Optical aberrations

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 Definition
The 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
Seidel Aberrations
Spherical Aberrations
Coma
Astigmatism and Curvature of Field
Astigmatism
Field Curvature
Distortion
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 Plane
Ideal Optics
ideal wavefront
parallel beam
=
plane wavefront
Image Plane
Non-ideal Optics
defocused wavefront
ideal wavefrontparallel beam
=
plane wavefront
Image Plane
Non-ideal Optics
aberrated beam
=
iregular wavefront
diverging beam
=
spherical wavefront
aberrated beam
=
irregular wavefront
Image Plane
Non-ideal Optics
ideal 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 in
straight lines.
. 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 the
surface normal is equal to the angle between the
reflected ray and the surface normal:
. Law of Refraction
An optical disturbance moving from a medium of
refractive index n1 into a medium of refractive index
n2 will have its incident ray, the surface normal between
the 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 the
reflected 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
21
21 =⋅







− rd
sd
rd
n
sd
rd
n
rayray 

In this case we have and21 nn =
( ) ( )
( ) 0ˆˆ
21
21 =⋅−=⋅







− 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
21
21 =⋅







− rd
sd
rd
n
sd
rd
n
rayray 

where is on the boundary between the two regions andrd

( ) ( )
sd
rd
s
sd
rd
s
rayray 2
:ˆ,
1
:ˆ 21

==
rd

22
ˆsn
11
ˆsn
1122
ˆˆˆ snsn −
( ) 0ˆˆˆ 1122 =⋅− rdsnsn

Refracted Ray
21
ˆ −n
2n
1n i
θ
t
θ
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 − 21
ˆ −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 Snell
1580-1626

10
SOLO
Huygens Principle
Christiaan Huygens
1629-1695
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
P1
Q1A1
A
P'1
iφ rφ
n
'n
Optical
axis
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 Malus
1775-1812
A surface passing through the end points of rays which have traveled equal optical path
lengths 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 is
the wavefront), they are said to form a normal congruence.
The Malus-Dupin Theorem (introduced in 1808 by Malus
and modified in 1812 by Dupin) states that:
“The set of rays that are orthogonal to a wavefront remain
normal to a wavefront after any number of refraction or reflections.”
Charles Dupin
1784-1873
n
'n
P
Q
VA
P’
A'
B B'
Wavefront
from 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
[ ] [ ] ( )2
εOBQAQ +=
[ ] [ ] ( )2
'' εOQBQA += ray BQ’ is normal to wave W’ at B’
Proof for Refraction:
Optics

13
Geometrical OpticsSOLO
Hamilton’s Point Characteristic Function and the Direction of a Ray
William Rowan
Hamilton
(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'
Wavefront
from P Wavefront
to P'
r

'r

( )rP
 ( )'' rP

Hamilton’s Point Characteristic
Function of a Ray as the
Optical Path Length along the ray:
( ) ( )( ) ( ) ∫==
'
','',
PtoP
Path
Optical
dsnrrVrPrPV

Consider as in the Figure bellow a neighboring point to ( )'' rP

( )''" rrP

δ+
n
Wavefront
from P
r
 'r

( )rP
 ( )'' rP

( )''' rrP

δ+
'' rr

δ+
'r

δ
( )
sd
PP
rd
s
ra y
',
:ˆ

=
( ) ( ) ( ) ( ) ( ) VrrrVrrrVPPVPPVPPV ''','',',",'," ⋅∇=−+=−=

δδ
According to the definition of optical path ( ) '''," rsnPPV

δ⋅=
Since those relations are true for every small :'r

δ ( ) Vsn PP '' ', ∇=

Direction of ray is normal to ( ) .', constrrV =

( )
sd
rd
s
ray
PP

=:ˆ ',where and ( ) 2/1
: rayray rdrdsd

⋅=

14
Geometrical OpticsSOLO
In 1828 Hamilton published
William Rowan
Hamilton
(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 rays
emitted 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.
( ) ( )( ) ( ) ∫==
'
','',
PtoP
Path
Optical
dsnrrVrPrPV

n
'n
Q
VA A'
B B'
Wavefront
from P Wavefront
to P'
r

'r

( )rP
 ( )'' rP

Vsn '' ∇=


15
SOLO
converging beam
=
spherical wavefront
parallel beam
=
plane wavefront
Image Plane
Ideal Optics
P'
Optical Aberration
converging beam
=
spherical wavefront
Image Plane
Ideal Optics
diverging beam
=
spherical wavefront
P
P'
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

16
SOLO
ideal wavefrontparallel beam
=
plane wavefront
Image Plane
Non-ideal Optics
aberrated beam
=
iregular wavefront
diverging beam
=
spherical wavefront
aberrated beam
=
irregular wavefront
Image Plane
Non-ideal Optics
ideal 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 reference
Wavefront.
SOLO Optical Aberration
Optical Aberration W (x,y)

18
Display of Optical Aberration W (x,y)
Rays Deviation3
Optical Path Length Difference2
wavefront shape W (x,y)1
Red circle denotes the pupile margin.
Arrows shows how each ray is deviated
as it emerges from the pupil plane.
Each of the vectors indicates the the
local slope of W (x,y).
The aberration W (x,y) is
represented in x,y plane by
color contours.
x
y
( )yxW ,
Wavefront
Error
x
y
( )yxW ,
Optical
Distance
Errors
x
y
Ray
Errors
The Wavefront error agrees with
Optical Path Length Difference,
But has opposite sign because a
long (short) optical path causes
phase retardation (advancement).
Aberration Type:
Negative vertical
coma
Reference

19
Display of Optical Aberration W (x,y)
Advanced phase <= Short optical
path
Retarded phase <= Long optical
path
Reference
Ectasia
x
y
Ray Errors
y
( )yxW ,
x
Optical Distance Errors
x
y
( )yxW ,
Wavefront Error

20
OpticsSOLO
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
OpticsSOLO
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 be
seen 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 the
elements preceding the Aperture Stop.
Exit Pupil: the image of the Aperture Stop as seen from an axial point on the
image plane.

22
OpticsSOLO
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 be
seen 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 the
elements preceding the Aperture Stop.
Exit Pupil: the image of the Aperture Stop as seen from an axial point on the
image plane.

23
OpticsSOLO
Principal Planes: the two planes defined by the intersection of the parallel incident rays
entering an optical system with the rays converging to the focal points
after 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
OpticsSOLO
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 that
passing 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 the
Cartezian 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
+==
+=
( ) ( )[ ]
( ) ( )[ ]
[ ] [ ] 2/1222/12
0
2
0
2/1222
2/12
0
22
0
gg
gg
zxnzxn
zzyxxn
zzyxxn
+++=
−++−
++++−
Optical Aberration
Aberrations (continue – 1)

26
SOLO
For homogeneous media (n = constant) the velocity of light is constant, therefore the
rays starting/arriving from/to a point are perpendicular to the spherical wavefronts.
Optical paths from P:
( ) ( )[ ] 2/12
0
22
0),( zzyxxnQPV +++−=
( ) ( )[ ] 2/1222
)',( gg zzyxxnPQV −++−=
Optical paths to P’:
Rays from P:
( ) ( )
( ) ( )
( ) ( )[ ] 2/12
0
22
0
00
,,),(
ˆ
,
1
zzyxx
zzzyyxxx
QPV
n
s zyxQP
+++−
+++−
=
∇=


Rays to P’:
( ) ( )
( ) ( )
( ) ( )[ ] 2/1222
,,)',(
ˆ
',
1
gg
gg
zyxPQ
zzyxx
zzzyyxxx
PQV
n
s
−++−
−++−
−=
∇=−=


Optical Aberration

27
OpticsSOLO
Real Imaging Systems – Aberrations (continue – 3)
Departures from the idealized conditions of Gaussian Optics in a real Optical System are
called 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 point
O (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 "PQ
Play it

28
SOLO
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, the
optical path length of the ray starting
at the object P and ending at
is identical to that of the Chief Ray
ending at O.
Q
Therefore the Wave Aberration is defined as
the 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
( ) ( ) ( )( )( ) ( ) ( )( )0,0,0,,0,,,,,,0,, 0000 OzxPVyxzyxQzxPVyxW −=
Optical Aberration

29
SOLO
Given the Wave Aberration function W (x,y)
he Gaussian Image P’(xg,0,zg)
f P and the point Q (x,y,z)
n the Reference Gaussian Sphere
we want to find the point P”(xi,yi,zg)
022222
( ) ( ) ( )( )( ) ( ) ( )( )0,0,0,,0,,,,,,0,, 0000 OzxPVyxzyxQzxPVyxW −=
Solution:
( ) ( )( )( ) ( )( )( )
Q
x
z
z
yxzyxQPV
x
yxzyxQPV
x
yxW
∂
∂
∂
∂
+
∂
∂
=
∂
∂ ,,,,,,,,,
( )
( ) ( ) ( )[ ] 2/1222
,,
,,
zzyyxx
zzyyxx
n
z
V
y
V
x
V
gii
gii
−+−+−
−−−
=





