optical aberrations

1

Optical Aberrations

SOLO HERMELIN

Updated: 16.01.10 4.01.15

http://www.solohermelin.com

http://www.solohermelin.com/

2

Table of Content

SOLO Optical Aberration

Optical Aberration DefinitionThe Three Laws of Geometrical Optics

Fermat’s Principle (1657)

Reflection Laws Development Using Fermat Principle Huygens Principle

Optical Path Length of Neighboring Rays

Malus-Dupin Theorem

Hamilton’s Point Characteristic Function and the Direction of a Ray

Ideal Optical System

Real Optical System

Optical Aberration W (x,y)

Lens Definitions

Real Imaging Systems – Aberrations

Defocus Aberration

Wavefront Tilt Aberration

Seidel Aberrations

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Table of Content (continue – 1)



Seidel Aberrations

Spherical Aberrations

Coma

Astigmatism and Curvature of Field

Astigmatism

Field CurvatureDistortion

Thin Lens Aberrations

Coddington Position Factor Coddington Shape Factor

Thin Lens Spherical Aberrations

Thin Lens Coma

Thin Lens Astigmatism

Chromatic Aberration

4

Table of Content (continue – 2)


Image Analysis

Two Dimensional Fourier Transform (FT)

Point Spread Function (PSF)

Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function

Convolution

Modulation Transfer Function (MTF)

Phase Transfer Function (PTF)

Other Metrics that define Image Quality

Strehl Ratio

Pickering Scale

Image Degradation Caused by Atmospheric Turbulence

Zernike’s Polynomials

Aberrometers

References

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SOLO

converging beam=

spherical wavefront

parallel beam=

plane wavefront

Image PlaneIdeal Optics

ideal wavefrontparallel beam

=plane wavefront

Image PlaneNon-ideal Optics

defocused wavefront

ideal wavefrontparallel beam=

plane wavefront


aberrated beam=

iregular wavefront

diverging beam=

spherical wavefront

aberrated beam=

irregular wavefront

Image Plane

Non-ideal Opticsideal wavefront

Optical Aberration Optical Aberration is the phenomenon of Image Distortion due to Optics Imperfection

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SOLO

The Three Laws of Geometrical Optics

1. Law of Rectilinear Propagation In an uniform homogeneous medium the propagation of an optical disturbance is instraight lines.

2. Law of Reflection

An optical disturbance reflected by a surface has the property that the incident ray, the surface normal, and the reflected ray all lie in a plane,and the angle between the incident ray and thesurface normal is equal to the angle between thereflected ray and the surface normal:

3. Law of Refraction

An optical disturbance moving from a medium ofrefractive index n1 into a medium of refractive indexn2 will have its incident ray, the surface normal betweenthe media , and the reflected ray in a plane,and the relationship between angle between the incident ray and the surface normal θi and the angle between thereflected ray and the surface normal θt given by Snell’s Law: ti nn θθ sinsin 21 ⋅=⋅

ri θθ =

“The branch of optics that addresses the limiting case λ0 → 0, is known as Geometrical Optics, since in this approximation the optical laws may be formulated in the language of geometry.”

Max Born & Emil Wolf, “Principles of Optics”, 6th Ed., Ch. 3

Foundation of Geometrical Optics

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SOLO Foundation of Geometrical Optics

Fermat’s Principle (1657)

The Principle of Fermat (principle of the shortest optical path) asserts that the optical length

of an actual ray between any two points is shorter than the optical ray of any other curve that joints these two points and which is in a certai neighborhood of it. An other formulation of the Fermat’s Principle requires only Stationarity (instead of minimal length).

∫2

1

P

P

dsn

An other form of the Fermat’s Principle is:

Princple of Least Time The path following by a ray in going from one point in space to another is the path that makes the time of transit of the associated wave stationary (usually a minimum).

The idea that the light travels in the shortest path was first put forward by Hero of Alexandria in his work “Catoptrics”, cc 100B.C.-150 A.C. Hero showed by a geometrical method that the actual path taken by a ray of light reflected from plane mirror is shorter than any other reflected path that might be drawn between the source and point of observation.

a, 08/16/2005

Hero proof is described in M.V.Klein, T.E.Furtak, "Optics", pp.3-5

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SOLO

1. The optical path is reflected at the boundary between two regions

( ) ( )0

2121 =⋅

− rd

sd

rdn

sd

rdn rayray

In this case we have and21 nn =( ) ( ) ( ) 0ˆˆ

2121 =⋅−=⋅

− rdssrd

sd

rd

sd

rd rayray

We can write the previous equation as:

i.e. is normal to , i.e. to the boundary where the reflection occurs.

21 ˆˆ ss − rd

( ) 0ˆˆˆ 2121 =−×− ssn

REFLECTION & REFRACTION

Reflection Laws Development Using Fermat Principle

This is equivalent with:

ri θθ = Incident ray and Reflected ray are in the same plane normal to the boundary.&

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SOLO

2. The optical path passes between two regions with different refractive indexes n1 to n2. (continue – 1)

( ) ( )0

2121 =⋅

− rd

sd

rdn

sd

rdn rayray

where is on the boundary between the two regions andrd

( ) ( )sd

rds

sd

rds rayray 2

:ˆ,1

:ˆ 21

==

rd

22 sn

11 sn

1122 ˆˆˆ snsn −

( ) 0ˆˆˆ 1122 =⋅− rdsnsn

Refracted Ray

21ˆ −n

2n

1n iθ

tθ

Therefore is normal to .

2211 ˆˆ snsn − rd

Since can be in any direction on the boundary between the two regions is parallel to the unit vector normal to the boundary surface, and we have

rd

2211 ˆˆ snsn −21ˆ −n

( ) 0ˆˆˆ 221121 =−×− snsnn

We recovered the Snell’s Law (1621)from Geometrical Optics

REFLECTION & REFRACTION

Refraction Laws Development Using Fermat Principle

ti nn θθ sinsin 21 = Incident ray and Refracted ray are in the same plane normal to the boundary.

&

Willebrord van Roijen Snell1580-1626

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SOLO

Huygens Principle

Christiaan Huygens1629-1695

Every point on a primary wavefront serves the source of spherical secondary wavelets such that the primary wavefront at some later time is the envelope o these wavelets. Moreover, the wavelets advance with a speed and frequency equal to that of the primary wave at each point in space.

“We have still to consider, in studying the spreading of these waves, that each particle of matter in which a wave proceeds not only communicates its motion to the next particle to it, which is on the straight line drawn from the luminous point, but it also necessarily gives a motion to all the other which touch it and which oppose its motion. The result is that around each particle there arises a wave of which this particle is a center.”

Huygens visualized the propagation of light in terms of mechanical vibration of an elastic medium (ether).

Optics 1678

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Optical Path Length of Neighboring Rays Consider the ray PQ incident on a spherical surface and refracted to QP’.

Optics

ir nn φφ sinsin 0=Snell’s Law

incident angle, between incident ray and the spherical surface

iφ

iφ

rφQ

C

P'

P

P1Q1A1

A

P'1

iφ rφ

n

'n

Opticalaxis

refracted angle, between refracted ray and the spherical surface

rφ

Consider now a neighboring ray P1Q1

incident on a spherical surface and refracted to Q1P’1, such that QQ1 is small. Assume that PP1 and P’P’1 are perpendicular to one of the rays. Define the optical path on a ray between points P and P’ as

( ) [ ] [ ] [ ] ''',,',', QPnPQnPQQPPPPPVpathoptical +=+=== From Figure

( ) ( ) ( ) ( ) ( ) 0sinsin',,',', 11111 =−≈−=− ir nnQQAQVAQVPPVPPV φφ The Optical Path lengths along two neighboring rays measured between planes that are perpendicular to one (ore both) of them are equal.

12

Malus-Dupin Theorem

SOLO

Étienne Louis Malus1775-1812

A surface passing through the end points of rays which have traveled equal optical pathlengths from a point object is called an optical wavefront. If a group of ray is such that we can find a surface that is orthogonal to each and every one of them (this surface isthe wavefront), they are said to form a normal congruence.

The Malus-Dupin Theorem (introduced in 1808 by Malusand modified in 1812 by Dupin) states that:“The set of rays that are orthogonal to a wavefront remainnormal to a wavefront after any number of refraction or reflections.”

Charles Dupin1784-1873

n 'n

P

Q

VAP’

A'

B B'

Wavefrontfrom P Wavefront

to P' Using Fermat principle

[ ] [ ]'' BQBAVApathoptical ==

[ ] [ ] ( )2'' εOAVAAQA +=

VQ=ε is a small quantity [ ] [ ] ( )2'' εOBQBAQA +=

Since ray BQ is normal to wave W at B [ ] [ ] ( )2εOBQAQ +=[ ] [ ] ( )2'' εOQBQA += ray BQ’ is normal to wave W’ at B’

Proof for Refraction:

Optics

a, 08/20/2005

M.V.Klein, T.Furtak,"Optics", pp.34-35E. Hecht, A. Zajac,"Optics",pp.8, pp.225-226

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Geometrical Optics SOLO

Hamilton’s Point Characteristic Function and the Direction of a Ray

William RowanHamilton

(1805-1855)

In 1828 Hamilton published“Theory of Systems of Rays”in which he introduced the concept of

n 'n

Q

VA A'

B B'


to P'

r

'r

( )rP ( )'' rP

Hamilton’s Point Characteristic Function of a Ray as theOptical Path Length along the ray:

( ) ( )( ) ( ) ∫==

'

','',

PtoPPathOptical

dsnrrVrPrPV

Consider as in the Figure bellow a neighboring point to ( )'' rP( )''" rrP

δ+

n

Wavefrontfrom P

r 'r

( )rP ( )'' rP

( )''' rrP δ+

'' rr δ+

'rδ

( )sd

PPrd

s

ra y

',

:ˆ

=

( ) ( ) ( ) ( ) ( ) VrrrVrrrVPPVPPVPPV ''','',',",'," ⋅∇=−+=−= δδ

According to the definition of optical path ( ) '''," rsnPPV δ⋅=

Since those relations are true for every small :'rδ ( ) Vsn PP '' ', ∇=

Direction of ray is normal to ( ) .', constrrV =

( ) sd

rds ray

PP

=:ˆ ',where and ( ) 2/1: rayray rdrdsd

⋅=

14

Geometrical Optics SOLO

In 1828 Hamilton published

William RowanHamilton

(1805-1855)

http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Optics.html

Theory of Systems of Rays

Supplement to an Essay on the Theory of Systems of Rays (1830)

Second Supplement to an Essay on the Theory of Systems of Rays (1831)

Third Supplement to an Essay on the Theory of Systems of Rays (1837)

followed by

The paper includes a proof of the theorem that states that the raysemitted from a point or perpendicular to a wavefront surface, and reflected one ore more times, remain perpendicular to a series of wavefront surfaces (Theorem of Malus and Dupin).

The paper also discussed the caustic curves and surfaces obtained when light rays are reflected from flat or curved mirrors. This is an enlargement of Caustics, a paper published in 1824. Hamilton introduced also the characteristic function , V, that, in an isotropic medium, the rays are perpendicular to the level surface of V.

This work inspired Hamilton’s work on Analytical Mechanics.

( ) ( )( ) ( ) ∫==

'

','',

PtoPPathOptical

dsnrrVrPrPV

n 'n

Q

VA A'

B B'


to P'

r

'r

( )rP ( )'' rP

Vsn '' ∇=

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SOLO

converging beam=

spherical wavefront

parallel beam=

plane wavefront


P'

Optical Aberration

converging beam=

spherical wavefront


diverging beam=

spherical wavefront

PP'

An Ideal Optical System can be defined by one of the three different and equivalent ways:

All the rays emerging from a point source P, situated at a finite or infinite distance from the Optical System, will intersect at a common point P’, on the Image Plane.

3

All the rays emerging from a point source P will travel the same Optical Path to reach the image point P’.

2

The wavefront of light, focused by the Optical System on the Image Plane, has a perfectly spherical shape, with the center at the Image point P.

1


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SOLO

ideal wavefrontparallel beam=

plane wavefront


aberrated beam=

iregular wavefront

diverging beam=

spherical wavefront

aberrated beam=

irregular wavefront

Image Plane

Non-ideal Opticsideal wavefront

Optical Aberration

Real Optical System

An Aberrated Optical System can be defined by one of the three different and equivalent ways:

The rays emerging from a point source P, situated at a finite or infinite distance from the Optical System, do not intersect at a common point P’, on the Image Plane.

3

The rays emerging from a point source P will not travel the same Optical Path to reach the Image Plane

2

The wavefront of light, focused by the Optical System on the Image Plane, is not spherical.

1

17

Optical Aberration W (x,y) is the path deviation between the distorted and referenceWavefront.

