Experimental Physics EP2a – Reflection and refraction –

Experimental Physics EP2a – Reflection and refraction –

Experimental Physics EP2a
Electricity and Wave Optics
– Reflection and refraction –
Rustem Valiullin
http://research.uni-leipzig.de/valiu/
Experimental Physics IIa - Reflection and refraction of light
1
Malus law


E

B
I0
1
2
I0
1
2
I0
I  12 I 0 cos 2 
Experimental Physics IIa - Reflection and refraction of light
2
The Brewster’s law
n1
n2
Sir David Brewster
B
n1
Born
11 December 1781
Canongate, Jedburgh, Roxburghshire
Died
10 February 1868 (aged 86)
Allerly House, Gattonside, Roxburghshire
n2
1  2  90
tan  B 
Experimental Physics IIa - Reflection and refraction of light
n2
n1
3
The boundary conditions
A2
l
2
1
ŝ
A
 
d  
 E  ds   dt  B  dA
 
d  
 B  ds  0 I  0ε0 dt  E  dA
A1
 
E, B
B
d
(1)
B
dA1
n

dt
d
( 2)
( 2)
E
ds


B
 t 1 dt  n dA1
(1)
E
t
 ds1  
Et(1)  Et( 2 )
(1)
t
  Qin
 E   EdA 
ε0
 
 B   B  dA  0
B
( 2)
t
En(1)  En( 2 )
(1)
n
B
B
( 2)
n
dBn(1) dBn( 2 )

dt
dt
Experimental Physics IIa - Reflection and refraction of light
B  B0ei (t kr )
4
The boundary conditions
E
(e )
E
 ( e )  i (t k r )
1
E  e
 ( r )  i ( ( r )t k ( r ) r )
1
E  Re
 ( d )  i ( ( d )t k r )
2
E  De
(r )
n1
n2
z
E (d )
y
reflektierte
durchgehende

 (r)
 (d )
A(r )eit  B( r )ei t  C (r )ei t  0
  ( r )  ( d )
x
Ae
entfallende
i ( k1 x x k1 y y )
 Be
i ( k1( xr ) x k1( yr ) y )
k k k k 
2
1
2
1x
2
1y
2
1z
i ( k2 x x  k2 y y )
 Ce
2
c2
 
(r)
n  k1
2
1
2
k1x  k1(xr )  k2 x
0
k1 y  k1(yr )  k2 y
k k k k 
2
2
Experimental Physics IIa - Reflection and refraction of light
2
2x
2
2y
2
2z
2
c2
n22
5
The boundary conditions
k k k 
2
1
E (e )
k1 y  0
E (r )
n1
n2
z
E (d )
y

k1
2
1z
k k k 
2
2
2
2x
k1(zr )  k12  k12x
k22  k12x
x
2
1x
2
2z
2
c2
2
c
2
n12  k1( r ) 
2
n22
k 2 z  () k 22  k12x
k2 z   k22  k12x
k1 sin 1  k1x = k2 sin 2  k2 x
z
(r)
k1
1
2
 x
k2
k22  k12x
n2  n1 sin 1
Snell’s law
k2 z  i k12x  k22 
i
2h
total internal reflection
z
 (d )

i ( t k2 x x )
E  Dei (t k2 x xk2 z z )  De 2 h e
Experimental Physics IIa - Reflection and refraction of light
6
To remember!
 Light beam reflected under the Brewster angle
looses its component polarized perpendicular to
the plane of incidence.
 At the interface between two media with different refractive
indexes tangential and normal components
of both electric and magnetic
fields are continuous.
 All properties relevant to reflection and
to refraction (like the Snell’s law) are
natural consequences of the boundary
conditions.
Experimental Physics IIa - Reflection and refraction of light
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The Fresnel equations: E(e)
û1
E
û1(r)
(e )
E
N̂ k 1

    û y   ||û1
(r )

R  R û y  R||û1(r)

D  D û y  D||û 2
n1
z0
n2
z
E (d )
y
û 2

  û x   x  || û1  û x   || cos 1
x

k1
 û x,y,z

  û y   y   
z
(r)
k1
1
2
 x
k2

  ûz   z  || û1  ûz   || sin 1
Ex( e )   || cos 1; E y( e )    ; Ez( e )   || sin 1

i (t k r )
e
i (t k x x k y y k z z )
e
Experimental Physics IIa - Reflection and refraction of light
8
The Fresnel equations: B(e)
û1
E
û1(r)
(e )
E
N̂ k 1
E
y
    û y   ||û1
(r )
z0
z


E

B

v

n1



1 
cB  N̂ k1  E  k k  E
n2

v1B    N̂ k1  û y   || N̂ k1  û1

v1B    û1   || - û y 
 û x,y,z
(d )
û 2
x

k1

N̂ k1 

 

v1Bx    û1  û x     cos 1
z
(r)
k1
1
2
 x
k2
v1By   ||
v1Bz    û1  ûz     sin 1
v1Bx( e )    cos 1; v1By( e )  || ; v1Bz( e )    sin 1
Experimental Physics IIa - Reflection and refraction of light
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The Fresnel equations: E(r) and B(r)
û1
E
û1(r)
(e )
E
N̂ k 1

