# Wavefront Aberrations

```Wavefront Aberrations
I. Seidel Aberrations Review
II. Ray Picture of Seidel Aberrations
III. Wavefront Picture of Seidel Aberrations
IV. Zernike Polynomials
V. Aberrations and Refractive Surgery
I. Seidel Aberrations Review
Ideal image formation: Only true if all rays make small angles
with the lens surface (paraxial approximation).
V=U+P
or
1/ v = 1/u + 1/ f
Seidel aberrations (or “third-order aberrations”) are
imperfections in the images formed by an optical system that
occur because the light rays are not paraxial. Unlike chromatic
aberration, these aberrations are not due to the type of lens
material and occur even if a single wavelength of light is used.
These aberrations are also not due to imperfections in the
manufacture of the lens. The Seidel aberrations are:
1
1. Spherical aberration
2. Coma
4. Curvature of field
5. Distortion
1. Spherical aberration
The effect: Paraxial rays and peripheral rays are
imaged at different distances from a spherical lens.
TSA
A
B
Question #1: Which set of the sets of rays shown are the paraxial
rays?
Correction:
1. Somewhat costly methods of correction: aspheric
lenses, GRadient INdex lenses (GRIN), lens
combinations.
2. Use spherical surfaces that minimize LSA. The
optimum occurs when the light bends equal amounts at
each surface. This, in turn, depends upon the object
and image positions.
2
Spherical aberration
is minimum for a
specific shape of lens
that depends on
object and image
distances.
Here S is called the “Coddington Shape Factor” and is a number
that describes the shape of the lens.
2. Coma
The effect: Off-axis rays spread into a comatic flare at the
paraxial image plane, similar to the shape of a comet’s tail
(hence the name).
Correction:
3
As with spherical aberration, use spherical surfaces that
minimize coma. Again, the best choice will depend on
the object and image positions.
Coma is zero for a lens
with S = 0.8 and an
object at optical infinity
Coma
4
The effect: off-axis rays in different meridians (planes) are
imaged at different distances from the lens. Basically, a
spherical lens behaves like a spherocylindrical lens for offaxis rays.
Tangential
rays
Sagittal
image
Tangential
image
Sagittal
rays
For the figure above, the object point is vertically offaxis. The tangential plane is the plane of the paper and
the sagittal plane is perpendicular to this plane. The
rays diverging from the object point in the tangential
plane are imaged as a line perpendicular to the plane of
the paper and those diverging in the sagittal plane are
imaged as a line parallel to the paper.
Correction:
As with other aberrations, the shape of the lens can
correct for RA. Corrected curve spectacles are
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Tscherning ellipses specify the front and back vertex
powers for a spherical lens that give zero RA for one
off-axis angle. These values of the vertex powers are
used in lens design to minimize RA (just as the
Coddington shape factors are used to minimize
spherical aberration and coma).
Tscherning ellipses specify the
appropriate front surface power
of a lens that corresponds the
necessary back vertex Rx to
give zero RA
4. Curvature of field
The effect: The image plane of a lens is not actually a plane –
it is a curved surface. Therefore, different pieces of an
extended object are imaged at different distances from the
lens. This leads to curved images of straight objects.
Paraxial
image plane
Actual image is
curved
Correction:
An appropriate combination of lenses can be used to
eliminate (or at least significantly reduce) curvature of
field.
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5. Distortion
The effect: The lateral magnification, m, changes with offaxis angle This distorts the image in the paraxial image
plane in one of two ways:
a) pincushion (m increases with increasing )
b) barrel (m decreases with increasing ).
Correction: combinations of lenses that exhibit barrel and
pincushion distortion
II. Ray Picture of Seidel Aberrations
One approach to understanding the source of the Seidel
aberrations is by considering rays of light incident on a
lens surface. As described earlier, these aberrations
occur because the light rays are not paraxial.
Paraxial approximation: assume that all angles between
light rays and lens surface are small so that we can use
the approximation sin  ≈ 
Lenses and the paraxial approximation:
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Snell’s Law: n1 sin1 = n2 sin2
Paraxial Snell’s Law:
n1 1 ≈ n2 2
The paraxial approximation allows us to find simple
relationships between objects and their ideal images
(like V = U + P). In reality, many of the rays that are
incident on a lens are NOT paraxial. So, ideal images
are NOT formed.
We can begin to see the effects of this if we expand the
sine function in a power series expansion:
sin  =   3/3! + 5/5! ....
