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ECEG105 & ECEU646
Optics for Engineers
Course Notes
Part 4: Apertures, Aberrations
Prof. Charles A. DiMarzio
Northeastern University
Fall 2003
July 2003
Chuck DiMarzio, Northeastern University
10351-4-1
Advanced Geometric Optics
• Introduction
• Stops, Pupils, and
Windows
• f-Number
• Examples
• Design Process
– From Concept through
Ray Tracing
– Finalizing the Design
– Fabrication and
Alignment
– Magnifier
– Microscope
• Aberrations
July 2003
Chuck DiMarzio, Northeastern University
10351-4-2
Some Assumptions We Made
• All lenses are infinite in diameter
– Every ray from every part of the object reaches
the image
• Angles are Small:
– sin(q)=tan(q)=q
– cos(q)=1
July 2003
Chuck DiMarzio, Northeastern University
10351-4-3
What We Have Developed
• Description of an Optical System in terms
of Principal Planes, Focal Length, and
Indices of Refraction
s, s’ are z coordinates
• These equations describe a mapping
– from image space (x,y,z)
– to object space (x’,y’,z’)
July 2003
B
Chuck DiMarzio, Northeastern University
H
V V’
H’
B’
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Lens Equation as Mapping
f = 10 cm.
50
July 2003
30
20
s', Image Dist., cm.
• The mapping can be
applied to all ranges of
z. (not just on the
appropriate side of the
lens)
• We can consider the
whole system or any
part.
• The object can be
another lens
40
10
0
-10
-20
-30
-40
-50
-60
-40
L4’
Chuck DiMarzio, Northeastern University
-20
0
20
s, Object Dist., cm.
L1 L2 L3
40
60
L4
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Stops, Pupils, and Windows (1)
• Intuitive Description
– Pupil Limits Amount of Light Collected
– Window Limits What Can Be Seen
Window
July 2003
Chuck DiMarzio, Northeastern University
Pupil
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Stops, Pupils and Windows (2)
Images in
Object Space
Entrance Pupil
Limits Cone of
Rays from Object
Entrance Window
Limits Cone of
Rays From Entrance
Pupil
July 2003
Physical
Components
Aperture Stop
Limits Cone of
Rays from Object
which Can Pass
Through the System
Field Stop
Limits Locations of
Points in Object
which Can Pass
Through System
Chuck DiMarzio, Northeastern University
Images in
Image Space
Exit Pupil Limits
Cone of Rays from
Image
Exit Window Limits
Cone of Rays From
Exit Pupil
10351-4-7
Finding the Entrance Pupil
• Find all apertures in
object space
L4’ is L4 seen through L1-L3
L1 L2 L3 L4
• Entrance Pupil
Subtends Smallest
Angle from Object
L3’ L4’
L1 L2’
L3’ is L3 seen through L1-L2
July 2003
Chuck DiMarzio, Northeastern University
10351-4-8
Finding the Entrance Window
• Entrance Window
Subtends Smallest
Angle from Entrance
Pupil
L3’ L4’
July 2003
L1 L2’
• Aperture Stop is the
physical object
conjugate to the
entrance pupil
• Field Stop is the
physical object
conjugate to the
entrance window
• All other apertures are
irrelevant
Chuck DiMarzio, Northeastern University
10351-4-9
f=28 mm
Field of View
f=55 mm
f=200 mm
July 2003
Film=
Exit
Window
Chuck DiMarzio, Northeastern University
10351-4-10
Example: The Telescope
Aperture
Stop
July 2003
Chuck DiMarzio, Northeastern University
Field
Stop
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The Telescope in Object Space
Secondary’
Primary
Secondary
Entrance
Window
July 2003
Entrance
Pupil
Chuck DiMarzio, Northeastern University
10351-4-12
The Telescope in Image Space
July 2003
Primary
Primary’’
Secondary
Exit
Pupil
Chuck DiMarzio, Northeastern University
Exit
Window
Stopped 26 Sep 03
10351-4-13
Chief Ray
Aperture
Stop
Exit
Pupil
Field
Stop
• Chief Ray passes through the center of every pupil
July 2003
Chuck DiMarzio, Northeastern University
10351-4-14
Hints on Designing A Scanner
• Place the mirrors at pupils
July 2003
Chuck DiMarzio, Northeastern University
Put Mirrors Here
10351-4-15
f-Number & Numerical Aperture
Numerical Aperture
f-Number
f
q
A
F’ A’
F
D is Lens Diameter
5
f#, f-number
4
3
2
1
0
0
0.2
0.4
0.6
NA, Numerical Aperture
July 2003
0.8
1
Chuck DiMarzio, Northeastern University
10351-4-16
Importance of Aperture
• ``Fast’’ System
–
–
–
–
Low f-number, High NA (NA1, f# 1)
Good Light Collection (can use short exposure)
Small Diffraction Limit (l/D)
Propensity for Aberrations (sin q  q)
• Corrections may require multiple elements
– Big Diameter 
• Big Thickness  Weight, Cost
• Tight Tolerance over Large Area
July 2003
Chuck DiMarzio, Northeastern University
10351-4-17
The Simple Magnifier
F
A’
July 2003
A
F’
Chuck DiMarzio, Northeastern University
10351-4-18
The Simple Magnifier (2)
• Image Size on Retina Determined by x’/s’
• No Reason to go beyond s’ = 250 mm
• Magnification Defined as
• No Reason to go beyond D=10 mm
• f# 1 Means f=10 mm
• Maximum Mm=25
For the Interested Student: What if s>f ?
