•turning ideal imaging syst ems into real optics...

ECE 5616 OE System Design

Robert McLeod 214

Conic mirrorsTwo aberration-free conjugate points

•Turning ideal imaging systems into real optics–Imaging in non-ideal systems

22

2

111 rC

rCrs

r

s(r)“sag” of the surface (z coordinate vs r):

F1, F20Circle. Both foci at

center of curvature.

F21Parabola. One focus at ∞, one at center of curvature.

01 Ellipse. Two real foci.F1 F2

1Hyperbola. One real, one virtual focus F1 F2


Robert McLeod 215

Mirror-based telescopes

•Turning ideal imaging systems into real optics–Imaging in non-ideal systems

Newtonian. Parabola to image infinity to real focus, then lens as eyepiece. (Keplerian)

Parabola

Gregorian. Parabola to form real image from infinity followed by ellipse to relay (Keplerian) ParabolaEllipse

Cassegrain. Parabola to form real image from infinity followed by hyperbola (Gallilean or telephoto) ParabolaHyperbola

Mersenne-Gregorian. Parabola to form real image from infinity followed by parabola to reimage to infinity (afocal Keplerian). Invert parabola to form Mersenne-Cassegrain (afocal Gallilean).

ParabolaParabola


Robert McLeod 216

Bending the lensSolution for coma and spherical

•Turning ideal imaging systems into real optics–Correction of aberrations

0

2

4

6

8

10

12

14

-0.02 0.00 0.02 0.04 0.06

First surface curvature [1/mm]

TS

A a

t p

up

il e

dg

e [

mm

]

•F=100 BK7 lens at ∞ conjugate.•40 mm diameter (F/2.5).•First surface curvature varied.•TSA of marginal ray measured.

Single degree of freedom can be exploited for minimum aberration.


Robert McLeod 217

Impact of indexHigh index always better

0

200

400

600

800

1000

1200

1400

1600

1800

1.2 1.7 2.2

Index

TS

A [

mic

ron

s]

•Biconvex BK7 lens at ∞ conjugate.•40 mm diameter •R1, R2 chosen for

• EFL=100 mm• Minimum wavefront error


Reduce refraction angle to reduce aberrations.


Robert McLeod

Computation of primary aberrationsParaxial invariant

218

•Turning ideal imaging systems into real optics– Calculation of aberrations

nnycnuynuun

ucynucynA

uycnuycnA

The paraxial transfer equation can be augmented with the lens maker’s equation

to produce the refraction invariant for the paraxial marginal and chief rays

M&M Chapter 8

These two quantities characterize the bending of an axial (PMR) or edge of field (chief) ray at any surface

nA

nnAuu

nA

nnAuu

111

111

A greater than ~0.5 is the approximate boundary of the non-paraxial regime.


Robert McLeod

Computation of primary aberrationsSeidel aberration formulae

219

iiiii

k

ii

iIViIII

k

i i

iV

k

i iiIV

k

i i

iiiIII

k

i i

iiiiII

k

i i

iiiI

ucyuyA

SSA

AS

ncHS

n

uhAS

n

uhAAS

n

uhAS

22

1

1

1

2

1

2

1

1

2

1


Seidel sums Wavefront error

Spherical 8IS

Coma 2IIS

Astigmatism 2IIIS

Fieldcurvature

4

IVIII SS

Distortion 2VS

if any A = 0

The (somewhat amazing) conclusion is that one can calculate the third order aberrations from first-order (paraxial) ray tracing of just the marginal and chief rays.

PMRonly

Note

System only


Robert McLeod

Seidel vs. wavefront vs. ray aberrations

220Zemax manual

n’ and u’ are index and marginal ray angle in the image plane.


un

W

y

W

n

R

y

W

n

R

y

yxW

n

Ry

ppp

pp

,

Motivation for form of transverse ray aberration equations:

…not the index and marginal ray angle at the particular surface.


Robert McLeod

Petzval (field curvature)

221


4

IVIII SS

Astigmatism Petzval

c1c2n

nH

n

nccH

nc

ncH

ncHS

i iiIV

2

212

212

2

1

2

1

111

1

1

For a single thin (d ~ 0) lens

• So Petzval for a complex lens is approximately proportional to the sum of the powers of the individual elements.

• Thus a lens with all positive elements has unavoidable field curvature.

