Upload
solo-hermelin
View
795
Download
21
Tags:
Embed Size (px)
Citation preview
1
Optical Aberrations
SOLO HERMELIN
Updated: 16.01.10 4.01.15
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
3
Table of Content (continue – 1)
SOLO Optical Aberration
Real Imaging Systems – Aberrations
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)
SOLO Optical Aberration
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
5
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
Image PlaneNon-ideal Optics
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
6
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
7
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.
8
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.&
9
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
10
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
11
SOLO
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
13
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'
Wavefrontfrom P Wavefront
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'
Wavefrontfrom P Wavefront
to P'
r
'r
( )rP ( )'' rP
Vsn '' ∇=
15
SOLO
converging beam=
spherical wavefront
parallel beam=
plane wavefront
Image PlaneIdeal Optics
P'
Optical Aberration
converging beam=
spherical wavefront
Image PlaneIdeal Optics
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
Ideal Optical System
16
SOLO
ideal wavefrontparallel beam=
plane wavefront
Image PlaneNon-ideal Optics
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
SOLO Optical Aberration
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
SOLO Optical Aberration
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
Definitions (continue – 2)
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
Definitions (continue – 3)
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
Real Imaging Systems – Aberrations
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
SOLO
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
Real Imaging Systems
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
Aberrations (continue – 2)
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
SOLO
Real Imaging Systems
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
Aberrations (continue – 4)
29
SOLO
Real Imaging Systems
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
Aberrations (continue – 5)
30
SOLO
Real Imaging Systems
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
Aberrations (continue – 6)
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
Real Imaging Systems
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
Real Imaging Systems
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
Real Imaging Systems
Defocus Aberration (continue – 2)
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
Real Imaging Systems
Defocus Aberration (continue – 3)
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
Defocus Aberration (continue – 4)
SOLO
36
SOLO
Real Imaging Systems
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
Real Imaging Systems
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
Real Imaging Systems
Optical Aberration
40
SOLO
Real Imaging Systems
Optical Aberration
41
SOLO
Real Imaging Systems
Optical Aberration
42
SOLO
Real Imaging Systems
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
Real Imaging Systems
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
Real Imaging Systems
Seidel Aberrations (continue – 2)
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:
On Axis Point Object
Optical Aberration
n
'n
CBR
0P '0P
( )θ,rQ
0V
r
z
( ) s− 's
Chief Ray
General Ray
ASEnPExP
45
SOLO
Real Imaging Systems
Seidel Aberrations (continue – 3)
+
−+
−+
−+−≈ 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=
−−
−−=
On Axis Point Object
Optical Aberration
n
'n
CBR
0P '0P
( )θ,rQ
0V
r
z
( ) s− 's
Chief Ray
General Ray
ASEnPExP
46
SOLO
Real Imaging Systems
Seidel Aberrations (continue – 4)
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
Real Imaging Systems
Seidel Aberrations (continue – 5)
Off-Axis Point Object
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
Real Imaging Systems
Seidel Aberrations (continue – 6)
Off-Axis Point Object
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
Real Imaging Systems
Seidel Aberrations (continue – 7)
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
Real Imaging Systems
Seidel Aberrations (continue – 8)
( ) θθθθ cos''cos'cos'';, 32222234 rhCrhCrhCrhCrChrW DiFCAsCoSp ++++=
Optical Aberration
51
SOLO
Real Imaging Systems – Aberrations
Optical Aberration
Seidel Aberrations (continue – 9)
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
Real Imaging Systems
Seidel Aberrations (continue – 10)
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
