HSS
Dissertation
submitted to the
Combined Faculties for the Natural Sciences and for Mathematics
of the Ruperto-Carola University of Heidelberg, Germany
for the degree of
Doctor of Natural Sciences
presented by
Dipl. Phys.:
Ulrich v. Pape
born in:
Hamburg
Oral examination:
30th January, 2002
Wavefront Sensing in the
Human Eye
Referees: Prof. Dr. Josef Bille
Prof. Dr. Karl-Heinz Brenner
Zusammenfassung: Wellenfrontmessungen am menschlichen Auge
Fortschritte in der chirurgischen Technik zur Korrektur von refraktionsbedingten Sehfehlern
haben es ermoglicht, dass heute die Form der Hornhautvorderache auch ortsaufgelost
geandert werden kann. Die Refraktionsmessungen am menschlichen Auge beschrankten sich
allerdings auf die Messung von Sphare und Astigmatismus, und zwar bei einer einzigen
Pupillengrosse. Im Zuge dieser Arbeit wurde - basierend auf einem Hartmann-Shack Sensor
- ein Wellenfrontmessgerat zur ortsaufgelosten Refraktionsmessung des Auges entwickelt, das
den diagnostischen Anforderungen gerecht wird, die sich aus den neuen Moglichkeiten der
Augenchirurgie ergeben.
Der Aufbau des Gerates wird beschrieben. Die Ergebnisse von Messungen an Test-Optiken
werden dargestellt und mit den theoretischen Moglichkeiten verglichen. Die Ergebnisse von
Messungen an menschlichen Augen fuhren zu Einschatzungen uber die tatsachliche und die
erforderliche notwendige Auosung des Gerats. Die Reproduzierbarkeit der Ergebnisse wird
gepruft. Weitergehend wurde noch ein aktiver Senk-Matrix-Spiegel in das Gerat implementiert, der die Wellenfront korrigieren und dem Patienten seine aberrationsfreie Sehfahigkeit
demonstrieren kann.
Es zeigt sich: Das Gerat ist in der Lage, die Refraktionsmessungen einfach, schnell und
reproduzierbar durchzufuhren, und zwar mit einer Genauigkeit, die die Erfordernisse noch
ubertrit.
abstract: Wavefront Sensing in the Human Eye
Most recent technical advancements in the refractive surgery for correcting refraction errors
of the eye allow a spatial-resolved reshaping of the cornea. The diagnostics up to now have
been restricted to sphere and cylinder giving a mean value for one pupil size only.
For this study a wavefront sensor for spatial-resolved measurement of the refraction of the
eye - using the Hartmann-Shack principle - was developed. To meet the diagnostic requirements of present day ophthalmology was the main goal.
The setup of the device is described. Measurements at test-optics are detailed and compared
to theory. Measurements on human eyes give evidence for the actual resolution of the device
and the requirements as well. In addition an active mirror was implemented. The use of
this mirror lies in correcting the wavefront error and presenting the patient with wavefront
corrected images to test the non-optical-limited capability of his vision.
The results show: The device is well suited for measuring the refraction of the eye - working
fast, with results reproducible, and a precision, that even surpasses the needs of ophthalmology.
