Download Lecture Note No. 3 Reflection and Refraction (Sections 4.3 to 4.8)

Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
Engineering Optics and Optical Techniques
Lecture Note No. 3
Reflection and Refraction (Sections 4.3 to 4.8)
[Reading Assignment: Sections 4.9 to 4.12]
TRANSMISSION: In a conventionally dense medium, scattered E-M wavelets (secondary
wave) cancelled each other in all but the forward direction, i.e., the sustained ongoing beam.
*Most ordinary air environment below 100 miles altitude, liquid, glass, crystals etc., for
example,  ~ 500 nm and l ~ 3 nm for STP air. (Reminder)
/2
REFLECTION: Back-scattered E-M wavelets based on the non-resonant ground level vibration
by a thin layer (~/2) of unpaired atomic oscillators near the surface, i.e., bounced portion of the
ongoing beam (see Fig. 4-15).
nti 
nt
ni
ni: refractive index of the medium where the primary E-M wave is incident.
nt: refractive index of the medium where the primary E-M wave is transmitted.
Reflection layer
thickness
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
For ni < nt, the reflection is called “external”
For ni > nt, the reflection is called “internal”
REFLECTION has no apparent color preference:
1
2
Reflection layer thickness
No. of reflectors (atoms)
Rayleigh scattering dependency
Reflection efficiency
*Ray
*Wavefront
*Plane-of-incidence
*Specular reflection
*Diffuse reflection
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
REFRACTION
Ferma’s Principle … The beam path between two points takes one that needs the least travel
time, or shortest optical path length (OPL   ns ds   n j  s j ). The resulting travel time is
b
a
given as
 n
j
j
 s j  / c with c = speed light in vacuum or free space.
j
*OPL is different from the shortest path.


OPL  min  n j s j 
 j



t  min  n j s j  /c

Snell’s Law … ni sin  i  nt sin  t
SO O P
t 


Vi
Vt
From
b 2  a  x 
h2  x2

Vi
Vt
2
d t 
 0 for least travel time condition:
dx
sin  i sin  t

Vi
Vt
 ni sin  i  nt sin  t
3
c
j

Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
QUICK REVIEW ON WAVE NATURE OF E-M RADIATION
1-D Wave Equation and Plane Waves (Revisit)
§ Def’n of plane:
k  r  ro   0
or k  r  const
(k is the wave propagation direction.)
  x, t   A sin kx  t     A  Ime i kxt   
k  2

V

…wave frequency
  2 …angular frequency
1

1

k
2


  …wave number
  …wave period
2
2  

   for the case of constant V
V /
V
V
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
REFLECTION/REFRACTION BY E-M WAVE APPROACH
Revisit of the Snell’s Law
Plane-of-Incidence view of Fig. 4.38
E i  E oi cosk i  r   i t 
E r  E or cosk r  r   r t   r 
E t  E ot cosk t  r   t t   t 
*The tangential [perpendicular to the page] component of the electric oscillation vector must
remain the same before and after the interface, i.e., Etangential on one side of the surface 
Etangential on the other side. This is required to satisfy no arbitrary slip of atoms at the interface:



un  E i  un  E r  un  E t
This leads to
k i  r   i t  y b  k r  r   r t   r  y b  k t  r   t t   t  y b
which requires an identity
i  r  t
*Set t = 0 for the moment of incident of the beam on the surface:
ki  r  kr  r   r  kt  r   t
 k i  k r   r   r … The interface vector r sweeps out a plane perpendicular to a vector
k i  k r  , i.e., k i  k r  is perpendicular to the interface and parallel to u n , or

equivalently, k i  k r   un  0 . This last equation gives,
k i sin  i  k r sin  r , and  i   r since k i  k r ( k  2 /    / V ).
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm

 k i  k t   r   t , or k i  k t  is normal to the interface, or parallel to un .
Now,
k i  k t   u n  0 and
k i sin  i  k t sin  t
2
i
sin  i 
2
t
sin  t , and by 
c
i
or 
c
t
gives
ni sin  i  nt sin  t … Snell’s Law
AMPLITUDE OF REFLECTION (r) and TRANSMISSION (t)
I. E  the Plane-of-Incidence (Refer to Fig. 4.39)
(1) Continuity of tangential E (electric)-field at the interface:
E oi  E or  E ot
(2) Continuity of tangential H (magnetic)  B/-field at the interface:

