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The principles of geometrical optics and Fourier optics, including Fermat's principle, Snell's law, and propagation matrices in G.O. It also covers topics such as total internal reflection and ray bending. equations and examples to illustrate the concepts. The University of Illinois at Urbana-Champaign is mentioned as the institution where the author, Gabriel Popescu, is affiliated with. The document could be useful as study notes or lecture notes for a course in optical imaging or a related field.
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Gabriel
Popescu
University
of^ Illinois
at^ Urbana
‐Champaign
y^
p^ g
Beckman
Institute
Quantitative Light Imaging Laboratory
Electrical and Computer Engineering, UIUC
Principles of Optical Imaging
Quantitative
Light
Imaging
Laboratory
http://light.ece.uiuc.edu
Imaging
^ Introduction
to^ g
eometrical
optics
and
Fourier
optics
g^
p^
p
precedes
Microscopy
Chapter
3:^ Imaging
Imaging
^ G.O.
predicts
image
location
trough
complicated
systems;
p^
g^
g^
p^
y^
accuracy
is^ fairly
good
^ Nowadays
there
are
software
programs
that
can
run
“ray
ti^
” t^
h^
bit^
t^ i l
propagation”
trough
arbitrary
materials
^ So,
what
are
the
laws
of^ G.O.?
Chapter
3:^ Imaging
Imaging
a) n
=^ constant
b)^ n
=^ n(
function
of
r
position
B L
B
ds
A
L
A
Time:
^ straight
AB
AB
A
(3.1)
Chapter
3:^ Imaging
Imaging
^ Definition:
optical
path
length
^ How
can
we^
predict
ray^
bending
(eg.
mirage)?
(3.2)
^ Fermat’s
Principle: ^ Light
connects
any
two
points
by^ a
path
of^ minimum
time
(the least time principle)(the
least
time
principle)
B n S dS
(3.3)
B^
n S dS A
A ^ If n=constant
in^ space,
AB=line,
of^ course
Chapter
3:^ Imaging
Imaging
^ Consider
an^ interface
between
2 media:
y
x B
θ^1
θ^2
X
A
n^2
n^1
X
^ The
rays
are
“bent”
such
that:
^ Snell’s law (3 4) can be easily derived from Fermat’s principle
1
1
2
2
(3.4)
^ Snell s
law
can
be^ easily
derived
from
Fermat s
principle
by^ minimizing:
total
path
‐length
1
2
^ Take
it^ as
an^ exercise
Chapter
3:^ Imaging
demo^ available
Imaging
^ The
angle
of^ incidence
for
which
g is^ called
critical
angle
(3.6)
1
2
c
^ This
is^ total
internal
reflection
y
t
r n^1
n^2
θ
c) law
of^ reflection
Snell’s
law
is:
2
1
t^ x
θ^2 θ^1 i
(reflection
law)
^ Energy conservation: P
i
1
1
2
2
1
2
(3.7)
Energy
conservation:
Pt^
Pr^
Pi
Chapter
3:^ Imaging
demo^ available
Imaging
^ Efficient
way
of^ p
ropagating
rays
through
optical
systems
y^
p^ p g
g^ y
g^
p^
y y
Optical
System
θ^
θ^
x O
θ^1
y^2 θ^2
OA^ ≡^ Optical
Axis
y^1
^ Any
given
ray is^ completely
determined
at^ a
certain
plane
by
the angle with OA
and height w r t OA
y^1
the^
angle
with
and
height
w.r.t
y^1
^ Let’s
propagate
(y,^1
assume
small
angles
Gaussian
approximation Chapter
3:^ Imaging
Imaging
a) Translation
We^
can^
re‐write
in^ compact
form:
2
1
2
1
2
1
(3.9)
Chapter
3:^ Imaging
Imaging
b)Refraction
‐spherical
dielectric
interface
p
x
y
θ^2
^ θ^1
α^2 y^1
α^1
y^1
C ^ R
OA
^ Snell’s
law: ^ Geometry:
1 1
2 2
(3 10)
2
(^21)
2
(3.10)
1
2
1 1
1
2 2
2
Chapter
3:^ Imaging
Imaging
b)Refraction
‐spherical
dieletric
interface
p ^ Important:
To^ avoid
confusion
between
Ѳ^ and
angles,
use^
“sign
convention”
convention
‐
OA
^ Counter
clock
‐wise
=^ positive
2 distance convention2. distance
convention Left
^ negative Right
^ positive
A^
OAB
‐^
Chapter
3:^ Imaging
Imaging
b)Refraction
‐spherical
dieletric
interface
p ^ Example:
R
R
R +^
‐
We found
and 1
0 n^ n
n ^
^
^
1
0 n^
n^
n ^
^
^
We^
found
and
Same
(^1) convention applies to spherical mirrors. Without 2 1 2 2 n^ n
n n R n ^
^
^
2 1
1 2
2 n^
n^
n n R^
n
^
^
Same
convention
applies
to^ spherical
mirrors.
Without
sign
convention,
it’s^
easy
to^ get
the
wrong
numbers.
Chapter
3:^ Imaging
Imaging
^ The
nice
thing
is^ that
cascading
multiple
optical
components
g^
g^
p^
p^
p
reduces
to^ multiplying
matrices
(linear
systems)
^ Example:
A^
B
n^1
n^2
n^3
n^4
A^
B T^4
T^3
R^3
T^2
R^2 R^1 T^14
3
3
2
2
1 1
B^
A
B^
A
^ Note
the
reverse
order
multiplication
(chronological
order)
B^
A
Chapter
3:^ Imaging
Imaging
^ Note
the
reverse
order
multiplication
(chronological
order)
p^
g^
=^ Translation
matrix
d 1 ^
^
(^0 1)
1 0 n^
n^
n ^
^
^
^ R=refraction
matrix
2 1
1 2
2 n^
n^
n n R^
n
^
^
Chapter
3:^ Imaging