In the paraxial approximation, we keep just the first
term on the right hand side. Seidel aberrations result
from the third-order (3) term.
Seidel Snell’s Law: n1(   3/3!) ≈ n2(   3/3!)
In fact, an exact analysis of all aberrations can be done
by not using any approximations, following a large
number of rays through the optical system, and using
Snell’s Law in it’s exact form for each ray at each lens
surface. The disadvantage is that this can be
computationally very difficult and take a great deal of
computing power.
In order to avoid having to do this and still get many of
the important features of the aberrations, we make
approximations about the shape of the aberrated
wavefront that leaves the optical system.
III. Wavefront Picture of Seidel Aberrations
8
In order to form an ideal (aberration free) image, the
wavefronts that form the image of a point object must
be spherical.
Real object point
Real image point
Spherical wavefronts
If there are aberrations present in an optical system, the
wavefronts that form the image are no longer spherical.
Goal: Quantify the ways in which the wavefronts are
not spherical in order to design the optical system to
alter the wavefronts so that they are as close to
spherical as possible (minimize aberrations).
exit pupil
of optical
system
ideal wavefront
y
paraxial
image plane



R
x
ideal image
point
9
Mathematically, we can describe the shape of the ideal
wavefront as:
x2  y2
W(x,y) 
2R
This is just the paraxial equation for a sphere of radius
R centered at the paraxial image point.

It is often convenient to use cylindrical coordinates
instead so we use x = cos() and y = sin() so that
we can express the wavefront as
2
W(x,y) 
2R
If the optical system suffers from aberrations, the
wavefront will not be perfectly spherical anymore. We
 to account for the case that the paraxial image
also need
location might not be on axis (off-axis object location).
exit pupil
of optical
system
ideal wavefront
y
paraxial
image plane



R
x
h
aberrated
wavefront
10
ideal image
point centered
a distance h
off axis
Mathematically, we try and express the aberrated
wavefront as the sum of the ideal wavefront plus some
correction terms that account for how far the real
wavefront is from ideal.
If the aberrations are all Seidel aberrations the
wavefront can be expressed as:
1
1
1
1
1
W(x, y) Wideal  S1 4  S 2 h 3 cos()  S 3 h 2 2 cos2 ()  (S 3  S 4 )h 2 2  S 5 h 3 cos()
8
2
2
4
2
spherical
aberration
coma
astigmatism
curvature
of field
Here S1-S5 are constants that depend on the
characteristics of the optical system. If any of them are
zero, that particular aberration does not exist for that
optical system.
Some things to notice:
1) Spherical aberration is the only “on axis”
aberration (no dependence on h)
2) All of the Seidel aberrations can be reduced by
limiting  and h (paraxial approximation)
3) If astigmatism is present (S3 not zero) than
curvature of field will also be present
4) Coma, astigmatism, and distortion are not
symmetric about the optic axis (depend on )
Question: Which Seidel aberration will likely have the biggest
effect on a patient’s vision?
A. spherical aberration
B. coma
C. astigmatism
D. distortion
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distortion
Practice with Seidel Wavefronts
An instructive visualization tool can be found at:
http://wyant.optics.arizona.edu/seidelWavefrontMaps/s
eidel.htm
Begin by setting all of the coefficients equal to zero.
Turn on the focus term to see how it produces an ideal
wavefront. Next turn on each Seidel aberration one at a
time to see how it affects the ideal wavefront.
IV. Zernike Polynomials
The Seidel aberrations do not describe all possible
aberrations of the wavefront!
Goal: Find a way to quantify the discrepancy between
an arbitrary aberrated wavefront and the ideal spherical
wavefront.
Idea: Find a set of mathematical functions that can be
added together in various combinations to form any
arbitrary waveform at the exit pupil of an optical
system.
* Look at analogous situation
We can get an understanding of how this works by
considering a different (and simpler) situation: making
1-dimensional waveforms using sines and cosines
(Fourier Analysis).
It turns out that we can make any arbitrary repeating
waveform by adding together appropriate amplitude
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and frequency sine and cosine waves (Fourier’s
Theorem).
In mathematical terms this means that we can write:
2n 
2n 
f(x)  a n cos
x   b n sin
x 
  
  
where f(x) is the waveform we are trying to produce, an
 and bn are constants that depend on the waveform we
are trying to produce, and  is the fundamental
wavelength of the waveform.
Practice with Fourier Analysis
Go to the Physics Education Technology website:
Click on “Play with sims”  “Math”  “Fourier: Making
Waves”. This is a java based program and has trouble
opening sometimes so you may need to close it and open
it again. Be patient.
