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Trusit Dave PhD, BSc, MCOptom, FAAO
Instruments insight
Wavefront aberrometry
Part 1: Current theories and concepts
T
he
visual system is a complex structure, not simply in
terms of its ability to focus light but in terms of its ability to
detect (capture) and interpret the image.
Even in a normal eye with zero subjective
refraction, a point object does not form a
point image. This is due to optical
aberrations such as spherical aberration,
coma and other so-called high order
optical aberrations. Clinically, these can
induce patient symptoms of glare and even
diplopia. The principal site of these
aberrations is the cornea; one would
naturally expect this as the cornea has the
highest refractive power and therefore any
error in its shape would induce a variety of
aberrations. However, the lens and tear
film also contribute to the optical
image-forming properties of an eye.
As well as the effects of optical
aberrations, another factor, diffraction,
induces image blur. Diffraction is the
deviation of a light path which is
essentially not due to refraction or
reflection. It occurs at a pupil stop and as
the eye has a pupil, diffraction will always
exert some effect on image quality.
However, it is well documented that when
the pupil size is greater than 4.5mm,
diffraction has little or no effect on the
retinal image quality, as any image blur
induced by diffraction is within the
resolving power of the retinal cones
(Nyquist’s sampling limit). For the human
eye, it has been shown to be around 0.5
minutes of arc or approximately 6/3 1 .
Contrary to diffraction, where image
quality deteriorates with smaller pupils,
high order optical aberrations increase
with pupil size.
Advances in refractive surgery have led
to the need for measuring more subtle
errors of optical image degradation.
However, clinicians have long-since
understood the need to measure these
optical characteristics in cases where acuity
has not been sufficiently improved using
basic sphero-cylindrical refractive
corrections, such as in keratoconus.
Numerous advances in our ability to
measure corneal topography have provided
a better understanding of surface shape.
However, practitioners require a direct
method of evaluating the optical
characteristics of an eye. Quantitative and
qualitative descriptors of the optics of an
eye have not been available until the
development of devices known as
aberrometers.
Many optometrists will have been
involved in refractive surgery
co-management, or examined patients
having undergone a refractive surgical
procedure. More importantly, many
patients are aware of refractive surgical
techniques which potentially result in
‘super vision’ or customised vision
correction. This wavefront-guided
procedure represents an exciting time for
vision correction, with many academic
experts and leading surgeons now
discussing optical properties of eyes in
terms of wavefront aberrations and using
terms like ‘Zernike polynomials’ and ‘RMS
wavefront error’. This two-part article aims
to provide the clinician with an
understanding of these terms, and how the
technology can be used in optometric
practice.
High and low
order aberrations
Routine optometric practice involves
performing a full subjective refraction.
Here the sphere, cylinder and axis of
astigmatism are measured. Although we
may feel that this is the total refractive
error, we are only correcting two
components of a whole host of refractive
components of the optics of an eye.
However, these two components (sphere
and cylinder) constitute by far the majority
of the optical aberration of an eye.
Nevertheless, we will all have encountered
the situation where a patient complains of
poorer vision at night when subjective
refraction is near zero. We often refer to
this as night myopia.
Although night myopia will be one
cause of the symptoms – where the shift in
wavelength at night induces myopia –
other monochromatic aberrations, such as
spherical aberration, also play a part as
they increase with pupil size. In the
majority of normal patients, these high
order aberrations play a minor role,
however, in cases of refractive surgery,
keratoconus and orthokeratology, they can
induce a number of visual disturbances
which can be explained by measurement
of the optical aberrations. Therefore, it is
useful to think of subjective refraction as a
procedure used to reduce image blur
optimally using spheres and cylinders
when the image is blurred due to
additional factors such as scatter,
diffraction and other more subtle higher
order aberrations. Furthermore, subjective
refraction takes into account neural
processing factors which have developed
because of the world we live in.
This article will not address the effects
of chromatic aberration, where the
magnitude of refraction varies with
wavelength of light. Discussion will be
limited to monochromatic aberrations
where the optical aberrations are
referenced to one wavelength (550nm).