SOLO Optical Aberration Optical Aberration W (x,y)

18


Display of Optical Aberration W (x,y)

Rays Deviation3

Optical Path Length Difference2

wavefront shape W (x,y) 1

Red circle denotes the pupile margin.Arrows shows how each ray is deviatedas it emerges from the pupil plane.Each of the vectors indicates the thelocal slope of W (x,y).

The aberration W (x,y) is represented in x,y plane by color contours.

xy

( )yxW ,Wavefront Error

x

y

( )yxW ,

OpticalDistanceErrors

x

y

RayErrors

The Wavefront error agrees withOptical Path Length Difference, But has opposite sign because a long (short) optical path causes phase retardation (advancement).

Aberration Type:Negative vertical

coma Reference

19


Display of Optical Aberration W (x,y)

Advanced phase <= Short optical path

Retarded phase <= Long optical path

Reference

Ectasia

x

y

Ray Errors

y

( )yxW ,

x

Optical Distance Errorsx

y

( )yxW ,

Wavefront Error

20

Optics SOLO

Lens Definitions

Optical Axis: the common axis of symmetry of an optical system; a line that connects all centers of curvature of the optical surfaces.

Lateral Magnification: the ratio between the size of an image measured perpendicular to the optical axis and the size of the conjugate object.

Longitudinal Magnification: the ratio between the lengthof an image measured along the optical axis and the length of the conjugate object.

First (Front) Focal Point: the point on the optical axis on the left of the optical system (FFP) to which parallel rays on it’s right converge.

Second (Back) Focal Point: the point on the optical axis on the right of the optical system (BFP) to which parallel rays on it’s left converge.

21

Optics SOLO

Definitions (continue – 1)

Aperture Stop (AS): the physical diameter which limits the size of the cone of radiation which the optical system will accept from an axial point on the object.

Field Stop (FS): the physical diameter which limits the angular field of view of an optical system. The Field Stop limit the size of the object that can beseen by the optical system in order to control the quality of the image.

Entrance Pupil: the image of the Aperture Stop as seen from the object through theelements preceding the Aperture Stop.

Exit Pupil: the image of the Aperture Stop as seen from an axial point on theimage plane.

22

Optics SOLO


Aperture Stop (AS): the physical diameter which limits the size of the cone of radiation which the optical system will accept from an axial point on the object.

Field Stop (FS): the physical diameter which limits the angular field of view of an optical system. The Field Stop limit the size of the object that can beseen by the optical system in order to control the quality of the image.

Entrance Pupil: the image of the Aperture Stop as seen from the object through theelements preceding the Aperture Stop.

Exit Pupil: the image of the Aperture Stop as seen from an axial point on theimage plane.

23

Optics SOLO


Principal Planes: the two planes defined by the intersection of the parallel incident raysentering an optical system with the rays converging to the focal pointsafter passing through the optical system.

Principal Points: the intersection of the principal planes with the optical axes.

Nodal Points: two axial points of an optical system, so located that an oblique ray directed toward the first appears to emerge from the second, parallel to the original direction. For systems in air, the Nodal Points coincide with the Principal Points.

Cardinal Points: the Focal Points, Principal Points and the Nodal Points.

24

Optics SOLO


Start from the idealized conditions of Gaussian Optics.

( )00 ,0, zxP − Object Point

( )0,0,0O Center of ExP

( )gg zxP ,0,' Gaussian Image

gzz = Gaussian Image plane

'POP Chief Ray

'PQP General Ray

[ ] ':' QPnPQnPQPpathOptical +==

( )zyxQ ,, General Point on Exit Pupil

The Gaussian Image is obtained from rays starting at the Object P thatpassing through the Optics and

intersect Gaussian Image Plane at P’.

We have an Ideal Optical System with the center of the Exit Pupil (ExP) at point O (0,0,0).The Optical Axis (OA) passes through O in the z direction. Normal to OA we defined theCartezian coordinates x,y. (x,z) is the tangential (meridional) plane and (y,z) the sagittal plane defined by P and OA.

Play it

25

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Real Imaging Systems

'POP Chief Ray

'PQP General Ray

For an idealized system all the optical paths are equal.

[ ] ':' QPnPQnPQPpathOptical +==

( )zyxQ ,, General Ray

[ ][ ] ''

''

OPnPOnPOP

QPnPQnPQP

+==

+=

( ) ( )[ ]( ) ( )[ ]

[ ] [ ] 2/1222/120

20

2/1222

2/120

220

gg

gg

zxnzxn

zzyxxn

zzyxxn

+++=

−++−

++++−

Optical Aberration

Aberrations (continue – 1)

26

SOLO


For homogeneous media (n = constant) the velocity of light is constant, therefore therays starting/arriving from/to a point are perpendicular to the spherical wavefronts.

Optical paths from P:

( ) ( )[ ] 2/120

220),( zzyxxnQPV +++−=

( ) ( )[ ] 2/1222)',( gg zzyxxnPQV −++−=

Optical paths to P’:

Rays from P:

( ) ( )( ) ( )( ) ( )[ ] 2/12

022

0

00

,,),(

ˆ

,1

zzyxx

zzzyyxxx

QPVn

s zyxQP

+++−+++−=

∇=

Rays to P’:

( ) ( )( ) ( )( ) ( )[ ] 2/1222

,,)',(

ˆ

',1

gg

gg

zyxPQ

zzyxx

zzzyyxxx

PQVn

s

−++−−++−

−=

∇=−=

Optical Aberration


27

Optics SOLO

Real Imaging Systems – Aberrations (continue – 3)

Departures from the idealized conditions of Gaussian Optics in a real Optical System arecalled Aberrations

( )00 ,0, zxP − Object

( )0,0,0O Center of ExP

( )gg zxP ,0,' Gaussian Image

gzz = Gaussian Image plane

The aberrated image of P in the Gaussian Image plane is

( )gii zyxP ,,"

Define the Reference Gaussian Sphere having the center at P’ and passing through O:

022222 =−−++ gg zzxxzyx

P” is the intersection of rays normal to the Aberrated Wavefront that passes trough pointO (OP” is a Chief Ray).

Choose any point on the Aberrated Wavefront. The Ray intersects the Reference Gaussian Sphere at Q (x, y, z).

Q "PQPlay it

28

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Choose any point on the Aberrated Wavefront. The Ray intersects the Reference Gaussian Sphere at Q (x, y, z).

Q "PQ

( ) ( )QPVQPVW ,, −=

By definition of the wavefront, theoptical path length of the ray startingat the object P and ending at is identical to that of the Chief Ray ending at O.

Q

Therefore the Wave Aberration is defined asthe difference in the optical paths from P to Q V (P,Q) to that from P to ( )QPVQ ,,

Define the Optical Path from P(x0,0,-z0) to Q (x,y,z) as: ( )

( )

( )

∫−

=zyxQ

zxP

raydnQPV,,

,0, 00

:,

Since by definition: ( ) ( )OPVQPV ,, =

( ) ( )( ) ( ) ( )( ) ( )( )zyxQWOzxPVzyxQzxPVW ,,0,0,0,,0,,,,,0, 0000 =−=Since Q (x,y,z) is constraint on the Reference Gaussian Sphere:we can assume that z is a function of x and y, and

022222 =−−++ gg zzxxzyx

( ) ( ) ( )( )( ) ( ) ( )( )0,0,0,,0,,,,,,0,, 0000 OzxPVyxzyxQzxPVyxW −=

Optical Aberration


29

SOLO


Given the Wave Aberration function W (x,y)the Gaussian Image P’(xg,0,zg) of P and the point Q (x,y,z) on the Reference Gaussian Sphere

we want to find the point P”(xi,yi,zg)

022222 =−−++ gg zzxxzyx

( ) ( ) ( )( )( ) ( ) ( )( )0,0,0,,0,,,,,,0,, 0000 OzxPVyxzyxQzxPVyxW −=

Solution:( ) ( )( )( ) ( )( )( )

Qx

z

z

yxzyxQPV

x

yxzyxQPV

x

yxW

∂∂

∂∂+

∂∂=

∂∂ ,,,,,,,,,

( )( ) ( ) ( )[ ] 2/1222

,,,,

zzyyxx

zzyyxxn

z

V

y

V

x

V

gii

gii

−+−+−

−−−=

∂∂

∂∂

∂∂

Compute relative to Q by differentiating relative to x:022222 =−−++ gg zzxxzyxQ

x

z

∂∂

g

g

Qzz

xx

x

z

−−

−=∂∂

( ) ( ) ( ) ( )gig

ggi xx

R

n

zz

xxzz

R

nxx

R

n

x

yxW −=−−

−−−=∂

∂'''

, ( )x

yxW

n

Rxx gi ∂

∂+= ,'

In the same way:( )y

yxW

n

Ryi ∂

∂= ,'

The ray from Q to P” is given by (see ):

Optical Aberration


30

SOLO


Object

Gaussian

Image

planeExit Pupil(ExP)

Optics

( )00 ,0, zxP − ( )zyxQ ,,

( )gg zxP ,0,'

( )gii zyxP ,,"

iy

iz

ReferenceGaussian

Spherecenter P'

AberratedWavefrontcenter P"

( )0,0,0O

gz

y

x Q

z

Gaussian Image

AberratedImage

ChiefRay

ChiefRay

( )x

yxW

n

Rxx gi ∂

∂+= ,'

( )y

yxW

n

Ryi ∂

∂= ,'

Optical Aberration


Forward to a 2nd way

The aberration is the deviation of the image P”(xi,yi,zg) from the Gaussian image P’(xg,yg,zg)

The image P”(xi,yi,zg) coordinates in image plane are:

( )x

yxW

n

Rxxx gii ∂

∂=−=∆ ,'

( )y

yxW

n

Ryyy gii ∂

∂=−=∆ ,'

31

SOLO


Defocus Aberration

Consider an optical system for which theobject P, the Gaussian image P’ and theaberrated image P” are on the Optical Axis.

The Gaussian Reference Sphere passing throughO (center of ExP) has the center at P’.

The Aberrated Wavefront Sphere passing throughO (center of ExP) has the center at P”.

Consider a ray ( on the Aberrated

Wavefront Sphere) that intersects the Gaussian Reference Sphere at Q, that is at a distance r

from the Optical Axis.

Q"PQ

( ) ( ) UBBnQQnQQVrW cos/, === The Wave Aberration is defined as

( ) ( ) ( )

−−−−−=−−= 12

221

222cos

'"'"cos

RRrRrRU

nPPBPPB

U

nrW

Optical Aberration

32

SOLO


Defocus Aberration (continue – 1)

Let make the following assumptions:

( ) ( ) ( )

−−−−−=−−= 12

221

222cos

'"'"cos

RRrRrRU

nPPBPPB

U

nrW

21,1cos RRrU <<≈

( ) ( )

( )

+

−−

−=

−−

++−−

++−≈

−−=

4

1111

2

821

821

'"'"cos

4

32

31

2

21

1241

4

21

2

142

4

22

2

2

r

RRr

RR

n

RRR

r

R

rR

R

r

R

rRn

PPBPPBU

nrW

11682

1132

<++−+=+ xxxx

x

Assume: RRRRRR =≈−=∆ 2112 &

( ) 222

rR

RnrW

∆≈we have: Δ R is called the Longitudinal Defocus.

Optical Aberration

33

SOLO



For a circular exit pupil of radius a we have:

( ) 222

#8ρρρ dA

f

RnW =∆=

a

Rf

2:# = F number:

Define: a

r=:ρ

Therefore

Where is the peak value of theDefocus Aberration

2#8

:f

RnAd

∆=

Optical Aberration

34

http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf

( ) ( )22, yxAyxW d +⋅=

Optical Aberration

Wave Aberration: Defocus

SOLO



Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley

35http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf

Optical Aberration

Wavefront Errors for Defocus


SOLO

36

SOLO


Wavefront Tilt Aberration

Assume an optical system that has one ore more optical elements tilted and/or decentered.

The object P is on the Optical Axes (OA), therefore the Gaussian image P1 is also on OA. Therefore theGaussian Reference Sphere that passes trough ExP center O has it’s center at P1. P2 is the aberrated image on the Gaussian image plane (that contains P1) is a distance xi from OA. The Aberrated Wavefront that passes through O has it’s center at P2. Therefore for small P1P2 the two surfaces are tilted by an angle β.

Consider the ray where:2QPQ( )zyxQ ,, on the Gaussian Reference Sphere 02 1

222 =−++ xRzyx

Q on the Aberrated Wavefront Sphere centered at P2 and radius R.

