Transcript Lecture-6-Optics

Stops: Finite nature of lenses affects the energy and information collected
and imaged by optical systems
Important Definitions:
Aperture Stop (A.S.) – Adustable leaf diaphragm that determines the light
gathering ability of the lens.
Field Stop (F.S.) – Limits the size or angular breadth of the object. In a
camera, the edge of the film itself bounds the image plane and becomes the
F.S.
Entrance Pupil – Image of the A.S. as seen from the object side of the lens
system (below).
Exit Pupil - Image of the A.S. as seen on the image plane. View is from an
axial point for both the entrance and exit pupils.
Chief Ray- Enters the optical system along a line directed towards the midpoint of the Entrance Pupil and leaves the system along a line passing though
the center of the exit pupil.
Aperture Stop (A.S.) – Adustable leaf diaphragm that
determines the light gathering ability of the lens.
Field Stop (F.S.) – Limits the size or angular breadth
of the object. In a camera, the edge of the film itself
bounds the image plane and becomes the F.S.
Entrance Pupil – Image of the A.S. as seen from the
object side of the lens system (below).
Exit Pupil - Image of the A.S. as seen on the image
plane. View is from an axial point for both the
entrance and exit pupils.
Chief Ray- Enters the optical system along a line
directed towards the mid-point of the Entrance Pupil
and leaves the system along a line passing though the
center of the exit pupil.
Another Example showing a more complex arrangement.
Marginal Ray – Ray from the outermost object point on the axis that
goes through the A.S.
Relative Aperture and f-number: f/# = f/D
A  4 ( 2 f )
See next slide
I 2 f  I o
E.P.
D
E.P.
f/2
D/2
2
Let Io·Aobj represent the power radiated spherically
by an object at the point 2f behind a lens.
 ( D / 2)
2
4 ( 2 f )
2
Io  D 
 

64  f 
2
2
D
1
I    
2
f
(
f
/#
)
 
f/4
f /# 
f
D
D
2f
f
f
2f
Therefore, the time necessary for
photographic exposure is proportional to
(f/#)2 = speed of the lens.
So, an f/1.4 lens is twice as fast as an f/2 lens
since the time necessary for the same
exposure (I·t) is ½.
On cameras, e.g., f/# = 1.4, 2, 2.8, 4, 5.6, 8,
11, 16, and 22; so exposure times of 1, 2, 4, 8,
16 msec for each consecutive setting are
necessary to obtain the same amount of
energy absorbed in the photographic film.
Plane Mirrors
i = r , So and Si < 0 when they lie to
the right of V.
The image is invariably virtual in
which it is (1) erect and (2) inverted.
Reflection from a single mirror
causes a right-handed (r-h) object to
appear as a left-handed (l-h) image.
The transverse magnification is
simply MT = -Si/So = +1
i
r
So
Si
Systems with more than one mirror
give rise to an odd or even number of
reflections, resulting in r-h  l-h or
r-h  r-h symmetry, respectively.
l-h  r-h
Inversion
Plane Mirrors
l-h  r-h
Inversion
Multiple mirrors can be used to
control the parity of the image by
creating more than one symmetry
inversion.
Aspherical Mirrors:
The symmetry of incoming plane waves along the optical axis
gives
OPL  W 1 A1  A1 F  W 2 A 2  A 2 F
and
A1 D1  A1 F

parabola

W 1 A1  A1 D1  W 2 A 2  A 2 D 2
with
e AD  AF
and
e 1
Parabola
Directrix
4py = x2
p = distance from vertex to focus (or directrix)
Eccentricity (e) is defined as follows:
e = FP/PQ = 1 for a Parabola
Other conic sections: hyperbola or hyperboloid (e > 1)
ellipse or ellipsoid (e < 1 )
There are many introductory web-sites that describe conic sections, e.g., http://math2.org/math/algebra/conics.htm,
but be sure to refer to your math text books.
Circle
Definition:
A conic
section is
the
intersection
of a plane
and a cone.
Ellipse (h)
Parabola (h) Hyperbola (h)
Ellipse (v)
Parabola (v) Hyperbola (v)
The General Equation for a Conic Section:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
The type of section can be found from the sign of: B2 - 4AC
If B2 - 4AC
is...
<0
=0
>0
then the curve is a...
ellipse, circle, point or no curve.
parabola, 2 parallel lines, 1 line or no curve.
hyperbola or 2 intersecting lines.
The Conic Sections. For any of the below with
a center (j, k) instead of (0, 0), replace each x
term with (x-j) and each y term with (y-k).
Point
Line
Double Line
By changing the angle and location of intersection,
we can produce a circle, ellipse, parabola or
hyperbola; or in the special case when the plane
touches the vertex: a point, line or 2 intersecting
lines.
Equation (horiz. vertex):
Circle
Ellipse
Parabola
Hyperbola
x2 + y2 = r2
x2 / a2 + y2 / b2 = 1
4px = y2
x2 / a2 - y2 / b2 = 1
y = ± (b/a)x
Equations of Asymptotes:
Equation (vert. vertex):
x2 + y2 = r2
y2 / a2 + x2 / b2 = 1
4py = x2
y2 / a2 - x2 / b2 = 1
x = ± (b/a)y
Equations of Asymptotes:
a = major radius (=
1/2 length major axis)
b = minor radius (=
1/2 length minor axis)
c = distance center to
focus
Variables:
r = circle radius
Eccentricity:
0
Relation to Focus:
p=0
a 2 - b 2 = c2
Definition: is the locus of all
points which meet the
condition...
distance to the
origin is
constant
sum of distances to
each focus is
constant
Related Topics:
Geometry
section on
Circles
c/a (<1)
p = distance from
vertex to focus (or
directrix)
a = 1/2 length major axis
b = 1/2 length minor axis
c = distance center to
focus
1
c/a (>1)
a2 + b 2 = c2
distance to focus =
distance to directrix
difference between
distances to each foci is
constant
Using spherical sections to approximate parabolic reflection:
y  x  R   R
2
2