∂
∂
∂
∂
∂
∂
Compute relative to Q by differentiating relative to x:022222
Q
x
z
∂
∂
g
g
Q
zz
xx
x
z
−
−
−=
∂
∂
( ) ( ) ( ) ( )gi
g
g
gi xx
R
n
zz
xx
zz
R
n
xx
R
n
x
yxW
−=
−
−
−−−=
∂
∂
'''
, ( )
x
yxW
n
R
xx gi
∂
∂
+=
,'
In the same way:
( )
y
yxW
n
R
yi
∂
∂
=
,'
The ray from Q to P” is given by (see ):
Optical Aberration

30
SOLO
Object
Gaussian
Image
planeExit
Pupil
(ExP)
Optics
( )00 ,0, zxP − ( )zyxQ ,,
( )gg zxP ,0,'
( )gii zyxP ,,"
iy
iz
Reference
Gaussian
Sphere
center P'
Aberrated
Wavefront
center P"
( )0,0,0O
gz
y
x Q
z
Gaussian
Image
Aberrated
Image
ChiefRay
Chief
Ray
( )
x
yxW
n
R
xx gi
∂
∂
+=
,'
( )
y
yxW
n
R
yi
∂
∂
=
,'
Optical Aberration
Forward to
a 2nd
way
The aberration is the deviation of the
image P”(xi,yi,zg) from the Gaussian
image P’(xg,yg,zg)
The image P”(xi,yi,zg) coordinates in
image plane are:
( )
x
yxW
n
R
xxx gii
∂
∂
=−=∆
,'
( )
y
yxW
n
R
yyy gii
∂
∂
=−=∆
,'

31
SOLO
Defocus Aberration
Consider an optical system for which the
object P, the Gaussian image P’ and the
aberrated image P” are on the Optical Axis.
The Gaussian Reference Sphere passing through
O (center of ExP) has the center at P’.
The Aberrated Wavefront Sphere passing through
O (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
22
1
22
2
cos
'"'"
cos
RRrRrR
U
n
PPBPPB
U
n
rW
Optical Aberration

32
SOLO
Defocus Aberration (continue – 1)
Let make the following assumptions:
( ) ( ) ( )


 −−−−−=−−= 12
22
1
22
2
cos
'"'"
cos
RRrRrR
U
n
PPBPPB
U
n
rW
21,1cos RRrU <<≈
( ) ( )
( )








+







−−





−=








−−







++−−







++−≈
−−=


4
1111
2
82
1
82
1
'"'"
cos
4
3
2
3
1
2
21
124
1
4
2
1
2
14
2
4
2
2
2
2
r
RR
r
RR
n
RR
R
r
R
r
R
R
r
R
r
Rn
PPBPPB
U
n
rW
1
1682
11
32
<++−+=+ x
xxx
x 
Assume: RRRRRR =≈−=∆ 2112 &
( ) 2
2
2
r
R
Rn
rW
∆
≈we have: Δ R is called the Longitudinal Defocus.
Optical Aberration

33
SOLO
For a circular exit pupil of radius a we have:
( ) 22
2
#
8
ρρρ dA
f
Rn
W =
∆
=
a
R
f
2
:# =F number:
Define: a
r
=:ρ
Therefore
Where is the peak value of the
Defocus Aberration
2
#
8
:
f
Rn
Ad
∆
=
Optical Aberration

34
http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf
( ) ( )22
, yxAyxW d +⋅=
Optical Aberration
Wave Aberration: Defocus
SOLO
Austin Roorda, “Review of Basic Principles in Optics, Wavefront and Wavefront Error”,
University of California, Berkley

35http://129.7.217.162/VOI/WavefrontCongress/2006/presentations/1ROORDAprincip.pdf
Optical Aberration
Wavefront Errors for Defocus
SOLO

36
SOLO
Wavefront Tilt Aberration
Assume an optical system that has one ore more
optical elements tilted and/or decentered.
The object P is on the Optical Axes (OA), therefore
the Gaussian image P1 is also on OA. Therefore the
Gaussian Reference Sphere that passes trough ExP
center O has it’s center at P1. P2 is the aberrated
image on the Gaussian image plane (that contains
P1) is a distance xi from OA. The Aberrated
Wavefront that passes through O has it’s center at
P2. Therefore for small P1P2 the two surfaces are
tilted by an angle β.
Consider the ray where:2QPQ
( )zyxQ ,, on the Gaussian Reference Sphere 02 1
222
=−++ xRzyx
Q on the Aberrated Wavefront Sphere centered at P2 and radius R.
βcos1 RR =
( )12 ,0, RxP i the aberrated image
ββ RRxi ≈= sin
( ) ( )θθ sin,cos, rryx =
Optical Aberration

37
SOLO
Wavefront Tilt Aberration (continue – 1)
We have
x
W
n
R
Rxi
∂
∂
== β
( ) ( ) QQnQQVrW == ,
The Wave Aberration is
βn
x
W
=
∂
∂
θββ cos
0
rnxnxd
x
W
W
x
==
∂
∂
= ∫
For a circular exit pupil of radius a we have:
a
r
=:ρ
( ) θρθρβθρ coscos, 1BanW ==
where:
βanB =:1
Optical Aberration

38
SOLO
Departures from the idealized conditions of Gaussian Optics in a real Optical System are
called Aberrations
Monochromatic Aberrations
Chromatic Aberrations
• Monochromatic Aberrations
Departures from the first order theory are embodied
in the five primary aberrations
1. Spherical Aberrations
2. Coma
3. Astigmatism
4. Field Curvature
5. Distortion
This classification was done in 1857
by Philipp Ludwig von Seidel (1821 – 1896)
• Chromatic Aberrations
1. Axial Chromatic Aberration
2. Lateral Chromatic Aberration
Optical Aberration

39
SOLO
Optical Aberration

40
SOLO
Optical Aberration

41
SOLO
Optical Aberration

42
SOLO
Seidel Aberrations
Consider a spherical surface of radius R, with an object P0 and the image P0’ on the
Optical Axis.
n
'n
CB
R
0
P '0P
( )θ,rQ
0
V
r
z
( ) s− 's
Chief Ray
General Ray
Aperture Stop
Enter Pupil
Exit Pupil
The Chief Ray is P0 V0 P0’ and a
General Ray P0 Q P0’.
The Wave Aberration is defined as
the difference in the optical path
lengths between a General Ray and
the Chief Ray.
( ) [ ] [ ] ( ) ( )snsnQPnQPnPVPQPPrW +−+=−= '''''' 00000000
On-Axis Point Object
The aperture stop AS, entrance pupil EnP,
and exit pupil ExP are located at the
refracting surface.
Optical Aberration

43
SOLO
Seidel Aberrations (continue – 1)
n
'n
CB
R
0
P '0
P
( )θ,rQ
0
V
r
z
( ) s− 's
Chief Ray
General Ray
AS
EnP
ExP








−−=−−= 2
2
22
11
R
r
RrRRz
Define:
( ) 2
2
11
2
2
R
r
xxf
R
r
x
−=+=
−=
( ) ( ) 2/1
1
2
1
'
−
+= xxf
( ) ( ) 2/3
1
4
1
"
−
+−= xxf ( ) ( ) 2/5
1
8
3
'"
−
+−= xxf
Develop f (x) in a Taylor series ( ) ( ) ( ) ( ) ( ) ++++= 0"'
6
0"
2
0'
1
0
32
f
x
f
x
f
x
fxf
1
1682
11
32
<++−+=+ x
xxx
x 
Rr
R
r
R
r
R
r
R
r
Rz <+++=








−−= 5
6
3
42
2
2
1682
11
On Axis Point Object
From the Figure:
( ) 222
rzRR +−= 02 22
=+− rRzz
Optical Aberration

44
SOLO
From the Figure:
( )[ ] [ ]
( )[ ] ( ) 2/1
2
2/12
2
2/12222/122
0
212
2
22





 −
+=+−=
++−=+−=
−=
z
s
sR
sszsR
rsszzrszQP
rzRz
( ) ( )






+
−
−
−
+−≈
<++−+=+


2
4
2
2
1
1682
11
2
1
1
32
z
s
sR
z
s
sR
s
x
xxx
x
( ) ( )








+





+
−
−





+
−
+−=
+≈

2
3
42
4
2
3
42
2
82
822
1
82
1
3
42
R
r
R
r
s
sR
R
r
R
r
s
sR
s
R
r
R
r
z
( )[ ] +














