Download Representation of Wavefront Aberrations

Slide
4th Wavefront Congress - San Francisco - February 2003
Representation of
Wavefront Aberrations
Larry N. Thibos
School of Optometry, Indiana University,
Bloomington, IN 47405
[email protected]
http://research.opt.indiana.edu/Library/wavefronts/index.htm
The goal of this tutorial is to provide a brief introduction
to how the optical imperfections of a human eye are
represented by wavefront aberration maps and how
these maps may be interpreted in a clinical context.
1
Slide
2
Lecture outline
• What is an aberration map?
– Ray errors
– Optical path length errors
– Wavefront errors
• How are aberration maps displayed?
– Ray deviations
– Optical path differences
– Wavefront shape
• How are aberrations classified?
– Zernike expansion
• How is the magnitude of an aberration specified?
– Wavefront variance
– Equivalent defocus
– Retinal image quality
• How are the derivatives of the aberration map interpreted?
My lecture is organized around the following 5 questions.
•First, What is an aberration map? Because the aberration
map is such a fundamental description of the eye’ optical
properties, I’m going to describe it from three different, but
complementary, viewpoints: first as misdirected rays of
light, second as unequal optical distances between object and
image, and third as misshapen optical wavefronts.
•Second, how are aberration maps displayed? The answer
to that question depends on whether the aberrations are
described in terms of rays or wavefronts.
•Third, how are aberrations classified? Several methods are
available for classifying aberrations. I will describe for you
the most popular method, called Zernike analysis.
lFourth, how is the magnitude of an aberration specified? I
will describe three simple measures of the aberration map
that are useful for quantifying the magnitude of optical error
in a person’s eye. Other measures based on the quality of the
retinal images are more sophisticated conceptually, but may
be more important for predicting the visual impact of ocular
aberrations.
lLastly, how may we interpret the spatial derivatives of the
aberration map?
Slide
3
Clinical Examples
• Prism (“first Zernike order”)
– Video-keratoscopic errors
• Sphero-cylindrical (“second Zernike order”)
– myopia, astigmatism
• Keratoconus (“third Zernike order”)
– Vertical coma
• LASIK (“fourth Zernike order”)
– Spherical aberration
• Dry eye (“irregular higher order”)
To illustrate the concepts associated with aberration maps
I will use a series of clinical examples of normal and
abnormal eyes. Our goal in analyzing these eyes using
aberrometry is to describe the nature of the optical
problem and how it may be diagnosed by inspection of the
aberration map and its associated aberration coefficients.
Slide
4
Optically perfect eye is the “Gold Standard”
Rays from distant point
source, P.
P
perfect
retinal
image
Key points:
•
All rays from P intersect at common point P on the retina.
•
The optical distance from object P to image P is the same for
all rays.
•
Wavefront converging on retina is spherical.
To begin with the question “What is an aberration map” it
is helpful to consider an emmetropic eye which is free of
aberrations. Such an eye is optically perfect and therefore
may be used as a “Gold Standard” for judging the optical
imperfections of real eyes. The key property of a perfect
eye is that it focuses a distant point source of light into a
perfect image on the retina.
We can account for this perfect retinal image three
different ways.
• Firstly, in a perfect eye, all of the rays emerging from a
point source of light at the eye’s far point P that pass
through the eye’s pupil will intersect at a common point Pprime on the retina.
• Secondly, the perfect eye has the property that the
optical distance from the object to the image is the same
for every point of entry in the eye’s pupil.
• Thirdly, the wavefront of light focused by the eye has a
perfectly spherical shape.
The gold standard of optical quality depicted here is for the
case of an emmetropic eye with relaxed accommodation.
Slide
General case: accommodating or ametropic eye
P
rays
P
Object point
perfect
retinal
image of
object point
Key points:
•
All rays from P intersect at common point P on the retina.
•
The optical distance from object P to image P is the same
for all rays.
•
Wavefront converging on retina is spherical.
In general, we would like to have a gold standard that also
works for an object point other than infinity so that we can
describe the aberrations of an accommodating eye or an
ametropic eye.
Fortunately, the same three conditions for perfection
devised for an emmetropic eye with a far-point at infinity
apply also in this more general case of a near-sighted or
far-sighted eye.
The last two ways of describing a perfect eye are based on
the concept of “optical distance”, which is an important
concept in optics that is very useful for interpreting the
aberration map of eyes. The meaning of the term “optical
distance” is explained in the next slide.
5
Slide
6
Definition: Optical path length
Optical Path Length specifies distances in wavelengths ( ).
Equal optical length => equal phase
Water
Air
Optical distance = physical distance X refractive index
In everyday life we measure distances in physical units of
meters. However, in optics it is common to measure distances
with a ruler that is calibrated in wavelengths of light. Such a
ruler effectively measures the number of times light must
oscillate in traveling from the object to the image. If all rays
oscillate the same number of times, then the light will arrive at
the retina with the same phase and therefore will interfere
constructively to produce a perfect image.
For example, light has a shorter wavelength when it
propagates through the watery medium of the eye compared to
air. Consequently, rays traveling only a short physical distance
in water undergo the same number of oscillations, and
therefore travel the same optical distance, as a longer path in
air.
A simple formula for computing optical distance is to multiply
the physical distance times refractive index.
This concept of optical path length is very useful for
understanding the imaging property of lenses as shown in the
next slide.
Spheres are perfect wavefronts
Point sources produce
spherical wavefronts.
Spherical wavefronts
collapse onto point images.
Lens
A lens forms an image by refracting rays. If the optical distance
taken by every ray that passes through the lens is the same, then all
the rays will arrive at the image plane with the same phase to form a
perfect image. Thus the perfect eye is one which provides the same optical
distance from object to image for all the rays passing through the eye’s
pupil.
This concept of optical distance is also useful for understanding
wavefronts of light. A wavefront is the locus of points which are the
same optical distance from their source point. When light
propagates in a homogeneous medium, equal optical distances are
equal physical distances. Consequently, the wavefronts produced by
a point source are perfect spheres because all points on a sphere are
the same distance from the center of the sphere.