Real Imaging Systems
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
Seidel Aberrations (continue – 11)
54
SOLO
Real Imaging Systems
Seidel Aberrations (continue – 12)
( )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
Real Imaging Systems
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
Real Imaging Systems
1. Spherical Aberrations (continue – 1)
Optical Aberration
Ray Errors
0
0.5
1
Optical Distance Errors Wavefront Error
57
SOLO
Real Imaging Systems
1. Spherical Aberrations (continue – 2)
Optical Aberration
58
SOLO
Real Imaging Systems
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
Real Imaging Systems
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
( ) θθθθ cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++= The general Wave Aberration Function is:
60
SOLO
Real Imaging Systems
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
Real Imaging Systems
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
Real Imaging Systems
Graphical Explanation of Coma Blur (continue – 1)
2. Coma (continue – 3)
Optical Aberration
64
SOLO
Real Imaging Systems
Optical Aberration
2. Coma (continue-5)
65
http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf
( ) ( ) ''''cos'';, 22cos'
sin'
3 xyxhCrhChrW Co
rx
ryCoCo +===
=
θ
θθθ
Wave Aberration: Coma
Optical Aberration
2. Coma (continue-6)
SOLO
66
SOLO
Real Imaging Systems – Aberrations
3.&4. Astigmatism and Curvature of Field
Optical Aberration
( )( ) 2222
cos'
sin'
2222222
'''
'cos'';,
yCxCChb
rhbCrhbChrW
FCFCAs
rx
ry
FCAsAs
++=
+==
=
θ
θ
θθ
( ) θθθθ cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++= The general Wave Aberration Function is:
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
Real Imaging Systems – Aberrations (continue – 3)
3.&4. Astigmatism and Curvature of Field
Sagittalplane
Objectpoint
OpticalSystem
Chiefray
Opticalaxis
'L 'L∆
( )( )θρθρ sin,cos
,
=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
Real Imaging Systems – Aberrations (continue – 3)
3.&4. Astigmatism and Curvature of Field
Optical Aberration
Meridionalplane
Sagittalplane
Objectpoint
OpticalSystem
Chiefray
SFTF
Opticalaxis
'L 'L∆
( )( )θρθρ sin,cos
,
=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
Real Imaging Systems – Aberrations (continue – 3)
3.&4. Astigmatism and Curvature of Field
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
Real Imaging Systems – Aberrations (continue – 3)
3.&4. Astigmatism and Curvature of Field
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
Real Imaging Systems
3. Astigmatism
Joseph Max Petzval1807 - 1891
72
SOLO
Real Imaging Systems
3. Astigmatism
73
SOLO
Real Imaging Systems
3. Astigmatism
74
SOLO
Real Imaging Systems
3. Astigmatism
r = radius
q = meridian
Optical Aberration
75
SOLO
Real Imaging Systems – Aberrations
4. Field Curvature
( ) 222 '';, rhbChrW FCFC =θ
Optical Aberration
76
4. Field Curvature
SOLO Optical Aberration
77
SOLO
Real Imaging Systems – Aberrations
5. Distortion
( )xhbC
rhbChrW
Di
DiDi
33
33
'
cos'';,
== θθ
Optical Aberration
( ) θθθθ cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++= The general Wave Aberration Function is:
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
Real Imaging Systems – Aberrations (continue – 5)
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
Real Imaging Systems – Aberrations (continue – 5)
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
Real Imaging Systems – Aberrations (continue – 5)
5. Distortion
( ) θθ cos'';, 33 rhbChrW DiDi =
Optical Aberration
( ) θθθθ cos''cos'cos'';, 33222222234 rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++= The general Wave Aberration Function is:
Consider only the DistorsionWave Aberration Function
81
http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf
( )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
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
Thin Lens Spherical Aberrations (continue – 2)
2R
1R
1C
IO
2C
Paraxialfocal plane2n
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:
Thin Lens Spherical Aberrations (continue – 4)
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
Paraxialfocal plane2n
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
Paraxialfocal plane2n
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
SOLO Optical Aberration
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.
Thin Lens Aberration
96
SOLO Optical Aberration
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
Thin Lens Aberration
3. Thin Lens Astigmatism
97
SOLO Optical Aberration
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
Thin Lens Aberration
3. Thin Lens Astigmatism
98
SOLO Optical Aberration
Thin Lens Aberration
3. Thin Lens Astigmatism
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.