Bi
i
cos i 
Br
r
cos r  
Bt
t
cos t
with v i  v r ,  i   r   t   o (for most transparent or optically thin materials), and  i   r ,
combining the related equations [ V j B j  E j , Eqs. (4.12), (4.13) & (4.14)] gives the amplitude
reflection coefficient and the amplitude transmission coefficient respectively as,
E 
n cos i  nt cos t
r   or   i
 E oi   ni cos i  nt cos t
E 
2ni cos i
t    ot  
 E oi   ni cos i  nt cos t
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
Substituting the Snell’s law, ni sin  i  nt sin  t
r  
t 
sin(  i   t )
sin(  i   t )
2 sin  t cos  i
sin(  i   t )
II. E  the Plane-of-Incidence (Refer to Fig. 4.40)
E
r   or
 E oi

n cos  i  ni cos  t
  t
  ni cos  t  nt cos  i
E 
2ni cos  i
t    ot  
 E oi   ni cos  t  nt cos  i
Substituting the Snell’s law gives,
r 
tan( i   t )
tan( i   t )
t 
2 sin  t cos  i
sin(  i   t ) cos( i   t )
For graphical results, refer to Figs. 4.41 and 4.42.
For normal incidence (  i  0 ),
r  r 
t  t 
nt  ni
nt  ni
2ni
nt  ni
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
By examining the sign conversion of r, the phase shift of E-field for internal and external
reflections (refer to Fig. 4.44).

ni
 nt  for External Reflection Figs. 4.44-a & b (air >> glass)
ni
 nt  for Internal Reflection: Fig. 4.44-c, d & e (glass >> air)
E 
n cos  i  nt cos  t
By examining r   or   i
 E oi   ni cos  i  nt cos  t
For example, for external reflection “-“ for all  i …    (Fig. 4.44-a)
E 
n cos  i  ni cos  t
By examining r   or   t
… (Fig. 4.44-b)
 E oi   ni cos  t  nt cos  i
For external reflection, “+” then “-“ beyond    p (Critical angle for total reflection)
No phase shift is occurred for transmission as t is always “+”.
E 
2ni cos  i
t    ot  
 E oi   ni cos  i  nt cos  t
E 
2ni cos  i
t    ot  
 E oi   ni cos  t  nt cos  i
.
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
REFLECTANCE (R) AND TRANSMITTANCE (T)

E2 
 I  c o o 

2 

Reflectance  reflected power/incident power:
2
I  A cos  r I r  E or 
  r2
R r


I i  A cos  i
I i  E oi 
for both  and  cases.
Transmittance  transmitted power/incident power
I  A cos  t nt cos  t
T t

I i  A cos  i ni cos  i
 E or

 E oi
For graphical results, refer to Fig. 4.48.
And
R + T =1
2

 n cos  t
   t

 ni cos  i
for both  and  cases.
9
 2
t

for both  and  cases.
Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
REFLECTANCE (R) AND TRANSMITTANCE (T)
When  i  0 (Normal incidence),
 n  ni 

R  R  R   t
n

n
t
i


T  T  T 
Dielectric substances
(TransmittedIncident)
Diamond-air
(Hardest and most
dense: highest
refractive index: most
sparkling)
2
4n t n i
nt  ni 2
nt
ni
R
T
2.41
1
0.17
0.83
Glass-air
1.5
1
0.04
0.96
Water-air
1.33
1
0.02
0.98
Diamond-water
(not as glittering as in
air)
2.41
1.33
0.083
0.917
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
TOTAL INTERNAL REFLECTION
From Snell’s law ni sin  i  nt sin  t ,  i   c when  t  90  , i.e.,
 c  sin 1 nti
[ nti  nt / ni ]
Visible light wave in glass (ni = 1.5) on air (nt = 1.0),  c  41.8 
Condition for total internal reflection: sin  i  nti
When total internal reflection is occurred, Ir = Ii and It = 0, R = 1.0 and T = 0.
EVANESCENT (short-lived) TRANSMITTED WAVES
[Existing under total internal reflection]
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
The E-M wave expression of the transmitted electric field is
E t  E ot exp i k t  r  t   E ot exp i k tx x  k ty y  t 
(1)
 E ot exp ikty y  exp i k tx x  t 
Using the Snell’s law [ ni sin  i  nt sin  t ] and the condition for total internal reflection
[ sin  i  nti ], we have
k sin  i
k tx  k t sin  t  t
nti
and
 sin 2  i
k ty  k t cos  t  k t 1 
nti2





