July 2003 Chuck DiMarzio, Northeastern University 10351-4-1 ECEG105 & ECEU646 Optics for Engineers...

July 2003 Chuck DiMarzio, Northeastern University 10351-4-1

ECEG105 & ECEU646 Optics for Engineers

Course NotesPart 4: Apertures, Aberrations

Prof. Charles A. DiMarzio

Northeastern University

Fall 2003


Advanced Geometric Optics

• Introduction• Stops, Pupils, and

Windows• f-Number• Examples

– Magnifier

– Microscope

• Aberrations

• Design Process– From Concept through

Ray Tracing

– Finalizing the Design

– Fabrication and Alignment


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()=tan()=– cos()=1


What We Have Developed

• Description of an Optical System in terms of Principal Planes, Focal Length, and Indices of Refraction

• These equations describe a mapping – from image space (x,y,z) – to object space (x’,y’,z’)

s, s’ are z coordinates

H H’V V’BB’


Lens Equation as Mapping

• 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

-60 -40 -20 0 20 40 60-50

-40

-30

-20

-10

0

10

20

30

40

50

s, Object Dist., cm.

s',

Imag

e D

ist.

, cm

.

f = 10 cm.

L1 L2 L3 L4L4’


Stops, Pupils, and Windows (1)

• Intuitive Description– Pupil Limits Amount of Light Collected

– Window Limits What Can Be Seen

PupilWindow


Stops, Pupils and Windows (2)

Aperture StopLimits Cone of Rays from Object which Can Pass Through the System

Field StopLimits Locations of Points in Object which Can Pass Through System

Physical Components

Images in Object Space

Entrance Pupil Limits Cone of Rays from Object

Entrance Window Limits Cone of Rays From Entrance Pupil

Images in Image Space

Exit Pupil Limits Cone of Rays from Image

Exit Window Limits Cone of Rays From Exit Pupil


Finding the Entrance Pupil

• Find all apertures in object space

L1 L2 L3 L4

L4’ is L4 seen through L1-L3

L3’ is L3 seen through L1-L2

• Entrance Pupil Subtends Smallest Angle from Object

L1 L2’L4’L3’


Finding the Entrance Window

• Entrance Window Subtends Smallest Angle from Entrance Pupil

• 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

L1 L2’L4’L3’


Field of View

Film=ExitWindow

f=28 mm

f=55 mm

f=200 mm


Example: The Telescope

ApertureStop

FieldStop


The Telescope in Object SpacePrimary

Secondary

Secondary’

EntrancePupilEntrance

Window


The Telescope in Image SpacePrimary

Secondary

Primary’’

ExitPupil

ExitWindow

Stopped 26 Sep 03


Chief Ray

ApertureStop

FieldStop

ExitPupil

• Chief Ray passes through the center of every pupil


Hints on Designing A Scanner• Place the mirrors at pupils

Put Mirrors Here


f-Number & Numerical Aperture

F’

FA

A’

f

D is Lens Diameter

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

NA, Numerical Aperture

f#, f

-num

ber

f-Number Numerical Aperture


Importance of Aperture

• ``Fast’’ System– Low f-number, High NA (NA1, f# 1)– Good Light Collection (can use short exposure)– Small Diffraction Limit (/D)– Propensity for Aberrations (sin

• Corrections may require multiple elements

– Big Diameter • Big Thickness Weight, Cost• Tight Tolerance over Large Area


The Simple Magnifier

FF’AA’


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# Means f=10 mm

• Maximum Mm=25

For the Interested Student: What if s>f ?


Microscope

• Two-Step Magnification– Objective Makes a Real Image

– Eyepiece Used as a Simple Magnifier

F’F

A’A

F’F


Microscope Objective

F’F

A’A

F’F


Microscope Eyepiece

F’F

A2A

F’F

A2’


Microscope Effective Lens

A

D’D

F F’

H H’

A

F

H

F’

H’

192 mm

19.2 mm 19.2 mm

f1=16mmf2=16mm

Barrel Length = 160 mm

Effective Lens:f = -1.6 mm


Microscope Aperture Stop

F’F

ApertureStop=EntrancePupil

ExitPupil

Put the Entrance Pupil of your eyeat the Exit Pupil of the System, Not at the Eyepiece, because1) It tickles (and more if it’s a rifle scope)2) The Pupil begins to act like a window

Image

Analysis in Image Space

Stopped here 30 Sep 03


Microscope Field Stop

F’F

Field Stop= Exit Window

EntranceWindow


Microscope Effective Lens


Aberrations

• Failure of Paraxial Optics Assumptions– Ray Optics Based On sin()=tan()=– Spherical Waves =0+2x2/

• Next Level of Complexity– Ray Approach: sin()=– Wave Approach: =0+2x2/c

• A Further Level of Complexity– Ray Tracing


Examples of Aberrations (1)

-10 -5 0 5 10-1

-0.5

0

0.5

1

R = 2,n=1.00, n’=1.50s=10, s’=10

Paraxial Imaging

In this example for a ray having height h at the surface, s’(h)<s’(0).

m4061_3


8.5 9 9.5 10 10.5-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

Example of Aberrations (2)

m4061_3

z(h=0.6)

z(h=1.0)LongitudinalAberration = z

Transverse Aberration = x

x(h=1.0)

Where Exactly is the image?

What is its diameter?


Spherical Aberration Thin Lens in Air


Transverse Spherical Aberration

sxh

s(0)


Evaluating Transverse SA


Coddington Shape Factors

q=0-1 +1

p=0-1 +1


Numerical Examples

-5 0 510

-6

10-5

10-4

10-3

10-2

Beam Size, m

q, Shape Factor

s=1m, s’=4cm

DL at 10 m

n=2.4 n=1.5

n=4

DL at 1.06 m 500 nm

1-1 -0.5 0 0.5

-5

0

5

p, Position Factor

q, S

hap

e F

acto

r

n=4

n=2.4

n=1.5


Phase Description of Aberrations

EntrancePupil Exit

Pupil

Object

Imagez

yv

• 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


Coordinates for Phase Analysis

2a a

0<<1

z

yv

v

u

y

xh

phase

v

Solid Line is phase of a spherical wave toward the image point.Dotted line is actual phase.Our goal is to find(,h).


Aberration Terms

Odd Terms involve tilt,not considered here.

cos


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.


Fourth Order (1)

Spherical Aberration Astigmatism and Field Curvature

S SagittalPlane

TangentialPlane T

Sample Images

At T

At S

2 is focus: depends on h2 and h2cos


Fourth Order (2)

Object

BarrelDistortion

PincushionDistortion

y

x

Coma

u

v

2cos is Tilt: Dependson h3


Example: Mirrors

r1

r2

By Fermat’s Principle,r1+r2 = constant

Result is an Ellipse with these foci.One focus at infinity requires a parabolaFoci co-located requires a sphere.

z

v


Mirrors for Focussing

-0.1 -0.08 -0.06 -0.04 -0.02 0-0.1

-0.05

0

0.05

0.1

EllipseParabola

Sphere

s=30 cm,s’=6cm

Note:These Mirrors have a very large NA to illustrate the errors. In most mirrors, the errors could not be seen on this scale.

v, cm.

z, cm.


Numerical Example


Optical Design Process


Ray Tracing Fundamentals


Ray Tracing (1)


Ray Tracing (2)


If One Element Doesn’t Work...

Add Another Lens “Let George Do It”

AsphericsDifferent Index?Smaller angles withhigher index. Thus germanium is better than ZnSe in IR. Not much hope in the visible.


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 10351-4-1 ECEG105 & ECEU646 Optics for Engineers...

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