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
SOLO Optics Chromatic Aberration
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
SOLO Optics Chromatic Aberration
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
SOLO Optics Chromatic Aberration
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
SOLO Optics Chromatic Aberration
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
SOLO Optics Chromatic Aberration
( ) ( )( ) ( )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
Return to Table of Content
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:
Optical Aberration
110
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
111
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
Optical Aberration
112
Image Analysis
SOLO
Two Dimensional Fourier Transform (FT)Fourier Transform Theorems (continue – 2)
( ){ } ( ){ } ( )yxgyxgFTFTyxgFTFT ,,, 11 == −−
7. Fourier Integral Theorem
Optical Aberration
113
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.
Optical Aberration
114
Image AnalysisSOLO
Two Dimensional Fourier Transform (FT)Fourier Transform for a Circular Symmetric Optical Aperture
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
Optical Aberration
115
SOLO
E. Hecht, “Optics”
Circular Aperture
Image Analysis
Two Dimensional Fourier Transform (FT)Fourier Transform for a Circular Symmetric Optical Aperture
( ) ( ){ } ( )( )ρπ
ρπρa
aJargFTG
2
2: 12
0 ==
Return to Table of Content
Optical Aberration
116
SOLO
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
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
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
SOLO
Return to Table of Content
Optics
119
Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function
• 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
Optical Aberration
120
• 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.
Optical Aberration
Point Spread Function (PSF)
121
Point Spread Function (PSF)
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.
Image AnalysisSOLO Optical Aberration
122
Point Spread Function (PSF)
As the pupil size gets larger, the Airy disc gets smaller.
Image AnalysisSOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
Return to Table of Content
123
Convolution
( ) ( ) ( )yxIyxOyxPSF ,,, =⊗( )[ ] ( )[ ]{ } ( )yxIyxOFTyxPSFFTFT ,,,1 =•−
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Image AnalysisSOLO
Convolution
Return to Table of Content
Optical Aberration
124 Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Modulation Transfer Function (MTF)
Image AnalysisSOLO Optical Aberration
125
Modulation Transfer Function (MTF)
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
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Image AnalysisSOLO
( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, =
MTF as a function of pupil size (diameter)
Optical Aberration
126
Modulation Transfer Function (MTF)
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Image AnalysisSOLO
http://voi.opt.uh.edu/voi/WavefrontCongress/2005/presentations/1-RoordaOpticsReview.pdf
( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, =
Return to Table of Content
Optical Aberration
127
Phase Transfer Function (PTF)
• PTF contains information about asymmetry in PSF • PTF contains information about contrast reversals (spurious resolution)
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Image AnalysisSOLO
Return to Table of Content
Optical Aberration
128
SOLO Optical Aberration
129
Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function
( ) ( )
=− yxWi
eFTyxPSF,
2
, λπ
( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, = ( ) ( ){ }[ ]iiyx yxPSFFTPhaseffPTF ,, =
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Image AnalysisSOLO
Ideal Optical System
Optical Aberration
130
Relations between Wave Aberration, Point Spread Function and Modulation Transfer Function
( ) ( )
=− yxWi
eFTyxPSF,
2
, λπ
( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, = ( ) ( ){ }[ ]iiyx yxPSFFTPhaseffPTF ,, =
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
SOLO
Real Optical System
Optical Aberration
131( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, =
( ) ( )
=− yxWi
eFTyxPSF,
2
, λπ
Point Spread Function
SOLO Optical Aberration
132
( ) ( )
=− yxWi
eFTyxPSF,
2
, λπ
( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, =
Point Spread Function
SOLO Optical Aberration
133
( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, =
( ) ( )
=− yxWi
eFTyxPSF,
2
, λπ
Point Spread Function
SOLO
Return to Table of Content
Optical Aberration
134
Other Metrics that define Image Quality
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.
SOLO Optical Aberration
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
Other Metrics that define Image Quality
Strehl Ratio
Expectation Notation∫∫
∫∫==dxdy
dxdyyxuuu
),(
SOLO Optical Aberration
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:
Other Metrics that define Image Quality
SOLO Optical Aberration
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.