βcos1 RR =( )12 ,0, RxP i the aberrated image

ββ RRxi ≈= sin

( ) ( )θθ sin,cos, rryx =

Optical Aberration

37

SOLO


Wavefront Tilt Aberration (continue – 1)

We have

x

W

n

RRxi ∂

∂== β

( ) ( ) QQnQQVrW == ,

The Wave Aberration is

βnx

W =∂∂

θββ cos0

rnxnxdx

WW

x

==∂∂= ∫

For a circular exit pupil of radius a we have:

a

r=:ρ

( ) θρθρβθρ coscos, 1BanW ==

where:

βanB =:1

Optical Aberration

38

SOLOReal Imaging Systems

Departures from the idealized conditions of Gaussian Optics in a real Optical System arecalled Aberrations

Monochromatic Aberrations

Chromatic Aberrations

• Monochromatic Aberrations

Departures from the first order theory are embodied in the five primary aberrations

1. Spherical Aberrations

2. Coma

3. Astigmatism

4. Field Curvature

5. Distortion

This classification was done in 1857 by Philipp Ludwig von Seidel (1821 – 1896)

• Chromatic Aberrations

1. Axial Chromatic Aberration

2. Lateral Chromatic Aberration

Optical Aberration

39

SOLO


Optical Aberration

40

SOLO


Optical Aberration

41

SOLO


Optical Aberration

42

SOLO


Seidel Aberrations

Consider a spherical surface of radius R, with an object P0 and the image P0’ on the Optical Axis.

n

'n

CBR

0P '0P

( )θ,rQ

0V

r

z

( ) s− 's

Chief Ray

General Ray

Aperture StopEnter PupilExit Pupil

The Chief Ray is P0 V0 P0’ and aGeneral Ray P0 Q P0’.

The Wave Aberration is defined asthe difference in the optical path lengths between a General Ray and the Chief Ray.

( ) [ ] [ ] ( ) ( )snsnQPnQPnPVPQPPrW +−+=−= '''''' 00000000

On-Axis Point Object

The aperture stop AS, entrance pupil EnP, and exit pupil ExP are located at the refracting surface.

Optical Aberration

43

SOLO


Seidel Aberrations (continue – 1)

n

'n

CBR

0P '0P

( )θ,rQ

0V

r

z

( ) s− 's

Chief Ray

General Ray

ASEnPExP

−−=−−=

2

222 11

R

rRrRRz

Define:

( )2

2

112

2

R

rxxf

R

rx

−=+=−=

( ) ( ) 2/112

1' −+= xxf

( ) ( ) 2/314

1" −+−= xxf ( ) ( ) 2/51

8

3'" −+−= xxf

Develop f (x) in a Taylor series ( ) ( ) ( ) ( ) ( ) ++++= 0"'6

0"2

0'1

032

fx

fx

fx

fxf

11682

1132

<++−+=+ xxxx

x

RrR

r

R

r

R

r

R

rRz <+++=

−−=

5

6

3

42

2

2

168211

On Axis Point Object

From the Figure:

( ) 222 rzRR +−= 02 22 =+− rRzz

Optical Aberration

44

SOLO



From the Figure:

( )[ ] [ ]( )[ ] ( ) 2/1

2

2/122

2/12222/122

0

212

222

−+=+−=

++−=+−=−=

zs

sRsszsR

rsszzrszQP

rzRz

( ) ( )

+−−−+−≈

<++−+=+

24

2

2

11682

11

2

11

32

zs

sRz

s

sRs

xxxx

x

( ) ( )

+

+−−

+−+−=

+≈

2

3

42

4

2

3

42

2

82

822

1

821

3

42

R

r

R

r

s

sR

R

r

R

r

s

sRs

R

r

R

rz

( )[ ] +

−+

−+

−+−≈+−= 4

2

2

22/122

0

11

8

111

8

111

2

1r

sRssRRr

sRsrszQP

( )[ ] +

−+

−+

−+≈+−= 4

2

2

22/122

0

1

'

1

'8

11

'

1

8

11

'

1

2

1''' r

RssRsRr

RssrzsPQ

In the same way:


Optical Aberration

n

'n

CBR

0P '0P

( )θ,rQ

0V

r

z

( ) s− 's

Chief Ray

General Ray

ASEnPExP

45

SOLO



+

−+

−+

−+−≈ 4

2

2

2

0

11

8

111

8

111

2

1r

sRssRRr

sRsQP

+

−+

−+

−+≈ 4

2

2

2

0

1

'

1

'8

11

'

1

8

11

'

1

2

1'' r

RssRsRr

RssPQ

Therefore:

( ) ( ) ( )4

22

2

42

000

11

'

11

'

'

8

1

82

'

'

'

''''

rsRs

n

sRs

n

R

rr

R

nn

s

n

s

n

snsnQPnQPnrW

−−

−−

+

−−−=

+−+=

Since P0’ is the Gaussian image of P0 we have( ) R

nn

s

n

s

n −=−

+ '

'

'

and:( ) 44

22

0

11

'

11

'

'

8

1rar

sRs

n

sRs

nrW S=

−−

−−=


Optical Aberration

n

'n

CBR

0P '0P

( )θ,rQ

0V

r

z

( ) s− 's

Chief Ray

General Ray

ASEnPExP

46

SOLO



Off-Axis Point Object

Consider the spherical surface of radius R, with an object P and its Gaussian image P’ outside the Optical Axis.

The aperture stop AS, entrance pupil EnP, and exit pupil ExP are located at the refracting surface.

Using the similarity of the triangles:

''~ 00 CPPCPP ∆∆ the transverse magnification

( ) ( )

s

n

s

nnn

s

s

n

s

nnn

s

Rs

Rs

h

hM t

−

−+−

−

−−

=+−

−=−

=

'

''

'

''

'

''

( ) sn

sn

nns

snn

nns

snn

M t −=

−+−

+−−=

'

'

''

'

''

'

n

'n

CBR

0P

'0P

( )θ,rQ

0V

r

z

( ) s−'s

Chief Ray

General R

ay

ASEnPExP

'P

Undeviated Ray

P

( ) h−

'h

θ

V

Optical Aberration

47

SOLO




The Wave Aberration is defined as the difference in the optical path lengths between the General Ray and the Undeviated Ray.

( ) [ ] [ ][ ] [ ]{ } [ ] [ ]{ }

( )4

0

4

0

00 ''''

'':

VVVQa

PPVPPVPPVPQP

PVPPQPQW

S −=

−−−=−=

For the approximately similar triangles VV0C and CP0’P’ we have:

CP

CV

PP

VV

''' 0

0

0

0 ≈ '''

'''

0

0

00 hbh

Rs

RPP

CP

CVVV =

−=≈

Rs

Rb

−=

':

−−

−−=

2211

'

11

'

'

8

1

sRs

n

sRs

naS

Optical Aberration

n

'n

CBR

0P

'0P

( )θ,rQ

0V

r

z

( ) s−'s

Chief Ray

General R

ay

ASEnPExP

'P

Undeviated Ray

P

( ) h−

'h

θ

V

48

SOLO




Wave Aberration.

( ) [ ] [ ] ( )4

0

4'' VVVQaPVPPQPQW S −=−=

θθ cos'2'cos2 222

0

2

0

22

hbrhbrVVrVVrVQ ++=++≈

'0 hbVV =

( ) [ ] [ ] ( )( )[ ]442222

4

0

4

'cos'2'

''

hbhbrhbra

VVVQaPVPPQPQW

S

S

−++=

−=−=

θ( ) ( )θθθθ cos'4'2cos'4cos'4';, 33222222234 rhbrhbrhbrhbrahrW S ++++=

Optical Aberration

n

'n

CBR

0P

'0P

( )θ,rQ

0V

r

z

( ) s−'s

Chief Ray

General R

ay

ASEnPExP

'P

Undeviated Ray

P

( ) h−

'h

θ

V

θ

r

y

x

0V

V

Q

Exit Pupil Plane

Define the polar coordinate (r,θ) of the projection of Q in the plane of exit pupil, withV0 at the origin, and assume (third order = Seidel approximation) that projected onexit pupil is equal to .

VQVQ

49

SOLO



General Optical Systems

( ) θθθθ cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++=

A General Optical Systems has more than one Reflecting orRefracting surface. The image of one surface acts as anobject for the next surface, therefore the aberration is additive.

We must address the aberration in the plane of the exit pupil, since the rays follow straight lines from the plane of the exit pupil.

The general Wave Aberration Function is:

1. Spherical Aberrations Coefficient SpC

2. Coma CoefficientCoC

3. Astigmatism Coefficient AsC

4. Field Curvature Coefficient FCC

5. Distortion Coefficient DiC

where:

n

'n

C O

0P

'0P

( )θ,rQ

0V

r

( ) s−'s

Chief Ray

General R

ay

Exit PupilExp

'P

Undeviated Ray

P

( ) h−

'h

θ

~

Optical Aberration

50

SOLO



( ) θθθθ cos''cos'cos'';, 32222234 rhCrhCrhCrhCrChrW DiFCAsCoSp ++++=

Optical Aberration

51

SOLO


Optical Aberration


52

n

C O '0P

RP

'L

Chief Ray

General Ray

Exit PupilExp

'P

Undeviated Ray

'h

~

True WaveFront

ReferenceSphere

α

α

TP

RP'TP'

α'Lr ≈∆r

r∆

Imageplane

True WaveFront

ReferenceSphere

r∆

l∆

RP

RP'

TP

TP'

α

GP

SOLO



nWPP TR /=

Assume that P’ is the image of P.

The point PT is on the Exit Pupil (Exp) and on theTrue Wave Front (TWF) that propagates toward P’.This True Wave Front is not a sphere because of theAberration. Without the aberration the wave front would be the Reference Sphere (RS) with radius PRPG.

W (x’,y’;h’) - wave aberrationn - lens refraction index

L’ - distance between Exp and Image plane

ά - angle between the normals to the TWF and RS at PT.

Assume that P’R and P’T are two points onRS and TWF, respectively, and on a ray closeto PRPT ray, converging to P’, the image of P.

lPPPP TRTR ∆+=''

Optical Aberration

53

SOLO


n

C O '0P

RP

'L

Chief Ray

General Ray

Exit PupilExp

'P

Undeviated Ray

'h

~

True WaveFront

ReferenceSphere

α

α

TP

RP'TP'

α'Lr ≈∆r

r∆

Imageplane

True WaveFront

ReferenceSphere

r∆

l∆

RP

RP'

TP

TP'

α

GP

Optical Aberration


54

SOLO



( )x

hyxW

n

Lxi ∂

∂=∆ ';,

'

' ( )y

hyxW

n

Lyi ∂

∂=∆ ';,

'

'

θθ

sin

cos

ry

rx

==

( ) nhyxWPP TR /';,= lPPPP TRTR ∆+=''

α=∆∆=

∆−=

∂∂

→∆→∆ r

l

r

PPPP

x

W

n r

TRTR

r 00lim

''lim

1

x

W

n

LLr

∂∂==∆ '

'α

Optical Aberration

n

C O '0P

RP

'L

Chief Ray

General Ray

Exit PupilExp

'P

Undeviated Ray

'h

~

True WaveFront

ReferenceSphere

α

α

TP

RP'TP'

α'Lr ≈∆r

r∆

Imageplane

True WaveFront

ReferenceSphere

r∆

l∆

RP

RP'

TP

TP'

α

GP

Deviation of image due to aberrations. We recovered the equations developed in

55

SOLO


1. Spherical Aberrations

( )( ) ( )';,

';,222

4

hyxWyxC

rChrW

SpSp

SpSp

=+=

=θ

( )xrC

n

L

x

hyxW

n

Lx Spi

2

'

'4

';,

'

' =∂

=∆

( )yrC

n

L

y

hyxW

n

Ly Spi

2

'

'4

';,

'

' =∂

=∆

( ) ( )[ ] 32/122

'

'4 rC

n

Lyxr Spiii =∆+∆=∆

Consider only the Spherical Wave Aberration Function

The Spherical Wave Aberration is aCircle in the Image Plane

Optical Aberration

( ) θθθθ cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++= The general Wave Aberration Function is:

56

SOLO


1. Spherical Aberrations (continue – 1)

Optical Aberration

Ray Errors

0

0.5

1

Optical Distance Errors Wavefront Error

57

SOLO


1. Spherical Aberrations (continue – 2)

Optical Aberration

58

SOLO


Assume an object point outside the Optical Axis.

Meridional (Tangential) plane isthe plane defined by the object point and the Optical Axis.

Sagittal plane is the plane normal toMeridional plane that contains theChief Ray passing through theObject point.