−+





−+





−+−≈+−= 4
2
2
22/122
0
11
8
111
8
111
2
1
r
sRssRR
r
sR
srszQP
( )[ ] +














−+





−+





−+≈+−= 4
2
2
22/122
0
1
'
1
'8
11
'
1
8
11
'
1
2
1
''' r
RssRsR
r
Rs
srzsPQ
In the same way:
Optical Aberration
n
'n
CB
R
0
P '0
P
( )θ,rQ
0
V
r
z
( ) s− 's
Chief Ray
General Ray
AS
EnP
ExP

45
SOLO
+














−+





−+





−+−≈ 4
2
2
2
0
11
8
111
8
111
2
1
r
sRssRR
r
sR
sQP
+














−+





−+





−+≈ 4
2
2
2
0
1
'
1
'8
11
'
1
8
11
'
1
2
1
'' r
RssRsR
r
Rs
sPQ
Therefore:
( ) ( ) ( )
4
22
2
42
000
11
'
11
'
'
8
1
82
'
'
'
''''
r
sRs
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
1
rar
sRs
n
sRs
n
rW S
=














−−





−−=
Optical Aberration
n
'n
CB
R
0P '0P
( )θ,rQ
0V
r
z
( ) s− 's
Chief Ray
General Ray
AS
EnP
ExP

46
SOLO
Off-Axis Point Object
Consider the spherical surface of radius R, with an object P and its Gaussian image P’
outside the Optical Axis.
The aperture stop AS, entrance pupil EnP, and
xit pupil ExP are located at the refracting surface.
Using the similarity of the triangles:
''~ 00 CPPCPP ∆∆
the transverse magnification
( ) ( )
s
n
s
n
nn
s
s
n
s
n
nn
s
Rs
Rs
h
h
Mt
−
−
+−
−
−
−
=
+−
−
=
−
=
'
'
'
'
'
'
'
''
( ) sn
sn
nn
s
s
nn
nn
s
s
nn
Mt
−
=
−+−
+−−
=
'
'
'
'
'
'
'
'
n
'n
CB
R
0P
'0P
( )θ,rQ
0V
r
z
( ) s−
's
Chief Ray
General Ray
AS
EnP
ExP
'P
Undeviated Ray
P
( ) h−
'h
θ
V
Optical Aberration

47
SOLO
The Wave Aberration is defined as the difference
n 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
0
0 hbh
Rs
R
PP
CP
CV
VV =
−
=≈
Rs
R
b
−
=
'
:














−−





−−=
22
11
'
11
'
'
8
1
sRs
n
sRs
n
aS
Optical Aberration
n
'n
CB
R
0P
'0P
( )θ,rQ
0V
r
z
( ) s−
's
Chief Ray
General Ray
AS
EnP
ExP
'P
Undeviated Ray
P
( ) h−
'h
θ
V

48
SOLO
Wave Aberration.
( ) [ ] [ ] ( )4
0
4
'' VVVQaPVPPQPQW S −=−=
θθ cos'2'cos2 222
0
2
0
2
2
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
CB
R
0
P
'0
P
( )θ,rQ
0
V
r
z
( ) s−
's
Chief Ray
General Ray
AS
EnP
ExP
'P
Undeviated Ray
P
( ) h−
'h
θ
V
θ
r
y
x
0
V
V
Q
Exit Pupil Plane
Define the polar coordinate (r,θ) of the projection of Q in the plane of exit pupil, with
V0 at the origin, and assume (third order = Seidel approximation) that projected on
exit pupil is equal to .
VQ
VQ

49
SOLO
eneral Optical Systems
( ) θθθθ cos''cos'cos'';, 33222222234
rhbCrhbCrhbCrhbCrChrW DiFCAsCoSp ++++=
A General Optical Systems has more than one Reflecting or
Refracting surface. The image of one surface acts as an
bject 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 CoefficientSpC
2. Coma CoefficientCoC
3. Astigmatism CoefficientAs
C
4. Field Curvature CoefficientFC
C
5. Distortion CoefficientDi
C
where:
n
'n
C O
0P
'0
P
( )θ,rQ
0
V
r
( ) s−
's
Chief Ray
General Ray
Exit Pupil
'P
Undeviated Ray
P
( ) h−
'h
θ
~
Optical Aberration

50
SOLO
( ) θθθθ cos''cos'cos'';, 32222234
rhCrhCrhCrhCrChrW DiFCAsCoSp ++++=
Optical Aberration

51
SOLO
Optical Aberration

52
n
C O '0
P
R
P
'L
Chief Ray
General Ray
Exit Pupil
Exp
'P
Undeviated Ray
'h
~
True Wave
Front
Reference
Sphere
α
α
TP
RP'
TP'
α'Lr ≈∆
r
r∆
Image
plane
True Wave
Front
Reference
Sphere
r∆
l∆
R
P
R
P'
T
P
TP'
α
G
P
SOLO
nWPP TR /=
Assume that P’ is the image of P.
The point PT is on the Exit Pupil (Exp) and on the
True Wave Front (TWF) that propagates toward P’.
This True Wave Front is not a sphere because of the
Aberration. Without the aberration the wave front
would be the Reference Sphere (RS) with radius PRPG.
W (x’,y’;h’) - wave aberration
n - 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 on
RS and TWF, respectively, and on a ray close
to PRPT ray, converging to P’, the image of P.
lPPPP TRTR ∆+=''
Optical Aberration

53
SOLO
n
C O '0P
RP
'L
Chief Ray
General Ray
Exit Pupil
Exp
'P
Undeviated Ray
'h
~
True Wave
Front
Reference
Sphere
α
α
T
P
RP'
T
P'
α'Lr ≈∆
r
r∆
Image
plane
True Wave
Front
Reference
Sphere
r∆
l∆
R
P
R
P'
T
P
TP'
α
G
P
Optical Aberration

54
SOLO
( )
x
hyxW
n
L
xi
∂
∂
=∆
';,
'
' ( )
y
hyxW
n
L
yi
∂
∂
=∆
';,
'
'
θ
θ
sin
cos
ry
rx
=
=
( ) nhyxWPP TR /';,= lPPPP TRTR ∆+=''
α=
∆
∆
=
∆
−
=
∂
∂
→∆→∆ r
l
r
PPPP
x
W
n r
TRTR
r 00
lim
''
lim
1
x
W
n
L
Lr
∂
∂
==∆
'
'α
Optical Aberration
n
C O '0P
R
P
'L
Chief Ray
General Ray
Exit Pupil
Exp
'P
Undeviated Ray
'h
~
True Wave
Front
Reference
Sphere
α
α
T
P
R
P'
TP'
α'Lr ≈∆
r
r∆
Image
plane
True Wave
Front
Reference
Sphere
r∆
l∆
RP
RP'
TP
TP'
α
G
P
Deviation of image due to aberrations. We recovered the equations developed in

55
SOLO
1. Spherical Aberrations
( )
( ) ( )';,
';,
222
4
hyxWyxC
rChrW
SpSp
SpSp
=+=
=θ
( ) xrC
n
L
x
hyxW
n
L
x Spi
2
'
'
4
';,
'
'
=
∂
=∆
( ) yrC
n
L
y
hyxW
n
L
y Spi
2
'
'
4
';,
'
'
=
∂
=∆
( ) ( )[ ] 32/122
'
'
4 rC
n
L
yxr Spiii =∆+∆=∆
Consider only the Spherical Wave Aberration Function
The Spherical Wave Aberration is a
Circle in the Image Plane
Optical Aberration
( ) θθθθ cos''cos'cos'';, 33222222234

56
SOLO
1. Spherical Aberrations (continue – 1)
Optical Aberration
Ray Errors
0
0.5
1
Optical Distance
Errors Wavefront Error

57
SOLO
1. Spherical Aberrations (continue – 2)
Optical Aberration

58
SOLO
Assume an object point outside the
Optical Axis.
Meridional (Tangential) plane is
the plane defined by the object point
and the Optical Axis.
Sagittal plane is the plane normal to
Meridional plane that contains the
Chief Ray passing through the
Object point.
Optical Aberration
Meridional and Sagittal Planes

59
SOLO
2. Coma
Consider only the Coma Wave Aberration Function
( ) ( ) ''''cos'';, 22
cos'
sin'
3
xyxhbCrhbChrW Co
rx
ryCoCo
+==
=
=
θ
θ
θθ
( ) ( ) ( )
( )θ
θ
2cos2
'
''
cos21
'
''
3
'
''';,
'
'
2
2222
+=
+=+=
∂
=∆
r
n
Lhb
C
r
n
Lhb
Cyx
n
Lhb
C
x
hyxW
n
L
x
Co
CoCoi
( ) ( ) θ2sin
'
''
2
'
''';,
'
' 2
r
n
Lhb
Cyx
n
Lhb
C
y
hyxW
n
L
y CoCoi ==
∂
=∆
1
'
''
2
'
''
2
2
2
2
=












