Transcript Ray matrices and geometrical optics

xin, qin
The Ray Vector
xout, qout
A light ray can be defined by two co-ordinates:
its position, x
q
its slope, q
x
Optical axis
These parameters define a ray vector,
which will change with distance and as
the ray propagates through optics.
 x
q 
 
Ray Matrices
For many optical components, we can define 2 x 2 ray matrices.
An element’s effect on a ray is found by multiplying its ray vector.
Ray matrices
can describe
simple and complex systems.
Optical system ↔ 2 x 2 Ray matrix
 xin 
q 
 in 
A
C

B
D 
 xout 
q 
 out 
These matrices are often called ABCD Matrices.
Ray matrices as
derivatives
Since the displacements and angles
are assumed to be small, we can
think in terms of partial derivatives.
spatial
magnification
xout
xin
xout
xout
xout

xin 
qin
xin
qin
qout
qout
qout

xin 
q in
xin
qin
xout
q in
 xout   A B   xin 
q   C D  q 
  in 
 out  
q out
xin
q out
q in
angular
magnification
We can write
these equations
in matrix form.
Ray matrix for free space or a medium
If xin and qin are the position and slope upon entering, let xout and
qout be the position and slope after propagating from z = 0 to z.
xout qout
xin, qin
xout  xin  z qin
qout  qin
Rewriting these expressions
in matrix notation:
z
z=0
Ospace
1 z 
= 

0 1 
 xout 
1 z   xin 
q   0 1 q 

  in 
 out 
Ray matrix for a lens
Olens
1/ f  (n 1)(1/ R1 1/ R2 )
 1
= 
-1/f
0

1
The quantity, f, is the focal length of the lens. It’s the single most
important parameter of a lens. It can be positive or negative.
In a homework problem, you’ll extend the Lens Maker’s Formula to
lenses of greater thickness.
R1 > 0
R2 < 0
f>0
If f > 0, the lens deflects
rays toward the axis.
R1 < 0
R2 > 0
f<0
If f < 0, the lens deflects
rays away from the axis.
A lens focuses parallel rays to a point
one focal length away.
For all rays
A lens followed by propagation by one focal length:
xout = 0!
f   xin   0 
 xout  1 f   1 0  xin   0
q   0 1  1/ f 1  0   1/ f 1   0    x / f 

  
    in 
 out  
f
f
Assume all input
rays have qin = 0
At the focal plane, all rays
converge to the z axis (xout = 0)
independent of input position.
Parallel rays at a different angle
focus at a different xout.
Looking from right to left, rays diverging from a point are made parallel.
Consecutive
lenses
Suppose we have two lenses
right next to each other (with
no space in between).
 1
Otot = 
-1/f 2
f1
f2
0  1
0
1
0

= 




1  -1/f1 1 
-1/
f

1/
f
1
 1
2

1/f tot = 1/f1 +1/f 2
So two consecutive lenses act as one whose focal length is
computed by the resistive sum.
As a result, we define a measure of inverse lens focal length, the
diopter.
1 diopter = 1 m-1
A system images an object when B = 0.
When B = 0, all rays from a
point xin arrive at a point xout,
independent of angle.
xout = A xin
 xout   A 0   xin   A xin 
q   C D  q   C x  D q 
  in   in
in 
 out  
When B = 0, A is the magnification.
The Lens Law
From the object to
the image, we have:
1) A distance do
2) A lens of focal length f
3) A distance di
0  1 d o 
1 d i   1
O
  1/ f 1  0 1 
0
1




do 
1 di   1

  1/ f 1  d / f 
0
1


o

1  di / f d o  di  d o di / f 



1/
f
1

d
/
f
o


B  d o  di  d o di / f 
do di 1/ do  1/ di  1/ f  
0 if
1 1 1
 
d o di f
This is the Lens Law.
Imaging
Magnification
If the imaging condition,
1 1 1
 