(2)
Combining (1) and (2) gives,
1/ 2
E t  E ot e
 y
e
i  k t x sin i / nti t 
 sin2  i

with    t , i , nt , ni   kt 
 1
2
 nti

The transmitted wave amplitude drops off exponentially as it penetrates the less
dense medium in the y-direction (the first exponential function). The amplitude
diminishes within a few -distance.
The transmitted wave advances in the x-direction as a surface wave or
evanescent wave (the second exponential function).
FTIR (Frustrated Total Internal Reflection) – energy flow creeping the gap (of
an order of -thickness) or tunneling … a principle of beam-splitters.
*Example applications: Total Internal Reflection Fluorescence Microscopy (TIRFM)
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
Substance Categories in view of Fluid Dynamics:
Fluid (gas, liquid), Solid
Substance Categories in view of Optics:
Free space (vacuum, air):
Di-electric medium (Non-conducting):
Metal (Conducting):
Metal:
E (E-M) V
I
i2 R
No electrons
All bound electrons
Free electrons + Bound electrons
Thermal diffusion … significant absorption
E-M WAVES ON METAL SURFACES
Since metal is conducting (abundant free electrons), i.e.,   0 ,
the Maxwell equation is given as,
2 E 
2E 2E 2E
2E
E




 
2
2
2
2
t
x
y
z
t
dx
] in the oscillator model.
dt
We then need to substitute the complex index to account for the absorption, i.e.,
The conduction term acts like a damping term [ 
n  nR  inI
Substituting n into
E  Eo cost  ky  Eo cos  t  ky /    Eo cos  t  y /    Eo cos  t  ny / c 
gives:



y
E  E o e nI y / c cos   t 
 c / nR
13



Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm


This depicts that the transmitted E-field amplitude is given as E o e nI y / c , whereas the
incident E-field Amplitude is specified as E o  .
2
I  E  Amplitude  tranmitted 
  e 2nI y / c  e y
And thus,
 
I o  E  Amplitude  incident 
where the attenuation coefficient,

2n I
c
y =1/  is called the skin depth through which the light intensity drops to
1/3 (~1/e = 1/2.7) of the incident beam intensity (extremely thin in the
order of nm for most metals)
*Since metal has abundant free electrons (  o  k e / m e  0 with k e  0 and   0 ), the
expression of refractive index based on the oscillator model, Eq. (3.72), should be extended to
account for the additional oscillation of free electrons:
Eq. (3.72)
n 2    nr  ini   1 
2
 1
Nqe2
 o me
Nqe2
 o me


fj

j   2   2  i  
j
 oj




fj
fe



j   2   2  i  
2
    i e
j
 oj

Assuming the contribution of bound electrons are negligibly small compared with that of free
electrons, i.e., fe = 1.0, e = 0, and fj = 0:
 p
Nqe2
n    1 
 1  
2
 o m e

2



2
with the plasma frequency  p  Nqe2 /  o me
For incident wave frequency    p , n-real and the metal is transparent to  , and for    p , ncomplex and the metal is opaque to  .
This is the reason why X-rays are penetrating most materials including
metals (exception: lead). >>> Table 4.3
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Engineering Optics and Optical Techniques Lecture Note No. 3
Spring 2007, Prof. Kenneth D. Kihm
Reflectance from a Metal:
The Amplitude of Reflection for  i  0 is given as (Eq. 4.47):
ro  r  r 
nt  ni
n t  ni
~  n  in ),
For normal incidence in air (ni = 1.0) on a metal surface (nt = n
R
I
R  ro
2
 ro ro* 
nR  12  nI2
nR  12  nI2
R becomes Rdielectric for nI ~ 0 (no free electron).
R ~ 1 for nI >> 0 (100% reflection).
* For o  589.3 nm ,
Metals
nR
nI
R
Sodium (Na)
0.04
2.4
0.9
Tin (Sn)
1.5
5.3
0.8
Galium (GA)
3.7
5.4
0.7
Glass
(dielectric)
1.5
0
0.04
Homework Assignment #3
E-o-C Problems:
4.19, 4.24, 4.40, 4.44, 4.57, 4.61, 4.77, 4.80
Due by 1 p.m. on Tuesday of February 6, 2006
15