−− ttirii EEE θµεθθ

µε

1

0000

=−+ tri EEE2

( ) 0ˆ 00021

=−+×− tri EEEn

[ ] 0ˆ/ˆ/ˆ/ˆ 02201101121


( ) 0ˆ 02010121 =−+⋅− tri EEEn

εεε


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 =



Boundary Conditions


23


( ) 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




24


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




Fresnel Equations

25


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 −=⋅−=⋅−=⋅ −−−


26


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


( ) 0ˆ 02010121 =−+⋅− tri EEEn

εεε


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


27



( )( )

( )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||

+=

=


28



( )( )

( )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||


Fresnel Equations

29


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)


Hendrik Antoon Lorentz1853-1928

a, 08/16/2005

The Lorentz derivation in 1875 is given in V.Novacu"Electrodinamica", (romanian), p.163. I must find a cross reference.

30


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


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


Discussion of Fresnel Equations

1

2

sin

sin

n

n

t

i =θθ

Snell’s law


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


Discussion of Fresnel Equations (continue)

n2 / n1 =1.5


Brewster Angle

n1 < n2

33


Discussion of Fresnel Equations (continue)

n2 / n1 =1/1.5


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


n1 > n2 Total Reflection


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


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


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


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


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



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


40


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



41


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


42


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


43


( )( )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


( )( )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



||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'

a, 08/20/2005

M.V.Klein, T.Furtak,"Optics", pp.34-35E. Hecht, A. Zajac,"Optics",pp.8, pp.225-226

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


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


48


( ) ( ) ( ) ( )( ) ( )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:


49


Thinks to complete

Photons and Laws of Reflection & Refraction (Hecht & Zajac pp. 93)

http://physics.nad.ru/Physics/English/index.htm

50


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


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


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



( ) ( ) 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



( ) ( ) 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


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