Contents
1
Introduction
1
2
The Human Eye
5
3
Basics of Wavefront Sensing
2.1 Anatomy of the Eye . . . . . . .
2.1.1 Tear Film . . . . . . . . .
2.1.2 Cornea . . . . . . . . . . .
2.1.3 Anterior Chamber . . . .
2.1.4 Iris . . . . . . . . . . . . .
2.1.5 Crystalline Lens . . . . . .
2.1.6 Vitreous . . . . . . . . . .
2.1.7 Retina . . . . . . . . . . .
2.2 The Dioptric System . . . . . . .
2.3 Styles-Crawford Eect . . . . . .
2.4 Magnication . . . . . . . . . . .
2.5 Eye Movements . . . . . . . . . .
2.6 Accommodation . . . . . . . . . .
2.7 Monochromatic Aberrations . . .
2.7.1 Myopia and Hyperopia . .
2.7.2 Astigmatism . . . . . . . .
2.7.3 Higher Order Aberrations
2.8 Chromatic Aberrations . . . . . .
2.8.1 LCA . . . . . . . . . . . .
2.8.2 TCA . . . . . . . . . . . .
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3.1 Wavefront Sensors in Ophthalmology . . . . . . . . . . . . . . 24
i
CONTENTS
3.2 Principle of a Hartmann-Shack Sensor . . . . . . . . .
3.2.1 Shape of the Microspots . . . . . . . . . . . . .
3.2.2 Dynamic Range . . . . . . . . . . . . . . . . . .
3.2.3 Resolution . . . . . . . . . . . . . . . . . . . . .
3.3 Zernike Polynomials . . . . . . . . . . . . . . . . . . .
3.4 Fourier Optics . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Fourier Transformation . . . . . . . . . . . . . .