It is important to understand, however,
that optical aberrations will affect all
wavelengths.
Measuring wavefront
aberration
A number of commercial devices are
available to measure ocular aberration.
Most are based on the Shack-Hartmann
principle. Before covering the basics of this
technique, the concept of wavefronts will
be discussed.
Traditionally, we have been taught to
think in terms of light rays for the purpose
of ray tracing. However, light rays only
provide information about one point of an
object in image space. To understand how
the optics have affected the image, we need
to observe and measure the bundle of all
light rays exiting the eye. This bundle is
often referred to as the ‘emergent
wavefront’. In scientific terms, a wavefront
may be defined as “the surface over which
an optical disturbance has constant phase”.
Put another way, the wavefront is the
surface which joins individual points on
rays which have the same optical path
length. The optical path length is simply
the distance a ray travels, multiplied by the
refractive index of the material it travels in.
Figures 1a and 1b show a plane and
circular wavefront respectively.
If a point object could be imaged on
the retina, the emergent wavefront for an
eye with perfect optics would be plane
(Figure 2a). As already discussed, the
reality is that this rarely occurs. Virtually
all eyes exhibit some level of aberration,
even in the absence of conventional
refractive error. As a result, the emergent
bundle of light will not be plane, but
rather it will be deformed. Wavefronts may
adopt any shape. However, in cases of
simple myopia or hyperopia, they will
always be spherical provided there is no
other ocular aberration (Figure 2b).
An aberrometer measures ocular
aberrations, including low and high order,
by principally measuring the wavefront
emerging from the eye. There are, however,
a number of different ways to achieve this.
This article only covers the most popular
type – the Shack-Hartmann aberrometer.
Shack-Hartmann aberrometer
The Shack-Hartmann sensor has been used
extensively in astronomy to correct the
refractive blur induced by atmospheric
41 | November 19 | 2004 OT

Instruments Insight
Trusit Dave PhD, BSc, MCOptom, FAAO
Figure 1a
Parallel light rays travelling the same distance over time. The wavefront
summarising the bundle of rays is plane
Figure 1b
Light rays travelling in different directions. The same distance is
travelled over time (same phase), and the wavefront summarising
the bundle is spherical
Figure 2a
The emergent wavefront of an eye with perfect optics, i.e. no refractive
error and no high order aberrations
Figure 2b
The wavefront in an eye with simple myopia and no other aberrations.
Here the emergent wavefront is perfectly spherical in shape
turbulence. Astronomers developed an
aberrometer to detect the aberrations
induced by changes in atmospheric
conditions and then corrected these using
a deformable mirror – an optical
engineering process known as adaptive
optics 2 . This science is now being applied
to the eye by virtue of an ocular
aberrometer. The object of the
Shack-Hartmann aberrometer is to sample
various points on the emerging wave
and in doing so, derive the shape of the
Figure 3
A diagrammatic representation of a
Shack-Hartmann plate
wavefront.
Essentially, a Shack-Hartmann plate is a
series of micro lenses arranged in a linear
fashion (Figure 3). These lenslets are
typically 0.25mm in diameter. Each lenslet
focuses a view of the point source through
various points of the entrance pupil
(Figure 4). In this way, the aberrometer
determines the shape of the wavefront
exiting the eye. Although infrared light is
used, the aberrations are measured with
respect to peak spectral sensitivity of the
eye (550nm) by using a correction
algorithm to account for the shift in focus
from the measurement wavelength to the
reference wavelength 3 .
Figure 5 shows the basic process
involved in capturing the wavefront. For a
perfect eye with no aberration, the
position of the spot on the CCD sensor
will be at the optical axis of each lenslet.
However, the reality is that in a normal
eye, there will be some level of aberration
and, as such, the image of the spot will be
Figure 4
The principle of the Shack-Hartmann aberrometer
42 | November 19 | 2004 OT

Instruments insight
displaced from the optical axis. A question
commonly asked is: “How is it that the
infrared laser forming a spot on the retina
is not affected by the ocular aberrations?”
