Download Lecture-8-Optics

(a)
Graded Index Optical Systems
Flat disk of glass with an index, n(r),
that varies as a function of distance r
from the center is an example of a
GRIN lens.
(b)
Based on the observation that rays
(wave-fronts) slow down in an
optically dense region and speed-up in
less dense regions.
The center of cylindrical lens has n =
nmax along its optical axis. So, along
the optical axis the optical path length
(OPL) is given as (OPL)O = nmaxd
At a height r, (OPL)r  n(r)d.
In order for the rays to converge at a
focus, the planar wavefront must bend
into a spherical wavefront, which
defines surfaces of constant phase.
In order to match the phase at all points
on the wavefront we must require
nmax
n(r)
n(r )  nmax 
2
r
2 fd
(OPL ) r  AB  (OPL ) O
and
n(r )d  AB  nmax d
also
AF  r 2  f 2
Therefore
r
AB  AF  f
 n(r )d  r 2  f 2  f  nmax d
r2  f 2  f
 n(r )  nmax 
d
Assume r  f
Then  n(r )  nmax
 nmax

f 1 r2 / f 2

d

1/ 2
f
r2

2 fd
Similar to a multimode-graded index core
optical fiber
The most common device is a GRIN
cylinder a few millimeters in
diameter. They are usually fabricated
using an ionic diffusion process in
which a homogeneous glass is
immersed in a molten salt bath for
many hours.
The focal length is determined by the
index change n < ~0.10. The profile
is usually expressed as
n(r) = nmax(1 – ar2/2)
Used in a
copy machine
Common Applications: Laser printers,
photocopiers, fax machines (i.e. devices
requiring image transfer between surfaces.)
Rays striking the surface in a plane
of incidence that contains the optical
axis travel in a sinusoidal path have
spatial period T = 2/a1/2 where a =
a().
It is possible to create real erect
images by changing the object
distance or length L of the lens.
Radial GRIN lenses are often specified in
terms of their pitch. A 1.0 pitch rod represents
L = T = 2/a1/2 (full sine wave). A pitch of
0.25 has a length of quarter sine wave T/4.
Note that the block of glass is formed so that n = n(z).
The block can then be grounded and polished into the
shape of a lens. The result is a reduced n for marginal
rays.
It is possible to use this
approach to significantly
reduce the effects of
spherical aberration in
comparison to a regular
lens in (c) to the right.
Superposition of Electromagnetic (E-M) Waves
Any E-M wave satisfies the wave equation:
The equation is linear since any linear
combination will also satisfy this equation:
2
1


2
  2 2
v t
N


 (r , t )   ci i (r , t )
i 1
This is called the principle of superposition.
Consider the addition of E-M waves possessing the same frequency but having
different phases:
E1  E01 sin t  1  and
E2  E02 sin t   2 
Then E  E1  E2  E0 sin t   
2
2
where E02  E01
 E02
 2 E01E02 cos 2  1 
and
E01 sin 1  E02 sin  2
tan  
E01 cos 1  E02 cos  2
Note that we can define a phase difference, , such that  = 2 - 1.
The composite wave is harmonic with the same frequency, but the amplitude
and phase are different.
Let  = -(kx + )   = 2 - 1 = (2/)(x1 - x2), where we let 1 = 2 for now.
x1 and x2 are the distances from the sources of the two waves to the point of
observation and  is index dependent, and so  = o/n . Then, we can write
  kx 
2
o
nx1  x2   ko   2
where   nx1  x2 ; Optical
Note that

o

x1  x2


o
Path Difference (OPD )  
 (# waves in medium)
If 1 = 2 = const. the two E-M waves are said to be coherent.
Suppose that we have a superposition of two waves that travel a small
difference in distance (x):
E1  E01 sin t  k ( x  x) and
and
let
E2  E02 sin t  kx
E01  E02 ;  2  1  kx
Use sin( A  B)  sin( A) cos( B)  sin( B) cos( A) and

kx  k ( x  x / 2)  kx / 2
E  E1  E2  2 E01 coskx / 2sin t  k ( x  x / 2)
and   kx
Consider two special cases: (1) x = (n + 1/2) (out-of-phase)
and (2) x = n (in-phase); n = 0, 1, 2, 3, …
Case 1
kx 2 1