July 2003
Chuck DiMarzio, Northeastern University
10351-4-19
Microscope
F’
F
F’
F
A’
A
• Two-Step Magnification
– Objective Makes a Real Image
– Eyepiece Used as a Simple Magnifier
July 2003
Chuck DiMarzio, Northeastern University
10351-4-20
Microscope Objective
F
F’
F’
F
A’
A
July 2003
Chuck DiMarzio, Northeastern University
10351-4-21
Microscope Eyepiece
F
F’
A2
A
A2’
July 2003
F’
F
Chuck DiMarzio, Northeastern University
10351-4-22
Microscope Effective Lens
H
H’
192 mm
Barrel Length = 160 mm
FA
f2=16mm
f1=16mm
D
19.2 mm
Effective Lens:
f = -1.6 mm
H’
H
A
F’
D’
19.2 mm
F’
F
July 2003
Chuck DiMarzio, Northeastern University
10351-4-23
Microscope Aperture Stop
Analysis in Image Space
F’
F
Exit
Pupil
Image
Aperture
Stop
=Entrance
Pupil
Put the Entrance Pupil of your eye
at the Exit Pupil of the System,
Not at the Eyepiece, because
1) It tickles (and more if it’s a rifle scope)
2) The Pupil begins to act like a window
July 2003
Chuck DiMarzio, Northeastern University
10351-4-24
Stopped here 30 Sep 03
Microscope Field Stop
F
F’
Entrance
Window
Field Stop
= Exit Window
July 2003
Chuck DiMarzio, Northeastern University
10351-4-25
Microscope Effective Lens
July 2003
Chuck DiMarzio, Northeastern University
10351-4-26
Aberrations
• Failure of Paraxial Optics Assumptions
– Ray Optics Based On sin(q)=tan(q)=q
– Spherical Waves f=f0+2px2/rl
• Next Level of Complexity
– Ray Approach: sin(q)=q+q3/3!
– Wave Approach: f=f0+2px2/rl+cr4+...
• A Further Level of Complexity
– Ray Tracing
July 2003
Chuck DiMarzio, Northeastern University
10351-4-27
Examples of Aberrations (1)
1
Paraxial Imaging
0.5
0
-0.5
-1
-10
m4061_3
July 2003
-5
0
5
10
Chuck DiMarzio, Northeastern University
R = 2,
n=1.00, n’=1.50
s=10, s’=10
In this example for a
ray having height h
at the surface,
s’(h)<s’(0).
10351-4-28
Example of Aberrations (2)
0.2
D z(h=1.0)
Longitudinal
Aberration = D z
D z(h=0.6)
0.15
0.1
0.05
Transverse
Aberration =D x
0
-0.05
Where Exactly is
the image?
-0.1
-0.15
2D x(h=1.0)
m4061_3
-0.2
8.5
July 2003
9
9.5
10
10.5
Chuck DiMarzio, Northeastern University
What is its
diameter?
