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Reflective Optics
Chapter 25
Reflective Optics
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Wavefronts and Rays
Law of Reflection
Kinds of Reflection
Image Formation
Images and Flat Mirrors
Images and Spherical Mirrors
The Paraxial Approximation and Aberrations
Wavefronts and Rays
wavefronts
(E = E0)
A wave is the propagation
of a condition or
disturbance.
ray

ray
A wavefront is a surface
over which the value of
that condition is
constant.
ray
ray
Wavefronts and Rays
wavefronts
(E = E0)
The direction of motion is
always locally normal to
the wavefront.
ray

ray
A line drawn in the
direction of advance is
called a ray.
ray
ray
Wavefronts and Rays
The directional distribution of these rays depends on the nature and
geometry of the source of the waves.
ray

ray
ray
wavefronts (E = E0)
Wavefronts and Rays
As distance from the point source increases, the radii of the spherical
wavefronts becomes larger, until the wavefronts approximate
planes. Waves from an infinitely-distant source are sometimes
called plane waves.
ray
ray
ray
wavefronts (E = E0)
Law of Reflection
When light encounters the surface of a material, three
things happen:

reflection

transmission

absorption
Law of Reflection
In reflection, the light
“bounces” off the surface.
The bounce occurs according
to the law of reflection:
 r  i
i
r
Law of Reflection
 r  i
Notice that:


Both angles are measured
from the surface normal
The incident ray, the
reflected ray, and the
surface normal all lie in a
single plane: the plane of
incidence
i
r
Law of Reflection
Notice that:


 r  i
If the surface normal is
rotated through an angle a
within the plane of
incidence, and the incident
direction is constant, the
reflected ray rotates through
twice the angle (2a)
If the plane of incidence
rotates, the reflected ray
rotates with it (“one for
one”)
i
r
Kinds of Reflection
We distinguish between two sorts of reflection:

Specular (from smooth surfaces)
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
mirror
polished metal
calm liquid
Diffuse (from rough or irregular surfaces)



white paper
projection screen
clouds or snow
Kinds of Reflection: Specular
A surface producing specular reflection has a constant, or a
“well-behaved” (slowly and continuously changing) normal
direction.
For a constant incident direction, the reflected direction is either
constant or changes continuously: “organized.”
Kinds of Reflection: Diffuse
A surface producing diffuse reflection has random surface
normal directions that change chaotically with location on the
surface.
The law of reflection is everywhere obeyed: but with random
results.
Image Formation
Consider an object that either produces light, or that scatters light
from its surroundings. Each point on its surface acts as a
spherically-symmetric source (“point source”), sending out
rays in many directions.
Image Formation
If something acts on some of the rays that originate at one point
on the object, and causes them to converge at a point
somewhere else, or to diverge from a point somewhere else,
then it has formed an image of that object.
Image Formation: Two Kinds of Image
Images may be sorted into two categories:
 virtual images: formed when the rays never
physically come back to one point, but instead
diverge as if they came from one point. The place
they appear to have come from is the image.