By the same line of reasoning, a converging spherical wavefront will
collapse down to a perfect point image. Thus we may conclude that
a perfect eye converts spherical expanding wavefronts into spherical
collapsing wavefronts. If we require that an eye be well focused for
distant objects, then the sphrerical wavefronts arriving at the eye will
be plane waves. Nevertheless, the perfect eye will focus these plane
waves into spherical wavefronts centered on the retinal
photoreceptors.
Notice in this drawing that the rays of light emanating from a point
source are always perpendicular to the wavefront surface. This
perpendicular relationship makes rays and wavefronts
interchangeable and complementary concepts. If you know where
the rays are, you can draw the wavefront, and visa-versa.
Slide
Aberrated eye
Rays from distant point
source, P.
P
flawed
retinal
image
Key points:
•
Rays do NOT intersect at the same retinal location.
•
The optical distance from object to retina is NOT the same
for all rays. OPD = Optical Path Difference
•
Wavefront is NOT spherical.
Now that we know what a perfect eye is like, we can
define an aberrated eye 3 ways that correspond to our 3
ways of defining optical perfection.
Firstly, the rays do NOT focus at a common retinal
location.
Secondly, the optical path distance from an object point
to the retinal image is NOT the same for all rays passing
through the pupil. The difference in optical distance
between any ray and the center ray is called the Optical
Path Difference, or OPD.
And thirdly, the wavefronts inside the eye are NOT
spherical, they are distorted.
8
What is an aberration map?
An aberration map is a visualization of how
the eye’s aberrations vary across the pupil.
3 Formats:
• Map #1: ray deviations
• Map #2: optical path length differences
• Map #3: wavefront shape
2 Viewpoints:
• Light propagating towards the retina
• Reflected light propagating away from the eye
With this background in optical theory, we can now answer the first
question posed: What is an aberration map?
By definition, an aberration map is a graphical display or
“visualization” of how aberrations vary across the eye’s pupil.
Since we have defined aberrations 3 different ways, there are 3
different maps we can draw.
Firstly, we can show how each ray deviates from a perfect ray.
Second, we can show how the optical distance from object to image
varies across the pupil.
And thirdly, we can show how the shape of the wavefront differs from
a sphere.
Although these three descriptions are equivalent, they represent
different ways of thinking about aberrations that are all valuable in
their own way for interpreting clinical cases of optical dysfunction.
Thus, when we interpret an aberration map of an eye we may do so
three different ways, in terms of where rays strike the retina, in terms
of the optical distance from the object to the retina through different
points of entry in the pupil, and in terms of the shape of the wavefront
of light produced by the eye.
Not only do we have 3 formats for displaying the aberration map, but
each of these 3 formats can be used from two different viewpoints.
The first viewpoint is of light propagating from the external world into
the eye towards the retina. The second viewpoint is of light reflected
by the fundus back out of the eye, propagating away from the
individual.
These two viewpoints are interchangeable conceptually, but they are
very different from a practical point of view. Fortunately, by having
two different viewpoints available for investigating the same
phenomenon, we are able to overcome a serious practical problem.
The perfect eye
P
object
point
rays
P
Image
reflected
point
wavefront
ER
S
LA
The problem is that the eye is a closed system so it is difficult for a
clinician to inspect the light rays and wavefronts once they enter the
eye to see if they are perfect. For this reason, objective methods have
been devised for measuring the optical aberrations of eyes using light
that is reflected out of an eye due to a laser beam that is focused onto
the retina at some point P -prime.
Optically, it doesn’t matter whether we think of point P as the object
and P-prime as the image, or visa-versa. The rays in this diagram do
not depend on the direction of propagation of the light. Similarly, the
optical distance between P and P’ is independent of the direction of
light propagation. This means we can avoid the practical problem of
having a closed system if we characterize the eye’s aberrations in
terms of the light reflected out of the eye from a small spot of light
placed on the retina with a narrow laser beam.
If the eye is perfect, then a point source located on the retina at point
P’ will emerge from the eye as a perfect spherical wavefront centered
on the far-point P outside of the eye. So, in principle, all we need to
do to characterize the optical quality of a real eye is to capture a
wavefront of light emerging from the eye and compare it to a sphere.
Any deviations from a sphere are called “aberrations” and a
quantitative description of these deviations across the pupil is called a
“wavefront aberration function” or simply an “aberration map”.
To say the same thing in terms of optical path lengths, all we need to
measure is the difference in optical path length between object point
and image point through every possible pupil location.
To say the same thing a third time, now in the language of ray optics,
all we need to measure is the direction of individual rays of light as
they emerge from the eye’s pupil. By comparing the actual rays to the
perfect rays we can construct a ray aberration map.
Slide
11
Aberrometers measure the wavefront reflected from point P
Reference sphere
Wavefront
y
Wavefront
Error
P
z
P
x
Deviations of the wavefront from a perfect sphere indicate
the presence of optical aberrations.
In practice, an instrument called an “aberrometer” is
used to measures the shape of the wavefront of light
reflected back out of the eye. If that wavefront is a
sphere centered on the eye’s far point, then the eye is
optically perfect except for a focusing error associated
with myopia or hyperopia. To eliminate this focusing
error as well and make the eye truly perfect, the far point
must be at infinity. Since a spherical wavefront with
infinite radius of curvature is a plane wave, the perfect
emmetropic eye is characterized by the plane wave it
reflects from a point source focused on the retina.
A plane wave is a very convenient reference wavefront
that makes it is easy to specify the aberrations of an eye.
Aberrations are given by the difference between the
actual wavefront and a reference wave in the x-y plane of
the pupil.
Slide
12
Specification of the wavefront error of an eye
y
Wavefront
Z(x,y)
Ideal reference
plane-wave
P
z
Error
W(x,y)
x
Pupil
circle
The surface Z(x,y) is also the error W(x,y) between the reflected
wavefront and the ideal reference plane wave in the pupil plane.