Optical Aberration

Meridional and Sagittal Planes

59

SOLO


2. ComaConsider only the Coma Wave Aberration Function

( ) ( ) ''''cos'';, 22cos'

sin'

3 xyxhbCrhbChrW Co

rx

ryCoCo +===

=

θ

θθθ

( ) ( ) ( )

( )θ

θ

2cos2'

''

cos21'

''3

'

''';,

'

'

2

2222

+=

+=+=∂

=∆

rn

LhbC

rn

LhbCyx

n

LhbC

x

hyxW

n

Lx

Co

CoCoi

( ) ( ) θ2sin'

''2

'

''';,

'

' 2rn

LhbCyx

n

LhbC

y

hyxW

n

Ly CoCoi ==

∂=∆

1

'

''2

'

''

2

2

2

2

=

∆

+

−∆

rn

LhbC

y

rn

LhbC

x

Co

i

Co

i

( )( ) ( ) ( ) 2222 rRyrRx CoiCoi =∆+−∆

( ) 2

'

'': r

n

LhbCrR CoCo =

Optical Aberration


60

SOLO


2. Coma (continue – 1)We obtained

2

'

'': MAXCoS r

n

LhbCC =

( )( ) ( ) ( ) 2222 rRyrRx CoiCoi =∆+−∆

( ) MAXCoCo rrrn

LhbCrR ≤≤= 0

'

'': 2

Define:

1

2

3 4

P

ImagePlane

O

SC

SC

ST CC 3=

Coma Blur Spot Shape in the Image Plane

TangentialComa

SagittalComa

30

'h

ix

iy

Optical Aberration

61

SOLO


Graphical Explanation of Coma Blur

1

1

2

2

3

3

4

4

Optical Axis

1

Meridional

(Tangential)

Plane

PImagePlane

TangentialRays 1

O

Lens

A Tangential Rays 1

Chief R

ay1

1

1

2

2

3

3

4

4

Optical Axis

1

SagittalPlane

P ImagePlane

SagittalRays 2

O

Lens

A

2

Sagittal Rays 2

Chief R

ay

2

1

1

2

2

3

3

4

4

Optical Axis

1

P ImagePlane

SkewRays 3

O

LensA

23

Skew Rays 3

Chief R

ay

3

1

1

2

2

3

3

4

4

Optical Axis

1

P ImagePlane

SkewRays 4

O

Lens

A

23

4

Skew Rays 4

Chief R

ay

4

2. Coma (continue – 2)

Optical Aberration

62

SOLO


Graphical Explanation of Coma Blur (continue – 1)

2. Coma (continue – 3)

Optical Aberration

64

SOLO


Optical Aberration

2. Coma (continue-5)

65


( ) ( ) ''''cos'';, 22cos'

sin'

3 xyxhCrhChrW Co

rx

ryCoCo +===

=

θ

θθθ

Wave Aberration: Coma

Optical Aberration

2. Coma (continue-6)

SOLO

66

SOLO


3.&4. Astigmatism and Curvature of Field

Optical Aberration

( )( ) 2222

cos'

sin'

2222222

'''

'cos'';,

yCxCChb

rhbCrhbChrW

FCFCAs

rx

ry

FCAsAs

++=

+==

=

θ

θ

θθ


Consider the Astigmatism and the Field Curvature Wave Aberration Function

( ) ( ) xCCn

Lhb

x

hyxW

n

Lx FCAsi +=

∂=∆

'

''2

';,

'

' 22

( )yC

n

Lhb

y

hyxW

n

Ly FCi '

''2

';,

'

' 22

=∂

=∆

Meridionalplane

SagittalplaneObject

point OpticalSystem

Chiefray

Opticalaxis

'L

( )( )θρθρ sin,cos

,

=yx

( )ii yx ∆∆ ,

Ellipse

Imageplane

Opticalaxis

( )1

'''

2'

''2

2

22

2

22=

∆

+

+

∆

FC

i

FCAs

i

Crn

Lhb

y

CCrn

Lhb

x

θθ

sin

cos

ry

rx

==Ellipse

67

SOLO



Sagittalplane

Objectpoint

OpticalSystem

Chiefray

Opticalaxis

'L 'L∆


,

=yx ( )ii yx ∆∆ , ( )'','' ii yx ∆∆

Optical Aberration

We want to see what happens if wemove the image plane from the OpticalSystem by a small distance Δ L’,to obtain (Δ xi”, Δ yi”)

From the Figure we found that:

''

'

"

"

"

"

LL

L

yy

yy

xx

xx

ii

ii

ii

ii

∆+∆=

∆−∆−∆

=∆−∆−∆

''

''

"'"

''

''

"'"

L

yLy

LL

yyLyy

L

xLx

LL

xxLxx

ii

iiii

ii

iiii

∆−∆≈∆+∆−

∆−∆=∆

∆−∆≈∆+∆−

∆−∆=∆

( ) ( ) xCCn

Lhb

x

hyxW

n

Lx FCAsi +=

∂=∆

'

''2

';,

'

' 22

( )yC

n

Lhb

y

hyxW

n

Ly FCi '

''2

';,

'

' 22

=∂

=∆

( ) iFCAsi xL

LCC

n

Lhbx

∆−+=∆'

'

'

''2"

22

iFCi yL

LC

n

Lhby

∆−=∆'

'

'

''2"

22 ( )1

''

'''

2''

'''

2

2

22

2

22=

∆−

∆+

∆−+

∆

LL

Crn

Lhb

y

LL

CCrn

Lhb

x

FC

i

FCAs

i

68

SOLO



Optical Aberration

Meridionalplane

Sagittalplane

Objectpoint

OpticalSystem

Chiefray

SFTF

Opticalaxis

'L 'L∆


,

=yx

( )ii yx ∆∆ ,

( )'','' ii yx ∆∆

1""

22

=

∆+

∆

y

i

x

i

b

y

b

x( )

'

'

'

''2:

'

'

'

''2:

22

22

L

LCr

n

Lhbb

L

LCCr

n

Lhbb

FCy

FCAsx

∆−=

∆−+=

When bx or by is zero, the ellipse degenerates to a straight line

0=yb 0"''

''2"

22

=∆=∆ SAsS yxCn

LhbxFCS Cr

n

LhbL

'

''2'

222

=∆ Sagittal or Radial image

0=xb ''

''2"0"

22

yCn

Lhbyx FCTT −=∆=∆( )FCAsT CCr

n

LhbL +=∆

'

''2'

222

Tangential image

When Δ L’ is halfway between the two values just defined ( ) ( )FCAsTST CCrn

LhbLLL 2

'

''''

2

1'

222

+=∆+∆=∆

then we obtain the Circle of Least ConfusionAsyx Cr

n

Lhbbb

'

''22

=−=

( ) xL

LCC

n

Lhbx FCAsi

∆−+=∆'

'

'

''2"

22

yL

LC

n

Lhby FCi

∆−=∆'

'

'

''2"

22

69

SOLO



Meridionalplane

Sagittalplane

Primaryimage Secondary

image

Circle of leastconfusion

Objectpoint

OpticalSystem

Chiefray

SFTF

Ray inSagittal plane

Ray inMeridional plane

Opticalaxis

Opticalaxis

Optical Aberration

0"''

''2"

22

=∆=∆ SAsS yxCn

Lhbx

FCS Crn

LhbL

'

''2'

222

=∆

Sagittal or Radial image

''

''2"0"

22

yCn

Lhbyx FCTT −=∆=∆

( )FCAsT CCrn

LhbL +=∆

'

''2'

222

Tangential image

( ) ( )FCAsTST CCrn

LhbLLL 2

'

''''

2

1'

222

+=∆+∆=∆

Circle of Least Confusion

( ) ( )222

22

'

''""

=∆+∆ AsCC Crn

Lhbyx

70

SOLO



Optical Aberration

If we rotate the object (and therefore the image) point about the optical axis then since

222

''

'2' hC

n

LrbL FCS

=∆

( ) 222

''

'2' hCC

n

rLbL FCAsT +=∆

Sagittal or Radial image position

Tangential image position

As the off-axis image distance h’ varies, the loci of these two image points, (Δ L’S,h’) and (Δ L’T,h’), sweep out two paraboloids of revolution σS and σT.

SσTσ

Exit Pupil

Optic Axis

When is no astigmatism CAs = 0,then σS and σT coincide to forma curved surface called thePetzval Surface.

71

SOLO


3. Astigmatism

Joseph Max Petzval1807 - 1891

72

SOLO


3. Astigmatism

73

SOLO


3. Astigmatism

74

SOLO


3. Astigmatism

r = radius

q = meridian

Optical Aberration

75

SOLO


4. Field Curvature

( ) 222 '';, rhbChrW FCFC =θ

Optical Aberration

76

4. Field Curvature


77

SOLO


5. Distortion

( )xhbC

rhbChrW

Di

DiDi

33

33

'

cos'';,

== θθ

Optical Aberration


Consider only the DistorsionWave Aberration Function

( ) ( )0

';,

'

'&

'

''';,

'

' 33

=∂

=∆=∂

=∆y

hyxW

n

LyC

n

Lhb

x

hyxW

n

Lx iDii

Meridionalplane

Sagittalplane

Objectpoint

OpticalSystem

Chiefray

Ray inSagittal plane

Ray inMeridional plane

Opticalaxis

Opticalaxis

gxix∆

We can see that the DistortionAberration is only in the object Meridional (Tangential) Plane.

78

SOLO


5. Distortion ( )xhbC

rhbChrW

Di

DiDi

33

33

'

cos'';,

== θθ

Optical Aberration

( ) ( )0

';,

'

'&

'

''';,

'

' 33

=∂

=∆=∂

=∆y

hyxW

n

LyC

n

Lhb

x

hyxW

n

Lx iDii

Objectpoints

OpticalSystem

Chiefray

Opticalaxis

Opticalaxis

1gx1ix∆

0θ

0r

0x

0y

5gx5ix∆

Gaussianimage

Distortedimage

4gx

4ix∆

1 23

4 5

Tangentialplane # 4

Let take instead of a point image, a line (multiple image points).

For each point we have a different tangentialplane and therefore a different x.

( ) '2/122 hyx ⇒+

To obtain the image we must substitute

( ) ( ) 2/1222/122sin&cos

yx

y

yx

x

+=

+= θθ

and we get:

( ) ( ) ( ) ( )233

2/122

2/3223

2/3223

'

'

'

'cos

'

'yxxC

n

Lb

yx

xyxC

n

LbyxC

n

Lbx DiDiDii +=

++=+=∆ θ

( ) ( ) ( ) ( )323

2/122

2/3223

2/3223

'

'

'

'sin

'

'yyxC

n

Lb

yx

yyxC

n

LbyxC

n

Lby DiDiDii +=

++=+=∆ θ

79

SOLO


5. Distortion ( )xhbC

rhbChrW

Di

DiDi

33

33

'

cos'';,

== θθ

Optical Aberration

Objectpoints

OpticalSystem

Chiefray

Opticalaxis

Opticalaxis

1gx1ix∆

0θ

0r

0x

0y

5gx5ix∆

Gaussianimage

Distortedimage

4gx

4ix∆

1 23

4 5

Tangentialplane # 4

Now consider a line object that yields a paraxial image x =a (see Figure).

( ) ( ) ( ) ( )233

2/122

2/3223

2/3223

'

'

'

'cos

'

'yxxC

n

Lb

yx

xyxC

n

LbyxC

n

Lbx DiDiDii +=

++=+=∆ θ

( ) ( ) ( ) ( )323

2/122

2/3223

2/3223

'

'

'

'sin

'

'yyxC

n

Lb

yx

yyxC

n

LbyxC

n

Lby DiDiDii +=

++=+=∆ θ

( )233

'

'yaaC

n

Lbx Dii +=∆

( )323

'

'yyaC

n

Lby Dii +=∆

80

SOLO


5. Distortion

( ) θθ cos'';, 33 rhbChrW DiDi =

Optical Aberration


Consider only the DistorsionWave Aberration Function

81


( )yxW ,

Optical Aberration SOLO

82

SOLO

Thin Lens Aberrations

Given a thin lens formed by twosurfaces with radiuses r1 and r2

with centers C1 and C2. PP0 is the object, P”P”0 is the Gaussian image formed by the first surface,P’P’0 is the image of virtual objectP’P”0 of the second surface.