∆
+












−
∆
r
n
Lhb
C
y
r
n
Lhb
C
x
Co
i
Co
i
( )( ) ( ) ( )222
2 rRyrRx CoiCoi =∆+−∆
( ) 2
'
''
: r
n
Lhb
CrR CoCo =
Optical Aberration
( ) θθθθ cos''cos'cos'';, 33222222234

60
SOLO
2. Coma (continue – 1)
We obtained
2
'
''
: MAXCoS r
n
Lhb
CC =
( )( ) ( ) ( )222
2 rRyrRx CoiCoi =∆+−∆
( ) MAXCoCo rrr
n
Lhb
CrR ≤≤= 0
'
''
: 2
Define:
1
2
3 4
P
Image
Plane
O
SC
SC
ST CC 3=
Coma Blur Spot Shape in the Image Plane
Tangential
Coma
Sagittal
Coma

30
'h
i
x
iy
Optical Aberration

61
SOLO
Graphical Explanation of Coma Blur
1
1
2
2
3
3
4
4
Optical Axis
1
Meridional
(Tangential)
Plane
P
Image
Plane
Tangential
Rays 1
O
Lens
A Tangential Rays 1
Chief Ray
1
1
1
2
2
3
3
4
4
Optical Axis
1
Sagittal
Plane
P
Image
Plane
Sagittal
Rays 2
O
Lens
A
2
Sagittal Rays 2
Chief Ray
2
1
1
2
2
3
3
4
4
Optical Axis
1
P
Image
Plane
Skew
Rays 3
O
Lens
A
2
3
Skew Rays 3
Chief Ray
3
1
1
2
2
3
3
4
4
Optical Axis
1
P
Image
Plane
Skew
Rays 4
O
Lens
A
2
3
4
Skew Rays 4
Chief Ray
4
Optical Aberration

62
SOLO
Graphical Explanation of Coma Blur (continue – 1)
Optical Aberration

64
SOLO
Optical Aberration
2. Coma (continue-5)

65
( ) ( ) ''''cos'';, 22
cos'
sin'
3
xyxhCrhChrW Co
rx
ryCoCo
+==
=
=
θ
θ
θθ
Wave Aberration: Coma
Optical Aberration
2. Coma (continue-6)
SOLO

66
SOLO
&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
Consider the Astigmatism and the Field Curvature
Wave Aberration Function
( ) ( ) xCC
n
Lhb
x
hyxW
n
L
x FCAsi
+=
∂
=∆
'
''
2
';,
'
' 22
( ) yC
n
Lhb
y
hyxW
n
L
y FCi
'
''
2
';,
'
' 22
=
∂
=∆
Meridional
plane
Sagittal
planeObject
point Optical
System
Chief
ray
Optical
axis
'L
( )
( )θρθρ sin,cos
,
=
yx
( )ii
yx ∆∆ ,
Ellipse
Image
plane
Optical
axis
( )
1
'
''
2
'
''
2
2
22
2
22
=












∆
+












+
∆
FC
i
FCAs
i
Cr
n
Lhb
y
CCr
n
Lhb
x
θ
θ
sin
cos
ry
rx
=
=Ellipse

67
SOLO
.&4. Astigmatism and Curvature of Field
Sagittal
plane
Object
point
Optical
System
Chief
ray
Optical
axis
'L 'L∆
( )
( )θρθρ sin,cos
,
=
yx
( )ii
yx ∆∆ , ( )'','' ii yx ∆∆
Optical Aberration
We want to see what happens if we
move the image plane from the Optical
System 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
y
Ly
LL
yy
Lyy
L
x
Lx
LL
xx
Lxx
i
i
ii
ii
i
i
ii
ii
∆−∆≈
∆+
∆−
∆−∆=∆
∆−∆≈
∆+
∆−
∆−∆=∆
( ) ( ) xCC
n
Lhb
x
hyxW
n
L
x FCAsi
+=
∂
=∆
'
''
2
';,
'
' 22
( ) yC
n
Lhb
y
hyxW
n
L
y FCi
'
''
2
';,
'
' 22
=
∂
=∆
( ) iFCAsi
x
L
L
CC
n
Lhb
x 




 ∆
−+=∆
'
'
'
''
2"
22
iFCi y
L
L
C
n
Lhb
y 




 ∆
−=∆
'
'
'
''
2"
22
( )
1
'
'
'
''
2
'
'
'
''
2
2
22
2
22
=












∆
−
∆
+












∆
−+
∆
L
L
Cr
n
Lhb
y
L
L
CCr
n
Lhb
x
FC
i
FCAs
i

68
SOLO
.&4. Astigmatism and Curvature of Field
Optical Aberration
Meridional
plane
Sagittal
plane
Object
point
Optical
System
Chief
ray
S
F
T
F
Optical
axis
'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
L
Cr
n
Lhb
b
L
L
CCr
n
Lhb
b
FCy
FCAsx
∆
−=
∆
−+=
When bx or by is zero, the ellipse degenerates to a straight line
0=y
b 0"'
'
''
2"
22
=∆=∆ SAsS
yxC
n
Lhb
xFCS
Cr
n
Lhb
L
'
''
2'
222
=∆ Sagittal or Radial image
0=x
b '
'
''
2"0"
22
yC
n
Lhb
yx FCTT
−=∆=∆( )FCAsT
CCr
n
Lhb
L +=∆
'
''
2'
222
Tangential image
When Δ L’ is halfway between the two values just defined ( ) ( )FCAsTST
CCr
n
Lhb
LLL 2
'
''
''
2
1
'
222
+=∆+∆=∆
then we obtain the Circle of Least Confusion
Asyx
Cr
n
Lhb
bb
'
''22
=−=
( ) x
L
L
CC
n
Lhb
x FCAsi 




 ∆
−+=∆
'
'
'
''
2"
22
y
L
L
C
n
Lhb
y FCi 




 ∆
−=∆
'
'
'
''
2"
22

69
SOLO
Meridional
plane
Sagittal
plane
Primary
image
Secondary
image
Circle of least
confusion
Object
point
Optical
System
Chief
ray
SF
T
F
Ray in
Sagittal plane
Ray in
Meridional plane
Optical
axis
Optical
axis
Optical Aberration
0"'
'
''
2"
22
=∆=∆ SAsS
yxC
n
Lhb
x
FCS
Cr
n
Lhb
L
'
''
2'
222
=∆
Sagittal or Radial image
'
'
''
2"0"
22
yC
n
Lhb
yx FCTT
−=∆=∆
( )FCAsT
CCr
n
Lhb
L +=∆
'
''
2'
222
Tangential image
( ) ( )FCAsTST
CCr
n
Lhb
LLL 2
'
''
''
2
1
'
222
+=∆+∆=∆
Circle of Least Confusion
( ) ( )
2
22
22
'
''
"" 





=∆+∆ AsCC
Cr
n
Lhb
yx

70
SOLO
Optical Aberration
If we rotate the object (and therefore the image) point
about the optical axis then since
2
22
'
'
'
2' hC
n
Lrb
L FCS 





=∆
( ) 2
22
'
'
'
2' hCC
n
rLb
L 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.
Sσ
Tσ
Exit Pupil
Optic Axis
When is no astigmatism CAs = 0,
then σS and σT coincide to form
a curved surface called the
Petzval Surface.

71
SOLO
3. Astigmatism
Joseph Max Petzval
1807 - 1891

72
SOLO
3. Astigmatism

73
SOLO
3. Astigmatism

74
SOLO
3. Astigmatism
r = radius
q = meridian
Optical Aberration

75
SOLO
4. Field Curvature
( ) 222
'';, rhbChrW FCFC
=θ
Optical Aberration

76
4. Field Curvature

77
SOLO
. Distortion
( )
xhbC
rhbChrW
Di
DiDi
33
33
'
cos'';,
=
= θθ
Optical Aberration
( ) θθθθ cos''cos'cos'';, 33222222234
Consider only the Distorsion
( ) ( ) 0
';,
'
'
&
'
''';,
'
' 33
=
∂
=∆=
∂
=∆
y
hyxW
n
L
yC
n
Lhb
x
hyxW
n
L
x iDii
Meridional
plane
Sagittal
plane
Object
point
Optical
System
Chief
ray
Ray in
Sagittal plane
Ray in
Meridional plane
Optical
axis
Optical
axis
gx
ix∆
We can see that the Distortion
Aberration is only in the object
Meridional (Tangential) Plane.