In summary, the wavefront aberration map of an eye indicates
the distance between the reflected wavefront and the ideal
plane wavefront located in the plane of the eye’s pupil. This
distance between the reflected wavefront and the pupil plane
represents an error of optical path length which varies from
point-to-point across the pupil. The aberration map can
therefore be quantified by a mathematical function W that
depends on the x- and y-coordinates of points inside the eye’s
pupil. Thus we see that the aberration map can be understood
and displayed using all 3 formats of ray optics, optical path
errors, and wavefront shape errors regarless of whether the
light is propagating into the eye or reflected out of the eye.
Slide
13
Coordinate System for Aberration Maps
Every point in
pupil plane has a
location in
rectangular (x,y)
coordinates and
in polar (r, )
coordinates.
y
r
x
The aberration
function W(x,y)
or W(r, ) may be
defined with
either coordinate
system.
This raises my second question, “How are aberration
maps displayed?” To display aberration maps we first
have to agree on a coordinate system for the eye’s pupil
plane. The natural coordinate system for use in
ophthalmic optics is the standard mathematical
coordinate system shown here. Every point in the pupil
plane can be located uniquely by its x-y coordinates of
the rectangular coordinate frame, or in terms of the polar
coordinates r and theta.
The aberration function W can therefore be described
either as a function of the x-y rectangular coordinates of
the eye, or as a function of the r-theta polar coordinates
of the eye.
Typically we use the same coordinate system for either
eye. However, to take account of the bi-lateral symmetry
of eyes, it is often best to flip the coordinate system about
the y-axis so that the positive x-axis points nasally rather
than to the right.
How are aberration maps displayed?
Ray Errors
Optical Distance
Errors
Wavefront Error
Aberration type: negative vertical coma
Having agreed on a coordinate system, we are able to display
aberration maps 3 ways using the three formats just described.
Here we show aberration maps for a particular type of aberration
called “coma”, plotted these 3 different ways.
On the left we have a graphical depiction of the ray errors produced
at various locations. The red circle denotes the pupil margin and the
arrows shows how each ray is deviated as it emerges from the pupil
plane.
This pattern of ray aberrations indicates that rays near the upper and
lower margins of the pupil are deflected upwards relative to the
perfect ray, while the rays near the middle of the pupil are deflected
downwards relative to the perfect ray.
In the center figure we see that the optical distance from object to
image is greater over most of the lower half of the pupil compared to
the upper half.
The figure at the right shows the shape of the wavefront aberration
function. This diagram shows that the phase of the light is retarded
over most of the lower half of the pupil and advanced in the upper
half.
Notice that the wavefront map on the right agrees with the optical
distance map in the middle, but has opposite sign. This is because a
long optical path causes phase retardation, and a short optical path
causes phase advancement.
Notice also that the ray map on the left agrees with the wavefront
map on the right. To connect these two maps, recall that each of the
vectors in the map at the left indicates the local slope of the surface on
the right. Consequently, these vectors are showing the local direction
of propagation of the wavefront, which is always perpendicular to the
surface of the wavefront.
Thus we see that although the three maps look quite different, they all
describe the same optical aberration from three different perspectives.
Slide
15
Wavefront Aberration in Keratoconus
Reference
Advanced phase <= Short optical path
Ectasia
Retarded phase <= Long optical path
Clinically, the preceeding example of vertical coma is
commonly associated with the corneal disease Keratoconus. If
the protruding cone of keratoconus is in the inferior cornea,
then the optical path of rays in the inferior half of the pupil will
be longer because rays will spend relatively more time in water
than in air. Because light travels slower in water, the
wavefront will be retarded in the vicinity of the ectasia.
This example shows how to conceive of optical path errors and
wavefront errors based on a simple model of the physical cause
of an aberration.
Slide
16
Clinical Example: LASIK-induced spherical aberration
n=1
P
Before surgery
P
After surgery
n>1
P′
P′
Here is another example showing how to interpret the
ocular aberrations of a clinical condition, in this case
LASIK refractive surgery.
The top of this slide shows an optical diagram for a
myopic eye that is otherwise free of optical aberrations.
For such an eye, light reflected from a point of light at
retinal point P-prime would form a perfect image at the
eye’s far-point, P. Consequently, the wavefront
converging on P will be a perfect sphere.
After LASIK surgery, the cornea has become flatter at the
apex and so the rays emerging from the center of the
pupil will be parallel and focused at infinity, as expected
for an emmetropic eye. However, outside this central
region the cornea is still highly curved and so the eye is
still myopic for marginal rays. This combination of
emmetropia for paraxial rays and myopia for marginal
rays is a classical case of positive spherical aberration.
The aberration maps for an eye with spherical aberration
are shown on the next slide.
Slide
17
Aberration maps for LASIK
Ray Errors
Optical Distance
Errors
Wavefront Error
1
0.5
0
Aberration type: Seidel spherical aberration
Looking first at the ray aberration map on the left, we see that
the central region of the pupil is largely free of aberrations.
However, the ray aberrations grow rapidly near the pupil
margin where corneal curvature is greater.
The map of optical path length for this example of spherical
aberration reveals that paraxial rays follow a longer path to a
distant far point than do the marginal rays, which implies that
the phase of the central rays will lag behind the phase of the
marginal rays. This explains why the wavefront error map on
the right shows a concave wavefront with the central area
lagging behind the marginal portion of the wavefront.
Please keep in mind that this example, and the previous one, are
idealized cases created for tutorial purposes. The aberration
structure of real human eyes is usually complicated by the
presence of a variety of different kinds of aberrations. In order
to describe the complicated aberration structure of real eyes, it
is very useful to have a method available for systematically
classifying aberrations. With such a method we may
decompose any aberration map, no matter how complicated,
into a combination of basic, elementary forms.
This leads to our third question, How are aberrations classified?