2

y  2 Rx  x  0
2
2
y  2 Rx  x  0
2

2
x
2R 
4R  4 y
2
2
 R
2
R  y
2
2
2

y 
 R  R  1  2 
R 

1/ 2
Consider |y/R| << 1 and x < R, use a binomial expansion 1   1 / 2  1   / 2
x
y
2
 O( y )
4

y  2 Rx  4 fx
2

f  R/2
2R
y
Paraxial region
y << R
y  4 fx
2
Equation for a
parabola
Concave spherical mirror
SC

CP
i  r
because
SA
PA
also
SC  s o  R
also
so , si  0,
and
R0
CP  R  s i

R  R
 SC  s o  R ,
SC

SA

CP

R
f

1


 1 

R
2
R
1
so

PA  s i
Paraxial Approx.
si

si
SA  s o
si  R 
so
so
1
so  R
PA
1
CP    s i  R 

1
si

2
R
f  fo  fi  
 lim s i  lim s o  f
so  
R
2

R
2
si  

1

so
1

1
si
f > 0 (concave mirror); f < 0 (convex mirror) Also, as before, M T 
f
yi

yo
Note: si on the left of V is now taken as positive; opposite to the lens system
Similar behaviors and properties: concave mirror  convex lens
convex mirror  concave lens
si
so
C
Fig. 5.55 The image-forming behavior
of a concave spherical mirror.
Consider a dispersing prism which can be used to perform spectroscopy as
shown on the left.
Dispersing Prism: The emerging ray will be deflected (refracted) by an angle 
from the original direction. Consider the basic geometry:
   i1   t 1    t 2   i 2 
    t1   i 2
    i1   t 2  
and
180      t 1   i 2  180 
Using Snell’s law:
 t 2  sin
use
and
1
 n sin
 i 2   sin
1
n sin 
  t 1 
sin    t 1   sin  cos  t 1  sin  t 1 cos 
n sin  t 1  (1) sin  i1

 t 2  sin
and
again
1
sin  n
2
Snell
' s law
 sin  i1
2

1/ 2

and
 sin  i1 cos 
cos  t 1 

   i1   t 2  
Thus, as n increases then  also increases.
So n decreases as  increases (see figure 3.40)
and n increases as  decreases.
(Red) <  (Blue) (can also be seen in the figure with colors).
The smallest value of  , the minimum deviation, m, is important.
1  sin  t 1
2
Setting
d
d  i1
0

i )  i1   t 2
and
ii )  t 1   i 2
The symmetry of these angles must therefore produce a ray parallel to its
base for the minimum of  (m).
The symmetrical ray transfer (parallel to its base) gives the following:
(1)    t 1   i 2  2 t 1
( 2 )    m   i 1   t 2    2 i 1  
(3) 

n sin  / 2   (1) sin  i1
n (  )  sin  / 2 
1
  m    
 sin 

2


  m (  )    
sin 

2


Therefore, we can experimentally determine n() by
measuring the minimum angle of deviation m() for
light rays having different .
First, we need to make a prism
out of the transparent material
having the cross-section of an
isosceles triangle.

m
nm
By finding m for each ray having a different , we can determine n().
r-h
r-h
Fig. 5.61 The right-angle
prism
Fig. 5.62 The Porro prism.
The orientation is reversed
but remains right-handed.
Reflecting Prisms shown here are used to
change orientation and handedness. Some of
the surfaces are silvered if TIR conditions
aren’t met, such as c = 42 for n = 1.5.
Notice that prisms can be
constructed according to
requirements of size, shape,
angle, etc…
Fig. 5.66 The penta prism and
its mirror equivalent. Same
orientation and remains r-h.
r-h
r-h
Fiber Optics: Rays reflected within a dielectric cylinder
Snell’s Law:
n f sin  t  (1) sin  i