78
SOLO
5. Distortion ( )
xhbC
rhbChrW
Di
DiDi
33
33
'
cos'';,
=
= θθ
Optical Aberration
( ) ( ) 0
';,
'
'
&
'
''';,
'
' 33
=
∂
=∆=
∂
=∆
y
hyxW
n
L
yC
n
Lhb
x
hyxW
n
L
x iDii
Object
points
Optical
System
Chief
ray
Optical
axis
Optical
axis
1gx
1ix∆
0
θ
0
r
0
x
0
y
5gx
5ix∆
Gaussian
image
Distorted
image
4gx
4i
x∆
1 2
3
4 5
Tangential
plane # 4
Let take instead of a point image, a line (multiple image points).
For each point we have a different tangential
plane and therefore a different x.
( ) '
2/122
hyx ⇒+
To obtain the image we must substitute
( ) ( ) 2/1222/122
sin&cos
yx
y
yx
x
+
=
+
= θθ
and we get:
( ) ( )
( )
( )23
3
2/122
2/322
3
2/322
3
'
'
'
'
cos
'
'
yxxC
n
Lb
yx
x
yxC
n
Lb
yxC
n
Lb
x DiDiDii
+=
+
+=+=∆ θ
( ) ( )
( )
( )32
3
2/122
2/322
3
2/322
3
'
'
'
'
sin
'
'
yyxC
n
Lb
yx
y
yxC
n
Lb
yxC
n
Lb
y DiDiDii
+=
+
+=+=∆ θ

79
SOLO
5. Distortion ( )
xhbC
rhbChrW
Di
DiDi
33
33
'
cos'';,
=
= θθ
Optical Aberration
Object
points
Optical
System
Chief
ray
Optical
axis
Optical
axis
1gx
1ix∆
0
θ
0
r
0
x
0
y
5gx
5ix∆
Gaussian
image
Distorted
image
4gx
4i
x∆
1 2
3
4 5
Tangential
plane # 4
Now consider a line object that yields a
paraxial image x =a (see Figure).
( ) ( )
( )
( )23
3
2/122
2/322
3
2/322
3
'
'
'
'
cos
'
'
yxxC
n
Lb
yx
x
yxC
n
Lb
yxC
n
Lb
x DiDiDii
+=
+
+=+=∆ θ
( ) ( )
( )
( )32
3
2/122
2/322
3
2/322
3
'
'
'
'
sin
'
'
yyxC
n
Lb
yx
y
yxC
n
Lb
yxC
n
Lb
y DiDiDii
+=
+
+=+=∆ θ
( )23
3
'
'
yaaC
n
Lb
x Dii
+=∆
( )32
3
'
'
yyaC
n
Lb
y Dii
+=∆

80
SOLO
5. Distortion
( ) θθ cos'';, 33
rhbChrW DiDi
=
Optical Aberration
( ) θθθθ cos''cos'cos'';, 33222222234
Consider only the Distorsion

81
( )yxW ,
Optical AberrationSOLO

82
SOLO
hin Lens Aberrations
Given a thin lens formed by two
urfaces with radiuses r1 and r2
with centers C1 and C2. PP0 is
he object, P”P”0 is the Gaussian
mage formed by the first surface,
P’P’0 is the image of virtual object
P’P”0 of the second surface.
( ) 



−
+
++= q
n
n
pn
sfn
CCo
1
1
12
'4
1
2
( )2
'2/1 sfCAs −=
( ) ( )2
'4/1 sfnnCFC +−=
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
CSp 14
1
2
123
1132
1 22
3
3
where:
f
s
OA
C11
r
F”
F
''f
''s
2
r
1=n
n
h
"h
D
0P
P
0'P
0"P
"P
'P
'h
's
CR
AS
EnP
ExP
r
( )θ,rQ
OC2
1=n
( ) [ ] [ ]0000 '', OPPQPPrW −=θ
Coddington position factor: '
2
11
2
'
'
s
f
s
f
ss
ss
p −=−=
−
+
=
Coddington shape factor:
12
12
rr
rr
q
−
+
=
From:
( ) 2222234
'cos'cos'';, rhCrhCrhCrChrW FCAsCoSp +++= θθθwe find:
Optical Aberration
( )
frr
n
ss
111
1
'
11
21
=





−−=+
Lens Maker’s
Formula

83
SOLO
oddington Position Factor
2R 1R f
1
C 2
FO 1
F
2C
2
n
1
n
s 's
'2 sfs ==
2R 1
R f
1C 2
F1
F
2C
2
n
1n
s 's
fss =∞= ',
2R 1R f
1C 2F1
F
2C
2
n
1n
s 's
fss <> ',0
2
R 1
R
f
1C 2
F1
F
2C
2
n
1n
s 's
∞== ', sfs
2R 1Rf
1C2
F1F 2C 2
n
1n
s 's
0', << sfs
CRCR
2R
1R
f1
C
2
F
1
F 2
C2
n
1n
s
's
0'0 <<> sfs
2R
1
R
f1C
2
F1
F 2
C2n
1n
s 's
fss =∞= ',
1=p
2R1
R f
1
C 2F1F
2
C
2n
1
n
s
's
∞== ', sfs
1>p
2R1
R f
1
C 2F1F
2
C
2
n
1n
s 's
0',0 ><< ssf
0=p
2R1
R f
1
C 2FO 1F
2
C
2
n
1
n
s 's
'2 sfs ==
1−=p1−<p
ss
ss
p
−
+
=
'
'
ss
ss
p
−
+
=
'
'
'
111
ssf
+=
'
2
11
2
s
f
s
f
p −=−=
Optical Aberration

84
SOLO
Coddington Position Factor
f f2f2− f− 0
Figure Object
Location
Image
Location
Image
Properties
Shape
Factor
Infinity
Principal
focus
'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, inverted
small
p = -1
Real, inverted
smaller
-1 < p <0
Real, inverted
same size
p = 0
Real, inverted
larger
0 < p <1
No image p = 1
Virtual, erect
larger
p>1
's
's
0<s fs <' p < -1
Imaginary,
inverted
small
Optical Aberration

85
SOLO
Coddington Shape Factor
1
02
1
−=
<
∞=
q
R
R
2
R
1
R
2
C 2
n
1
n
Plano
Convex
2
n
1
0,0
21
21
−<
>
<<
q
RR
RR
1
C 2
C
1
n
1
R
2
R
Positive
Meniscus
2
R
1
R f
1
C 2
F1
F 2
C 2
n
1
n
0
0,0
21
21
=
=
<>
q
RR
RR
Equi
Convex
2
R
1
R
1C2n
1n
Plano
Convex
1
0
2
1
=
∞=
>
q
R
R
2
R1
R
f
1
C 2
F 2
C
2
n
1
n
1
0,0
21
21
>
<
>>
q
RR
RR
Positive
Meniscus
12
12
RR
RR
q
−
+
=
2
R
1
R f
2F1
F
2C
2n
1
n
1C
Negative
Meniscus
1
0,0
21
21
−<
>
>>
q
RR
RR
1
0, 21
−=
>∞=
q
RR
Plano
Concave
2
R
1
R
f
2
F1
F
2
C
2n
1
n
2
R1
R f
1
C 2F1
F
2C
2
n
1
n
0
0,0
21
21
=
=
><
q
RR
RR
Equi
Concave
2
R
1
R
f
1F 2F
1C
2
n
1
n
1
,0 21
=
∞=<
q
RR
Plano
Concave
Negative
Meniscus
1
0,0
21
21
>
<
<<
q
RR
RR
2
R
1
R
f
2
F1
F 2C
2n
1
n
1C
Optical Aberration

86
REFLECTION & REFRACTIONSOLO
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 the
Wave 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
rr
q
−
+
=
ss
ss
p
−
+
=
'
'
Coddington Lens
http://www.eyeantiques.com/MicroscopesAndTelescopes/Coddington%20microscope_thick_wood.htm

87
SOLO
Thin Lens Spherical Aberrations
( ) 4
rCrW SpSA =
Given a thin lens and object O on the
ptical Axis (OA). A paraxial ray will cross
e OA at point I, at a distance s’p from
e lens. A general ray, that reaches the lens
a distance r from OA, will cross OA at
int E, at a distance s’r.
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
CSp 14
1
2
123
1132
1 22
3
3
where:
Define:
2
R
1
R
1
C
IO
2C
Paraxial
focal plane2
n
1n
s
ps'
E
rs' Long. SA
Lat. SA
φ
Paraxial
Ray
General
Ray
'φ
r
rp ssSALongAberrationSphericalalLongitudin ''. −==
( ) rrp srssSALatAberrationSphericalLateral '/''. −==
We have:
Optical Aberration

88
SOLO
12
12
RR
RR
qK
−
+
==
( )
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
r
rWSp 14
1
2
123
1132
22
3
3
4
Thin Lens Spherical Aberrations (continue – 1)
Optical Aberration
Shape factor