• Inclusion of negative elements can cancel.

Field curvature is the sum of two terms:


Robert McLeod

Aberration-free surfaces

222


uuAi 0Marginal ray

Marginal ray along radius = object is at center of curvature

0III SS No spherical or coma

uuAi 0Chief ray

Chief ray along radius = pupil conjugate to center of curvature

0IVIIIII SSS No coma, astigmatism, distortion

0hObject conjugate to surface

0IIIIII SSS No spherical coma, astigmatism

Marginal ray


Robert McLeod 223

Field flattenerSolution for field curvature

www.phys.unsw.edu.au/ astro/dmt/


Douglas Mawson Telescope2-m infrared telescope for Antarctica

Petzval does not depend on space of elements and is proportional to element power. Thus a negative lens placed near the image can minimize field curvature. If this lens is placed near an image plane, it has little impact on other imaging properties.


Robert McLeod 224

Field lenses

http://www-optics.unine.ch/education/optics_tutorials/field_lens.html

Field lens

A lens placed at the an image plane introduces only field curvature and distortion. We have already used this as a field flattener. Another important use is the field lens, as shown below. • Magnification and other paraxial imaging properties unchanged.• Vignetting significantly altered• Often used when total OD is constrained (e.g. eyepieces)• Lens is at an image plane, thus is very sensitive to dust or scratches.

No field lens



Robert McLeod

Seidel design example

225


Goal: Diffraction-limited 1:1 imaging of a 6 mm field with 1000 Rayleigh-resolved spots (separated by radius) at 0.55 microns.

0 1 2 3First design: Symmetric BK7 singlet. Object space NA = 0.0499

Note: Transverse ray aberrations per surface calculated to match Zemax via where n’ and u’ are the local index and marginal ray angle. This is WRONG. All entries use the final image plane n’ and u’. THUS, the “total” column is not the total of the contributions.

un

STSA I

2


Robert McLeod

Seidel design exampleDesign 1: Zemax

226


100 m

-100 m


Robert McLeod

Spherical aberration of slab at focus

227


4

022

0

111NA

t

nnnr

TSA

t

n nuth

nunu

unA

0

4

022

0 8

111

8

1

r

TSANA

t

nnn

Wspherical

What is SA due to focusing into slab?

0rTSA

Snell’s law

Paraxial invariant of plane

0r

TSA

0W

81

641

641

181

1

181

81

81

8

SR

95.99.

99.

95.


Robert McLeod

Seidel design exampleDesign 2: Seidel

228

Second design: Add BK7 field flatteners.

0 2 31 4 5



Robert McLeod

Seidel design exampleDesign 2: Zemax

229

50 m

-50 m



Robert McLeod 230

Symmetrical systems

No coma or distortion. Reduces them for M ≠1.

2 mm

2 mm

SA+FC+AST

SA+FC+AST+COMA


SA+AST

Asymmetrical system

Symmetrical system

Symmetrical system + best form lenses +

field flatteners200 um

SA

SA

SA

0VII SSSums linear in = 0.A


Robert McLeod

Aplanatic condition

231


0/ nu

0IIIIII SSS No spherical, coma or astigmatism

Higher-order spherical and coma also zero.

Since n and n´ >0, u and u´ must have same sign and thus one of object or image must be virtual.

Q: So what’s it good for? A: Changing NA with minimum aberrations.

The solid immersion lens, Kino, ICOSN ‘99

http://www.optics-online.com/acp.asp

Aplanatic meniscus. One surfacehas A = 0, second is aplanatic, sono spherical or coma, minimalastigmatism. Add an aplanaticmeniscus to an achromat tochange the NA by factor of n´/n

Solid immersion lens. Partialsphere, often of high indexmaterial (diamond, GaP). Left isA=0, increases NA by n. Right is“superSIL”, aplanatic condition,increases NA by n2


Robert McLeod 232

Stop shiftSolution for coma, astigmatism, distortion

0

10

20

30

40

50

60

0.00 20.00 40.00 60.00

Stop distance from first surface [mm]

RM

S s

po

t s

ize

[u

m]

•Biconvex BK7 lens at ∞ conjugate.•40 mm diameter R1=-R2=100 mm•10 mm diameter stop moved•RMS spot size measured

0

30

70


•turning ideal imaging syst ems into real optics...

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