real images: formed when the rays converge, so that
they physically arrive at the same point. That point
of physical reconvergence is the image.
Image Formation: Real Image
The rays here physically converge: real image.
Image Formation: Virtual Image
The rays here diverge as if they came from an image
point: virtual image.
Image Formation by a Flat Mirror
The image formed by a flat mirror:
 virtual
 upright
 same size
 same distance
on the other
side of the
mirror
Image Formation by a Flat Mirror
Two flat mirrors: the image formed by one mirror
acts as an object for the second mirror.
Image Formation by Spherical Mirrors
A spherical mirror is one whose surface is a portion
of a sphere.
The radius of the sphere at the mirror’s center is
called the optical axis. The center of the sphere
is the center of curvature.
Image Formation by Spherical Mirrors
Necessary terms:
focal
length
vertex
axis
center of curvature
focal point
vertex
focal point
axis
center of curvature
focal
length
R2f
Spherical Mirrors: Special Rays
Chief ray: a ray striking the vertex reflects
symmetrically about the axis.
center of curvature
axis
focal point
center of curvature
axis
focal point
Spherical Mirrors: Special Rays
Axial ray: a ray parallel to the axis passes through the
focal point after being reflected (or appears to have)
center of curvature
axis
focal point
axis
center of curvature
focal point
Spherical Mirrors: Special Rays
This means that the image (real or virtual) of an
infinitely-distant object is formed at the focal point.
center of curvature
axis
focal point
axis
center of curvature
focal point
Spherical Mirrors: Special Rays
A ray passing through the center of curvature passes
through it again after reflection.
center of curvature
axis
focal point
axis
center of curvature
focal point
Finding Images by Ray Tracing
We can use these special ray properties to find the
locations where images are formed.
We can also find out:
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
the image size
the image orientation
whether the image is real or virtual
Finding Images by Ray Tracing
Example: concave mirror, object outside the center of
curvature
C
F
Image: real, inverted, between focal point and center of curvature
Finding Images by Ray Tracing
Concave mirror, object at the center of curvature
C
F
Image: real, inverted, at center of curvature
Finding Images by Ray Tracing
Object between center of curvature and focal point:
C
F
Image: real, inverted, outside center of curvature
Finding Images by Ray Tracing
Object at the focal point:
C
F
Image: real, inverted, located at infinity
Finding Images by Ray Tracing
Object inside the focal point:
C
F
Image: virtual, upright
Image Formation Mathematics
Trace a single chief ray from object to image:
do
ho
hi
di
 hi
di
magnification: m 

ho
do
do and di are the conjugate distances
Image Formation Mathematics
Trace a single axial ray:
C
ho
F
hi
f
di
hi
ho

di  f
f
Image Formation Mathematics
hi
h
 o
di  f
f
m
 hi
d
 i
ho
do

hi f  ho d i  ho f

ho  hi

do
di
do
hi  d i
hi d o
do
do di
di
f 



do
d

d
do  di
i
o
hi
 hi hi 1  d o 


di
di
d
i 

f 
ho d i
ho  hi
Image Formation Mathematics
Mirror equation:
do di
f 
d o  di

1
1
1


f do di
The mirror equation relates the conjugate distances and
the focal length. With the definition of magnification
 hi
di
m

ho
do
it can be used generally to characterize images formed
by mirrors.
Image Formation Mathematics
In solving problems, we must keep a standard set of
sign conventions in mind.
do
ho
hi
di
In the picture, all dimensions shown are positive except
for hi, which is negative.
Image Formation Mathematics
Sign convention summary:

f : + for a concave mirror; - for a convex mirror

conjugate distances (do and di): + if object or image is in
front of mirror … - if behind

magnification, m : + if image is upright; - if image is
inverted
The Paraxial Approximation
Did you notice a “stolen base?”
C
ho
F
hi
f
di
f, which is the distance from the focal point to the
vertex, isn’t quite the base of the green triangle.
The Paraxial Approximation
C
ho
F
hi
f
di
The larger the height of the axial ray, the more
difference there is between f and the length of the
triangle’s base.
The Paraxial Approximation
The mirror equation is valid only as a paraxial
approximation. It applies to a threadlike cylinder of
infinitesimal diameter, centered on the axis.
The difference between the paraxial approximation and
the consequences of exact spherical geometry cause
what is called spherical aberration.
The larger the height of an axial ray, the closer to the
vertex it passes through the axis.
Spherical Aberration
focus error vs. zone radius
2.5
f = 100 mm
focus error, mm
2.0
1.5
1.0
0.5
0.0
0
5
10
15
20
zone radius, mm
25
30
35
40
Spherical Aberration
5
4
paraxial
focus
3
2
1
0
-1
-2
circle of
least confusion
-3
-4
-5
90
92
94
96
98
100
102
104
The Paraxial Approximation
So: what good is the mirror equation, since it is only an
approximation?
Optical system design:
 paraxial layout (approximation)
 computer modeling and optimization (exact)