89
SOLO
2
R
1
R
1
C
IO
2C
Paraxial
focal plane2
n
1
n
s
ps'
E
rs' Long. SA
Lat. SA
φ
Paraxial
Ray
General
Ray
'φ
r
12
12
RR
RR
q
−
+
=
F.A. Jenkins & H.E. White, “Fundamentals of Optics”,
4th
Ed., McGraw-Hill, 1976, pg. 157
Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm
In Figure we can see a comparison
of the Seidel Third Order Theory
with the ray tracing.
Optical Aberration

90
SOLO
We can see that the Thin Lens Spherical Aberration WSp is a parabolic function of the
Coddington Shape Factor q, with the vertex at (qmin,WSp min)
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
r
WSp 14
1
2
123
1132
22
3
3
4
Thin Lens Spherical Aberrations (continue -3)
The minimum Spherical Aberration for a given Coddington Position Factor p is obtained
by:
( )
( ) 014
1
2
2
132 3
4
=



++
−
+
−
−=
∂
∂
pnq
n
n
fnn
r
q
W
p
Sp
1
1
2
2
min
+
−
−=
n
n
pq








+
−





−
−= 2
2
3
4
min
2132
p
n
n
n
n
f
r
WSp
The minimum Spherical Aberration is zero for ( )
( )
1
1
2
2
2
>
−
+
=
n
nn
p
Optical Aberration

91
SOLO
In order to obtain the radii of the lens for a given focal length f and given Shape Factor
and Position Factor we can perform the following:
Those relations were given by Coddington.
'
2
11
2
s
f
s
f
p −=−=
( )
fRR
n
ss
111
1
'
11
21
=





−−=+
p
f
s
p
f
s
−
=
+
=
1
2
'&
1
2
12
12
RR
RR
q
−
+
=
( ) ( )12
21
1 RRn
RR
f
−−
=
12
1
12
2 2
1&
2
1
RR
R
q
RR
R
q
−
=−
−
=+
( ) ( )
1
12
&
1
12
21
−
−
=
+
−
=
q
nf
R
q
nf
R
2
R
1
R
1
C
IO
2
C
Paraxial
focal plane2
n
1
n
s
ps'
E
rs' Long. SA
Lat. SA
φ
Paraxial
Ray
General
Ray
'φ
r
Optical Aberration

92
SOLO
Thin Lens Coma
( ) ( )
( ) ( ) 



−
+
++
+
=
+==
q
n
n
pn
sfn
xyxh
xyxhCrhChrW CoCoCo
1
1
12
'4
''''
''''cos'';,
2
22
223
θθ
or thin lens the coma factor is given by:
where:we find:
( ) 2
22
2
1
1
12
4
'''
: MAXMAXCoS rq
n
n
pn
fn
h
r
n
sh
CC 



−
+
++==
1
2
3 4
P
Image
Plane
O
SC
SC
ST CC 3=
Coma Blur Spot Shape
Tangential
Coma
Sagittal
Coma

30
'h
'x
'y
( )( ) ( ) ( )222
'2' rRyrRx CoCo =∆+−∆ ( ) MAXCoCo rrr
n
sh
CrR ≤≤= 0
''
: 2
Define:
( ) ( ) ( ) ( )θθ 2cos2
''
cos21
''
''3
''
'
';',''
' 2222
+=+=+=
∂
=∆ r
n
sh
Cr
n
sh
Cyx
n
sh
C
x
hyxW
n
s
x CoCoCo
( ) ( ) θ2sin
''
''2
''
'
';',''
' 2
r
n
sh
Cyx
n
sh
C
y
hyxW
n
s
y CoCo ==
∂
=∆
Optical Aberration
2
R
1R
1
C
IO
2
C
Paraxial
focal plane2
n
1n
s
ps'
E
rs' Long. SA
Lat. SA
φ
Paraxial
Ray
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. 165
Lens thickness = 1cm, f = 10cm, n = 1.5, h = 1cm, y = 2 cm
( ) 2
22
1
1
12
4
'
: MAXS rq
n
n
pn
fn
h
C 





−
+
++=
oma is linear in q
( ) ( )
( )
p
n
nn
qCS
1
112
0
+
−+
−=⇐=
n Figure 800.00 =⇐= qCS
The Spherical Aberration is
arabolic in q
( )
( ) ( ) ( ) 





++
−
+
+−++
−−
−= qpnq
n
n
pnn
n
n
fnn
CSp 14
1
2
123
1132
1 22
3
3
1
1
2
2
min
+
−
−=
n
n
pq








+
−





−
−= 2
2
3min
2132
1
p
n
n
n
n
f
CSp
In Figure
714.0min =q
Optical Aberration

94
Thin Lens Aberration

95
SOLO
Optical Aberration
The astigmatic lens may be focussed to yield a sharp image of either the sagittal or the
tangential detail, but not simultaneously. This is illustrated in Fig. 1 with the archetypal
example of astigmatism: a spoked wheel. A well-corrected lens delivers an all-sharp
image (left wheel). On the other hand, an astigmatically aberrated lens may be focussed
to yield a sharp image of the spokes (middle wheel), but at the expense of blurring of the
rims, which have a tangential orientation. Vice versa, when the rim is in focus the spokes
are blurred. It is customary to speak of the sagittal focus and tangential focus,
respectively, as indicated in Fig. 1.
Figure 1. Classic example of astigmatism. Left wheel: no astigmatism. In
the presence of astigmatism (middle and right wheels) one discriminates
between the sagittal and tangential foci.

96
Although the wheels in Fig. 1 are instructive, they are an oversimplification of
astigmatism as it occurs with photographic lenses. Where the figure suggests that the
amount of blurring in either the sagittal or radial direction is constant across the field,
this is not the case in practice. Unless a lens is poorly assembled, there will be no
astigmatism near the image center. The aberration occurs off-axis. With a real lens, the
sagittal and tangential focal surfaces are in fact curved. Fig. 2 displays the astigmatism
of a simple lens. Here, the sagittal (S) and tangential (T) images are paraboloids which
curve inward to the lens. As a consequence, when the image center is in focus the image
corners are out of focus, with tangential details blurred to a greater extent than sagittal
details. Although off-axis stigmatic imaging is not possible in this case, there is a
surface lying between the S and T surfaces that can be considered to define the
positions of best focus.
Fig. 2

97
Lens designers have a few degrees of freedom, such as the position of the aperture stop and the
choice of glass types for individual lens elements, to reduce the amount of astigmatism, and, most
desirably, to manoeuvre the S and T surfaces closer to the sensor plane. A complete elimination of
astigmatism is illustrated in the left sketch of Fig. 3. Although astigmatism is fully absent, i.e., the
S and T surfaces coincide, there is a penalty in the form of a pronouncedly curved field. When the
image center is in focus on the sensor the corners are far out of focus, and vice versa. In the late
nineteenth century, Paul Rudolf coined the word anastigmat to describe a lens for which the
astigmatism at one off-axis position could be reduced to zero [2]. The right sketch in Fig. 3 depicts
a typical photographic anastigmat. As a slight contradiction in terms, the anastigmat has some
residual astigmatism, but more importantly, the S and T surfaces are more flat than those in the
uncorrected scheme of Fig. 2 and the strictly stigmatic left scheme in Fig. 3. As such, the
anastigmat offers an attractive compromise between astigmatism and field curvature
Fig. 2 Fig. 3

98

99
SOLO
Chester Moor Hall (1704 – 1771) designed in secrecy the achromatic lens.
He experienced with different kinds of glass until he found in 1729 a combination of
convex component formed from crown glass with a concave component formed from
flint glass, but he didn’t request for a patent.
http://microscopy.fsu.edu/optics/timeline/people/dollond.html
In 1750 John Dollond learned from George Bass on Hall achromatic lens and designed
his own lenses, build some telescopes and urged by his son
Peter (1739 – 1820) applied for a patent.
Born & Wolf,”Principles of Optics”, 5th
Ed.,p.176
In 1733 he built several telescopes with apertures of 2.5” and 20”. To keep secrecy
Hall ordered the two components from different opticians in London, but they
subcontract the same glass grinder named George Bass, who, on finding that both
Lenses were from the same customer and had one radius in common, placed them
in 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
SOLO
Optical Aberration
Chromatic Aberrations arise in
Polychromatic IR Systems because
the material index n is actually
a function of frequency. Rays at
different frequencies will traverse
an optical system along different paths.

