introduction to aberrations...opti 518 structural stop shifting parameter s’ is the distance from...

Prof. Jose SasianOPTI 518

Introduction to aberrations

OPTI 518Lecture 14


Topics

• Structural aberration coefficients• Examples


Structural coefficients

Ж

Requires a focal systemAfocal systems can be treated with Seidel sums


Structural stop shifting parameter

s’ is the distance from the rear principal plane to exit pupil

s is the distance from the front principal plane to entrance pupil

2

2 2P P Py y yS SЖ Ж

'

2 1 2 1 ' 2 'P Py y s sSЖ Y s n Y s n

1 / 2nu Y y

Using ω on we can express:

2P Py ySЖ


Review of concepts

• Thin lens as the thickness tends to zero

• Shape of a lens and shape factor• Conjugate factor to quantify how the lens

is used. Related to transverse magnification

• Must know well first-order optics

1 2 1 2tn


Shape and Conjugate factors

nu ' 1' 1

mYm

1 2 1 2

1 2 1 2

c c R RXc c R R

Lens bending concept


Shape X

X=-1

X=0

X=-1.7

X=-3.5X=3.5

X=1.7

X=1

X=0


Shape

X=0

X=0

X=-1X=-2X=-3

X=1X=2X=3


Shape or bending factor X• Quantifies lens shape• Optical power of thin lens is maintained• Not defined for zero power, R1=R2

1 2 1 2

1 2 1 2

c c R RXc c R R


Example:Refracting surface free from spherical aberration

Object at infinity Y=1

2 2 22 2 2 2

2 2 2 2 2 2

1 ' ' ' 1 2 2 2 112 ' ' ' 2 ' ' ' 'In n n n n n n nn n n n n n n n n n n n

44 3 4 3 4 3

23

1 1 14 4 '

PI P I P I P

yS y n K y y Kr n n

2 4 3 24 3

2 22 2

1 2 1 14 ' ' '' '

PI P

yn nS y K Kn n n nn n n n

22

20'InS Kn

Parabola for reflectionEllipse for air to glassHyperbola for glass to air


Note

24 3 4 3 4 3

21 1 1 24 4 ''

I P I P P IS y y K y Kn nn n

The contribution to the structural coefficient from the aspheric cap is

22'Icap Kn n

For a reflecting surface is just the conic constant K


Icap K


Spherical MirrorA spherical mirror can be treated as a convex/concave

plano lens with n=-1. The plano surface acts as an unfoldingflat surface contributing no aberration.

2

1

11

000

I

II

III

IV

V

L

T

XYY

14

011401

A

BC

D

EF

n=-1

n=1 n=1


Field curves


Field curves

Rt~f/3.66Rm~f/2.66Rs~f/1.66Rp~f/0.66


Thin lensSpherical aberration and coma

I II

Fourth-order


Spherical aberration of a F/4 lens

•Asymmetry•For high index 4th order predicts well the aberration


Thin lens spherical aberration

n=1.517

I

Y

X


Thin lens coma aberration

n=1.517

II

X

Y


This lensAplanatic solutions

X=4

Y=5n=1.5

' 1' 1

mYm


Thin lensSpherical and coma @ Y=0

III

N=1.5N=2N=2.5N=3N=3.5N=4

Strong index dependence


Thin lens special casesstop at lens


Thin lens special cases stop at lens



For double convex lens (CX)For double concave lens (CC)


Achromatic doubletTwo thin lenses in contactThe stop is at the doublet

1 2

1 2

11

22

y y

1 2

21

1

12

2

1

YY

YY


Achromatic doubletCorrection for chromatic change of focus

1 2

1 2

11 2

1 2

22 1

1 2

1

1

L

L

L

0L

For an achromatic doublet:

2,

,0

kP kk

L L ki P

yy


Achromatic doubletCorrection for spherical aberration

3 2 2 3 2 21 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2I A X B X Y CY D A X B X Y C Y D

43,

,0

kP kk

I I ki P

yy

For a given conjugate factor Y, spherical aberration is a functionOf the shape factors X1 and X2 . For a constant value of spherical

aberration we obtain a hyperbola as a function of X1 and X2.