Slide
How are aberrations classified?
Mathematical descriptions of wavefront aberrations:
• Seidel aberrations
• Taylor series expansion of aberration map
• Zernike expansion of aberration map
Classical aberration theory was developed by Seidel for
symmetrical optical systems. Unfortunatley, the lack of
symmetry in human eyes requires a more general
treatment, such as Taylor series expansion of the
wavefront map, or Zernike expansion of the aberration
map.
Each method has its advantages and disadvantages. For
today’s lecture I’ve selected the Zernike method for
illustrating the concept of systematic classification of
aberrations into fundamental forms.
18
Slide
19
Wavefront error for defocus
Some of these fundamental forms are familiar to
everyone in the field of ophthalmic optics.
For example, the wavefront error for a myopic or
hyperopic eye has a spherical shape that is defined
mathematically in terms of a simple equation that
contains the key expression x-squared plus y-squared.
There are some other constants in the equation which
aren’t essential, but which provides some convenient
features. For example, the minus-1 in the equation forces
the average error to be zero. All of the fundamental
aberrations in Zernike’s expansion except the first one
share this property of having zero mean error.
Slide
20
Wavefront error for astigmatism
r = radius
= meridian
The equations Zernike used to represent wavefront
errors often look much simpler if written not in terms of
the X-Y coordinates of a rectangular coordinate system,
but instead in terms of the polar coordinates r and theta.
An eye with astigmatism, for example, will reflect a
saddle-shaped wavefront which has a rather simple
algebraic equation, when written in polar form.
Now that we see how to visualize the eye’s wavefront
aberration function and how to describe it
mathematically with an equation, the next step is to
combine simple wavefronts like those I”ve been showing
to make more complex wavefronts that can describe the
aberration structure of real eyes.
Anatomy of Zernike basis functions
order
+2
Z2 = 6 r ⋅c o s ( 2)
Astigmatism:
normalization constant
Coma:
frequency
−1
2
polynomial
harmonic
Z3 = 8 (3r − 2r) ⋅sin( )
3
To do this we need a catalog of basic shapes that we can add
together. Since the basic building blocks of Zernike will be the
basis for describing the aberration structure of eyes, they are
known as “basis functions”.
Each Zernike basis function is the product of two other functions,
one of which depends only on radius and the other which depends
only on meridian.
For example, the astigmatism wavefront is described by the
formula:
square-root of 6 times r-squared times cosine of 2 theta
Notice that the term involving the radius variable “r” is a
polynomial. The largest exponent in this polynomial is 2, so it is
called a “second-order” polynomial. The trigonometric term
involving the meridian variable theta is a co-sinusoidal harmonic,
in this case with a frequency of 2. The normalization constant
square-root of 6 is not absolutely necessary but its inclusion
provides a mathematically convenient property that the variance of
the waveform will be 1.
This basic pattern of a normalizing constant times a polynomial
times a harmonic occurs for all of the Zernike functions. Another
example is a basis function called “coma” which is the product of a
3rd order polynomial and a sinusoidal harmonic of frequency 1.
By convention, the superscript is given a minus sign when the
trigonometrical harmonic is a sine function, and is given a plus
sign when the harmonic is a cosine function.
Periodic table
of Zernike
functions
0
2
Z
f
n
3
Radial order
1
4
f = frequency
n = order
5
-5 -4 -3 -2 -1 0 1 2 3 4 5
Meridional frequency
The first 21 Zernike basis functions, or “modes” as they are
often called, are shown in this catalog.
The Zernike catalog is best viewed as a pyramid, rather like the
periodic table of the elements. Each row in the pyramid
corresponds to a given order of the polynomial component of
the function and each column corresponds to a different
meridional frequency. Again, by convention, co-sinusoidal
harmonics are assigned positive frequencies and sinusoidal
harmonics are assigned negative frequencies.
Some of these aberrations have names like “spherical
aberration” and “coma” but most are simply identified by their
frequency and order. Although it is possible to number these
modes in sequence starting from the top of the pyramid, this is
not the most intuitive naming scheme. Instead, the Optical
Society of America has recommended a double-script notation
which designates each basis function according to its order and
frequency. The order is used as a subscript and the frequency is
used as a superscript to unambiguously and conveniently
identify each mode.
Traditionally, ophthalmology and optometry have been
concerned solely with Zernike aberrations of the first and
second orders. The first order terms are called vertical and
horizontal prism, and the second order terms are called
astigmatism and defocus. Only these 5 aberrations of the first
and second order can be corrected by conventional spectacle
lenses or contact lenses. The current resurgence of interest in
visual optics research and refractive surgery is focused
primarily on the measurement and correction of the higher
order aberrations in the 3rd, 4th,and 5th orders.
Slide
23
OSA Standards for Reporting Ocular Aberrations*
• Double -script notation
– Subscript = radial order
– Superscript = meridional frequency
• Meridian measured counter-clockwise
from horizontal
• Normalizing constants produce unit
variance
* Thibos, Applegate, Schwiegerling, & Webb (2000). Standards for reporting
the optical aberrations of eyes. In Lakshminarayanan, V. (Ed.), Trends in
Optics and Photonics. Washington, DC: Optical Society of America.
Reprinted in Customized Corneal Ablation (2001) by MacRae, Krueger, and
Applegate. Slack Inc.
It is an unfortunate fact of history that a variety of ways
for defining the Zernike polynomials exists in the optics
literature. To avoid potential confusion, a taskforce was
formed in the year 2000 under the auspices of the Optical
Society of America to recommend standards for
reporting optical aberrations of the eye. The full
recommendations of the taskforce are published in the
series “Trends in Optics and Photonics” and was
reprinted in the recent book by MacRae, Krueger, and
Applegate.
The main recommendations are:
(1) the use of a double-script notation for identifying
each Zernike polynomial, with a subscript indicating
radial order and a sperbscript indicating meridional
frequency
(2) the meridian angle be measured counter-clockwise
from the horizonatal, which is the standard ophthalmic
convention for specifying the axis of astigmatism
(3) and the use of normalization constants which make
all of the Zernike polynomials have unit variance.