The answer is that as the spot is very small
and enters through the centre of the
entrance pupil to the fovea, the influence
of the ocular aberrations is negligible.
Thus, the CCD camera is used to
capture the displacement of the spot from
the optical axis of each lenslet.
Displacements may occur in the horizontal
and vertical meridian; Figure 6 shows the
displacement in the y meridian. The slope
of the wavefront is easily calculated as the
displacement/the focal length of the
lenslet. After examining the slope at each
micro lenslet in the x and y meridia, the
entire wavefront can be plotted in 3D.
One of the objectives in measuring the
wavefront is to calculate the wavefront
error (Figure 7). This describes the optical
path difference between the measured
wavefront and the reference wavefront
(in microns) at the entrance pupil. The
wavefront error is derived mathematically
from the reconstructed wavefront.
However, these computations result in
many values of wavefront error. They are
therefore plotted as a 2D or 3D wavefront
error map for qualitative analysis (Figure
7). These maps are analogous to colour
coded corneal topography maps, where
colours denote varying corneal dioptric
power. In wavefront error maps, each
colour represents a specific degree of
wavefront error in microns. It is important
to note the range and interval of the scale.
Generally, in order to see as much detail as
possible, choose the narrowest scale which
permits the maximum number of colour
changes within the map. However, when
comparing pre and post treatments, make
sure the same scale and step sizes are used
to enable a like-for-like comparison. There
is currently no absolute scale which has
constant range and step size, as in corneal
topography maps.
Zernike polynomials
The wavefront error map only provides
qualitative information about the
wavefront. The shape of the wavefront
takes on many forms and it is difficult for
the clinician to evaluate the effects of the
shape on vision. In order to analyse the
wavefront quantitatively, it needs to be
broken down into terms which are
clinically meaningful. Conventional
refraction breaks the wavefront down into
only basic terms – sphere, cylinder and
cylinder axis. A more sophisticated analysis
is required to understand the subtle
aberrations. Zernike polynomials are
equations which are used to fit the
wavefront data in three dimensions. These
polynomials have unique qualities, the
principal one being that they decompose
the shape of the wavefront into terms
which describe optical aberrations such as
spherical aberration, coma etc. Figure 8
Figure 5
The process involved in capturing the wavefront from the entrance pupil. Each lenslet captures a
very small portion of the wavefront. The displacement of the spot is compared to that of a
perfect wavefront. For a calibrated sensor, a perfect (or plane) wavefront would image the laser
spot at the optical axis of the lenslet
shows the Zernike modes commonly used
to describe the measured wavefront.
The individual modes, or terms, in the
polynomial have two variables, ρ (rho)
and θ (theta). ρ is the normalised distance
from the pupil centre. Normalising ρ
means it has a maximum value at the edge
of the pupil of one. Therefore, for a 6mm
pupil, a point 3mm from the centre would
have a ρ value of 0.5. θ is the angular
subtense of the imaginary line joining the
pupil centre and the point of interest to
the horizontal (Figure 9).
The key point here is that aberrations
are dependent on pupil size. Therefore, all
aberrometry measures must be related to
the patient’s pupil diameter. Wavefront
measures must be referenced to a pupil
size.
Each term is sequentially added to
describe the entire wavefront. Thus, the
wavefront may be described as:
W(ρ,θ)=C 1
-1
Z 1
-1
+C 11
Z 11
+C 2
-2
Z 2
-2
+C 20
Z 20
+C 22
Z 2
2
+C 3
-3
Z 3
-3
+C 3
-1
Z 3
-1
+C 31
Z 31
+C 33
Z 33
+C 4
-2
Z 4
-2
etc
Zernike terms are defined using a double
index notation as proposed by the Optical
Figure 6
Calculation of the slope of the wavefront from the
displacement of the image at each individual lenslet
43 | November 19 | 2004 OT

Instruments Insight
Trusit Dave PhD, BSc, MCOptom, FAAO
Figure 7
The wavefront error map shows the difference between the
aberrated and reference (plane) wavefront. Once this information is
processed for all points over the pupil, a 2D wavefront error map
can be displayed (right) Figure 8
Society of America 4 . This nomenclature
groups each term according to the radial
order (n) and angular frequency (m), thus
each term is written in the form Z m n. The
radial order (n) groups Zernike modes in
terms ρ (rho), whereas the angular
frequency (m) groups the modes in terms
of θ (theta). The coefficient, C, in each
term varies according to the number of
modes in the Zernike polynomial. C
defines the level of a particular mode of
aberration in microns and can have a
positive or negative value.