( )

−+++= q

n

npn

sfnCCo 1

112

'4

12

( )2'2/1 sfCAs −=( ) ( )2'4/1 sfnnCFC +−=

( ) ( ) ( ) ( )

++

−++−++

−−−= qpnq

n

npnn

n

n

fnnCSp 14

1

2123

1132

1 223

3

where:

f

s

OAC11r

F”

F

''f

''s

2r

1=nn

h

"h

D

0P

P

0'P0"P

"P'P

'h

's

CR

ASEnPExP

r

( )θ,rQ

OC2

1=n

( ) [ ] [ ]0000 '', OPPQPPrW −=θ

Coddington position factor: '

211

2

'

'

s

f

s

f

ss

ssp −=−=

−+=

Coddington shape factor:12

12

rr

rrq

−+=

From:

( ) 2222234 'cos'cos'';, rhCrhCrhCrChrW FCAsCoSp +++= θθθwe find:

Optical Aberration

( )frr

nss

1111

'

11

21

=

−−=+ Lens Maker’s

Formula

83

SOLO

Coddington Position Factor

2R1R f

1C2FO 1F

2C

2n

1n

s 's

'2 sfs ==

2R1R f

1C 2F1F

2C

2n

1n

s 's

fss =∞= ',

2R 1R f

1C 2F1F

2C

2n

1n

s 's

fss <> ',0

2R 1Rf

1C 2F1F

2C

2n

1n

s 's

∞== ', sfs

2R1Rf

1C2F1F 2C

2n

1n

s 's

0', << sfs

CRCR

2R1R

f1C

2F1F 2C

2n

1n

s's

0'0 <<> sfs

2R1R

f1C

2F1F2C

2n

1n

s 's

fss =∞= ',

1=p

2R1R f

1C 2F1F

2C

2n

1n

s's

∞== ', sfs

1>p

2R1R f

1C 2F1F

2C

2n

1n

s 's

0',0 ><< ssf

0=p

2R1R f

1C 2FO 1F

2C

2n

1n

s 's

'2 sfs ==

1−=p1−<p

ss

ssp

−+='

'

ss

ssp

−+='

'

'

111

ssf+=

'

211

2

s

f

s

fp −=−=

Optical Aberration

84

SOLO

Coddington Position Factor

f f2f2− f− 0

Figure ObjectLocation

ImageLocation

ImageProperties

ShapeFactor

InfinityPrincipalfocus

'ss

fs 2> fsf 2'<<

fs 2= fs 2'=

fsf 2<< fs 2'>

's

's

s

s

fs = ∞='s

s

s's

fs < fs <'

Real, invertedsmall p = -1

Real, invertedsmaller

-1 < p <0

Real, invertedsame size

p = 0

Real, invertedlarger

0 < p <1

No image p = 1

Virtual, erectlarger

p>1

's

's0<s fs <' p < -1

Imaginary,invertedsmall

Optical Aberration

85

SOLO

Coddington Shape Factor

1

02

1

−=<

∞=

q

R

R

2R

1R

2C2n

1n

PlanoConvex

2n

1

0,0

21

21

−<>

<<

q

RR

RR

1C2C

1n

1R

2R

PositiveMeniscus

2R1R f

1C2F1F

2C 2n

1n

0

0,0

21

21

==

<>

q

RR

RR

EquiConvex

2R

1R

1C2n

1n

PlanoConvex

1

0

2

1

=∞=

>

q

R

R

2R1R

f

1C 2F 2C2n

1n

1

0,0

21

21

><

>>

q

RR

RR

PositiveMeniscus

12

12

RR

RRq

−+=

2R1R f

2F1F

2C

2n

1n

1C

NegativeMeniscus

1

0,0

21

21

−<>

>>

q

RR

RR

1

0, 21

−=

>∞=

q

RR

PlanoConcave

2R1R

f

2F1F

2C

2n1n

2R1R f

1C 2F1F

2C

2n1n

0

0,0

21

21

==

><

q

RR

RR

EquiConcave

2R1Rf

1F 2F

1C

2n1n

1

,0 21

=

∞=<

q

RR

PlanoConcave

NegativeMeniscus

1

0,0

21

21

><

<<

q

RR

RR

2R

1R

f

2F1F 2C2n

1n

1C

Optical Aberration

86

REFLECTION & REFRACTION SOLO

http://freepages.genealogy.rootsweb.com/~coddingtons/15763.htm

History of Reflection & Refraction

Reverent Henry Coddington (1799 – 1845) English mathematician and cleric.

He wrote an Elementary Treatise on Optics (1823, 1st Ed., 1825, 2nd Ed.). The book was displayed the interest on Geometrical Optics, but hinted to the acceptance of theWave Theory.

Coddington wrote “A System of Optics” in two parts:1. “A Treatise of Reflection and Refraction of Light” (1829), containing a

thorough investigation of reflection and refraction. 2. “A Treatise on Eye and on Optical Instruments” (1830), where he explained

the theory of construction of various kinds of telescopes and microscopes.

He recommended the use of the grooved sphere lens, first described by David Brewster in 1820 and in use today as the

“Coddington lens”.

Coddington introduced for lens:

Coddington Shape Factor: Coddington Position Factor:

12

12

rr

rrq

−+=

ss

ssp

−+='

'Coddington Lens

http://www.eyeantiques.com/MicroscopesAndTelescopes/Coddington%20microscope_thick_wood.htm

87

SOLO

Thin Lens Spherical Aberrations

( ) 4rCrW SpSA =

Given a thin lens and object O on theOptical Axis (OA). A paraxial ray will crossthe OA at point I, at a distance s’p from the lens. A general ray, that reaches the lensat a distance r from OA, will cross OA at point E, at a distance s’r.

( ) ( ) ( ) ( )

++

−++−++

−−−= qpnq

n

npnn

n

n

fnnCSp 14

1

2123

1132

1 223

3

where:

Define:

2R

1R

1C

IO

2C

Paraxialfocal plane2n

1n

sps'

E

rs' Long. SA

Lat. SA

φ ParaxialRay

General

Ray

'φ

r

rp ssSALongAberrationSphericalalLongitudin ''. −==

( ) rrp srssSALatAberrationSphericalLateral '/''. −== We have:

Optical Aberration

a, 06/24/2006

Jurgen R. Meyer,"Introduction to Classical and Modern Optics", 3d Ed., Prentice Hall, p.113

88

SOLO

12

12

RR

RRqK

−+==

( ) ( ) ( ) ( ) ( )

++

−++−++

−−−= qpnq

n

npnn

n

n

fnn

rrWSp 14

1

2123

113222

3

3

4

Thin Lens Spherical Aberrations (continue – 1)

Optical Aberration

Shape factor

89

SOLO


2R

1R

1C

IO

2C


1n

sps'

E

rs' Long. SA

Lat. SA

φ ParaxialRay

General

Ray

'φ

r

12

12

RR

RRq

−+=

F.A. Jenkins & H.E. White, “Fundamentals of Optics”, 4th Ed., McGraw-Hill, 1976, pg. 157Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm

In Figure we can see a comparisonof the Seidel Third Order Theorywith the ray tracing.

Optical Aberration

90

SOLO

We can see that the Thin Lens Spherical Aberration WSp is a parabolic function of theCoddington Shape Factor q, with the vertex at (qmin,WSp min)

( ) ( ) ( ) ( )

++

−++−++

−−−= qpnq

n

npnn

n

n

fnn

rWSp 14

1

2123

113222

3

3

4

Thin Lens Spherical Aberrations (continue -3)

The minimum Spherical Aberration for a given Coddington Position Factor p is obtained by:

( ) ( ) 0141

22

132 3

4

=

++

−+

−−=

∂∂

pnqn

n

fnn

r

q

W

p

Sp

1

12

2

min +−−=

n

npq

+

−

−−= 2

2

3

4

min 2132p

n

n

n

n

f

rWSp

The minimum Spherical Aberration is zero for ( )( ) 1

1

22

2 >−+=

n

nnp

Optical Aberration

91

SOLO

In order to obtain the radii of the lens for a given focal length f and given Shape Factorand Position Factor we can perform the following:


Those relations were given by Coddington.

'

211

2

s

f

s

fp −=−=

( )fRR

nss

1111

'

11

21

=

−−=+

p

fs

p

fs

−=

+=

1

2'&

1

2

12

12

RR

RRq

−+=

( ) ( )12

21

1 RRn

RRf

−−=

12

1

12

2 21&

21

RR

Rq

RR

Rq

−=−

−=+

( ) ( )1

12&

1

1221 −

−=+

−=q

nfR

q

nfR

2R

1R

1C

IO

2C


1n

sps'

E

rs' Long. SA

Lat. SA

φ ParaxialRay

General

Ray

'φ

r

Optical Aberration

92

SOLO

Thin Lens Coma

( ) ( )( ) ( )

−++++=

+==

qn

npn

sfn

xyxh

xyxhCrhChrW CoCoCo

1

112

'4

''''

''''cos'';,

2

22

223 θθ For thin lens the coma factor is given by:

where:we find:

( ) 2

22

2

1

112

4

''': MAXMAXCoS rq

n

npn

fn

hr

n

shCC

−+++==

1

2

3 4

P

ImagePlane

O

SC

SC

ST CC 3=

Coma Blur Spot Shape

TangentialComa

SagittalComa

30

'h

'x

'y

( )( ) ( ) ( ) 222 '2' rRyrRx CoCo =∆+−∆ ( ) MAXCoCo rrrn

shCrR ≤≤= 0

'': 2

Define:

( ) ( ) ( ) ( )θθ 2cos2''

cos21''

''3''

'

';',''' 2222 +=+=+=

∂=∆ r

n

shCr

n

shCyx

n

shC

x

hyxW

n

sx CoCoCo

( ) ( ) θ2sin''

''2''

'

';',''' 2r

n

shCyx

n

shC

y

hyxW

n

sy CoCo ==

∂=∆

Optical Aberration

2R

1R

1C

IO

2C


1n

sps'

E

rs' Long. SA

Lat. SA

φ ParaxialRay

General

Ray

'φ

r

93

SOLO

Thin Lens (continue – 1)

F.A. Jenkins & H.E. White, “Fundamentals of Optics”, 4th Ed., McGraw-Hill, 1976, pg. 165Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm, y = 2 cm

( ) 2

22 1

112

4

': MAXS rq

n

npn

fn

hC

−+++=

Coma is linear in q

( ) ( )( ) pn

nnqCS 1

1120

+−+−=⇐=

In Figure 800.00 =⇐= qCS

The Spherical Aberration is parabolic in q

( ) ( ) ( ) ( )

++

−++−++

−−−= qpnq

n

npnn

n

n

fnnCSp 14

1

2123

1132

1 223

3

1

12

2

min +−−=

n

npq

+

−

−−= 2

2

3min 2132

1p

n

n

n

n

fCSp

In Figure

714.0min =q

Optical Aberration

94


Thin Lens Aberration

95

SOLO

3. Thin Lens Astigmatism

Optical Aberration

The astigmatic lens may be focussed to yield a sharp image of either the sagittal or the tangential detail, but not simultaneously. This is illustrated in Fig. 1 with the archetypal example of astigmatism: a spoked wheel. A well-corrected lens delivers an all-sharp image (left wheel). On the other hand, an astigmatically aberrated lens may be focussed to yield a sharp image of the spokes (middle wheel), but at the expense of blurring of the rims, which have a tangential orientation. Vice versa, when the rim is in focus the spokes are blurred. It is customary to speak of the sagittal focus and tangential focus, respectively, as indicated in Fig. 1.

Figure 1. Classic example of astigmatism. Left wheel: no astigmatism. In the presence of astigmatism (middle and right wheels) one discriminates between the sagittal and tangential foci.


96


Although the wheels in Fig. 1 are instructive, they are an oversimplification of astigmatism as it occurs with photographic lenses. Where the figure suggests that the amount of blurring in either the sagittal or radial direction is constant across the field, this is not the case in practice. Unless a lens is poorly assembled, there will be no astigmatism near the image center. The aberration occurs off-axis. With a real lens, the sagittal and tangential focal surfaces are in fact curved. Fig. 2 displays the astigmatism of a simple lens. Here, the sagittal (S) and tangential (T) images are paraboloids which curve inward to the lens. As a consequence, when the image center is in focus the image corners are out of focus, with tangential details blurred to a greater extent than sagittal details. Although off-axis stigmatic imaging is not possible in this case, there is a surface lying between the S and T surfaces that can be considered to define the positions of best focus.

Fig. 2



97


Lens designers have a few degrees of freedom, such as the position of the aperture stop and the choice of glass types for individual lens elements, to reduce the amount of astigmatism, and, most desirably, to manoeuvre the S and T surfaces closer to the sensor plane. A complete elimination of astigmatism is illustrated in the left sketch of Fig. 3. Although astigmatism is fully absent, i.e., the S and T surfaces coincide, there is a penalty in the form of a pronouncedly curved field. When the image center is in focus on the sensor the corners are far out of focus, and vice versa. In the late nineteenth century, Paul Rudolf coined the word anastigmat to describe a lens for which the astigmatism at one off-axis position could be reduced to zero [2]. The right sketch in Fig. 3 depicts a typical photographic anastigmat. As a slight contradiction in terms, the anastigmat has some residual astigmatism, but more importantly, the S and T surfaces are more flat than those in the uncorrected scheme of Fig. 2 and the strictly stigmatic left scheme in Fig. 3. As such, the anastigmat offers an attractive compromise between astigmatism and field curvature

Fig. 2 Fig. 3



98




99

SOLO

Chester Moor Hall (1704 – 1771) designed in secrecy the achromatic lens. He experienced with different kinds of glass until he found in 1729 a combination of convex component formed from crown glass with a concave component formed from flint glass, but he didn’t request for a patent.

http://microscopy.fsu.edu/optics/timeline/people/dollond.html

In 1750 John Dollond learned from George Bass on Hall achromatic lens and designedhis own lenses, build some telescopes and urged by his sonPeter (1739 – 1820) applied for a patent.