Begin by playing the “Wave Game” for the first few game
levels (1-4) to begin to get some experience for how sine
waves can be added together to make a more complicated
waveform.
Next, go to “Discrete” and look at how sine waves of the
appropriate amplitudes can be added to produce a triangle
wave and a square wave.
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Of particular imporatance for later discussions: Which
waveform (triangle or square) can be best represented by
adding together a small number of sine waves?
* End of analogy
We want to find a set of mathematical functions (like the
sines and cosines are for 1-D waveforms) that can be
added together over a 2-D circular aperture (the exit pupil)
to produce any arbitrary waveform.
It turns out that the appropriate functions for this situation
are the “Zernike Polynomials”. The first nine are listed
below (there are an infinite number of them).
Zo = 1
Z1 = cos()
Z2 = sin()
Z3 = 22 – 1
Z4 = 2cos2(2)
Z5 = 2sin2(2)
Z6 = (32 – 2)cos()
Z7 = (32 – 2)sin()
Z8 = 64 - 62 + 1
So, an arbitrary waveform at the exit pupil of an optical
system can be written as:
W(,)  Wn Z n
where Wn are constants that depend on the shape of the
waveform
we are trying to represent (like the a’s and b’s

in the Fourier Analysis case).
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Note: When analyzing waveforms using Zernike
polynomials the reference case is typically a plane wave
leaving the exit pupil (flat wavefront).
The first 4 Zernike polynomials (Z0-Z3) correspond to an
ideal spherical wavefront centered on the paraxial image
point and are NOT aberrations:
Z0 is a constant or “piston” term that changes the overall
phase of the wave but does not affect the image
Z1 and Z2 are “tilt” or “prism” terms that shift the center
of the spherical wavefront to the correct paraxial image
location
Z3 is a “focus” term that gives the correct curvature to the
spherical wavefront to cause the paraxial image
Practice with Zernike Polynomials
An instructive visualization tool can be found at:
http://wyant.optics.arizona.edu/zernikes/zernikes.htm
Begin by setting all of the coefficients equal to zero.
Turn each of the first 4 on one at a time (set the
coefficients to some small non-zero number) to see how
these produce an ideal wavefront.
Zernike Polynomials and Seidel Aberrations
Suppose we write out the full wavefront expansion for an
arbitrary aberrated wavefront using the first 9 Zernike
polynomials. We obtain:
15

W(,) W0  W1 cos()  W2 sin()  W3 (2 2 1)  W4 2 cos(2)  W5 2 sin(2) 
W6 (3 2  2) cos()  W7 (3 2  2) sin()  W8 (6 4  6 2 1)
Take a moment to compare this expression with that for
the Seidel aberrated wavefront that we wrote down earlier.
Some important things to note:
a. There are many terms that look pretty similar (as they
should since we are hoping to be able to express any
aberrated wavefront using appropriate Zernike
polynomials).
b. The Zernike wavefront has no h dependence! That is
clearly a problem since we know the actual Seidel
aberrations do depend on the off-axis distance. That turns
out to be ok because we can determine the off-axis
dependence by doing a Zernike expansion of the
wavefront corresponding to each off-axis image location.
c. The first 4 terms (along with contributions from some
higher order terms) determine the paraxial (ideal)
wavefront. If we do a little math and ignore some angular
shifts (for the sake of clarity of this discussion) the
relationships are approximately:
Focus 2W3  6W8  W42  W52
Tilt  (W1  2W6 )2  (W2  2W7 )2

Additionally, the Seidel coefficients (S) can be related to

the Zernike
coefficients (W). Again, ignoring some
angular shifts the relationships are approximately:
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S1 = 48W8
(spherical aberration)
S 2  6 W62  W72
(coma)
S 3  4 W42  W52
(astigmatism)
Note: Because the Zernike wavefront expansion does not
give us any off axis height (h) dependence for a single
image location the Seidel aberrations of Curvature of field
and Distortion look like additional tilt and focus (same 
and  dependence if you compare those Seidel and
Zernike terms).
More Practice with Zernike Polynomials
Go back to the website that allows us to visualize the
Zernike abberated wavefronts. Try turning on appropriate
Zernike coefficients to make a wavefront that could be
described by spherical aberration, coma, and astigmatism
(be careful – remember that the focus and tilt terms might
be altered!)
Of course, that is not the whole story! The Zernike
expansion can describe many more aberrations than the
Seidel aberrations can! Try turning on some of the higherorder aberration coefficients and see what they do to the
wavefront.