Slide
24
Zernike expansion = weighted sum of modes
W(r, ) = a00 ⋅ Z00
+a1−1 ⋅ Z1−1 + a1+1 ⋅ Z1+1
+a2−2 ⋅ Z 2−2 + a20 ⋅ Z20 + K
=
∑ ∑
anf ⋅ Znf
order frequency
a
f
n
= aberration coefficient (weight)
Z nf = Zernike basis function (mode)
To summarize Zernike analysis, we describe the aberration
structure of an eye mathematically as the weighted sum of
Zernike basis functions. Such a description is called a “Zernike
expansion” of the wavefront aberration.
The weight which must be applied to each basis function, or
mode, when computing the sum is called an aberration
coefficient.
Each aberration coefficient is just a number, with physical
units typically specified in microns, or sometimes they are
reported in the units of the wavelength of light.
The aberration coefficients of a Zernike expansion are
analogous to the Fourier coefficients of a Fourier expansion,
which are in turn analogous to the energy spectrum of a light
source. Thus it is common language to speak of chromatic
spectrum of light, or the “Fourier spectrum” of a waveform,
and in the same way we may speak of the “Zernike spectrum”
of the eye’s optical aberrations.
Slide
25
Statistics of Zernike modes
Mean( Znf ) = 0
for all n > 0, all f
Variance( Znf ) = 1,
for all n, f
Variance(a ⋅ Znf ) = a 2 , a = aberration coefficient
Total variance = sum of individual variances =
∑a
2
modes
One convenient feature of the Zernike expansion I’ve just
described is that every mode except the zero-order mode
has zero mean, and they all are scaled so as to have unit
variance. This puts all of the modes on a common basis
so their relative magnitudes can be compared easily.
Furthermore, the variance of a weighted Zernike mode is
just the square of the aberration coefficient.
An attractive feature of the set of Zernike functions is
that they are mutually orthogonal, which means they are
independent of each other mathematically. The practical
advantage of orthogonality is that we can determine the
amount of defocus, or astigmatism, or any other Zernike
mode occurring in an aberration function without having
to worry about the presence of other modes.
Furthermore, the orthogonality of the Zernike basis
functions makes it easy to calculate the total variance in a
wavefront as the sum of the variances in the individual
components.
Normal eye, weak higher-order aberrations
Phase = - OPD
6 mm pupil
z=d24o8i.od.sh.ex01.sr1.zc;
I can imagine that some of you are thinking that this lecture
has had more than enough optical theory for such an early
hour of the morning. However, the reward for your
perseverance is that we now have all of the tools we need to
begin interpreting aberration maps of real human eyes.
Over the course of the next several slides I will show you some
examples of aberration maps from normal, healthy eyes. It
would be a mistake to think that normal eyes are not aberrated.
In fact, all eyes are aberrated to some degree. What
distinguishes the normal from abnormal eye is the magnitude
of these aberrations. Thus it is important to gain experience
with normal eyes first so that we will be able to recognize eyes
that are outside the normal range.
In order to focus our attention on the higher-order aberrations,
I have assumed that the lower-order aberrations of prism,
defocus, and astigmatism can be corrected perfectly with
spectacle lenses. Thus, all we are seeing in this aberration map
is the effect of the higher order aberrations.
The first example is of an eye with very little aberration in the
higher order modes. The maximum difference between peak
and valley measurements is less than 1 micron for this eye over
a 6mm pupil. When I look at this map, I am struck by the
prominent combination of a single peak and a single valley.
This is characteristic of the aberration coma and since the axis
connecting peak to valley is nearly horizontal, this eye appears
to be dominated by horizontal coma.
Slide
27
Normal eye, weak higher-order aberrations
Zernike Spectrum
Order Distribution
Radial Order
0
1
H. coma
2
3
4
5
6
7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
0
0.1
0.2
Meridional Frequency RMS Error ( µ m)
A quantitative Zernike expansion of this aberration map
provides a large number of aberration coefficients to
assess. A convenient way to visualize this 2-dimensional
Zernike spectrum graphically is to build a pyramid of
boxes in the shape of the periodic table of Zernike basis
functions as shown on the left. Each box is colored using a
grey scale of intensity to indicate the value of the
corresponding Zernike coefficient. White signifies a large
positive value, black signifies a large negative value, and
grey signifies zero.
The most prominent mode in the spectrum of this eye
occurs for the third order, with meridional frequency of +1.
This is indeed the mode called horizontal coma, just as we
suspected.
The Zernike spectrum is sometimes simplified by
combining all of the information contained in a given row
of the pyramid. The correct way to combine aberration
coefficients on the same row of the pyramid is to sum their
squared values. The square root of this sum is called RMS
error, which stands for “root mean squared”. For this eye
we clearly see an exponential decline in RMS error with
Zernike order that is a common feature of normal eyes.
Slide
Normal eye, moderate higher-order aberrations
Phase = - OPD
6 mm pupil
z=d24oa5.od.sh.ex02.sr1.zc;
This next example is of a normal eye with a moderate
level of higher-order aberrations. The pattern in the
aberration map for this eye is very different from the
previous eye. Here we see 4 prominent peaks in the
aberration function, which suggest the presence of a
large amount of a 4th order aberration called
“quadrafoil”.
28
Slide
29
Normal eye, moderate higher-order aberrations
Zernike Spectrum
Radial Order
0
1
Trefoil
Quadrafoil
V. coma
Order Distribution
0
1
2
2
3
3
4
4
5
5
6
6
7
7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
0
0.1
0.2
Meridional Frequency RMS Error ( µ m)
This visual inspection is confirmed by a quantitative
analysis that reveals a strong negative mode in the 4th
row, 4th column. The spectrum also shows two
prominent modes in the third order. The one with
frequency -1 is vertical coma and the one with frequency
-3 is called “trefoil”. The distribution by order shows
that this eye has equal amounts of third and fourth order
aberrations, which is rather unusual for normal eyes.