Born & Wolf,”Principles of Optics”, 5th Ed.,p.176

Chromatic Aberration

In 1733 he built several telescopes with apertures of 2.5” and 20”. To keep secrecyHall ordered the two components from different opticians in London, but they subcontract the same glass grinder named George Bass, who, on finding that bothLenses were from the same customer and had one radius in common, placed themin contact and saw that the image is free of color.

The other London opticians objected and took the case to court, bringing Moore-Hall as a witness. The court agree that Moore-Hall was the inventor, but the judge Lord Camden, ruled in favor of Dollond saying:”It is not the person who locked up his invention in the scritoire that ought to profit by a patent for such invention, but he who brought it forth for the benefit of the public”

Optical Aberration

100

SOLOChromatic Aberration

Optical Aberration

Chromatic Aberrations arise inPolychromatic IR Systems because

the material index n is actuallya function of frequency. Rays atdifferent frequencies will traverse an optical system along different paths.

a, 06/24/2006

1. Jurgen R. Meyer-Arendt,"Introduction to Classical & Modern Optics",Prentice Hall, 3th Ed.,1989,p.123

101

SOLOChromatic Aberration

Optical Aberration

102

SOLO Optics Chromatic Aberration

Every piece of glass will separate white light into a spectrum given the appropriate angle. This is called dispersion. Some types of glasses such as flint glasses have a high level of dispersion and are great for making prisms. Crown glass produces less dispersion for light entering the same angle as flint, and is much more suited for lenses. Chromatic aberration occurs when the shorter wavelength light (blue) is bent more than the longer wavelength (red). So a lens that suffers from chromatic aberration will have a different focal length for each color To make an achromat, two lenses are put together to work as a group called a doublet. A positive (convex) lens made of high quality crown glass is combined with a weaker negative (concave) lens that is made of flint glass. The result is that the positive lens controls the focal length of the doublet, while the negative lens is the aberration control. The negative lens is of much weaker strength than the positive, but has higher dispersion. This brings the blue and the red light back together (B). However, the green light remains uncorrected (A), producing a secondary spectrum consisting of the green and blue-red rays. The distance between the green focal point and the blue-red focal point indicates the quality of the achromat. Typically, most achromats yield about 75 to 80 % of their numerical aperture with practical resolution

103


In addition, to the correction for the chromatic aberration the achromat is corrected for spherical aberration, but just for green light. The Illustration shows how the green light is corrected to a single focal length (A), while the blue-red (purple) is still uncorrected with respect to spherical aberration. This illustrates the fact that spherical aberration has to be corrected for each color, called spherochromatism. The effect of the blue and red spherochromatism failure is minimized by the fact that human perception of the blue and red color is very weak with respect to green, especially in dim light. So the color halos will be hardly noticeable. However, in photomicroscopy, the film is much more sensitive to blue light, which would produce a fuzzy image. So achromats that are used for photography will have a green filter placed in the optical path.

104


As the optician's understanding of optical aberrations improved they were able to engineer achromats with shorter and shorter secondary spectrums. They were able to do this by using special types of glass call flourite. If the two spectra are brought very close together the lens is said to be a semi-apochromat or flour. However, to finally get the two spectra to merge, a third optical element is needed. The resulting triplet is called an apochromat. These lenses are at the pinnacle of the optical family, and their quality and price reflect that. The apochromat lenses are corrected for chromatic aberration in all three colors of light and corrected for spherical aberration in red and blue. Unlike the achromat the green light has the least amount of correction, though it is still very good. The beauty of the apochromat is that virtually the entire numerical aperture is corrected, resulting in a resolution that achieves what is theoretically possible as predicted by Abbe equation.

105


With two lenses (n1, f1), (n2,f2) separated by a distance

d we found

2121

111

ff

d

fff−+=

Let use ( ) ( ) 222111 1/1&1/1 ρρ −=−= nfnf

We have

( ) ( ) ( ) ( ) 22112211 11111 ρρρρ −−−−+−= nndnnf

nF – blue index produced by hydrogen wavelength 486.1 nm.

nC – red index produced by hydrogen wavelength 656.3 nm.

nd – yellow index produced by helium wavelength 587.6 nm.

Assume that for two colors red and blue we have fR = fB

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) 22112211

22112211

1111

11111

ρρρρ

ρρρρ

−−−−+−=

−−−−+−=

FFFF

CCCC

nndnn

nndnnf

106


Let analyze the case d = 0 (the two lenses are in contact)

nd – yellow index produced by helium wavelength 587.6 nm.

( ) ( ) ( ) ( ) 22112211 11111 ρρρρ −+−=−+−= FFCC nnnnf

We have ( )( )

( )( )1

1

1

1

1

2

1

2

2

1

−−−=

−−−=

F

F

C

C

n

n

n

n

ρρ ( )

( )CF

CF

nn

nn

11

22

2

1

−−−=

ρρ

For the yellow light (roughly the midway between the blue and red extremes) the compound lens will have the focus fY:

( ) ( )

YY f

d

f

dY

nnf

21 /1

22

/1

11 111 ρρ −+−= ( )

( ) Y

Y

d

d

f

f

n

n

1

2

1

2

2

1

1

1

−−=

ρρ

( )( )

( )( )

( ) ( )( ) ( )1/

1/

1

1

111

222

2

1

11

22

1

2

−−−−−=

−−

−−−=

dCF

dCF

d

d

CF

CF

Y

Y

nnn

nnn

n

n

nn

nn

f

f

107


( ) ( )( ) ( )1/

1/

111

222

1

2

−−−−−=

dCF

dCF

Y

Y

nnn

nnn

f

f

The quantities are called

Dispersive Powers of the two materials forming the lenses.

( )( )

( )( )1

&1 2

22

1

11

−−

−−

d

CF

d

CF

n

nn

n

nn

Their inverses are called

V-numbers or Abbe numbers.

( )( )

( )( )CF

d

CF

d

nn

nV

nn

nV

22

22

11

11

1&

1

−−=

−−=

Return to Table of Content

108

SOLO

Image Analysis

Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA


Optical Aberration

109

Image Analysis

SOLO

( ) ( ) ( )[ ] { }gFTydxdyfxfjyxgffG yxyx =+−= ∫∫Σ

π2exp,:,

The two dimensional Fourier Transform F of the function f (x, y)

The Inverse Fourier Transform is

( ) ( ) ( )[ ] { }GFTfdfdyfxfjyxgyxgF

yxyx12exp,, −=+= ∫∫ π

( ) ( ) ( ) ( )[ ] { }Σ

Σ

=+−= ∫∫ fFTddkkjfkkF yxyx ηξηξηξπ

exp,2

1:, 2

Two Dimensional

Fourier Transform

Two Dimensional Fourier Transform (FT)

Fraunhofer Diffraction and the Fourier Transform

In Fraunhofer Diffraction we arrived two dimensional Fourier Transform of the field within the aperture

P

0P

Q 1x

0x1y

0yη

ξ

Sr '

Sr

ρr

O

Sθ θ

Screen

Image

plane

Source

plane

0O

1O

Sn1

Σ

Σ - Screen Aperture

Sn1 - normal to Screen

1r0r

SSS rn θcos11 =⋅θcos11 =⋅ rnS

z

Sn1

'r Fr

FrPP

=0

SrQP

=0

rQP

=

SrOP '0

=

'1 rOO

=

Using kx = 2 π fx and ky = 2 π fy we obtain:

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Image Analysis

SOLO

( ) ( ){ } ( ){ } ( ){ }yxhFTyxgFTyxhyxgFT ,,,, βαβα +=+1. Linearity Theorem

Two Dimensional Fourier Transform (FT)Fourier Transform Theorems

( ){ } ( )yx ffGyxgFT ,, =

2. Similarity Theorem

( ){ }

=

b

f

a

fG

baybxagFT yx ,

1,If then

( ){ } ( )yx ffGyxgFT ,, =

3. Shift Theorem

( ){ } ( ) ( )[ ]bfafjffGbyaxgFT yxyx +−=−− π2exp,,

If

then

Optical Aberration

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Image AnalysisSOLO

Two Dimensional Fourier Transform (FT)Fourier Transform Theorems (continue – 1)

( ){ } ( )yx ffGyxgFT ,, =

4. Parseval’s Theorem

( ) ( )∫∫∫∫ = yxyx fdfdffGydxdyxg22

,,

If

then

( ){ } ( )yx ffGyxgFT ,, =

5. Convolution Theorem

( ) ( ){ } ( ) ( )yxyx ffHffGddyxhgFT ,,,, =−−∫∫ ηξηξηξ

If

then

( ){ } ( )yx ffHyxhFT ,, =and

( ){ } ( )yx ffGyxgFT ,, =

6. Autocorrelation Theorem

( ) ( ){ } ( ) 2* ,,, yx ffGddyxggFT =−−∫∫ ηξηξηξ

If

then

similarly ( ){ } ( ) ( )∫∫ −−= ηξηξηξηξ ddffGGgFT yx ,,, *2

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Image Analysis

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Two Dimensional Fourier Transform (FT)Fourier Transform Theorems (continue – 2)

( ){ } ( ){ } ( )yxgyxgFTFTyxgFTFT ,,, 11 == −−

7. Fourier Integral Theorem

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Image AnalysisSOLO

Two Dimensional Fourier Transform (FT)Fourier Transform for a Circular Symmetric Optical Aperture

To exploit the circular symmetry of g (g (r,θ) = g (r) ) let make the following transformation

( )

( ) φρφ

φρρ

θθ

θ

sin/tan

cos

sin/tan

cos

1

22

1

22

==

=+=

==

=+=

−

−

yxy

xyx

fff

fff

ryxy

rxyxr

{ } ( ) ( )[ ] ( ) ( )[ ]( ) ( )

( ) ( )[ ]∫ ∫

∫ ∫∫∫

−−=

+−=+−=

=

=

Σ

a

o

rgrg

a

o

drdrydxd

yx

drjrdrrg

drjrgrdrydxdyfxfjyxggFT

πθ

πθ

θφθρπ

θθφθφρπθπ

2

0

,

2

0

cos2exp

sinsincoscos2exp,2exp,

Use Bessel Function Identity ( ) ( )[ ]∫ −−=π

θφθ2

00 cosexp dajaJ

( ) ( ){ } ( ) ( )∫==a

ordrrgrJrgFTG ρπρ 2: 00

to obtain

J0 is a Bessel Function of the first kind, order zero.

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Image AnalysisSOLO


For a Circular Pupil of radius a we have

( )

>≤

=ar

arrg

0

1

Use Bessel Function Identity

J1 is a Bessel Function of the first kind, order one.

( ) ( ){ } ( )∫==a

ordrrJrgFTG ρπρ 2: 00

( ) ( )xJxdJx

o 10 =∫ ςςς

( ) ( ){ } ( ) ( ) ( )

( ) ( ) ( )ρπρπ

ςςςρπ

ρπρπρπρπ

ρ

ρπaJ

adJ

rdrrJrgFTG

a

o

a

o

222

1

2222

1:

1

2

02

020

==

==

∫

∫

Bessel Functions of the first kind

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SOLO

E. Hecht, “Optics”

Circular Aperture

Image Analysis


( ) ( ){ } ( )( )ρπ

ρπρa

aJargFTG

2

2: 12

0 ==


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Resolution of Optical Systems Airy Rings

In 1835, Sir George Biddell Airy, developed the formula for diffraction pattern, of an image of a point source in an aberration-free optical system, using the wave theory.

E. Hecht, “Optics”

Optics

117

Resolution – Diffraction Limit


Image AnalysisSOLO Optics

118

Diffraction limit to resolution of two close point-object images: best resolution is possible when the two are of near equal, optimum intensity. As the two PSF merge closer, the intensity deep between them rapidly diminishes. At the center separation of half the Airy disc diameter - 1.22λ/D radians (138/D in arc seconds, for λ=0.55μ and the aperture diameter D in mm), known as Rayleigh limit - the deep is at nearly 3/4 of the peak intensity. Reducing the separation to λ/D (113.4/D in arc seconds for D in mm, or 4.466/D for D in inches, both for λ=0.55μ) brings the intensity deep only ~4% bellow the peak. This is the conventional diffraction resolution limit, nearly identical to the empirical double star resolution limit, known as Dawes' limit. With even slight further reduction in the separation, the contrast deep disappears, and the two spurious discs merge together. The separation at which the intensity flattens at the top is called Sparrow's limit, given by 107/D for D in mm, and 4.2/D for D in inches (λ=0.55μ).

