TOTAL INTERNAL REFLECTION
Kinematics properties

Since the k vectors are coplanar, let’s consider the incident plane coincides with the
XZ plane; hence kiy  kry  kty  0 .
Consider the case in which the light is incident from the medium of higher index of
refraction ni > nt. For incident angles greater than the critical angle C  sin 1 (nt / ni ) , the

horizontal component of ki ( i.e. the component kix of the incident wave) will be greater

than the magnitude of the transmitted light’s vector kt = (2/) nt  k' . This is shown

in the figure below; notice kt is too small to satisfy the required kinematics boundary
condition,
ktx  krx
(19)

Thus, there is no real vector k t to satisfy the condition (19) when i > C.
Casei=C
Z
Z
 2
ki  ni  k
Casei>C
Z

t)

ki

kr
c

kt
k'
k' 
i
Slower medium
ni
nt
faster medium
2

nt
nt < ni

ki

kr
c
k'

kt
X
Snell’s law
not being
fulfilled
Fig. 20 Total internal reflection. For i > C the length of the real-variable wavevector

k t (transmitted wave) turns out to be too small to satisfy the boundary condition ktx  krx .

That is, it does not exist a real-variable k t wave-vector able to satisfy the kinematic conditions
(Snell’s law) at the boundary. The implication is that the incident wave is fully reflected (there
is no propagating transmitted wave penetrating in the –Z direction).

Er

There is a way to get around this limitation. We allow the kt to be a complex vector.

Indeed, even though the magnitude of kt is fixed and equal to,
kt
2
2
 ( ktx )  ( ktz )
2
 2 

nt 
  
  k' 
2
2
2
Notice, ktx  could be greater than k'  if ( ktz ) where negative; that is, if ktz were a
pure complex number,
2
2
ktz  i
z
*
Accordingly, let’s consider a transmitted wave of complex vector kt ,
*
kt  (ktx , 0,  i )
complex vector
z
(20)
2
2
whose modulus square kt  ktx  (0)  ( j  )  ktx   is required to satisfy,
z
z
2
 2
2
k *  ktx  
t
2
2
z
2
 2 

nt 
  
  k' 
2
2
(20)’
2
*
The complex wave-vector kt  (ktx , 0,  j
z
)
is then able to satisfy the Snell’s law for
incident angles i > C,
ktx  krx or kix ,
(21)

ktx  kt*  k'
(21)’
Notice also
Casei=C Z Z

ki

Slower medium

kr
c

ki
ni

kt
nt
Z
i

kr
c
2
Casei>C
nt
2
faster medium

X
nt
ktx
*
kt  (ktx , 0,  i
nt < ni
z
)
Complex vector
Fig. 21 Notice on the right-side diagram that the evanescent wave propagates along the
interface with wave-vector component ktx, ( ktx > (2/)nt ).
Next, for a given incident angle i >C, let’s figure out the corresponding value of  .
z
The kinematics condition ktx  krx  kix implies, ktx  kix  k Sin i 
ktx  kix

2

ni Sin i
2

ni Sin i .
(22)
Using this value in (21), gives,

2
z
 ktx
2
 kix
2
 2 

nt 
  
2
 2 

nt 
  
 2

  ni Sini 
 

2
2
2
 2 

nt 
  

2
 2 
2
2
   ni Sini   nt 
  
 2 
2
2
 ni Sini   nt 
  
 
z
Using C  sin 1 (nt / ni )

(23)

Er 
 2 
ni
  
 
z
In summary,
Sini 2  Sinc 2
for i >C
(23)’

Incident wave
Reflected wave

 
j (ki r -  t )
Ei  Eoi e
 


j ( kr  r -  t )
Er  Eor e
(24)
(24)’


*
 
j (kt * r -  t )
Transmitted wave Et  Eot e
, where kt  (ktx , 0,  j )
z

j(ktx x  ktz z - ω t)
 Eot e

j(k x  j γz z - ω t)
 Eot e tx
 
  z j (k x - ωt )
Et  Eot e z e t x
For the particular case at hands, we take the
positive sign (otherwise we would have a
wave carrying infinite energy.


γz z j(kt x x - ωt t)
Et  Eot
 
Ei  Eoi e
ni
nt
nt < ni
e
e
Z
 
j (ki  r -  t )

ki
c
k'
(24)”
(for z < 0)
 


j ( kr  r -  t )
Er  Eor e

kr
k
ktx

kt*  (ktx , 0,  j z )
k' 
2

2

ni
nt
X
 
γ z j(k x - ω t)
Et  Eot e z e t x
Fig. The refracted wave propagates only parallel to the surface (with wave-vector ktx) and it
is attenuated exponentially (decay factor z) along the vertical direction in the medium of
lower index of refraction.
So, it does exist electromagnetic field in the medium of lower index of refraction, but it is
attenuated exponentially beyond the interface. No energy flow (no propagating energy)
crosses vertically the interface.
Dynamics properties
*
How to incorporate the complex vector kt into the Fresnel equations?
 First, Snell’s law states ni sin i  nt sint . In terms of the critical angle
sin C  nt / ni one has,
Notice,
Hence,
sin i
 sint
sinC
(25)
sint  1
(25)’
for  i >  C
t is a complex angle when  i > C .
[ Notice, for a complex angle   /2  j one obtains
sin (/2  j )  1  12 ( j )2  1  12  2 >1 ]
 Let’s evaluate also the cos of the complex angle t.
2
ktz  k' Cos t  nt Cos t ,
On one hand, we have

ktz  i ,
on the other hand
z
2
which gives
Using the expression for 
2
z
nt Cost  i
nt Cost  i z

given in (23),

2
2
 ni Sini   nt 
  

which gives,
2
2
Cost  i
 n Sin  
 i
i 
1
 n 
t


for  i >  C
2
 i
 Sini 

  1
Sin

c 

(26)
Notice from (24) and (26) that the complex angle t satisfies,
sin2t  cos 2t  1
(27)
The expression for cos t, obtained above for the case when i>C are inserted now in
the Fresnel’s equations:

For TE or s- polarization
 Ero 
niCosi  nt Cost


 E   n Cos  n Cos  r
t
i
t
 io 
i
Amplitude reflection coefficient
 Eto 
2niCosi


 E   n Cos  n Cos  t 
t
i
t
 io 
i
Amplitude transmission coefficient
Notice that for a complex Cost , as given in (26), the magnitude of r is equal to1,
as expected in a total internal reflection case.
r  1

For TM or p- polarization
 Ero 
nt Cosi  niCost


 E   n Cos  n Cos  r//
t
i
t
 io  //
i
 Eto 
2niCosi


 E   n Cos  n Cos  t //
t
i
t
 io  //
i
(28)
Amplitude reflection coefficient
Amplitude transmission coefficient
Notice also that (26) implies,
r//  1
(26)
Notice that in both cases, TE and TM polarization, the complex Cost given in (24)
results in complex amplitude reflection coefficients. It implies that,
under total internal reflection conditions, there is
a change in the phase of the reflected wave.
(27)
Fresnel’s rhombus. These phase changes can be utilized to convert one type of
polarization into another. For example in a Fresnel’s rhombus , linearly polarized light
with equal TE and TM polarization amplitudes is converted by two successive internal
reflections (each involving a relative phase change of 45o) into circularly polarized light.