Achromatic doubletCorrection for coma aberration

22

,, ,

0

kP kk

II II k k I ki P

yS

y

2 21 1 1 1 1 2 2 2 2 2II E X FY E X F Y

For a given conjugate factor Y and a constant amount of coma the graph of X1 and X2 is a straight line.


Achromatic doubletAstigmatism aberration

2, , ,

0

2k

kIII III k k II k k I k

i

S S

1 21 1III

Astigmatism is independent of the relative lens powers, shape factors,or conjugate factors.


Achromatic doublet

1 2

1 2IV n n

2 1

2 1

n n

Field curvature aberration

,0

kk

IV IV ki

0IV For an achromatic doublet there is no field curvature if


Achromatic doubletDistortion and chromatic change of magnification

, ,0

k

T T k k L ki

S

2

2 3, , , , ,

0 ,

3 3k

PV V k k IV k III k k II k k I k

i P k

y S S Sy

00

V

T


Cemented achromatic doublet

1 1 2 212 21

1 2

1 1 2 2

1 2

2 1

1 12 1 2 1

1 11 1

1 1

X Xc c

n n

X Xn n

X X

2 1

1 2

2 1

1 2

11

11

nn

nn

For achromat

For a cemented achromatic lens the graph of X1 and X2 is a straight line.


Crown in front: BK7 and F8


Flint in front: BK7 and F8


Cemented achromatic doublet


Cemented doublet solutions

Crown in front

Flint in front


Doublets with no spherical aberration but with varying coma

Crown in front, no spherical aberration, F/5, f=100 mm, no chromatic aberration

Coma=0Coma=-10 waves Coma=10 waves


Lister objective


Lister objective• Two achromatic doublets that are spaced• Telecentric in image space• Normalized system

11100

A

A

IA

IB

Жyu


Lister objective

The aperture stop is at the first lens.The system is telecentric

111

B

A B

B

yy


Lister objective


Condition for zero coma

22,

, ,0

2 2 2 2

2 2

2

2

0

1 0

1

kP kk

II II k k I ki P

II A A IIA B B IIB

IB

II B IIA B IIB

BIIA IIB

B

yS

y

y y

y y

yy


Condition for zero astigmatism

2, , ,

0

2

2

1 0

1 1 0

2 02

12

2

1

kk

III III k k II k k I ki

III A B B IIB

III B B IIB

B B IIB

BIIB

BIIB

B

B BIIA

B

S S

y

y y

y y

y

yy

y y

y

, ,

2k P k P k

k

y yS

Ж


Lister Objective

2

2 2

2 2

221

1

1 212123

3

IIB IIA

B BB

B B

B B

B B B

B

A

IIA

IIB

y yyy y

y y

y y y

y

Choose:


1Petzval IV

Petzval

Cn

N-BK7 : Petzval radius -151.7 mm

Plano convex lens


Wollaston meniscus lens

222 220/ 0.8PW W

• Artificially flattening the field• Periscopic lenses


Periskop lens

• Principle of symmetry• No distortion


1 2 1 2

1 2 1 2 1 2

1Petzval IV

Petzval

v vCn n v v n n

N-BAK1 and N-LLF6: Petzval radius -185 mm

N-BK7 : Petzval radius -151.7 mm

N-BK7 and N-F2: Petzval radius -139.99 mm (+139.99 for negative doublet)

Field curvature

• Old achromat• New achromat


Chevalier landscape lens

222 220/ 0.8PW W • F/5 telescope doublet used in reverse

and with an aperture stop in front


• F/8• Glass selection is keyto minimize spherical aberrationwhile artificially flattening the field

Rapid rectilinear

RMS spot size


• Telecentric

Lister microscope objective

2 2 2 2II A A IIA B B IIBy y

2

21B

IIA IIBB

yy

12By

IIA IIB

1 1 0III B B IIBy y

0IA IB

* 22III III II IS S SS S S


Lister microscope objectivePractical solution

• Two identical doublets• Spherical aberration and coma are corrected• Astigmatism is small• Telecentric• Less vignetting