3.4.2 Optical Imaging in Fourier Representation . . .
3.5 Propagating Wavefronts . . . . . . . . . . . . . . . . .
3.5.1 Correcting Aberrations in the Conjugate Plane
3.5.2 Using a Telescope for Correcting Sphere . . . .
3.5.3 Correcting Cylinder . . . . . . . . . . . . . . . .
3.5.4 Spatial Filtering . . . . . . . . . . . . . . . . . .
3.6 Single Pass Measurement . . . . . . . . . . . . . . . . .
3.7 Describing Optical Imaging Quality . . . . . . . . . . .
3.7.1 Root Mean Square . . . . . . . . . . . . . . . .
3.7.2 Optical Aberration Index . . . . . . . . . . . . .
3.7.3 Modulation Transfer Function . . . . . . . . . .
3.7.4 Point Spread Function . . . . . . . . . . . . . .
4
Setup
4.1 Specication of the System . . . . . . . . . . . . . . . .
4.2 The Optical Setup . . . . . . . . . . . . . . . . . . . .
4.3 The Observation Unit . . . . . . . . . . . . . . . . . .
4.3.1 Determination of the Axial Position of the Eye .
4.4 The Target and Vision-Chart Unit . . . . . . . . . . .
4.5 The Active Mirror . . . . . . . . . . . . . . . . . . . .
4.6 The Measurement Unit . . . . . . . . . . . . . . . . . .
4.6.1 Light Source . . . . . . . . . . . . . . . . . . . .
4.6.2 Hartmann-Shack Sensor . . . . . . . . . . . . .
4.7 Software . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Precompensation of Lower Order Aberrations . . . . .
4.8.1 Pre-Correction of Sphere . . . . . . . . . . . . .
4.8.2 Pre-Correction of Astigmatism . . . . . . . . . .
ii
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62
CONTENTS
4.8.3 Calculating Sphero-Cylindrical Lenses . . . . .
4.8.4 The Use of Power Vectors . . . . . . . . . . . .
4.9 Speckles . . . . . . . . . . . . . . . . . . . . . . . . . .
4.10 Test Measurements on Articial Eyes . . . . . . . . . .
4.10.1 Testing Sphero-Cylindrical Measurements . . .
4.10.2 Testing Higher Order Aberration Measurements
4.10.3 Performance Test of the Active Mirror . . . . .
5
The Hartmann-Shack Sensor at the Human Eye
6
Visual Acuity
7
Conclusion and Outlook
5.1 Measurements at the Human Eye . . . . . . . . . . .
5.1.1 Comparing the Sphero-Cylindrical Refraction
5.1.2 Reproducibility of the Results . . . . . . . . .
5.2 Standard Deviation of Sphere and Cylinder . . . . . .
5.3 Change of Higher Order Aberration . . . . . . . . . .
5.3.1 Age . . . . . . . . . . . . . . . . . . . . . . .
5.3.2 Accommodation . . . . . . . . . . . . . . . . .
5.3.3 Daily Fluctuations . . . . . . . . . . . . . . .
5.4 Perfect Vision Study . . . . . . . . . . . . . . . . . .
5.5 Excimer Study . . . . . . . . . . . . . . . . . . . . .
5.5.1 The Excimer Laser System . . . . . . . . . . .
5.5.2 Refractive Surgery Methods . . . . . . . . . .
5.5.3 Study Group . . . . . . . . . . . . . . . . . .
5.5.4 Results . . . . . . . . . . . . . . . . . . . . . .
6.1 Vision Charts . . . . . . . . . . . . . . . . . . . . .
6.1.1 Conditions For Visual Acuity Measurements
6.2 Fundamental Limits to Visual Performance . . . . .
6.2.1 Optical Limits . . . . . . . . . . . . . . . . .
6.2.2 Retinal Limits . . . . . . . . . . . . . . . . .
6.3 Predicting Visual Performance . . . . . . . . . . . .
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List of Figures
107
iii
CONTENTS
Bibliography
111
iv
Chapter 1
Introduction
In the last few years new techniques in refractive laser surgery have been developed and the eld has progressed rapidly. Prior to these developments we
had excimer lasers with a potential to remove cornea tissue in a symmetric
way only, correcting sphere and - with limitations - cylinder.
Nowadays ying spot laser or laser scanning systems are capable to reshape
the cornea spatially resolved by sizes smaller than 1mm.
This development has created new challenges for diagnostics. So far there
were subjective and objective methods like manifest refraction and autorefractometer, useful for measuring sphere and cylinder only. For the new
refractive surgery methods they were no support.
New eorts have been made - or older ones intensied - to avoid a gap between the techniques of surgery and the tools for diagnostics. A combination
of corneal topography and manifest refraction was a rst approach. On the
assumption that most of the higher order aberrations originate in the cornea,
it was thought that dealing with the cornea should be suÆcient. However it
turned out that the whole optical system has to be taken into account. At
this point wavefront sensors came in consideration (e.g. see[Kl98]).
These sensors are based on a number of principles, the main ones being
Tscherning, Ray-Tracing and Hartmann-Shack or - as it is called in America
- Shack-Hartmann.
The wavefront sensor used in our study is based on the Hartmann-Shack
principle. The idea behind this approach was developed in astronomy in the
1
CHAPTER 1. INTRODUCTION
seventies with the objective of determining the quality of telescope optics.
Liang in 1991, in our group in Heidelberg, was the rst to make a case for
using this method for measurements on the human eye [Li91]. In his dissertation he layed out the theory and described how the rst experimental
setup was built.
Another main part of the waveform device originates in astronomy as well:
This is the adaptive optic, which is a combination of wavefront devices, one
for measuring (the HSS) and the other one for correction (active mirror,
ASKM). In astronomy it is used for correcting the aberrations caused by
the atmosphere in order to sharpen the images of objects in space. In our
device it is used to correct the higher order aberrations of the eye. This gives
a chance to demonstrate a patient the quality of vision he will achieve by
correction the higher order aberrations.
Compared to man made optics the optic of the eye is very poor. So the
standard for the measurement at eyes is low too, especially compared to
astronomy. The problem here lies in the fact that the eye is alive and gets
damaged very easily. These circumstances set the frame for the development.
The setup for the measurements is detailed. In testing the device several
steps were taken. A rst set of tests was designed to determine the precision
of our measurements. Sphere, cylinder and higher order aberrations were
measured for a well known rigid test optic.
Following this, tests were performed for human eyes. The results were compared to those obtained from classical methods for determining refraction.
The reproducibility of the values is established.
Higher order aberrations vary on their own in short or long time periods and
in processing accommodation as well. The range of these aberrations gives a
suggestion of the bound of precision still making sense. The results also give
an idea of the minimum amount of aberration a laser surgery may be helpful
for.
Two applications of our device bring the study to a close.
\Perfect Vision\ and its optical prerequisite is object of the rst study. The
2
Visual Acuity of about 70 eyes was measured and the result was compared
to their higher order aberrations.
What results can be expected from excimer refractive surgery? This is the
question a second study centers around. 42 patients were treated both ways,
with wavefront guided laser surgery on one eye and the traditional way on
the other. A glimpse at the potential of the new method is given.