Other Metrics that define Image Quality
SOLO Optical Aberration
138
Other Metrics that define Image Quality
Strehl Ratio
dl
eye
H
HRatioStrehl =
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
SOLO Optical Aberration
139
Other Metrics that define Image Quality
Strehl Ratio
( )∑= 2m
nCrmswhen rms is small
( ) 2
22
1 rmsStrehl
−≈
λπ
SOLO Optical Aberration
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
SOLO Optical Aberration
141
FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled.
(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/
Image AnalysisSOLO Optical Aberration
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.
FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled.
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.
FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in ~0.80 Strehl (~1/14 wave RMS), and the error doubled.
http://www.telescope-optics.net/
Image AnalysisSOLO
Return to Table of Content
Optical Aberration
144
Other Metrics that define Image Quality
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
Optical Aberration
145
Other Metrics that define Image Quality
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
SOLO
Return to Table of Content
Optical Aberration
146
SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
147
SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
148
SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
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
Optical Aberration
150
SOLOZernike’s Polynomials
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
151
SOLOZernike’s Polynomials
In 1934 Frits Zernike introduces a complete set of orthonormal polynomialsto describe aberration of any complexity.
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
SOLO Optical Aberration
153
SOLOZernike’s Polynomials
( ) ( )
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
154
SOLOZernike’s Polynomials
In 1934 Frits Zernike introduces a complete set of orthonormal polynomialsto describe aberration of any complexity.
Optical Aberration
155
Zernike’s Decomposition
SOLO Optical Aberration
156
SOLOZernike’s Decomposition
Geounyoung Yoon, “Aberration Theory”
Optical Aberration
157
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Zernike’s Polynomials SOLO Optical Aberration
158
Zernike’s Polynomials SOLO Optical Aberration
159
Zernike’s Polynomials SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
160
Zernike’s Polynomials SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
161
Zernike’s Polynomials SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
162
Zernike’s Polynomials SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
163
Zernike’s Polynomials SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
164
Zernike’s Polynomials SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Return to Table of Content
Optical Aberration
165
SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
166
SOLO
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations affect vision”, University of Houston, TX, USA
Optical Aberration
167
SOLO Optical Aberration
168
Tilt (n=1, m=1) ( )0cos θθρ +
The wavefront: Contour plot and 3D
The spot diagram in the focal plane
SOLO Optical Aberration
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.
SOLO Optical Aberration
170
Coma (n=3, m=1)
SOLO
Return to Table of Content
Optical Aberration
171
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
Optical Aberration
172
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.
Optical Aberration
173
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
174
SOLO
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.
Hartmann - Shack Aberrometer
Optical Aberration
176
SOLO
Hartmann - Shack Aberrometer
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.
Return to Table of Content
Optical Aberration
177
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
Optical Aberration
178
SOLO
References Optical Aberration
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
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://www.optics.arizona.edu/jcwyant/
http://voi.opt.uh.edu/voi/WavefrontCongress/2005/presentations/1-RoordaOpticsReview.pdf
Optical Aberration
179
SOLO
References
OPTICS
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
180
SOLO
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,
OPTICS
181
SOLO
References
Optics
[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
Return to Table of Content
January 6, 2015 182
SOLO
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
183http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf
( ) ( )22, yxAyxW d +⋅=
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Optical Aberration
184http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf
( ) ( ) ''''cos'';, 223 xyxhCrhChrW CoCoCo +== θθ
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Optical Aberration
185
Optical Aberration
186 Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,University of California, Berkley
Optical Aberration
187http://research.opt.indiana.edu/Library/HVO/Handbook_NF.html#3.2RayAberration
Optical Aberration
188http://www.optics.arizona.edu/jcwyant/
Optical Aberration
189
http://www.optics.arizona.edu/jcwyant/
Optical Aberration
190
http://www.optics.arizona.edu/jcwyant/
Optical Aberration
191
http://www.optics.arizona.edu/jcwyant/
Optical Aberration
192
http://www.optics.arizona.edu/jcwyant/
Optical Aberration
193http://www.optics.arizona.edu/jcwyant/
Optical Aberration
194
SOLO
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
Optics History
195
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
Optical Aberration
196
SOLO Optics
197
SOLO Optics
198
SOLO Optics
199
SOLO Optics
200
SOLO Optics