• RMS wavefront error in waves


Aplanatic concentric meniscus lens

• Optical speed is increased by an N factor


Petzval portrait objective

222 220/ 0.8PW W

• Chromatic aberration and spherical aberrationcorrected at each doublet

• Positive coma in the first doublet corrected with negative coma of aberration of the second doublet

• Negative astigmatism introduced by the negative coma of the second doublet toartificially flatten the field of view.

f’=144 mm; F/3.7; FOV=+/- 16.5⁰.



Concentric lens

• Use of new glasses• Reduced Petzval sum• Nearly flat field• Surfaces nearly concentric• Limited by spherical aberration

due to strong curvatures.

1 2 1 2

1 2 1 2 1 2

1Petzval IV

Petzval

v vCn n v v n n

N-BAK1 and N-LLF6: Petzval radius -185 mm


Anastigmatic lens

• Corrected for spherical aberration, coma, astigmatism, and field curvature• Distortion is negligible• Combination of an old achromat and a new achromat

RMS spot size


Telephoto lens

• Corrected for spherical aberration, coma, astigmatism, and field curvature• Distortion is not corrected• Telephoto ratio=TTL/f

Telephoto lens with BK7 and SF5 glasses. f’=100 mm, F/4, FOV=+/- 6.2⁰, TTL/F=0.8.

2

131 31112

W W Ж u


* 2 2III III B IIIB BS S Ж Ж

A B


Telephoto lens

• Corrected for spherical aberration, coma, astigmatism, and field curvature• Distortion is also corrected

Telephoto lens with BK7 and F6 glasses. f’=100 mm, F/4, FOV=+/- 6.2⁰, TTL/F=0.8


Reverse telephoto lens

• Corrected for spherical aberration, coma, astigmatism, and field curvature• Distortion is small ~-1.5%• Large back focal length/distance

Reverse telephoto lens with BK7 and SF5 glasses. f’=100 mm, BFL=200 mm, TTL= 324 mm, FOV=+/-12⁰, F/4


Ray diffractive law (1D)


Grating linear phase change

sin ' sin mn I n Id

sin ' sin my n I y yn I yd


Diffractive opticshigh-index model

1/d


Diffractive lens (n very large @ X=0)

A=0B=0C=3D=1E=0F=2

23 1I Y

2II Y 1III

0IV

0V 1

Ldiffractive

0T


Mirror Systems

'

2 1 2 1 ' 2 'P Py y s sSЖ Y s n Y s n


Two mirror afocal system


Two mirror systems111

PyЖ

2

2

2

2

/ 1 11/

111

2 111

P

M ML

y yMLML

SML

MYM

1

1

1

1

/ 1/ 1

00

Py y

SY

2

2

2

2

1

yLy

yMy


Two mirror systems


Schmidt camera


Merssene afocal systemAnastigmatic

Confocal paraboloids


Paul-Baker systemAnastigmatic-Flat field

AnastigmaticParabolic primarySpherical secondary and tertiaryCurved fieldTertiary CC at secondary

Anastigmatic, Flat fieldParabolic primaryElliptical secondary Spherical tertiaryTertiary CC at secondary


Meinel’s two stage optics concept (1985)

Large DeployableReflector

Second stage correctsfor errors of first stage;fourth mirror is at the

exit pupil.


Aplanatic, Anastigmatic, Flat-field, Orthoscopic (free from distortion, rectilinear, JS 1987)

Spherical primary telescope.The quaternary mirror is near the exit pupil. Spherical aberration and

Coma are then corrected with a single aspheric surface. The Petzval sum is zero.If more aspheric surfaces are allowed then more aberrations

can be corrected.


Summary

• Structural coefficients• Basic treatment• Analysis of simple systems

introduction to aberrations...opti 518 structural stop shifting parameter s’ is the distance from...

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