Slide
Normal eye, strong higher-order aberrations
Phase = - OPD
6 mm pupil
z=d24o9l.os.sh.ex03.sr1.zc;
My third example of normal eyes has a relatively large
amount of higher order aberration. This pattern is more
difficult to assess visually. The prominent pairing of
peak and valley suggests the presence of vertical coma,
but there evidently are other modes as well that
complicate the aberration map.
30
Slide
31
Normal eye, strong higher-order aberrations
Zernike Spectrum
Order Distribution
Radial Order
0
1
2
3
4
5
6
7
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7
0
0.4
0.8
Meridional Frequency RMS Error ( µm)
These observations are supported by the Zernike
spectrum, which shows several stong components, one
of which is vertical coma.
Notice that the magnitude of these aberrations is still less
than 1 micro meter of RMS error for any single Zernike
order.
Slide
Irregular aberrations: tear film disruption
Phase = - OPD
6 mm pupil
a1_AB_40sec.m
A clinical example of a highly irregular aberration map
is shown here for a case of a dry eye patient with a
disrupted tear film. The red and yellow areas of the
phase map on the left indicate areas where optical path
length is relatively short and the phase is relatively
advanced. These are regions where the corneal tear film
has been replaced by air as it becomes thinner and
eventually breaks up.
32
Slide
Irregular aberrations: tear film disruption
Radial Order
Zernike Spectrum
0
1
2
3
4
5
6
7
8
9
10
-10 -8 -6 -4 -2 0 2
4
6
Order Distribution
8 10
0
0.2
0.4
Meridional Frequency RMS Error ( µ m)
Because of the complicated shape of the aberration map,
the Zernike spectrum for this dry eye example contains
many significant terms with much higher radial order
and meridional frequency than usually occurs in the
normal eye.
33
Slide
Clinical Applications: LASIK refractive surgery
Before surgery
After surgery
3
3
0
2
2
1
-3
-2
1
-1
-1
-1
0
0
0
0
-1
-1
-1
-2
-2
-1
-2
-3
-3
-3
-2
-3
-1
0
1
2
3
-3
-2
-1
0
1
2
3
1.2
1.0
0
RMS error (µm)
RMS error (µm)
1
3
3
4
4
5
5
0
6
6
1.0
7
7
8
8
Zernike order
9
9
10
10
3
4
5
6
7
8
Zernike order
9
1
0
One last example shows the aberration map before and
after LASIK surgery to correct 3 diopters of myopia.
Before surgery this eye had excellent optical quality as
indicated by the nearly flat aberration map and small
Zernike coefficients. After surgery the eye was nearly
emmetropic but the optical quality was not as good.
Now the aberration map is not as flat and the Zernike
analysis shows significant increase in the magnitude of
spherical aberration.
34
Slide
35
How is the magnitude of an aberration specified?
4 common metrics:
§ Wavefront variance
§ RMS error
§ Equivalent defocus
§ Retinal image quality
These clinical examples give a brief glimpse of the great
variety of aberration maps encountered in normal
human eyes. Although the experienced clinician can
make a qualitative assessment of the eye’s optical system
by inspecting the aberration map visually, some
simplification is needed to answer questions like:
“Which eye is more aberrated?”
This issue of judging the degree of aberration of an eye
leads me to my fourth question, “How is the magnitude
of an aberration specified?”
Of the four common metrics listed here, the first three
are closely related to each other but the fourth represents
a completely different approach that requires a little
more optical theory to understand fully.
Slide
36
Wavefront Variance =
2
1
Z (x, y) − Z )
(
∑
# points x, y
WFE = Z(x,y)
Y
X
RMS Error =
Wavefront Variance
The first, and most popular metric of the strength of an
aberration is wavefront variance. The variance of a
wavefront is a measure of how much it differs from a
plane wave. It is easily computed by summing the
squared heights of every point on the curve. However, if
you know the aberration coefficients for an eye you don’t
need to carry out this calculation because the wavefront
variance is just the sum of squared Zernike coefficients.
The quantity called “RMS error” is another name for the
square root of wavefront variance. It is a popular metric
because the physical units are in microns.
One of the awkward features of wavefront variance is
that it changes when pupil diameter changes. To get
around this problem, we can use an alternative metric
called “equivalent defocus” which factors out pupil
diameter.
Slide
37
Clinical interpretation of wavefront variance
Equivalent defocus is:
the amount of defocus required to produce
the same RMS wavefront error produced
by one or more higher-order aberrations.
Equivalent Defocus (D) = 4
3
RMS Error ( m)
2
Pupil Area (mm )
“Equivalent defocus” is defined as the amount of
defocus required to produce the same wavefront
variance as found in one or more higher-order
aberrations.
A simple formula allows us to compute equivalent
defocus in diopters if we know the pupil size and the
total wavefront variance in other Zernike modes.
One of the appealing features of equivalent defocus as an
aberration metric is that it factors out pupil size by
normalizing RMS error by pupil area. Empirically we
have found that equivalent defocus is largely
independent of pupil diameter in normal eyes. This
simplifies the quantification of a person’s higher-order
aberrations because it becomes possible to summarize a
patient’s aberrations by saying, for example, that the
patient has 1 diopter of equivalent defocus without
having to specify pupil size, in the same way that we can
say a myopic eye is defocused by 1 diopter, without
having to specify pupil size.
Slide
38
Retinal image quality: step 1
Aberration map
Auto-correlation
Optical Transfer
Function
−1
c2 = c3 = 0.25 m
0
A completely different approach to specifying the
magnitude of aberrations is to assess their effect on
retinal image quality. Optical theory tells us that if we
know the aberration map, then we can compute retinal
image quality by a three step process.
Step 1 is to perform a mathematical operation called
“autocorrelation” on the aberration map which yields the
optical transfer function for the eye.