Path length traversed by the ray (l): l  L / cos  t  n f L n
Number of reflections
(N) in the cylinder:
t
D
t
x
N 
l
x
1 
D 
sin  t
x
l
D / sin  t

 sin  i 
2
n f sin  t L
1 
D n  sin  i
2
f
L sin  i
D n  sin  i
2
f
2
f
2
1
1 / 2
1
2
The  sign indicates that
N depends on where the
ray strikes the face of
the cylinder.
For example, if nf = 1.6, i = 30, D = 50 m
Then # reflections per length = N/L = 2000/feet
= 6,600/meter
To prevent cross-talk, must use a cladding.
Typically the fiber core has nf = 1.6 and
the cladding has nc = 1.5.
Rays that are incident (picture on the
right) with angles greater than max will
strike the interior wall at angles less
than c, and will leak out of the fiber.
Therefore, max , which is the
acceptance angle, defines the half-angle
of the acceptance cone of the fiber.
core
cladding
i = max
c - Critical angle at the core-cladding interface for TIR
tc – Critical angle for the transmitted ray at the core-air interface )c + tc = 90)
max- Maximum angle for the incident ray, beyond which there is no TIR at the
core-cladding interface
n f sin  c  n c sin 90 


sin  c 
nc

nf
nc
nf
 sin 90    tc   cos  tc
1  sin  tc
n i sin  max  n f sin  tc
n i sin  max


n f  nc
NA 

2

Numerical
2
2
sin  tc 
sin  max 
Aperture
 n 
1  c 
n 
 f 
n n
2
f
ni
( NA )
2
c
2
NA2 is a measure of the
light-gathering power of
the system and originates
in microscopy where the
equivalent expression
describes the capability of
the objective lens.
The NA is related to the
speed of the system by
f /# 
1
2 ( NA )
Consider the acceptance cone for an optical fiber:
L
max rmax
x
max
The collected intensity (irradiance) I  rmax2  sin2max  NA2  (f/#)-2
Because

D max  2 rmax ,
f /# 
x
2 rmax
use
ni  1
for
 max  90  ,
and

sin  max 
1
2 sin  max
NA 
rmax

L
rmax
x
1

2 NA
n f  nc
2
NA  1 (max)
2
usually
0 . 2  NA  1
Consider attenuation in optical fibers
Decibel (dB) is the customary unit for indicating ratio of two power
levels:
dB= - 10 log (Po/Pi ( and e.g. 1/100  20 dB
Attenuation () is in dB/km of fiber length L
 L = -10 log(Po/Pi (  Po  10   L / 10
Pi
Quality of Optical Glass
1965 1000 dB/km ()
1970
20 dB/km
1982
0.16 dB/km
Improvements in fabrication of fibers and
removal of metallic impurities like Fe, Ni, Cu
and removal of H20 and OH- have led to the
dramatic reduction in fiber attenuation ().
Today, reamplification after every ~80 km is
needed after a ~50 dB drop in signal.
Intermodal Dispersion (IMD)
The minimum and maximum transit times for a ray traversing a distance L
in the fiber are
t min 
while
L

vf

c/nf
t max 
Therefore ,
for
L
l
vf
Ln
f
( axial
ray )
c

L / cos  t

L /( n c / n f )
vf
 t  t max  t min

c/nf
Ln
2
f
cn c
nf  nf


 L
 1 

c  nc

n f  1 . 500 , n c  1 . 489

t
 37 ns / km
L
The spreading out of signals due to this effect is known as IMD
Three basic types of Optical Fibers each with different IMDs
Multi-mode Suffers
greatly from IMD; core
is 50-200 m and
cladding is 20 m thick.
Graded core diameter is about 20-90 m.
Core diameter is less than ~10 m and typically from 2-9 m.
Low attenuation of 0.2 dB/km; currently ideal choice for
long-haul communication networks today.
Graded index cores can
reduce IMD by factor of
100, due to the spiraling
effect and indexdependent speed of the
rays. IMD of 2 ns/km is
typical here.
Single mode fibers are
the best solution for IMD
since only one horizontal
mode with its ray
traveling parallel to the
axis can propagate.
Imaging “coherent” fiber bundle
Collection of individual parallel fibers for
image transfer applications
Fig. 5.73 Intermodal dispersion
in a stepped-index multimode
fiber. Higher-angle rays travel
longer paths.
Fig. 5.75 The spreading of an
input signal due to intermodal
dispersion (IMD).