10351-4-29
Spherical Aberration
Thin Lens in Air
July 2003
Chuck DiMarzio, Northeastern University
10351-4-30
Transverse Spherical Aberration
h
Dx
Ds
s(0)
July 2003
Chuck DiMarzio, Northeastern University
10351-4-31
Evaluating Transverse SA
July 2003
Chuck DiMarzio, Northeastern University
10351-4-32
Coddington Shape Factors
-1
-1
July 2003
+1
p=0
q=0
Chuck DiMarzio, Northeastern University
+1
10351-4-33
Numerical Examples
Beam
Size, m
-2
10
5
s=1m, s’=4cm
q, Shape Factor
10
n=1.5
0
n=2.4
n=4
-5
-1
-0.5
0
p, Position Factor
0.5
10
10
-3
n=2.4
-4
-5
1 -6
10
-5
n=1.5
n=4
DL at 10 mm
DL at 1.06 mm
500 nm
0
5
q, Shape Factor
July 2003
Chuck DiMarzio, Northeastern University
10351-4-34
Phase Description of Aberrations
v
y
Image
Object
z
Entrance
Pupil
Exit
Pupil
• Mapping from object space to image space
• Phase changes introduced in pupil plane
– Different in different parts of plane
– Can change mapping or blur images
July 2003
Chuck DiMarzio, Northeastern University
10351-4-35
Coordinates for Phase Analysis
v
y
v
D
z
phase
v
2a
ra
Solid Line is phase of a
spherical wave toward
the image point.
Dotted line is actual
phase.
Our goal is to find
D(r,f,h).
y
u
h
0<r<1
July 2003
Chuck DiMarzio, Northeastern University
x
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Aberration Terms
Odd Terms involve tilt,
not considered here.
D  r cos f
July 2003
Chuck DiMarzio, Northeastern University
10351-4-37
Second Order
Image Position Terms:
The spherical wave is
approximated by a second-order
phase term, so this error is
simply a change in focal length.
July 2003
Chuck DiMarzio, Northeastern University
10351-4-38
Fourth Order (1)
r2 is focus: depends on h2 and h2cosf
Spherical Aberration
Astigmatism and Field Curvature
Tangential
Plane
S
T
Sagittal
Plane
Sample Images
At T
At S
July 2003
Chuck DiMarzio, Northeastern University
10351-4-39
Fourth Order (2)
r2cosf is Tilt: Depends
on
Coma
h3
y
Object
Barrel
Distortion
Pincushion
Distortion
v
x
u
July 2003
Chuck DiMarzio, Northeastern University
10351-4-40
Example: Mirrors
By Fermat’s Principle,
r1+r2 = constant
v
r1
r2
z
Result is an Ellipse
with these foci.
One focus at infinity
requires a parabola
Foci co-located requires
a sphere.
July 2003
Chuck DiMarzio, Northeastern University
10351-4-41
Mirrors for Focussing
v, cm.
0.1
0.05
Sphere
Parabola
Ellipse
0
Note:
These Mirrors
have a very large
NA to illustrate
the errors. In
most mirrors, the
errors could not
be seen on this
scale.
s=30 cm,
s’=6cm
-0.05
-0.1
-0.1
July 2003
-0.08
-0.06
-0.04
-0.02
0
Chuck DiMarzio, Northeastern University
z, cm.
10351-4-42
Numerical Example
July 2003
Chuck DiMarzio, Northeastern University
10351-4-43
Optical Design Process
July 2003
Chuck DiMarzio, Northeastern University
10351-4-44
Ray Tracing Fundamentals
July 2003
Chuck DiMarzio, Northeastern University
10351-4-45
Ray Tracing (1)
July 2003
Chuck DiMarzio, Northeastern University
10351-4-46
Ray Tracing (2)
July 2003
Chuck DiMarzio, Northeastern University
10351-4-47
If One Element Doesn’t Work...
“Let George Do It”
Add Another Lens
Different Index?
Smaller angles with
higher index. Thus
germanium is better than
ZnSe in IR. Not much
hope in the visible.
July 2003
Aspherics
Chuck DiMarzio, Northeastern University
10351-4-48
Summary of Concepts So Far
•
•
•
•
Paraxial Optics with Thin Lenses
Thick Lenses (Principal Planes)
Apertures: Pupils and Windows
Aberration Correction
– Analytical
– Ray Tracing
• What’s Missing? Wave Optics
July 2003
Chuck DiMarzio, Northeastern University
10351-4-49