Slide
Retinal image quality: step 2
Optical Transfer
Function
Fourier Transform
Point-spread
Function
3.75 arcmin
Step 2 is to perform a different mathematical operation
called a Fourier transform on the optical transfer
function to produce the eye’s point-spread function,
which is a simulation of the retinal image formed by a
point source of light.
39
Slide
40
Retinal image quality: step 3
Point-spread
Function
Convolution
Retinal Image
3.75 arcmin
10 ′
Step 3 is to reproduce the point-spread function for every
point source in the object. This corresponds to yet
another mathematical operation called convolution,
which yields a simulation of the retinal image for any
object of interest, in this case an eye chart.
As you might imagine, there are many ways to assess the
quality of the retinal image for a complex object in the
real world. Currently there is active debate in the vision
science community about which metric of image quality
is the best predictor of visual performance.
Slide
41
How are derivatives of aberration maps interpreted?
P
Reference ray
∆y
∆x
P
object
point
Chief ray
R
SE
LA
Wavefront slope determines
ray aberrations
The final question I wish to address is how to interpret
the spatial derivatives of the aberration map.
Recall that rays are always perpendicular to wavefronts
so if a wavefront is tilted, then the direction of the ray
will change relative to a perfect reference ray drawn
parallel to the chief ray emerging from the center of the
pupil. In other words, the amount of horizontal
displacement, delta X, and the amount of vertical
displacement, delta Y, of a ray is determined by the local
slope of the wavefront.
Since the slope of the wavefront is computed
mathematically by taking the spatial derivative
horizontally, and vertically, we can conclude that the ray
aberrations are directly proportional to the first
derivatives of the wavefront aberration function. Indeed,
this is how many aberrometers, including the ShackHartmann wavefront sensor works: it measures ray
aberrations, which are then interpreted as wavefront
slopes, and these slopes are then integrated to obtain the
wavefront.
Wavefront Slope and Curvature: Defocus (z20)
2nd
deriv.
1st derivative
collapse
tails
(RMS=0.5 m, pupil=6mm, S= 0.385 D)
Let us now apply this line of thinking to some concrete examples,
such as defocus, shown here. The conventional aberration map for
defocus is shown in the upper left figure.
The first derivative of the aberration map gives the slope of the
wavefront at every point in the pupil. We can visualize the slope
information with the ray aberration map shown in the lower left
figure. Each arrow in this field of arrows shows the direction of
propagation of the light ray at the corresponding point in the pupil.
In this example all of the arrows are pointing towards the center,
which indicates that the rays of light are all converging to a common
focal point.
One of the nice feataures of the ray aberration map is that it can be
used to derive a geometrical-optics approximation to the pointspread function called a “spot diagram”. We do this by collapsing all
of the arrows so their tails coincide. The tips of the arrows, shown by
red spots in the figure at the bottom right, indicate where each ray of
light reflected out of the eye will intersect a distant screen. Or, to
change our viewpoint, the spots show where the rays from a distant
point source will intersect the retina in this myopic eye.
To go one step further in this kind of analysis, the second spatial
derivative may be used to determine the curvature of the wavefront
aberration map. Curvature, and its close cousin vergence, are a
familiar concepst to clinicains and are calibrated in the familiar units
of diopters. The figure in the upper right corner shows the average
of the horizontal and vertical curvatures, or vergences, in diopters,
for this same, defocused eye. Notice that this map shows only one
color, which means the curvature is the same everywhere in the
pupil. This makes sense because a defocused eye produces a
spherical wavefront and a spherical wavefront has the same
curvature everywhere.
Slide
Wavefront Slope and Curvature: Astigmatism (z 22)
(RMS=0.5 m, pupil=6mm, Jo= 0.27D)
Another example of the first and second derivatives of a
wavefront aberration function is shown here for Zernike
astigmatism. In this case the ray aberration map shows
that some rays are converging towards the optical axis
and some are diverging. However, the spot diagram is
still circular, as we would expect for an astigmatic
system with zero spherical equivalent.
Note that the mean curvature is zero everywhere in the
pupil, which indicates that this wavefront is similar to a
plane wave in that it will come to focus only at infinity.
This result occurs because the wavefront has a saddle
shape everywhere, with equal but opposite curvature in
the horizontal and vertical directions. The average of
these curvatures is zero everywhere in the pupil because
equal but opposite curvatures cancel out.
43
Slide
44
Wavefront Slope and Curvature: Coma (z 31)
(RMS=0.5 m, pupil=6mm, Meq= 0.385 D)
My last example of spatial derivativesof the aberration
map is for coma. Note the complicated ray aberration
map, which leads to a complex spot diagram which is the
geometrical approximation to the exact point-spread
function that can be computed by the Fourier optics
methods described earlier.
Currently we don’t have enough experience looking at
these spatial derivative maps to know whether they
provide valuable insight into clinical problems.
However I am optimistic that they will prove their worth
in the end. The easy transition from the map of
wavefront slopes to a spot diagram of the eye’s point
spread function is certainly a useful feature that is easy
to understand. Also, curvature maps have a long
tradition in ophthalmic optics for understanding the
shape of the cornea and the optical properties of eyes so I
suspect that these maps will also provide valuable
insight, in the fullness of time.
Slide
Reference materials
OSA Standards for Reporting Optical
Aberrations of eyes
http://research.opt.indiana.edu/Library/OSAStandard.pdf
Feature issues (2000, 2001, 2002)
J. Refractive Surgery (Sept. issue)
Monograph
Customized Corneal Ablation: The Quest for Super Vision,
MacRae, Krueger, & Applegate (eds.) Slack, Inc. (2001)
Lecture slides
http://research.opt.indiana.edu/Library/wavefronts.htm
This concludes my tutorial.
Details of the OSA standards for reporting optical
aberrations of eyes is available on the web at the address
shown here.
For further reading I can recommend recent feature
issues the Journal of Refractive Surgery which contain a
number of tutorial papers as well as current research
reports in the general area of visual optics.