(n  1 / 2)  (n  1 / 2)
2
 2
   kx  2 (n  1 / 2)
Destructive Interference E = 0, which is a minimum in Intensity.
Case 2
kx 2 1

n  n
2
 2
In both cases n = 0, 1, 2, 3…
   kx  (2 )n
Constructive Interference E0 = 2E01, which
is a maximum in Intensity.
For the more general case
in which E01  E02
Case 2: Constructive
interference  = 2n
Case 1: Partial
destructive interference
 = 2(n + 1/2)
Fig. 7.3 Waves out-of-phase by kx radians.
Fig. 7.4 The French fighter Rafale uses
active cancellation to confound (frustrate)
radar detection. It sends out a nearly equal
signal that is out-of-phase by /2 with the
radar wave that it reflects. Therefore, the
reflected and emitted waves cancel in the
direction of the enemy receiver.
In general, the sum of N such E-M waves is
N
E   E0i cos i  t   E0 cos  t 
i 1
N
N
N
where E   E  2 E0i E0 j cos( i   j )
2
0
2
0i
i 1
j i i 1
N
and
tan    E0i sin  i
i 1
N
E
i 1
0i
cos  i
Suppose that we have N random sources (e.g. a light bulb).
Then cos(i-j)t = 0  E02t =NE201 if each atom emits waves of equal E01.
This result is for an incoherent source of emitters.
For a coherent source, we have i = j and the sources are in-phase which gives
 E02  E02
t
2


2
   E0i   N 2 E01
Each atom emits waves of equal E01.
 i 1

N
Complex Method for Phasor Additon
E1  E01 coskx  t  1   E01 cos1  t    kx   
~
~
 E1  E01 exp i1  t  with E1  Re E1
The addition of N E-M waves becomes
~ N ~ N
i  j   i t
E   E j    E 0 j e  e  E 0 e i  e  i t
j 1
 j 1

N
 E0 e   E0 j e
i
j 1


E02  E0 ei E0 ei

i j
The complex amplitude can be expressed as a
vector in the complex plane, and is known as a
phasor. The resultant complex amplitude is the
sum of all constituent phasors.
*
Which can be used to calculate the resulting
irradiance from the complex amplitudes of the
constituent waves.
Imaginary
Axis
Phasor Addition
E1
Real Axis
Consider the sum of two E-M waves: E = E1 + E2
E1  E01 sin t  1  and
E2  E02 sin t   2 
Then E  E1  E2  E0 sin t   
2
2
where E02  E01
 E02
 2 E01E02 cos 2  1 
and
E01 sin 1  E02 sin  2
tan  
E01 cos 1  E02 cos  2
From the law of
cosines we can easily
calculate E02 and
further analysis of
the geometry gives
tan .
E1=5sint
E2=10sin(t+45º)
E3=sin (t-15º)
E4= 10sin(t+120º)
E5=8sin(t+180º)
Consider the addition
of these five E-M
waves using the
phasor addition below.
In electrical engineering, these
phasors can also be written with
the following notation:
50, 1045, 1-15,
10120, and 8180
E1  E01 sin( kx  t )
E2  E02 sin( kx  t   )
Fig. 7.7 The phasor sum
of E1, E2, E3, E4 and E5.
The summation of two sinusoidal functions of the same
frequency using phasor additon. Here E1 is taken as the
reference phasor, and since E2 leads E1 (i.e. its peak occurs at
an earlier location) the angle  is positive. Thus  is positive
and the resultant E also leads to E1.
E1  E01 sin( kx  t   )
E2  E02 sin( kx  t  2 )
E3  E03 sin( kx  t  3 )
E4  E04 sin( kx  t  4 )
E01  E02  E03  E04
Standing Waves: Consider reflection of E-M waves of a mirror
EI  E0 sin kx  t  and
ER  E0 sin kx  t 
E  EI  ER  E0 sin kx  t   sin kx  t 
 2 E0 sin kx cos t  2 E0 cos t sin kx
Description of standing wave:
Standing wave: Time-varying
amplitude with sinusoidal spatial
variation (see previous slide).
Nodes: x = 0, /2, , 3/2, 2...
Anti-Nodes: x = /4, 3/4, 5/4….
Now, consider the addition of two E-M waves having different frequencies:
E1  E01 cosk1 x  1t  and
k1  k 2 , 1  2 , v ph 

