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

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 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

1 2 1 2

c c R RXc c R R

Lens bending concept

Shape X

X=-1.7

X=-3.5X=3.5

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

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

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 ' ' '' '

yn nS y K Kn n n nn n n n

20'InS Kn

Parabola for reflectionEllipse for air to glassHyperbola for glass to air

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.

011401

n=1 n=1

Field curves

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

Thin lensSpherical aberration and coma

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

Thin lens coma aberration

n=1.517

This lensAplanatic solutions

Y=5n=1.5

' 1' 1

Thin lensSpherical and coma @ Y=0

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

Achromatic doubletCorrection for chromatic change of focus

For an achromatic doublet:

L L ki P

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

I I ki P

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

II II k k I ki P

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, , ,

kIII III k k II k k I k

1 21 1III

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

Achromatic doublet

1 2IV n n

Field curvature aberration

IV IV ki

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

Achromatic doubletDistortion and chromatic change of magnification

T T k k L ki

2 3, , , , ,

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

y S S Sy

Cemented achromatic doublet

1 1 2 212 21

1 1 2 2

1 12 1 2 1

1 11 1

X Xc c

X Xn n

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

Lister objective

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

Lister objective

Condition for zero coma

2 2 2 2

II II k k I ki P

II A A IIA B B IIB

II B IIA B IIB

BIIA IIB

Condition for zero astigmatism

2, , ,

III III k k II k k I ki

III A B B IIB

III B B IIB

B B IIB

B BIIA

2k P k P k

Lister Objective

1 212123

IIB IIA

y yyy y

Choose:

1Petzval IV

Petzval

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

IIA IIBB

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

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.

131 31112

W W Ж u

* 2 2III III B IIIB BS S Ж Ж

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

Diffractive lens (n very large @ X=0)

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

23 1I Y

2II Y 1III

Ldiffractive

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

/ 1 11/

y yMLML

/ 1/ 1

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

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