Acknowledgments

1
Engineering Optics

• Understanding light?
• Reflection and refraction
• Geometric optics (l ltlt D) ray tracing, matrix
  methods
• Physical optics (l D) wave equation,
  diffraction, interference
• Polarization
• Interaction with resonance transitions
• Optical devices?
• Lenses
• Mirrors
• Polarizing optics
• Microscopes and telescopes
• Lasers
• Fiber optics
• and many more

2
Engineering Optics
In mechanical engineering, a primary motivation
for studying optics is to learn how to use
optical techniques for making measurements.
Optical techniques are widely used in many areas
of the thermal sciences for measuring system
temperatures, velocities, and species on a time-
and space-resolved basis. In many cases these
non-intrusive optical devices have significant
advantages over physical probes that perturb the
system that is being studied. Lasers are finding
increasing use as machining and manufacturing
devices. All types of lasers from
continuous-wave lasers to lasers with femtosecond
pulse lengths are being used to cut and process
materials.
3
Huygens Wavelet Concept
At time 0, wavefront is defined by line (or
curve) AB. Each point on the original wavefront
emits a spherical wavelet which propagates at
speed c away from the origin. At time t, the new
wavefront is defined such that it is tangent to
the wavelets from each of the time 0 source
points. A ray of light in geometric optics is
found by drawing a line from the source point to
the tangent point for each wavelet.
Collimated Plane Wave
Spherical Wave
4
Geometric Optics The Refractive Index
The refractive index n fundamental property of
all optical systems, a measure of the effective
speed of propagation of light in a medium
The optical path length in a medium is the
integral of the refractive index and a
differential geometric length
ds
b
a
5
Fermats Principle Law of Reflection
Fermats principle Light rays will travel from
point A to point B in a medium along a path that
minimizes the time of propagation. Law of
reflection
(x3, y3)
qr
(0, y2)
qi
y
(x1, y1)
x
6
Fermats Principle Law of Refraction
Law of refraction
(x1, y1)
A
y
qi
ni
(x2, 0)
x
nt
qt
(x3, y3)
7
Imaging by an Optical System
Light rays are emitted in all directions or
reflected diffusely from an object point.
Spherical wavefronts diverge from the object
point. These light rays enter the (imaging)
optical system and they all pass through the
image point. The spherical wavefronts converge
on a real image point. The optical path length
for all rays between the real object and real
image is the same. Later we will discuss
scattering, aberrations (a geometric optics
concept), and diffraction (a physical optics
concept) which cause image degradation.
8
Imaging by Cartesian Surfaces
Consider imaging of object point O by the
Cartesian surface S. The optical path length for
any path from Point O to the image Point I must
be the same by Fermats principle.
The Cartesian or perfect imaging surface is a
paraboloid in three dimensions. Usually, though,
lenses have spherical surfaces because they are
much easier to manufacture.
9
Reflection at Spherical Surfaces I
Reflection from a spherical convex surface gives
rise to a virtual image. Rays appear to emanate
from point I behind the spherical reflector.
Use paraxial or small-angle approximation for
analysis of optical systems
10
Reflection at Spherical Surfaces II
Considering Triangle OPC and then Triangle OPI we
obtain
Combining these relations we obtain
Again using the small angle approximation
11
Reflection at Spherical Surfaces III
Now find the image distance s' in terms of the
object distance s and mirror radius R
At this point the sign convention in the book is
changed and the imaging equation becomes
The following rules must be followed in using
this equation 1. Assume that light propagates
from left to right. Image distance s is positive
when point O is to the left of point V.
2. Image distance s' is positive when I is to
the left of V (real image) and negative when to
the right of V (virtual image). 3. Mirror
radius of curvature R is positive for C to the
right of V (convex), negative for C to left of V
(concave).
12
Reflection at Spherical Surfaces IV
The focal length f of the spherical mirror
surface is defined as R/2, where R is the radius
of curvature of the mirror. In accordance with
the sign convention of the previous page, f gt 0
for a concave mirror and f lt 0 for a convex
mirror. The imaging equation for the spherical
mirror can be rewritten as
13
Reflection at Spherical Surfaces V
2
1
O'
3
I'
V
C
F
O
I
Ray 1 Enters from O' through C, leaves along
same path Ray 2 Enters from O' through F,
leaves parallel to optical axis Ray 3 Enters
through O' parallel to optical axis, leaves along
line through F and intersection of ray with
mirror surface
14
Reflection at Spherical Surfaces VI
1
O'
2
3
I
F
O
C
V
I'
15
Reflection at Spherical Surfaces VII
Real, Inverted Image
Virtual Image, Not Inverted
16
Geometrical Optics

• Index of refraction for transparent optical
  materials
• Refraction by spherical surfaces
• The thin lens approximation
• Imaging by thin lenses
• Magnification factors for thin lenses
• Two-lens systems