A list of the Zernike terms and their
optical equivalent up to the fourth order is
shown in Table 1. There are, in fact, an
infinite number of Zernike terms which
can be used to fit an individual wavefront.
In practice, however, Zernike terms up to
the 4th radial order are measured.
It is worth understanding the actual
terms in the equation. Let us take the
example of secondary astigmatism or Z 2 4.
−1
Z1 = 4ρ
sin( θ )
1
Z1 = 4ρ
cos( θ )
−
Z
2 6 2
2
= ρ sin(2θ
)
0
2
Z2 = 3(2ρ
−1)
Z
2 = 6ρ
2 cos(2 θ )
2
Individual modes of Zernike polynomials up to fourth order. ‘n’
represents the radial order and ‘m’ the angular frequency
Table 1
The double index notation of Zernike term, the equation of each mode in the Zernike polynomial
and the appearance of each mode in a colour-coded wavefront error map
Zernike term WFE Map Optical equivalent
Vertical prism
Horizontal prism
Astigmatism
Defocus
Astigmatism
The Zernike equation is
Z 2 4 = √10(4ρ 4 −3ρ 2 )cos(2θ)
What does
this mean?
As you can see, there are only two
variables in all the modes in Table 1.
Broadly speaking, you can categorise each
equation into three areas.
Figure 9
Wavefront error map defining ρ and θ
−
Z
3 8 3
3
= ρ sin(3θ
)
−1
3
Z3 = 8(3ρ
− 2ρ)sin(
θ )
1
3
Z3 = 8(3ρ − 2ρ)
cos( θ )
Z
3 = 8ρ
3 cos(3 θ )
3
−
Z
4 10 4
4
= ρ sin(4θ
)
−2
4 2
Z4 = 10(4ρ
−3ρ
) sin(2θ
)
0
4 2
Z4 = 5(6ρ
− 6ρ
+ 1)
2
4 2
Z4 = 10(4ρ
− 3ρ
)cos(2θ
)
Z
4 = 10ρ
4 cos(4 θ )
4
Trefoil
Coma
Coma
Trefoil
Tetrafoil
Secondary astigmatism
Spherical aberration
Secondary astigmatism
Tetrafoil
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Instruments insight
Z 2 4 = √10(4ρ 4 −3ρ 2) cos(2θ)
Normalisation
term
Radial
term
Angular
term
Normalisation of each mode means that
observation of the coefficients
immediately gives an indication of the
level of influence each type of aberration
has on the total aberration 5 . The radial
term, in this case of the fourth order,
describes the variation of the wavefront
error with distance from pupil centre. The
largest power ρ is raised to is the radial
order, i.e. to the power 4. The angular
frequency describes the number of repeat
cycles which are made over 360 degrees.
For example, in secondary astigmatism,
there are two repeat cycles over the
circular pupil, thus the radial order is 2.
Also, notice that where the radial order is
assigned a negative value, the equation
uses the sine of θ to describe the angular
variation.
The first order term, prism, is not
relevant to the wavefront as they represent
tilt and are corrected using a prism. The
second order terms represent low order
aberrations, namely defocus and
astigmatism. Defocus represents the
spherical component of the wavefront.
The astigmatic terms conversely describe
the cylinder. Using these three terms, any
sphero-cylindrical lens can be described.