d o di f
is satisfied, then:
1  di / f
O
 1/ f

1  do / f 
1 1
A  1  di / f  1  di   
 d o di 
di

M 
do
0 
1/ M 
1 1
D  1  do / f  1  do   
d o di 

do
 
 1/ M
di
0
So:
 M
O
 1/ f
magnification
• Linear or transverse magnification — For real images,
such as images projected on a screen, size means a
linear dimension
• Angular magnification — For optical instruments with
an eyepiece, the linear dimension of the image seen in
the eyepiece (virtual image in infinite distance) cannot be
given, thus size means the angle subtended by the
object at the focal point (angular size). Strictly speaking,
one should take the tangent of that angle (in practice,
this makes a difference only if the angle is larger than a
few degrees). Thus, angular magnification is defined as
Depth of Field
Only one plane is imaged (i.e., is in focus) at a time. But we’d like
objects near this plane to at least be almost in focus. The range of
distances in acceptable focus is called the depth of field.
It depends on how much of the lens is used, that is, the aperture.
Out-of-focus
plane
Image
Object
f
Aperture
Focal
plane
The smaller the aperture, the more the depth of field.
Size of blur in
out-of-focus
plane
Gaussian Beams
F-number
The F-number, “f / #”, of a lens is the ratio of its focal length and its
diameter.
f/# = f/d
f
d1
f
f/# =1
f
d2
f
f/# =2
Large f-number lenses collect more light but are harder to engineer.
Numerical Aperture
Another measure of a lens size is the numerical aperture. It’s the
product of the medium refractive index and the marginal ray angle.
NA = n sin(a)
a
f
High-numerical-aperture lenses are bigger.
Why this
definition?
Because the
magnification
can be shown to
be the ratio of
the NA on the
two sides of the
lens.
Numerical Aperture
• The maximum angle at which the incident
light is collected depends on NA
Remember diffraction
Numerical Aperture
• The maximum angle at which the incident
light is collected depends on NA
Remember diffraction
Scattering of Light
• What happens when particle size becomes less
than the wavelength?
• Rayleigh scattering :
– The intensity I of light scattered by a single small
particle from a beam of unpolarized light of
wavelength λ and intensity I0 is given by:
–
– where R is the distance to the particle, θ is the
scattering angle, n is the refractive index of the
particle, and d is the diameter of the particle.
Scattering of Light
• What happens when particle size
becomes less than the wavelength?
• MIE scattering : http://omlc.ogi.edu/calc/mie_calc.html
– Use a series sum to calculate scattered
intensity at arbitrary particle diameter
Sphere Diameter
0.10
microns
Refractive Index of Medium
1.0
Real Refractive Index of Sphere
1.5
Imaginary Refractive Index of Sphere
0
Wavelength in Vacuum
0.6328
microns
Concentration
0.1
spheres/micron3
MIE scattering
Sphere Diameter
0.2
microns
Refractive Index of Medium
1.0
Real Refractive Index of Sphere
1.5
Imaginary Refractive Index of Sphere
0
Wavelength in Vacuum
0.6328
microns
Concentration
0.1
spheres/micron3
Telescopes
Keplerian telescope
Image
plane #1
Image
plane #2
M1
M2
A telescope should image an object, but, because the object will have
a very small solid angle, it should also increase its solid angle
significantly, so it looks bigger. So we’d like D to be large. And use
two lenses to square the effect.
Oimaging
Otelescope
 M

 1/ f
0 
1/ M 
where M = - di / do
0   M1
0 
 M2

  1/ f 1/ M 

1/
f
1/
M

2
2
1
1
M 1M 2
0 




M
/
f

M
/
f
1/
M
M
 1 2
2
1
1
2
Note that this is
easy for the first
lens, as the object
is really far away!
So use di << do
for both lenses.
Telescope Terminology
Microscopes
Image
plane #1
Objective
M1
Eyepiece
Image
plane #2
M2
Microscopes work on the same principle as telescopes, except that
the object is really close and we wish to magnify it.
When two lenses are used, it’s called a compound microscope.
Standard distances are s = 250 mm for the eyepiece and s = 160 mm
for the objective, where s is the image distance beyond one focal
length. In terms of s, the magnification of each lens is given by:
|M| = di / do = (f + s) [1/f – 1/(f+s)] = (f + s) / f – 1 = s / f
Many creative designs exist
for microscope objectives.
Example: the Burch reflecting
microscope objective:
Object
To eyepiece
Example: Magnifying Glass