k
E2  E01 cosk 2 x  2t 

1
k1

2
k2
Then E  E01cosk1 x  1t   cosk 2 x  2t 
       
and with cos   cos   2 cos
 cos

 2   2 
Then E  2 E01 cosk m x  mt  cosk x   t 
where   1  2  / 2
and
and m  1  2  / 2
k  k1  k 2  / 2; k m  k1  k 2  / 2
The resultant wave is a traveling wave of frequency and wave number:
Then E  E0 x, t  cosk x   t 
(traveling wave)
, k
with E0 x, t   2 E01 coskm x  mt 
(modulated or time varying amplitude)
Note that the Irradiance is given by the following
2
2
1  cos2km x  2mt 
I  E02  4E01
cos 2 km x  mt   2 E01
2m  1  2
(beat frequency)
1
2
Detector
with fast
response
B  2m  1  2
(beat frequency)
Beats can also be observed through the superposition of E-M waves
possessing different amplitudes, as well as different frequencies. The
phasor method can be used to help illustrate the formation of beats.
Heterodyne Principle
Heterodyning is a method for transferring a broadcast
signal from its carrier to a fixed local intermediate
frequency in the receiver so that most of the receiver does
not have to be retuned when you change channels. The
interference of any two waves will produce a beat
frequency, and this technique provides for the tuning of a
radio by forcing it to produce a specific beat frequency
called the "intermediate frequency" or IF.
The carrier wave exhibits a high frequency
The phase velocity is given by v ph   / k
The group velocity is given by v g 
m
c    1  2  / 2
1  2



km
k1  k 2
k
This is the rate at which the modulation envelope or energy of the
wave advances or propagates. For a general dispersion  =  (k)
 d 
vg  

 dk 
and this speed is usually less than the speed of light c.
Using  = vk and v = c/n
d d
dv
d
vg 

(vk)  v  k
vk
(c / n)
dk dk
dk
dk

c dn c
c dn
 k dn 
vk 2
 k 2
 v 1 

n dk n
n dk
 n dk 
For normal dispersion, dn/dk > 0 and therefore vg < v.
Polarization of Light
Linear Polarization: Begin by defining individual components:


E x ( z, t )  iˆE0 x coskz  t  and E y ( z, t )  ˆjE0 y coskz  t   
 

E  E x  E y If   2n , n  0, 1, 2, 3, ...

 E  iˆE0 x  ˆjE0 y coskz  t  " in  phase" components
 

E  E x  E y If   2n  1 , n  0, 1, 2, 3, ...

 E  iˆE0 x  ˆjE0 y coskz  t  "180 out  of  phase" components




It is therefore possible to define any polarization orientation with a
constant vector in the x-y plane for the case of linear polarization.
For linear polarization, the state of polarization is often referred to
as a P - state. This is the symbol for a script P.
Circular polarization:


ˆ
E x ( z , t )  i E0 x coskz  t  and E y ( z , t )  ˆjE0 y coskz  t   
 

E  E x  E y If    / 2  2n , n  0, 1, 2, 3, ...

 
ˆ
ˆ
 E  E0 i coskz  t   j sin kz  t  with E  E  E02

 E  E0  const .


E0

The electric field vector clearly rotates clockwise while looking back at the
source from the direction of propagation. The frequency of rotation is  with
a period of T = 2/. This is the case of “right-circularly polarized light”. It is
often expressed as an R – state. This is a script R.
Right circular polarization
(R – state)
E0

Left circular polarization, i.e.
an L– state.
If    / 2  2n , n  0, 1, 2, 3, ...