Image Analysis

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Optics

119


• The Point Spread Function (PSF) is the Fourier Transform (FT) of the pupil function

• The Modulation Transfer Function (MTF) is the amplitude component of the FT of the PSF

• The Phase Transfer Function (PTF) is the phase component of the FT of the PSF

• The Optical Transfer Function (OTF) composed of MTF and PTF can also be computed as the autocorrelation of the pupil function.

( ) ( ) ( )

=− yxWi

yx eyxPFTffPSF,

2

,, λπ

( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, =

( ) ( ){ }[ ]iiyx yxPSFFTPhaseffPTF ,, =

( ) ( ) ( )[ ]yxyxyx ffPTFiffMTFffOTF ,exp,, =

Image AnalysisSOLO

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• The Point Spread Function (PSF) is the Fourier Transform (FT) of the pupil function

( ) ( ) ( )

=− yxWi

yx eyxPFTffPSF,

2

,, λπ

Image AnalysisSOLO

The Point Spread Function, or PSF, is the image that an optical system forms of a point source.

The point source is the most fundamental object, and forms the basis for any complex object.

The PSF is analogous to the Impulse Response Function in electronics.

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The Point Spread Function, or PSF, is the image that an Optical System forms of a Point Source. The PSF is the most fundamental object, and forms the basis for any complex object. PSF is the analogous to Impulse Response Function in electronics.

( )[ ] 2, yxPFTPSF =

The PSF for a perfect optical system (with no aberration) is the Airy disc, which is the Fraunhofer diffraction pattern for a circular pupil.

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As the pupil size gets larger, the Airy disc gets smaller.

Image AnalysisSOLO


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Convolution

( ) ( ) ( )yxIyxOyxPSF ,,, =⊗( )[ ] ( )[ ]{ } ( )yxIyxOFTyxPSFFTFT ,,,1 =•−


Image AnalysisSOLO

Convolution


Optical Aberration

124 Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley



125


The Modulation Transfer Function (MTF) indicates the ability of an Optical Systemto reproduce various levels of details (spatial frequencies) from the object to image. Its units are the ratio of image contrast over the object contrast as a function of spatial frequency.

λ⋅=

3.57

afcutoff


Image AnalysisSOLO


MTF as a function of pupil size (diameter)

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Image AnalysisSOLO

http://voi.opt.uh.edu/voi/WavefrontCongress/2005/presentations/1-RoordaOpticsReview.pdf



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Phase Transfer Function (PTF)

• PTF contains information about asymmetry in PSF • PTF contains information about contrast reversals (spurious resolution)


Image AnalysisSOLO


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129


( ) ( )

=− yxWi

eFTyxPSF,

2

, λπ

( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, = ( ) ( ){ }[ ]iiyx yxPSFFTPhaseffPTF ,, =


Image AnalysisSOLO


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( ) ( )

=− yxWi

eFTyxPSF,

2

, λπ

( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, = ( ) ( ){ }[ ]iiyx yxPSFFTPhaseffPTF ,, =


SOLO

Real Optical System

Optical Aberration

131( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, =

( ) ( )

=− yxWi

eFTyxPSF,

2

, λπ

Point Spread Function


132

( ) ( )

=− yxWi

eFTyxPSF,

2

, λπ




133


( ) ( )

=− yxWi

eFTyxPSF,

2

, λπ


SOLO


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Strehl Ratio

Strehl, Karl 1895, Aplanatische und fehlerhafte Abbildung im Fernrohr, Zeitschrift für Instrumentenkunde 15 (Oct.), 362-370.

Dr. Karl Strehl 1864 -1940

One of the most frequently used optical terms in both, professional and amateur circles is the Strehl ratio. It is the simplest meaningful way of expressing the effect of wavefront aberrations on image quality. By definition, Strehl ratio - introduced by Dr. Karl Strehl at the end of 19th century - is the ratio of peak diffraction intensities of an aberrated vs. perfect wavefront. The ratio indicates image quality in presence of wavefront aberrations; often times, it is used to define the maximum acceptable level of wavefront aberration for general observing - so-called diffraction-limited level - conventionally set at 0.80 Strehl.


135

The Strehl ratio is the ratio of the irradiance at the center of the reference sphere to the irradiance in the absence of aberration.

Irradiance is the square of the complex field amplitude u

0E

EStrehl =

2uE =

∫∫= dxdyyxWjUu )),(2exp(0 π


Strehl Ratio

Expectation Notation∫∫

∫∫==dxdy

dxdyyxuuu

),(


136

Derivation of Strehl Approximation

( ) 2

0

21 WE

EStrehl πσ−==

),(20

yxWjeUu π=

( ) 220 ),(2

2

1),(21 yxWyxWjUu ππ −+=

( ) 20

200 ),(2

2

1),(2 yxWUyxWUjUu ππ −+=

series expansion

( ) ( ) 2

022

02

0 ),(2),(2 yxWEyxWEEE ππ +−=

multiply by complex conjugate

2222 ),(),(),(),( yxWyxWyxWyxWW −=−=σ

wavefront variance:



137

( ) 2

0

21 WE

EStrehl πσ−==

222 ),(),( yxWyxWWWW −=−=σ

where σW is the wavefront variance:

( ) 22 WeStrehl πσ−=Another approximation for the Strehl ratio is

Strehl Approximation

Diffraction Limit

8.0≥Strehl

A system is diffraction-limited when the Strehl ratio is greater than or equal to 0.8

Maréchal’s criterion:

This implies that the rms wavefront error is less than λ /13.3 or that the total wavefront error is less than about λ /4.



138


Strehl Ratio

dl

eye

H

HRatioStrehl =



139


Strehl Ratio

( )∑= 2m

nCrmswhen rms is small

( ) 2

22

1 rmsStrehl

−≈

λπ


140

FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled.

(a) the effect of 1/4 and 1/2 wave P-V wavefront error of defocus on the PSF intensity distribution (left) and image contrast (right). Doubling the error nearly halves the peak diffraction intensity, but the average contrast loss nearly triples (evident from the peak PSF intensity).

(b) 1/4 and 1/2 wave P-V of spherical aberration. While the peak PSF intensity change is nearly identical to that of defocus, wider energy spread away from the disc results in more of an effect at mid- to high-frequency range. Central disc at 1/2 wave P-V becomes larger, and less well defined. The 1/2 wave curve indicates ~20% lower actual cutoff frequency in field conditions.

http://www.telescope-optics.net/Image Analysis


141


(c) 0.42 and 0.84 wave P-V wavefront error of coma. Both, intensity distribution (PSF) and contrast transfer change with the orientation angle, due to the asymmetric character of aberration. The worst effect is along the axis of aberration (red), or length-wise with respect to the blur (0 and π orientation angle), and the least is in the orientation perpendicular to it (green).

(d) 0.37 and 0.74 wave P-V of astigmatism. Due to the tighter energy spread, there is less of a contrast loss with larger, but more with small details, compared to previous wavefront errors. Contrast is best along the axis of aberration (red), falling to the minimum (green) at every 45° (π/4), and raising back to its peak at every 90°. The PSF is deceiving here: since it is for a linear angular orientation, the energy spread is lowest for the contrast minima.

http://www.telescope-optics.net/


142

(e) Turned down edge effect on the PSF and MTF. The P-V errors for 95% zone are 2.5 and 5 waves as needed for the initial 0.80 Strehl (the RMS is similarly out of proportion). Lost energy is more evenly spread out, and the central disc becomes enlarged. Odd but expected TE property - due to the relatively small area of the wavefront affected - is that further increase beyond 0.80 Strehl error level does almost no additional damage.

f) The effect of ~1/14 and ~1/7 wave RMS wavefront error of roughness, resulting in the peak intensity and contrast drop similar to those with other aberrations. Due to the random nature of the aberration, its nominal P-V wavefront error can vary significantly for a given RMS error and image quality level. Shown is the medium-scale roughness ("primary ripple" or "dog biscuit", in amateur mirror makers' jargon) effect.

(g) 0.37 and 0.74 wave P-V of wavefront error caused by pinching having the typical 3-sided symmetry (trefoil). The aberration is radially asymmetric, with the degree of pattern deformation varying between the maxima (red MTF line, for the pupil angle θ=0, 2π/3, 4π/3), and minima (green line, for θ=π/3, π, 5π/3); (the blue line is for a perfect aperture). Other forms do occur, with or without some form of symmetry.


143

h) 0.7 and 1.4 wave P-V wavefront error caused by tube currents starting at the upper 30% of the tube radius. The energy spreads mainly in the orientation of wavefront deformation (red PSF line, to the left). Similarly to the TE,further increase in the nominal error beyond a certain level has relatively small effect Contrast and resolution for the orthogonal to it pattern orientation are as good as perfect (green MTF line).

(i) Near-average PSF/MTF effect of ~1/14 and ~1/7 wave RMS wavefront error of atmospheric turbulence. The atmosphere caused error fluctuates constantly, and so do image contrast and resolution level. Larger seeing errors (1/7 wave RMS is rather common with medium-to-large apertures) result in a drop of contrast in the mid- and high-frequency range to near-zero level.


http://www.telescope-optics.net/

Image AnalysisSOLO


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FIGURE 34: Pickering's seeing scale uses 10 levels to categorize seeing quality, with the level 1 being the worst and level 10 near-perfect. Its seeing description corresponding to the numerical seeing levels are: 1-2 "very poor", 3 "poor to very poor", 4 "poor", 5 "fair", 6 "fair to good" 7 "good", 8 "good to excellent", 9 "excellent" and 10 "perfect". Diffraction-limited seeing error level (~0.8 Strehl) is between 8 and 9.

Pickering 1 Pickering 2 Pickering 3 Pickering 4 Pickering 5

Pickering 6 Pickering 7 Pickering 9 Pickering 10Pickering 8

William H. Pickering (1858-1938)

SOLOReturn to Table of Content

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FIGURE: Illustration of a point source (stellar) image degradation caused by atmospheric turbulence. The left column shows best possible average seeing error in 2 arc seconds seeing (ro~70mm @ 550nm) for four aperture sizes. The errors are generated according to Eq.53-54, with the 2" aperture error having only the roughness component (Eq.54), and larger apertures having the tilt component added at a rate of 20% for every next level of the aperture size, as a rough approximation of its increasing contribution to the total error (the way it is handled by the human eye is pretty much uncharted territory). The two columns to the right show one possible range of error fluctuation, between half and double the average error. The best possible average RMS seeing error is approximately 0.05, 0.1, 0.2 and 0.4 wave, from top to bottom (the effect would be identical if the aperture was kept constant, and ro reduced). The smallest aperture is nearly unaffected most of the time. The 4" is already mainly bellow "diffraction-limited", while the 8" has very little chance of ever reaching it, even for brief periods of time. The 16" is, evidently, affected the most. The D/ro ratio for its x2 error level is over 10, resulting in clearly developed speckle structure. Note that the magnification shown is over 1000x per inch of aperture, or roughly 10 to 50 times more than practical limits for 2"-16" aperture range, respectively. At given nominal magnification, actual (apparent) blur size would be smaller inversely to the aperture size. It would bring the x2 blur in the 16" close to that in 2" aperture (but it is obvious how a further deterioration in seeing quality would affect the 16" more).

Eugène Michel Antoniadi (1870 –, 1944)

The scale, invented by Eugène Antoniadi, a Greek astronomer, is on a 5 point system, with one being the best seeing conditions and 5 being worst. The actual definitions are as follows:

I. Perfect seeing, without a quiver.II. Slight quivering of the image with moments of calm lasting several seconds.III. Moderate seeing with larger air tremors that blur the image.IV. Poor seeing, constant troublesome undulations of the image.V. Very bad seeing, hardly stable enough to allow a rough sketch to be made.

Image Degradation Caused by Atmospheric Turbulence

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Return to Table of ContentOptical Aberration

149

SOLOZernike’s Polynomials

In 1934 Frits Zernike introduces a complete set of orthonormal polynomialsto describe aberration of any complexity.

( ) ( ) ( ) ( )θρθρθρ mN

mn

mn

mnN YRaZZ == ,,

,2,12

813min =

++−= N

NIntegern

( ) ( ) { }

−=

−++−=

oddN

evenNmsign

NnnIntegernm

1

1

4

212min2

Each polynomial of the Zernike set is a product of three terms.

where

( )

≠+

=+=

012

01

mifn

mifna m

n

( ) ( ) ( )( )[ ] ( )[ ]

( )sn

mn

s

sm

n smnsmns

snR 2

2/

0 !2/!2/!