I can also recommend the new monograph by MacRae,
Krueger, and Applegate as an excellent introduction to
the general topic of wavefront aberrations of the eye.
Lastly, a copy of my lecture slides and speaker notes is
available on the WEB at the address shown above.
45
Slide
Visual Optics Group at Indiana University
Larry Thibos, PhD
Arthur Bradley, PhD
Donald Miller, PhD
Support
National Institutes of
Health (R01-EY05109)
Borish Center for
Ophthalmic Research
Carolyn Begley, OD
Nikole Himebaugh, OD
Xin Hong, PhD
Vision Research at
Xu Cheng, MD
Fan Zhou, BS
Pete Kollbaum, OD
Kevin Haggerty, BS
The end
http://www.opt.indiana.edu
46
Slide
Supplementary slides follow
47
Slide
How are aberrations measured?
• Subjective
–Vernier alignment
–Spatially resolved refractometer
–Subjective aberroscope
• Objective
–Objective aberroscope
–Laser ray tracing
–Shack-Hartmann wavefront sensor
A variety of techniques have been invented over the
years for measuring the eye’s aberrations. In the next
talk, Ron Kruger will discuss these various technologies
for measuring aberrations. However, it is worth
mentioning one of these techniques now to demonstrate
how any given method can be described 3 ways: in
terms of ray aberrations, optical path length differences,
and wavefront shape. Consider, for example, the ShackHartmann wavefront sensor.
48
Slide
49
Scheiner Optometer (1619)
Ametropic eye
Reference ray
∆y
Retinal
point
source
∆x
Scheiner’s
disk
The principle of the Shack-Hartmann aberrometer is easily
understood in terms of ray optics as an elaboration of the 17th
century Scheiner Disk which Thomas Young later used to make
his famous optometer.
Imagine that we are able to place a point source of light on the
patient’s fundus. Light reflected back out of the eye from this
source will tell us everything we need to know about the eye’s
aberrations if we can determine the direction of each ray of light
as it emerges from the eye’s pupil. Scheiner’s disk allows us to
isolate individual rays to determine their direction of
propagation. If the pair of rays isolated by Scheiner’s disk do
not intersect at the eye’s far point, then the eye is optically
aberrated and we may quantify the magnitude of this aberration
in terms of the horizontal and vertical deviation of a ray from
the perfect reference ray.
Slide
50
Objective Scheiner/Hartmann Aberrometer
Point
source
d
CCD
sensor
Hartmann
screen
In principle we can track the direction of many rays
simultaneously by adding more holes to Scheiner’s disk to make
what is known as a Hartmann screen. By capturing these
isolated rays with photographic film or the CCD sensor inside a
video camera, we can determine the direction of propagation of
each ray as it passes through a specific point in the pupil. The
end result is a map of the ray aberrations of the eye.
To convert Scheiner’s subjective method into an objective
optometer, we reverse the direction of light propagation. In
other words, we put a spot of light on the retina which then
becomes a point source which reflects light back out of the eye.
Next, drill some more holes in Scheiner’s disk so it becomes a
Hartmann screen. That way each aperture in the Hartmann
screen isolates a narrow pencil of rays emerging from the eye
through a specific part of the pupil. The rays then intersect a
video sensor which tells us the horizontal and vertical
displacement of the ray from the non-aberrated, reference
position.
Thus we have a Hartmann aberrometer for objectively
measuring the ray aberrations of the eye. Now fill the individual
apertures of the Hartmann screen with tiny lenses and you have
a Hartmann-Shack aberrometer, or as I would prefer to say, a
Scheiner-Hartmann-Shack aberrometer.
Slide
51
Scheiner-Hartmann-Shack Aberrometer
Hartmann-Shack
wavefront sensor
Retinal
point
source
CCD
sensor
Relay
lenses
Lenslet
array
The modern version of this idea suggested by Dr. Roland
Shack and Ben Platt in 1965 replaces the Hartmann
screen with an array of tiny lenses which are more
efficient at capturing light and focusing it onto the CCD
sensor.
Having described the principle of operation of the
Scheiner-Hartmann-Shack aberrometer in terms of ray
aberrations, I will now repeat my description using the
language of wave optics so that you will appreciate why
this instrument is often called a wavefront sensor rather
than a ray sensor.
Slide
52
Principle of Wavefront Sensor
Sub-divide the wavefront
with micro-lenslets.
Local slope determines spot
position on video sensor.
Front
view
Video sensor
Micro-lenslet
array
Perfect wavefront
We can see how the wavefront sensor works by considering a
side view of the lenslet array as shown on the right-hand side
of this figure.
For a perfect, emmetropic eye the reflected wavefront will be a
plane wave perpendicular to a parallel bundle of rays. This
wavefront is sub-divided by the lenslet array into smaller
wavefronts that are focused by the lenses in the array into a
perfect lattice of point images. When viewed from the front,as
shown on the left, each image falls on the optical axis of the
corresponding lenslet. The overall result is a regular grid of
evenly spaced spots of light.
Slide
53
Principle of Wavefront Sensor
Displacement of spots from
reference grid indicates local
slope of aberrated wavefront.
Front
view
Video sensor
Micro-lenslet
array
Aberrated wavefront
If the eye is aberrated, then the reflected wavefront will be
distorted and the individual rays will not be parallel to each
other. Because rays are perpendicular to wavefronts, the
direction of propagation of each ray is determined by the local
slope of the wavefront over each lenslet. It is wavefront slope
which determines where the spot is focused.. Thus an
aberrated wavefront produces a disordered collection of spot
images.
By analyzing the displacement of each spot from its
corresponding lenslet axis, we can deduce the slope of the
aberrated wavefront when it entered the corresponding lenslet.
Mathematical integration of this slope information yields the
shape of the aberrated wavefront.
Slide
Slope of Wavefront: Defocus
This is a more detailed display of wavefront slopes.
54