 E  E0 iˆ coskz  t   ˆj sin kz  t 




E  E0  const .
The electric field vector clearly rotates counter-clockwise while looking back at
the source from the direction of propagation. The frequency of rotation is  with a
period of T = 2/. This is the case of “left-circularly polarized light”. It is often
expressed as an L– state. This is a script L.



ER  EL  E
It is easy to understand that for general parameters E0x, E0y, and ,
we have elliptical polarization.


E x ( z, t )  iˆE0 x coskz  t  and E y ( z, t )  ˆjE0 y coskz  t   
 

E  Ex  E y
Actual polarizers. Irradiance is
independent of the rotation angle 
for the conversion of natural light
(unpolarized) to linear polarization.
In the figure below, only the
component E01cos is transmitted 
I() = I(0)cos2, which is known as
Malus’s Law.
If Iu = I (natural or unpolarized light),
then I(0) = Iu <cos2>t = Iu/2.
Dichroism –selective absorption
of one of two orthogonal E
components.
1. Absorption of E-field in the ydirection causes e’s to flow.
2. Re-radiation of waves that
cancel incident waves polarized in
the y-direction.
This results in transmission of
waves with E-fields perpendicular
to the wires (i.e., along the xdirection).
Polaroid sheet (H-Sheet), most commonly used linear polarizer.
Contains a molecular analogue of the wire grid.
1. Sheet of clear polyvinyl alcohol is heated and stretched.
2. Then it is dipped in an ink solution rich in Iodine.
3. Iodine is incorporated into straight long-chain polymeric molecules
allowing electron conduction along the chain, simulating a metal wire.
HN-50 is the designation of a hypothetical, ideal H-sheet that transmits 50%
of the incident natural light while absorbing the other 50%. In practice,
about 4% of the light will be reflected at each surface leaving a maximum
transmittance of 92% for linearly polarized light incident on the sheet.
Thus, HN-46 would transmit 46% of incident natural light, and might be the
optimal polarizer. In general, for HN-x, the irradiance of polarized light
transmitted would be I=Io(x/50), where Io is the irradiance for the ideal case.
In practice, it is possible to purchase HN-38, HN-32, and HN-22 in large
quantities for reasonable prices, each differing in the amount of iodine
present.
Tourmaline
Crystal System: Hexagonal (trigonal)
Habit: As well-formed, elongate, trigonal prisms, with
smaller, second order prism faces on the corners. Prism
faces are often striated parallel to direction of elongation
(c-axis). The rounded triangular cross-sectional shape of
tourmaline crystals is diagnostic; no other gem mineral has
such a shape.
Hardness: 7-7.5
Cleavage: none
High birefringence (Two differenct indices of
refraction)
Strong Dichroism
Any transparent gem having a mean R.I. of 1.63 and a
birefringence of 0.015-0.020 is tourmaline.
Tourmaline is widespread in metamorphic, igneous and
sedimentary rocks. Gem Elbaite is, however, nearly
restricted to pegmatites. Literally thousands of tourmalinebearing pegmatites are known; only a few hundred
apparently contain gem quality material in mineable
quantities.
Found in Brazil, Sri Lanka, U.S., Southern California
Tourmaline )Boron Silicates)
Chemical
Formula
Specific
Gravity
XY3Al6B3Si6OH
(X = Na or Ca)
Hardness
7.5 - 7
Refractive
Index
1.621.65
Y = Mg, Li, Al or Fe
3.3 - 3.0
1) There is a specific direction within
the crystal known as the principal or
optic axis.
2) The E-field component of an
incident wave that is perpendicular to
the optic axis is strongly absorbed.
3) The thicker the crystal the more
complete will be the absorption.
4) A plate cut from a tourmaline
crystal parallel to its principle axis
and several mm thick will serve as a
linear polarizer.
5) Absorption depends on .
6) Advantages over H-sheet
polarizers with regard to maximum
irradiance permitted and can be used
with high power lasers.