!1 −−

=∑ −−−+

−−= ρρ

( )

≠≠=

=oddisNandmif

evenisNandmif

mif

Y mN

0sin

0cos

01

θθθ

radial index

meridional index

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Properties of Zernike’s Polynomials.

( ) ( )∑ ∑=n m

mn

mn ZCW θρθρ ,,

W (ρ,θ) – Waveform Aberration

Cnm (ρ,θ) – Aberration coefficient (weight)

Znm (ρ,θ) – Zernike basis function (mode)

( ){ } ( ) mallnallforZZMean mn

mn 00,, >== θρθρ1

( ){ } mnallforZVariance mn ,1, =θρ2

3 Zernike’s Polynomials are mutually orthogonal, meaning that they are independentof each other mathematically. The practical advantage of the orthogonality is that we can determine the amount of defocus, or astimagtism, or any other Zernike mode occurring in an aberration function without having to worry about the presence of the other modes.

4 The aberration coefficients of a Zernike expansion are analogous to the Fouriercoefficients of a Fourier expansion.

( ){ } ( ) ( )[ ] ( )∑ ∑∑ ∑ =

−=

n m

mn

n m

mn

mn

mn CZZCMeanWVariance

22

,,, θρθρθρ

( ) ( ) ( ) '

1

0' 12

1nn

m

n

m

n ndRR δρρρρ

+=∫ ( ) '0

2

0

1'coscos mmmdmm δδπθθθπ

+=∫

Optical Aberration

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Astigmatism

{ }4,4,,2 22 −− ayax

Coma1

{ }3,5,,2 2 −+ axaxρComa2

{ }4,4,,2 2 −+ ayaxρ

Spherical &Defocus

( ) { }3,5,,3.12 22 −+ aaρρ

36 Zernikes

Geounyoung Yoon, “Aberration Theory”

Optical Aberration

152

Surface of Revolution StereogramZernike Polynomials

http://www.optics.arizona.edu/jcwyant/

Play it


153


( ) ( )

mastigmatis

defocus

tilty

tiltx

piston

YRamnN mN

mn

mn

452sin6225

1123024

sin4113

cos4112

111001

2

2

θρ

ρ

θρ

θρ

θρ

−

−

−−

−

sphericalbalanced

shamrock

shamrock

comaxbalanced

comaybalanced

mastigmatis

)(116650411

3cos83310

3sin8339

)(cos238138

)(sin238137

902cos6226

24

3

3

2

2

2

+−

−

−−

−−−

ρρ

θρ

θρ

θρρ

θρρ

θρ

clover

clover

θρ

θρ

θρρ

θρρ

4sin104415

4cos104414

2sin34102413

2cos34102412

4

4

24

24

−

−−

−

Optical Aberration

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Zernike’s Decomposition


156

SOLOZernike’s Decomposition

Geounyoung Yoon, “Aberration Theory”

Optical Aberration

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Zernike’s Polynomials SOLO Optical Aberration

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Zernike’s Polynomials SOLO Optical Aberration

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Zernike’s Polynomials SOLO


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168

Tilt (n=1, m=1) ( )0cos θθρ +

The wavefront: Contour plot and 3D

The spot diagram in the focal plane


169

Defocus (n=2, m=0) 2ρThe wavefront: Contour plot and 3D

The spot diagram in the focal plane

The hole in the center of the figures is inthe optical element.


170

Coma (n=3, m=1)

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SOLO

Aberrometers

A number of technical and practical parameters that may be useful in choosing an aberrometer for daily clinical practice.

The main focus is on wavefront measurements, rather than on their possible application in refractive surgery. The aberrometers under study are the following:

1. Visual Function Analyzer (VFA; Tracey): based onray tracing; can be used with the EyeSys Vista corneal topographer.

2. OPD-scan (ARK 10000; Nidek): based on automatic retinoscopy; provides integrated corneal topography and wavefront measurement in 1 device.

3. Zywave (Bausch & Lomb): a Hartmann-Shack system that can be combined with the Orbscan corneal topography system.

4. WASCA (Carl Zeiss Meditec): a high-resolution Hartmann-Shack system.

5. MultiSpot 250-AD Hartmann-Shack sensor: a custom-made Hartmann-Shack system, engineered by the Laboratory of Adaptive Optics at Moscow State University, that includes an adaptive mirror to compensate for accommodation

6. Allegretto Wave Analyzer (WaveLight): an objective Tscherning device

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SOLOAberrometers

Figure 1. The principles of the wavefront sensors: Top: Skew ray. Center Left: Ray tracing. Center Right: Hartmann-Shack. Bottom Left: Automatic retinoscope. Bottom Right: Tscherning.

Single-head arrows indicate direction of movement for beams.

Figure 2. Reproductions of the fixation targets for the patient: A: VFA.B: OPD-scan. C: Zywave. D: WASCA.E: MultiSpot. F: Allegretto.

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SOLOAberrometers

Johannes Hartmann1865 - 1936

In 1920, an astrophysicist named Johannes Hartmann deviseda method of measuring the ray aberration of mirrors and lenses.He wanted to isolate rays of light so that they could be traced and anyimperfection in the mirror could be seen. The Harman Test consist on using metal disk in which regulary spaced holes had been drilled.

The disk or screen was then placed over the mirror that was to be testedand a photographic plate was placed near the focus of the mirror. Whenexposed to light, a perfect mirror will produce an image of regularyspaced dots. If the mirror does not produce regularly spaced dots, the irregularities, or aberrations, of the mirror can be determined.

Figure 1. Optical schematic for an early Hartmann test.

Schematic from Santa Barbara Instruments Group (SBIG) software for analysis of Hartmann tests.

1920Optical Aberration

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Optical schematic for first Shack-Hartmann sensor.

Around 1971 , Dr. Roland Shack and Dr. Ben Platt advanced the concept replacingthe screen with a sensor based on an array of tiny lenselets. Today, this sensor is known as the Hartmann - Shack sensor. Hartmann – Shack sensors are used in a variety of industries: military, astronomy, ophthalmogy.

Schematic showing Shack-Hartmann CCD output.

Schematic of Shack-Hartmann data analysis process.

Hartmann - Shack Aberrometer

Roland Shack

1971Optical Aberration

175

SOLO

Lenslet array made by Heptagon for ESO. The array has 40 x 40 lenslets, each 500 μm (0.5 mm) insize.

Part of lenslet array made by WaveFront Sciences. Each lens is 144 μm in diameter.


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SOLO


Recent image from Adaptive Optics Associates (AOA) shows the optical set-up used to test the first Shack-Hartmann sensor.

Upper left) Array of images formed by the lens array from a single wavefront.

Upper right) Graphical representation of the wavefront tilt vectors.

Lower left) Zernike polynomial terms fit to the measured data.

Lower right) 3-D plot of the measured wavefront.


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SOLO

References Optical Aberration

A. Yariv, P. Yeh, “Optical Waves in Crystals”, John Wiley & Sons, 1984

M. Born, E. Wolf, “Principles of Optics”, Pergamon Press,6th Ed., 1980

E. Hecht, A. Zajac, “Optics”, Addison-Wesley, 1979, Ch.8

C.C. Davis, “Lasers and Electro-Optics”, Cambridge University Press, 1996

M.V.Klein, T.E. Furtak, “Optics”, 2nd Ed., John Wiley & Sons, 1986

V.N. Mahajan, “Aberration Theory Made Simple”, SPIE, Tutorial Texts, Vol. TT6,1991

V.N. Mahajan, “Optical Imaging and Aberrations”, Part I, Ray Geometrical Optics, SPIE, 1998V.N. Mahajan, “Optical Imaging and Aberrations”, Part II, Wave Diffraction Optics, SPIE, 2001

http://grus.berkeley.edu/~jrg/Aberrations

http://grus.berkeley.edu/~jrg/Aberrations/BasicAberrationsandOpticalTesting.pdf

Jurgen R. Meyer-Arendt, “Introduction to Classic and Modern Optics”, Prentice Hall, 1989

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References Optical Aberration


Larry N. Thibos:”Representation of Wavefront Aberration”, http://research.opt.indiana.edu/Library/wavefronts/index.htm

Geounyoung Yoon, “Aberration Theory”http://www.imagine-optic.com/downloads/imagine-optic_yoon_article_optical-wavefront-aberrations-theory.pdf

J.C. Wyant, Optical Science Center, University of Arizona,


http://voi.opt.uh.edu/voi/WavefrontCongress/2005/presentations/1-RoordaOpticsReview.pdf

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References

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1. Waldman, G., Wootton, J., “Electro-Optical Systems Performance Modeling”, Artech House, Boston, London, 1993

2. Wolfe, W.L., Zissis, G.J., “The Infrared Handbook”, IRIA Center, Environmental Research Institute of Michigan, Office of Naval Research, 1978

3. “The Infrared & Electro-Optical Systems Handbook”, Vol. 1-7

4. Spiro, I.J., Schlessinger, M., “The Infrared Technology Fundamentals”, Marcel Dekker, Inc., 1989

5. Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley

6. Charles S. Williams, Orville A. Becklund, “Introduction to Optical Transfer Function”, SPIE Press, 2002

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References

[1] M. Born, E. Wolf, “Principle of Optics – Electromagnetic Theory of Propagation, Interference and Diffraction of Light”, 6th Ed., Pergamon Press, 1980,

[2] C.C. Davis, “Laser and Electro-Optics”, Cambridge University Press, 1996,

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[3] E.Hecht, A. Zajac, “Optics ”, 3th Ed., Addison Wesley Publishing Company, 1997,

[4] M.V. Klein, T.E. Furtak, “Optics ”, 2nd Ed., John Wiley & Sons, 1986


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TechnionIsraeli Institute of Technology

1964 – 1968 BSc EE1968 – 1971 MSc EE

Israeli Air Force1970 – 1974

RAFAELIsraeli Armament Development Authority

1974 –2013

Stanford University1983 – 1986 PhD AA


( ) ( )22, yxAyxW d +⋅=


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( ) ( ) ''''cos'';, 223 xyxhCrhChrW CoCoCo +== θθ


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186 Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley

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187http://research.opt.indiana.edu/Library/HVO/Handbook_NF.html#3.2RayAberration

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188http://www.optics.arizona.edu/jcwyant/

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193http://www.optics.arizona.edu/jcwyant/

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Camera Lens 1840

In 1840 Joseph Max Petzval (1807 – 1891) made the first portrait camera lens.

http://www.thespectroscopynet.com/Educational/Masson.htm

A Hungarian optician, Petzval was professor of Mathematics at the University of Vienna. He played a leading part in early photography by devising a portrait lens with an aperture of approximately f3.6 - gathering sixteen times more light than lenses currently in use at the time. which brought exposure times down to less than a minute, therefore began to pave the way for portraiture. This lens, which was made by his compatriot Peter Friedrich

Voigtlander in 1841, was popularly used well into this century. Sadly Petzval did not profit from this invention, unlike Voigtlander, with whom he had fallen out because he felt he had been cheated. Petzval died an embittered and impoverished man; Voigtlander old and rich two years later, having seen his firm expand from a small optical shop to a major industrial enterprise thanks to the success

of the Petzval lens.

© Robert Leggat, 1998. http://www.rleggat.com/photohistory/history/petzval.htm

Petzval portrait lens

Joseph Max Petzval1807 - 1891

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FIGURE: Aberrations of a concave mirror:

(A) ray spot plot for a 6" f/8.15 Newtonian with spherical and paraboloidal primary (SPEC'S).Diffraction images, which include 0.2D central obstruction effect are reduced in size by a factor of 3). At 0.28° off-axis, coma of the paraboloid has identical RMS wavefront error to the center-field spherical aberration of the sphere - 0.075 wave, for the 0.80 Strehl. The combined error of the sphere at 0.28° degrees off-axis is 0.12 wave RMS (0.56 Strehl).

B) coma in a paraboloid as it changes with the focal ratio number F. While the linear blur size varies inversely to F squared, the Airy disc changes in proportion to F, resulting in the wavefront error for given field height to vary inversely to the cube of F. Hence, quality field size of the paraboloidal mirror changes with the inverse square of its F number angularly, and with the inverse cube of it linearly (diffraction images are for 3mm off-axis; view from ~12" corresponds to ~40x/inch magnification).

(C) Geometric blurs of paraboloidal mirror at f/4.5 and f/9, typical conventional eyepiece aberration, introducing significant astigmatism and some spherical aberration with the f/4.5 mirror, and the combined mirror/eyepiece aberrated blur.

http://www.telescope-optics.net

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