101
SOLO
Optical Aberration

102
SOLO Optics
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
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
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
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 1111
1
ρρρρ −−−−+−= nndnn
f
nF – blue index produced by hydrogen
wavelength 486.1 nm.
nC – red index produced by hydrogen
nd – yellow index produced by helium
Assume that for two colors red and blue we have fR = fB
( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) 22112211
22112211
1111
1111
1
ρρρρ
ρρρρ
−−−−+−=
−−−−+−=
FFFF
CCCC
nndnn
nndnn
f

106
SOLO Optics
Let analyze the case d = 0 (the two lenses are in contact)
nd – yellow index produced by helium
( ) ( ) ( ) ( ) 22112211 1111
1
ρρρρ −+−=−+−= FFCC nnnn
f
We have ( )
( )
( )
( )1
1
1
1
1
2
1
2
2
1
−
−
−=
−
−
−=
F
F
C
C
n
n
n
n
ρ
ρ ( )
( )CF
CF
nn
nn
11
22
2
1
−
−
−=
ρ
ρ
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
d
Y
nn
f
21 /1
22
/1
11 11
1
ρρ −+−= ( )
( ) Y
Y
d
d
f
f
n
n
1
2
1
2
2
1
1
1
−
−
=
ρ
ρ
( )
( )
( )
( )
( ) ( )
( ) ( )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
( ) ( )
( ) ( )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
n
V
nn
n
V
22
2
2
11
1
1
1
&
1
−
−
=
−
−
=
Return to Table of Content

108
SOLO
Image Analysis
Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How
aberrations affect vision”, University of Houston, TX, USA
Optical Aberration

109
Image Analysis
SOLO
( ) ( ) ( )[ ] { }gFTydxdyfxfjyxgffG yxyx =+−= ∫∫Σ
π2exp,:,
The two dimensional Fourier Transform F of the function f (x, y)
The Inverse Fourier Transform is
( ) ( ) ( )[ ] { }GFTfdfdyfxfjyxgyxg
F
yxyx
1
2exp,, −
=+= ∫∫ π
( )
( )
( ) ( )[ ] { }Σ
Σ
=+−= ∫∫ fFTddkkjfkkF yxyx ηξηξηξ
π
exp,
2
1
:, 2
Two Dimensional
Fourier Transform
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
0x
1y
0y
η
ξ
Sr'

Sr

ρ
 r

O
Sθ θ
Screen
Image
plane
Source
plane
0O
1O
Sn1
Σ
Σ - Screen Aperture
Sn1 - normal to Screen
1r
0r

SSS rn θcos11 =⋅
θcos11 =⋅ rnS
z
Sn1
'r
 Fr

F
rPP

=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
Fourier Transform Theorems
( ){ } ( )yx ffGyxgFT ,, =
2. Similarity Theorem
( ){ } 





=
b
f
a
f
G
ba
ybxagFT
yx
,
1
,If then
( ){ } ( )yx ffGyxgFT ,, =
3. Shift Theorem
( ){ } ( ) ( )[ ]bfafjffGbyaxgFT yxyx +−=−− π2exp,,
If
then
Optical Aberration

111
Image Analysis
SOLO
Fourier Transform Theorems (continue – 1)
( ){ } ( )yx ffGyxgFT ,, =
4. Parseval’s Theorem
( ) ( )∫∫∫∫ = yxyx fdfdffGydxdyxg
22
,,
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
Fourier Transform Theorems (continue – 2)
( ){ } ( ){ } ( )yxgyxgFTFTyxgFTFT ,,, 11
== −−
7. Fourier Integral Theorem
Optical Aberration

113
Image Analysis
SOLO
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
θφθ
2
0
0 cosexp dajaJ
( ) ( ){ } ( ) ( )∫==
a
o
rdrrgrJrgFTG ρπρ 2: 00
to obtain
J0 is a Bessel Function of the first kind, order zero.
Optical Aberration

114
Image Analysis
SOLO
For a Circular Pupil of radius a we have
( )



>
≤
=
ar
ar
rg
0
1
Use Bessel Function Identity
J1 is a Bessel Function of the first kind, order one.
( ) ( ){ } ( )∫==
a
o
rdrrJrgFTG ρπρ 2: 00
( ) ( )xJxdJ
x
o
10 =∫ ςςς
( ) ( ){ }
( )
( ) ( )
( )
( ) ( )ρπ
ρπ
ςςς
ρπ
ρπρπρπ
ρπ
ρ
ρπ
aJ
a
dJ
rdrrJrgFTG
a
o
a
o
2
22
1
222
2
1
:
1
2
02
020
==
==
∫
∫
Bessel Functions of the first kind
Optical Aberration

115
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E. Hecht, “Optics”
Circular Aperture
Image Analysis
( ) ( ){ } ( )
( )ρπ
ρπ
ρ
a
aJ
argFTG
2
2
: 12
0 ==
Optical Aberration

116
SOLO
Resolution of Optical SystemsAiry 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 Analysis
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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).
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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
,, λ
π
( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, =
( ) ( ){ }[ ]iiyx yxPSFFTPhaseffPTF ,, =
( ) ( ) ( )[ ]yxyxyx ffPTFiffMTFffOTF ,exp,, =
Image Analysis
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Optical Aberration

120
• The Point Spread Function (PSF) is the Fourier Transform (FT) of the pupil function
( ) ( )
( )






=
− yxWi
yx eyxPFTffPSF
,
2
,, λ
π
Image Analysis
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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

121
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 Analysis

122
As the pupil size gets larger, the Airy disc gets smaller.
Image Analysis
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Optical Aberration

123
Convolution
( ) ( ) ( )yxIyxOyxPSF ,,, =⊗
( )[ ] ( )[ ]{ } ( )yxIyxOFTyxPSFFTFT ,,,1
=•−
Image Analysis
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Convolution
Optical Aberration

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

125
The Modulation Transfer Function (MTF) indicates the ability of an Optical System
to 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
a
fcutoff
Image AnalysisSOLO
MTF as a function of pupil size (diameter)
Optical Aberration

126
Image Analysis
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http://voi.opt.uh.edu/voi/WavefrontCongress/2005/presentations/1-RoordaOpticsReview.pdf
Optical Aberration

127
Phase Transfer Function (PTF)
• PTF contains information about asymmetry in PSF
• PTF contains information about contrast reversals (spurious resolution)
Image Analysis
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Optical Aberration

129
( )
( )






=
− yxWi
eFTyxPSF
,
2
, λ
π
( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, = ( ) ( ){ }[ ]iiyx yxPSFFTPhaseffPTF ,, =
Austin Roorda, “Review of Basic Principles in Optics,
Wavefront and Wavefront Error”,
niversity of California, Berkley
Image Analysis
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Optical Aberration

130
( )
( )






=
− yxWi
eFTyxPSF
,
2
, λ
π
( ) ( ){ }[ ]iiyx yxPSFFTAmplitudeffMTF ,, = ( ) ( ){ }[ ]iiyx yxPSFFTPhaseffPTF ,, =
ustin Roorda, “Review of Basic Principles in Optics,
vefront and Wavefront Error”,
versity of California, Berkley
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Real Optical System
Optical Aberration

131
( )
( )






=
− yxWi
eFTyxPSF
,
2
, λ
π
Point Spread Function

132
( )
( )






=
− yxWi
eFTyxPSF
,
2
, λ
π

133
( )
( )






=
− yxWi
eFTyxPSF
,
2
, λ
π
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Optical Aberration

134
Strehl Ratio
Strehl, Karl 1895, Aplanatische und fehlerhafte Abbildung im
Fernrohr, Zeitschrift für Instrumentenkunde 15 (Oct.), 362-370.
Dr. Karl Strehl
1864 -1940
One of the most frequently used optical terms in both,
professional and amateur circles is the Strehl ratio. It is
the simplest meaningful way of expressing the effect of
wavefront aberrations on image quality. By definition,
Strehl ratio - introduced by Dr. Karl Strehl at the end of
19th century - is the ratio of peak diffraction intensities of
an aberrated vs. perfect wavefront. The ratio indicates
image quality in presence of wavefront aberrations; often
times, it is used to define the maximum acceptable level of
wavefront aberration for general observing - so-called
diffraction-limited level - conventionally set at 0.80 Strehl.

135
The Strehl ratio is the ratio of the irradiance at the center of the reference
sphere to the irradiance in the absence of aberration.
Irradiance is the square of the complex field amplitude u
0
E
E
Strehl =
2
uE =
∫∫= dxdyyxWjUu )),(2exp(0 π
Strehl Ratio
Expectation Notation
∫∫
∫∫==
dxdy
dxdyyxu
uu
),(

136
Derivation of Strehl Approximation
( )2
0
21 W
E
E
Strehl πσ−==
),(2
0
yxWj
eUu π
=
( ) 22
0 ),(2
2
1
),(21 yxWyxWjUu ππ −+=
( ) 2
0
2
00 ),(2
2
1
),(2 yxWUyxWUjUu ππ −+=
series expansion
( ) ( ) 2
0
22
0
2
0 ),(2),(2 yxWEyxWEEE ππ +−=
multiply by
complex conjugate
2222
),(),(),(),( yxWyxWyxWyxWW −=−=σ
wavefront variance:

137
( )2
0
21 W
E
E
Strehl πσ−==
22
2
),(),( yxWyxWWWW −=−=σ
where σW is the wavefront variance:
( )2
2 W
eStrehl πσ−
=Another approximation for the Strehl ratio is
Strehl Approximation
Diffraction Limit
8.0≥Strehl
A system is diffraction-limited when the Strehl ratio is greater
than or equal to 0.8
Maréchal’s criterion:
This implies that the rms wavefront error is less than λ/13.3 or
that the total wavefront error is less than about λ/4.