V. Abberations and Refractive Surgery
The ultimate goal of refractive surgery would be to
completely eliminate refractive error including all
wavefront aberrations by resculpting the cornea to have
the correct shape. The first steps toward this are being
taken through “custom wavefront” refractive surgery.
17
How might this be accomplished?
Idea: Measure the aberrated wavefront that exits the eye
when a small region of the retina is illuminated (acts as a
point source). By the principle of reversibility if we can
send a wavefront into the eye that has this same aberrated
shape it will form a perfect point image on the retina
(except for diffraction).
So the goal becomes to take a plane wave (from an object
at optical infinity) and intentionally aberrate it so that it
has the correct shape for the eye to form an ideal image on
the retina.
How do we measure the aberrated wavefront? There are
several techniques. But the most popular is probably:
Shack-Hartmann Wavefront Sensor
http://www.lzr.com/centers/seattle/what_history.fxml
If the wavefront entering the lenslet array is a perfect
plane wave, each lens in the array will form a point
image at its focal point on the sensor. This is the
reference situation.
18
If the wavefront entering the lenslet array is aberrated,
the point image formed by each lens will be shifted on
the sensor by an amount proportional to the slope of the
wavefront at that lens. This information from all the
lenses can be combined to give the shape of the
wavefront entering the lenslet array.
J. Schwiegerling, “LASIK and Beyond,” Optics and
Photonics News, January 2002.
The aberrated wave is essentially used as a template to
etch the cornea with the same shape!
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Issues to consider (as of 2004):
1. Do higher order aberrations really impact vision?
Yes and no. In situations where the pupil size is large
(~ 5-6mm) or for highly misshaped corneas (such as
keratoconus) the correction of higher order aberrtions
can provide significant visual benefit. Also, some
studies show that normal LASIK surgery tends to
increase the amount of higher order aberrations
(particularly spherical) compared to pre-surgery.
2. Does the benefit of wavefront correction apply to all
viewing conditions?
No! Remember that the correction was done for
distance vision (object at optical infinity). If the eye
accommodates to view a nearby object, the overall
aberrations of the eye change!
Why? The refractive power of the eye comes from both
the cornea and the crystalline lens and both of these
contribute to the overall aberrations of the eye. As the
crystalline lens changes power in accommodation, the
aberrations it produces change and so do the overall
aberrations of the eye. So, the cornea that has been
altered to correct for aberrations at distance viewing
will not typically give aberration correction for near
viewing (and may actually make the aberrations worse
for this case).
3. Is the Zernike representation of higher order
aberrations the best method for wavefront correction?
20
Unclear. Remember the lesson of trying to make a
square wave using sines and cosines! Not all
waveforms are practical to make with a small number
of sines and cosines. Similarly, not all wavefront
shapes are practical to make with a small number of
Zernike polynomials!
Current practice is to correct a 3.4mm pupil size up to
4th order Zernike aberrations (mostly Seidel) and up to
8th order for a 7.3mm pupil size. However, some
studies show that this is not sufficient for all cases and
that up to 36 or more Zernike terms may be needed to
describe the aberrated wavefront in some instances.
That may make the Zernike expansion of the aberrated
wavefront impractical in some cases.
References
R.R. Krueger, R.A. Applegate, and S.M. MacRae, Wavefront
Customized Visual Correction: The Quest for Super Vision II,
SLACK Inc. (2004).
J.C. Wyant and K. Creath, Applied Optics and Optical
Engineering, Vol. XI, Chapter 1, “Basic Wavefront Aberration
Theory for Optical Metrology,” Academic Press (1992).
L.A. Carvalho, “Accuracy of Zernike Polynomials in
Characterizing Optical Aberrations and the Corneal Surface of the
Eye,” Investigative Ophthalmology & Visual Science, Vol. 46, No.
6 (2005).
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M.K. Smolek and S.D. Klyce, “Zernike Polynomial Fitting Fails to
Represent All Visually Significant Corneal Aberrations,”
Investigative Ophthalmology & Visual Science, Vol. 44, No. 11
(2003).
S.D. Klyce, M.D. Karon, and M.K. Smolek, “Advantages and
Disadvantages of the Zernike Expansion for Representing Wave
Aberration of the Normal and Aberrated Eye,” Journal of
Refractive Surgery, Vol. 20 (2004).
J. Schwiegerling, “LASIK and Beyond,” Optics and Photonics
News, January 2002.
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