    û y   ||û1
(r )
n1
z0
n2
z
E (d )
y
û 2
û
(r)
1
z
(r)
k1
1
2
N̂ k1 
û1  ûx   cos 1
x

k1

R  R û y  R||û1(r)

D  D û y  D||û 2
 û x,y,z
 x
k2
 û x    cos 1
E x( r )   R|| cos 1 v1Bx( r )   R cos 1
E y( r )  R
v1B y( r )   R||
E z( r )  R|| sin 1
v1Bz( r )  R sin 1
Experimental Physics IIa - Reflection and refraction of light
10
The Fresnel equations: E(d) and B(d)
û1
E
û1(r)
(e )
E
N̂ k 1

    û y   ||û1
(r )
n1
z0
n2
z
E
y
(d )
 û x,y,z
N̂ k1 
û 2
û2  ûx   cos 2
x

k1

R  R û y  R||û1(r)

D  D û y  D||û 2
z
(r)
k1
1
2
 x
k2
E x( d )  D|| cos  2
v2 Bx( d )  D cos  2
E y( d )  D
v2 B y( d )   D||
E z( d )  D|| sin  2
v2 Bz( d )  D sin  2
Experimental Physics IIa - Reflection and refraction of light
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The Fresnel equations
   cos 1
E x( e )  E x( r )  E x( d )
  ||
E y( e )  E y( r )  E y( d )
v1Bz( e )    sin 1
Bx( e )  Bx( r )  Bx( d )
E
(e)
x
  || cos 1 v1B
(e)
x
E
(e)
y
 
v1B
(e)
y
E
(e)
z
  || sin 1
B y( e )  B y( r )  B y( d )
E x( r )   R|| cos 1 v1Bx( r )   R cos 1
E y( r )  R
v1B y( r )   R||
E z( r )  R|| sin 1
v1Bz( r )  R sin 1
E x( d )  D|| cos  2
v2 Bx( d )  D cos  2
(d )
y
  D||
E y( d )  D
v2 B
E z( d )  D|| sin  2
v2 Bz( d )  D sin  2

||
 R|| cos 1  D|| cos 2
   R  D
   R v1 cos 1  Dv2 cos 2

Experimental Physics IIa - Reflection and refraction of light
||
 R|| v1  D||v2
12
The Fresnel coefficients
   R  D
   R v1 cos 1  Dv2 cos 2
n1 cos 1  n2 cos 2
r 

  n1 cos 1  n2 cos 2
R
d 
D


2n1 cos 1
n1 cos 1  n2 cos 2
r  
sin(1  2 )
sin(1  2 )
2 cos 1 sin 2
d 
sin(1  2 )


||
||
 R|| v1  D||v2
 R|| cos 1  D|| cos 2
r|| 
R||
 ||

n2 cos 1  n1 cos 2
n2 cos 1  n1 cos 2
2n1 cos 1
d|| 

 || n2 cos 1  n1 cos 2
D||
r|| 
tan(1  2 )
tan(1  2 )
2 cos 1 sin 2
d|| 
sin(1  2 ) cos(1  2 )
Experimental Physics IIa - Reflection and refraction of light
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Analysis of the Fresnel equations
sin(1  2 )
r  
sin(1  2 )
1  0
tan(1  2 )
r|| 
tan(1  2 )
r   r|| 
1   / 2
r 
R


n1 cos 1  n2 cos 2
n1 cos 1  n2 cos 2
n1  n2
n1  n2
r|| 
1  2   / 2
R||
 ||

n2 cos 1  n1 cos 2
n2 cos 1  n1 cos 2
r||  0; r  0
r  r||  1
E (e )

k1
z
(r)
k1
1
 x
2
k2
E (r )
n1
z0
n2
z
E (d )
y
x
Experimental Physics IIa - Reflection and refraction of light
14
Analysis of the Fresnel equations
1,0
0,8

||
0,6
I (r)
  (e)
I
Brewster's angle
Reflection coefficient
Air-Water
0,4
Reflection
Transmission
coefficients
I (d )
b  (e)
I
   b  1; ||  b||  1
iso  12 ||    
0,2
0,0
0
10
20
30
40
50
60
70
80
90
Angle of incidence
1,0

||
0,6
photoelement
0,8
Critical angle
Reflection coefficient
Water-Air
0,4
Total internal reflection
0,2
0,0
0
10
20
30
40
50
60
70
80
90
membrane
Angle of incidence
Experimental Physics IIa - Reflection and refraction of light
15
To remember!
 The Fresnel equations allow to quantify the
fractions of reflected and refracted light intensities.
 They also allow to trace the intensities for different
light polarizations.
 The Brewster’s law is naturally
predicted by the Fresnel equations.
 At low angles of incidence the
reflectance coefficient is close to one.
 In practice, the reflectance and
transmission coefficients are used.
Experimental Physics IIa - Reflection and refraction of light
16