138
Strehl Ratio
dl
eye
H
H
RatioStrehl =

139
Strehl Ratio
( )∑= 2m
nCrms
when rms is small
( )2
2
2
1 rmsStrehl 





−≈
λ
π

140
FIGURE : PSF and MTF for nominal wavefront errors of selected aberrations resulting in
~0.80 Strehl (~1/14 wave RMS), and the error doubled.
(a) the effect of 1/4 and 1/2 wave P-V
wavefront error of defocus on the
PSF intensity distribution (left)
and image contrast (right).
Doubling the error nearly halves
the peak diffraction intensity, but
the average contrast loss nearly
triples (evident from the peak PSF
intensity).
(b) 1/4 and 1/2 wave P-V of spherical
aberration. While the peak PSF
intensity change is nearly identical
to that of defocus, wider energy
spread away from the disc results in
more of an effect at mid- to high-
frequency range. Central disc at 1/2
wave P-V becomes larger, and less
well defined. The 1/2 wave curve
indicates ~20% lower actual cutoff
frequency in field conditions.
http://www.telescope-optics.net/
Image Analysis

141
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.
Image Analysis

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.
Image Analysis
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Optical Aberration

144
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)
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Optical Aberration

145
FIGURE: Illustration of a point source (stellar) image degradation caused by
atmospheric turbulence. The left column shows best possible average
seeing error in 2 arc seconds seeing (ro~70mm @ 550nm) for four
aperture sizes. The errors are generated according to Eq.53-54, with the
2" aperture error having only the roughness component (Eq.54), and
larger apertures having the tilt component added at a rate of 20% for
every next level of the aperture size, as a rough approximation of its
increasing contribution to the total error (the way it is handled by the
human eye is pretty much uncharted territory). The two columns to the
right show one possible range of error fluctuation, between half and
double the average error. The best possible average RMS seeing error is
approximately 0.05, 0.1, 0.2 and 0.4 wave, from top to bottom (the effect
would be identical if the aperture was kept constant, and ro reduced). The
smallest aperture is nearly unaffected most of the time. The 4" is already
mainly bellow "diffraction-limited", while the 8" has very little chance of
ever reaching it, even for brief periods of time. The 16" is, evidently,
affected the most. The D/ro ratio for its x2 error level is over 10, resulting
in clearly developed speckle structure. Note that the magnification shown
is over 1000x per inch of aperture, or roughly 10 to 50 times more than
practical limits for 2"-16" aperture range, respectively. At given nominal
magnification, actual (apparent) blur size would be smaller inversely to
the aperture size. It would bring the x2 blur in the 16" close to that in 2"
aperture (but it is obvious how a further deterioration in seeing quality
would affect the 16" more).
Eugène Michel Antoniadi
(1870 –, 1944)
The scale, invented by Eugène Antoniadi, a Greek astronomer, is on a 5 point system, with one being
the best seeing conditions and 5 being worst. The actual definitions are as follows:
I. Perfect seeing, without a quiver.
II. Slight quivering of the image with moments of calm lasting several seconds.
III. Moderate seeing with larger air tremors that blur the image.
IV. Poor seeing, constant troublesome undulations of the image.
V. Very bad seeing, hardly stable enough to allow a rough sketch to be made.
Image Degradation Caused by Atmospheric Turbulence
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Optical Aberration

146
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Optical Aberration

147
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148
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Optical Aberration

149
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In 1934 Frits Zernike introduces a complete set of orthonormal polynomials
to describe aberration of any complexity.
( ) ( ) ( ) ( )θρθρθρ
m
N
m
n
m
n
m
nN YRaZZ == ,,
,2,1
2
813
min =







 ++−
= N
N
Integern
( ) ( ) { }



−
=




 −++
−=
oddN
evenN
msign
Nnn
Integernm
1
1
4
212
min2
Each polynomial of the Zernike set is a product of
three terms.
where
( )



≠+
=+
=
012
01
mifn
mifn
a
m
n
( ) ( ) ( )
( )[ ] ( )[ ]
( )
sn
mn
s
s
m
n
smnsmns
sn
R 2
2/
0 !2/!2/!
!1 −
−
=
∑ −−−+
−−
= ρρ
( )





≠
≠
=
=
oddisNandmif
evenisNandmif
mif
Y
m
N
0sin
0cos
01
θ
θθ
radial index
meridional
index
Optical Aberration

150
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Properties of Zernike’s Polynomials.
( ) ( )∑∑=
n m
m
n
m
n ZCW θρθρ ,,
W (ρ,θ) – Waveform Aberration
Cn
m
(ρ,θ) – Aberration coefficient (weight)
Zn
m
(ρ,θ) – Zernike basis function (mode)
( ){ } ( ) mallnallforZZMean
m
n
m
n 00,, >== θρθρ1
( ){ } mnallforZVariance
m
n ,1, =θρ2
3 Zernike’s Polynomials are mutually orthogonal, meaning that they are independent
of 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 Fourier
coefficients of a Fourier expansion.
( ){ } ( ) ( )[ ] ( )∑∑∑∑ =














−=
n m
m
n
n m
m
n
m
n
m
n CZZCMeanWVariance
2
2
,,, θρθρθρ
( ) ( )
( ) '
1
0
'
12
1
nn
m
n
m
n
n
dRR δρρρρ
+
=∫ ( ) '0
2
0
1'coscos mmmdmm δδπθθθ
π
+=∫
Optical Aberration

151
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Astigmatism
{ }4,4,,2 22
−− ayax
Coma1
{ }3,5,,2 2
−+ axaxρ
Coma2
{ }4,4,,2 2
−+ ayaxρ
Spherical&
Defocus
( ) { }3,5,,3.12 22
−+ aaρρ
36Zernikes
Geounyoung Yoon, “Aberration Theory”
Optical Aberration

152
Surface of Revolution Stereogram
Zernike Polynomials
http://www.optics.arizona.edu/jcwyant/
Play it

153
SOLO
( ) ( )
mastigmatis
defocus
tilty
tiltx
piston
YRamnN
m
N
m
n
m
n

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
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Optical Aberration

155
Zernike’s Decomposition

156
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Zernike’s Decomposition
Optical Aberration

157

158

159
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Raymond A. Applegate, OD, PhD, “Aberration and Visual Performance: Part I: How aberrations
affect vision”, University of Houston, TX, USA
Optical Aberration

160
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Optical Aberration

161
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Optical Aberration

162
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Optical Aberration

163
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Optical Aberration

164
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affect vision”, University of Houston, TX, USA Return to Table of Content
Optical Aberration

165
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Optical Aberration

166
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Optical Aberration

168
Tilt (n=1, m=1) ( )0cos θθρ +
The wavefront: Contour plot and 3D
The spot diagram in the focal plane

169
Defocus (n=2, m=0) 2
ρ
The wavefront: Contour plot and 3D
The spot diagram in the focal plane
The hole in the center of the figures is in
the optical element.

170
Coma (n=3, m=1)
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Optical Aberration

171
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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 on
ray 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
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Aberrometers
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
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Aberrometers
Johannes Hartmann
1865-1936
In 1920, an astrophysicist named Johannes Hartmann devised
a method of measuring the ray aberration of mirrors and lenses.
He wanted to isolate rays of light so that they could be traced and any
imperfection 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 tested
and a photographic plate was placed near the focus of the mirror. When
exposed to light, a perfect mirror will produce an image of regulary
spaced 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.
1920
Optical Aberration

174
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Optical schematic for first Shack-Hartmann sensor.
Around 1971 , Dr. Roland Shack and Dr. Ben Platt advanced the concept replacing
the 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
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Lenslet array made by Heptagon
for ESO. The array has 40 x 40
lenslets, each 500 μm (0.5 mm) in
size.
Part of lenslet array made by WaveFront Sciences.
Each lens is 144 μm in diameter.
Optical Aberration

176
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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.
Optical Aberration

177
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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, 1998
V.N. Mahajan, “Optical Imaging and Aberrations”, Part II, Wave Diffraction Optics,
SPIE, 2001
http://grus.berkeley.edu/~jrg/Aberrations
http://grus.berkeley.edu/~jrg/Aberrations/BasicAberrationsandOpticalTesting.pdf
Jurgen R. Meyer-Arendt, “Introduction to Classic and Modern Optics”, Prentice
Hall, 1989
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178
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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
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

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