Wavefront maps with defocus and
astigmatism only can be perfectly
corrected using spectacles and contact
lenses. Caution, however, needs to be
exercised when interpreting these terms as
in practice, patient wavefront error maps
have high order terms. As a result, defocus
and astigmatism terms do not represent
the sphere and cylinder terms found in
subjective refraction. An estimate of
subjective refraction can be made if only
these three terms are fit to the wavefront
data. Second order terms are referred to as
low order aberrations.
Every mode after second order is a high
order aberration. These terms cannot be
measured in conventional refraction or
autorefraction. The only method of
quantifying these terms is by using an
aberrometer. The third order terms
describe coma and trefoil astigmatism.
These terms cannot be corrected using
spectacles, soft or RGP contact lenses.
Attempts have been made to correct
spherical aberration in contact lenses
using aspheric designs. However, this
correction only corrects for lens specific
spherical aberration and not the combined
effect of eye and lens. Therefore, the
optical benefits of such a design may
improve image quality for some patients
but equally there may be some
deterioration in image quality in other
patients 6 .
Root mean square error
In order to summarise the wavefront error,
scientists have tried to develop metrics to
try and best describe the wavefront error as
a numeric index. At the present time, the
most widely used metric is the root mean
square (RMS) error. All aberrometers
display a RMS error of high and low order
aberrations. This term describes the
weighted mean of the individual Zernike
modes. In other words, the RMS value
describes the overall aberration and gives a
‘feel’ of how bad an individual’s
aberrations are. The RMS error is simply:
RMS=√(C2 -2 ) 2 +(C 0 2) 2 +(C 2 2) 2 +(C -3 3) 2 +(C -1 3) 2 ...etc
There are number of disadvantages in
using the RMS value. The most significant
is that increases in RMS value do not
necessarily result in decreases in image
quality 5 . For example, if a subject has some
defocus and spherical aberration, then it is
possible that the spherical aberration will
help reduce some of the image blur
induced on the retina. However, because
of the way RMS is calculated, the
cancelling effects of one aberration over
another are not considered 7 . As research
continues, better predictors of visual
function will develop 5,8 .
Studies have evaluated normative
values of ocular aberration. Porter et al 9
found a random variation in the wavefront
aberration of 109 normal subjects with an
artificial pupil diameter of 5.7mm. Chalita
et al 10 performed aberrometry in 60 eyes
and found mean values of 0.35±0.29
microns for coma; 0.36±0.31 for spherical
aberration; and 0.31±0.14 microns for
other high order terms for a pupil
diameter of 7mm. It has also been
documented that 91% of the RMS
wavefront error can be accounted for by
the second order and 99% if Zernike
terms from the first four orders are used to
fit the wavefront for a pupil diameter of
5mm 11 . Wang and Koch 12 measured ocular
aberrations in subjects screened for
refractive surgery. High order aberrations
were found to have a mean RMS of
0.305±0.095 in 532 eyes across a 6mm
pupil. Spherical aberration was found to
be 0.128±0.074 and coma 0.170±0.089
microns.
Any instrument involved in clinical
measurement must be both accurate and
repeatable. Cheng et al 13 measured the
accuracy and repeatability of the COAS
(Wavefront Sciences) Shack-Hartmann
aberrometer on model eyes. They found
the instrument to be both accurate and
repeatable on model eyes. In another
study, Cheng et al 14 evaluated the variation
of monochromatic aberrometry
measurements over a range of timescales
(five measures in one second without
realignment, five measures in one hour
with realignment at the same time over
five days and on five days at monthly
intervals). The authors concluded that
variations in aberrometry maps between
days and months was due to biological
changes to the eye, and that this
fluctuation would prevent practitioners
from achieving perfect optical correction,
i.e. an aberration-free eye. Wavefront maps
are also known to vary with
accommodation 15 , tear film quality 16 and
media opacities 17 .
Part 2 of this article, to be
published on December 3, will look
at applications of wavefront
aberrometry in clinical practice.
About the author
Dr Trusit Dave is Director of Optimed
(Ophthalmic Medical Instruments),
Clinical Consultant for Topcon GB and a
partner in private practice in Coventry.
References
For a full set of references, email
nicky@optometry.co.uk.
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45 | November 19 | 2004 OT