17
Optical Materials
Source Catalog, CVI Laser Optics and Coatings.
18
Optical Materials
Source Catalog, CVI Laser Optics and Coatings.
19
Optical Materials
Source Catalog, CVI Laser Optics and Coatings.
20
Optical Materials
Source Catalog, CVI Laser Optics and Coatings.
21
Refractive Index of Optical Materials
Source Catalog, CVI Laser Optics and Coatings.
22
Refractive Index of Optical Materials
Source Catalog, CVI Laser Optics and Coatings.
23
Refraction by Spherical Surfaces
At point P we apply the law of refraction to
obtain
Using the small angle approximation we obtain
Substituting for the angles q1 and q2 we obtain
Neglecting the distance QV and writing tangents
for the angles gives
n2 gt n1
24
Refraction by Spherical Surfaces II
Rearranging the equation we obtain
Using the same sign convention as for mirrors we
obtain
n2 gt n1
25
Refraction at Spherical Surfaces III
O'
I
q1
V
C
O
q2
I'
26
The Thin Lens Equation I
n1
n1
n2
O'
C1
V1
O
C2
V2
For surface 1
s1
t
s'1
27
The Thin Lens Equation II
For surface 1
For surface 2
Object for surface 2 is virtual, with s2 given by
For a thin lens
Substituting this expression we obtain
28
The Thin Lens Equation III
Simplifying this expression we obtain
For the thin lens
The focal length for the thin lens is found by
setting s 8
29
The Thin Lens Equation IV
In terms of the focal length f the thin lens
equation becomes
The focal length of a thin lens is gt0 for a
convex lens and lt0 a concave lens.
30
Image Formation by Thin Lenses
Convex Lens
Concave Lens
31
Image Formation by Convex Lens
Convex Lens, focal length 5 cm
ho
F
RI
F
hi
32
Image Formation by Concave Lens
Concave Lens, focal length -5 cm
ho
hi
F
F
VI
33
Image Formation Summary Table
34
Image Formation Summary Figure
35
Image Formation Two-Lens System I
60 cm
36
Image Formation Two-Lens System II
7 cm
37
Matrix Methods

• Development of systematic methods of analyzing
  optical systems with numerous elements
• Matrices developed in the paraxial (small angle)
  approximation
• Matrices for analyzing the translation,
  refraction, and reflection of optical rays
• Matrices for thick and thin lenses
• Matrices for optical systems
• Meaning of the matrix elements for the optical
  system matrix
• Focal planes (points), principal planes (points),
  and nodal planes (points) for optical systems
• Matrix analysis of optical systems

38
Translation Matrix
39
Refraction Matrix
40
Refraction Matrix
41
Reflection Matrix
42
Thick Lens Matrix I
43
Thick Lens Matrix II
44
Thin Lens Matrix
The thin lens matrix is found by setting t 0
nL
45
Summary of Matrix Methods
46
Summary of Matrix Methods
47
System Ray-Transfer Matrix
Introduction to Matrix Methods in Optics, A.
Gerrard and J. M. Burch
48
System Ray-Transfer Matrix
Any paraxial optical system, no matter how
complicated, can be represented by a 2x2 optical
matrix. This matrix M is usually denoted
A useful property of this matrix is that
where n0 and nf are the refractive indices of
the initial and final media of the optical
system. Usually, the medium will be air on both
sides of the optical system and
49
Summary of Matrix Methods
50
Summary of Matrix Methods
51
System Ray-Transfer Matrix
The matrix elements of the system matrix can be
analyzed to determine the cardinal points and
planes of an optical system.
Lets examine the implications when any of the
four elements of the system matrix is equal to
zero.
52
System Ray-Transfer Matrix
Lets see what happens when D 0.
When D 0, the input plane for the optical
system is the input focal plane.
53
Two-Lens System
f1 50 mm
f2 30 mm
Input Plane
Output Plane
F1
F1
F2
F2
r
s
q 100 mm
T1
R1
T3
T2
R2
54
Two-Lens System
55
Two-Lens System Input Focal Plane
56
System Ray-Transfer Matrix
Lets see what happens when A 0.
When A 0, the output plane for the optical
system is the output focal plane.
57
Two-Lens System Output Focal Plane
58
System Ray-Transfer Matrix
Lets see what happens when C 0.
When C 0, collimated light at the input plane
is collimated light at the exit plane but the
angle with the optical axis is different. This
is a telescopic arrangement, with a magnification
of D af/a0.
59
Telescopic Two-Lens System
f1 50 mm
f2 30 mm
Input Plane
Output Plane
F1
F1
F2
F2
r
s
q 80 mm
T1
R1
T3
T2
R2
60
System Ray-Transfer Matrix
Lets see what happens when B 0.
When B 0, the input and output planes are
object and image planes, respectively, and the
transverse magnification of the system m A.
61
Two-Lens System Imaging Planes
62
Cardinal Points (Planes) of an Optical System
Distances measured to the right of the respective
reference plane are positive, distances measured
to the left are negative. As shown p lt 0
q gt 0
f1 lt 0
f2 gt 0 r gt 0
s lt 0 v gt 0
w lt 0
63
Cardinal Points (Planes) of an Optical System
64
Thick Lens Analysis
RP1
RP2

• Find for the lens
• Principal Points
• Focal Points
• Focal Length
• Nodal Points

n0 1.0
n0 1.0
nL 1.8
V1
V2
R2 45 mm
R1 30 mm
t 50 mm
65
Thick Lens Analysis
In Lecture 4 we found the matrix for a thick lens
with the same refractive index on either side of
the lens
66
Thick Lens Analysis
67
Thick Lens Analysis
RP1
PP2
RP2
PP1
n0 1.0
n0 1.0
nL 1.8
F1
V2
V1
H2
F2
H1
R2 45 mm
R1 30 mm
t 50 mm
68
Thick Lens Analysis
In general, for any optical system
si 86.7 mm
so -95 mm