Ray%20matrices%20and%20geometrical%20optics

1
The Ray Vector
xin, qin
xout, qout
A light ray can be defined by two co-ordinates
its position, x its slope, q
optical ray
q
x
Optical axis
These parameters define a ray vector,
which will change with distance and as the ray
propagates through optics.
2
Ray Matrices
For many optical components, we can define 2 x 2
ray matrices. An elements effect on a ray is
found by multiplying its ray vector.
Ray matrices can describe simple and com- plex
systems.
Optical system ? 2 x 2 Ray matrix
These matrices are often called ABCD Matrices.
3
Ray matrices as derivatives
Since the displacements and angles are assumed to
be small, we can think in terms of partial
derivatives.
We can write these equations in matrix form.
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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.
Rewriting these expressions in matrix notation
5
Ray matrix for a lens
The quantity, f, is the focal length of the lens.
Its the single most important parameter of a
lens. It can be positive or negative. In a
homework problem, youll extend the Lens Makers
Formula to lenses of greater thickness.
R1 gt 0 R2 lt 0
R1 lt 0 R2 gt 0
If f gt 0, the lens deflects rays toward the axis.
If f lt 0, the lens deflects rays away from the
axis.
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A lens focuses parallel rays to a point one focal
length away.
A lens followed by propagation by one focal
length
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.
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Consecutive lenses
Suppose we have two lenses right next to each
other (with no space in between).
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
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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
When B 0, A is the magnification.
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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

This is the Lens Law.
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Imaging Magnification
If the imaging condition,
is satisfied, then
So
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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

12
Depth of Field
Only one plane is imaged (i.e., is in focus) at a
time. But wed 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.
Object
Size of blur in out-of-focus plane
Image
f
Aperture
The smaller the aperture, the more the depth of
field.
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Gaussian Beams
14
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 / 2
Large f-number lenses collect more light but are
harder to engineer.
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Numerical Aperture
Another measure of a lens size is the numerical
aperture. Its the product of the medium
refractive index and the marginal ray angle.
NA n sin(a)
Why this definition? Because the magnification
can be shown to be the ratio of the NA on the two
sides of the lens.
High-numerical-aperture lenses are bigger.
16
Numerical Aperture

• The maximum angle at which the incident light is
  collected depends on NA

Remember diffraction
17
Numerical Aperture

• The maximum angle at which the incident light is
  collected depends on NA

Remember diffraction
18
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 ? and intensity I0 is given by
• where R is the distance to the particle, ? is the
  scattering angle, n is the refractive index of
  the particle, and d is the diameter of the
  particle.

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Scattering of Light

• What happens when particle size becomes less than
  the wavelength?
• MIE scattering
• Use a series sum to calculate scattered intensity
  at arbitrary particle diameter

http//omlc.ogi.edu/calc/mie_calc.html
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
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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
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Telescopes
Keplerian telescope
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 wed like
D to be large. And use two lenses to square the
effect.
where M - di / do
Note that this is easy for the first lens, as the
object is really far away!
So use di ltlt do for both lenses.
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Telescope Terminology
23
Eye- piece
Image plane 1
Micro-scopes
Objective
M1
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, its 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/(fs) (f
s) / f 1 s / f
Many creative designs exist for microscope
objectives. Example the Burch reflecting
microscope objective
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Example Magnifying Glass