3
CHAPTER 1. INTRODUCTION
4
Chapter 2
The Human Eye
In dealing with the anatomy of the human eye we have to consider two main
parts: The optical unit and the retina.
The optical unit depicts the world around us upside down onto the retina.
The retina records the image, converts it into an electrical signal, does a
rst step of image-processing, and transmits the signal to other parts of the
brain. (The retina is a part of the brain itself). Compared to man-made
optical apparatus the optical properties of the eye are quite inadequate even in a normal emmetropic eye. If the vision is still of a high quality,
this is mainly due to the excellent performance of the brain in analyzing the
received image. However: The job of reconstructing a picture can't be done
without a minimum of information. The causes for loss in the quality of
vision vary over a wide range: Age-related loss of accommodation, Myopia,
Hyperopia, accidental damages of the cornea, opacity of the lens or retinal
damage. The deciencies can be grouped as below:
1. the optical imaging quality of the eye
2. the light scattering in cornea and lens
3. the light scattering at the retina
4. the neuronal use of the retina
The Hartmann-Shack Sensor is designed primarily as a tool for determining
the quality of the optical image. Moreover it is - via the brightness of the
5
CHAPTER 2. THE HUMAN EYE
vitreous
fovea
iris
lens
cornea
yellow
spot
visual axis
papilla
anterior chamber
zonula
fibers
visual
nerve
lamina
cribrosa
posterior chamber
retina
uvea
sclera
ciliary muscle
Figure 2.1: Cross section of the eye
HSS spots - suitable to give information about eects of group 2 and 3. To
the neuronal use there is no access by Hartmann-Shack Sensors. Most of the
problems related to the optical unit can dealt with by an intervention at the
cornea - even if the problem does not originate in this very place.
2.1 Anatomy of the Eye
This section gives a description of the elements of the eye with special emphasis on their optical properties. The interaction between these elements is
the main subject of the next section.
The normal adult eye is approximately spherical with an anterior-posterior
diameter averaging 24.5mm.
The outer protective coating of the eye is the sclera. It is dense, white and
continues with the cornea anteriorly and the dural sheath of the optic nerve
posteriorly. The cornea is a transparent tissue inserted in the sclera at the
limbus.
The uveal tract is composed of the iris, the ciliary body and the choroid. It
6
2.1. ANATOMY OF THE EYE
is the middle vascular layer of the eye and contributes blood supply to the
retina. The lens is a biconvex structure suspended behind the iris by the
zonules which connects it with the ciliary body. The retina is a thin multilayered semitransparent sheet of neural tissue that lines the inner aspect of
the posterior two-thirds of the wall of the globe.
2.1.1
Tear Film
A very thin lm - measuring about 10 m - covers the cornea. For the optic
it is quite important: It is responsible for a smooth surface by compensating
rough parts of the cornea.
The lm consists of three layers. The outer lipid layer (0.02 m-0.4 m) mainly dierent fats - prevents evaporation. The middle aqueous layer mainly water (98%) and anorganic salts (1 %) - is the thickest part. The
internal mucin - an extremely thin mucous layer of 0.2 m - ensures the
adherence to the cornea. Abnormalities of the tear lm cause uctuations
and impair the optical properties of the eye. A tear in the tear lm leads to
strong reections and aberrations to the eye.
2.1.2
Cornea
The cornea is the most important optical part of the eye. The front may be
thought of as a section of a sphere with diameter 16mm, the base-circle being
12mm in diameter, the curvature slightly diminishing towards the periphery.
The width of the layer increases from about 0.5mm in the center to about
0.8mm at the periphery.
The cornea is built in several layers: The epithelium in front, the Bowman Membrane, the stroma, the Descemets Membrane and the endothelium.
Making up 90% in thickness, the stroma is the dominating part. The importance of the cornea lies in the fact that with about 43D ( 75%) the impact
on the total refraction of the eye is the largest. The refraction at the front
(49D) goes along with a much smaller refraction with a reverse sign at the
back (-6D).
The eect of the front is due to the extent of the dierence between the
refraction index n of air (n=1.0) and cornea tissue (n=1.376), the dierence
7
CHAPTER 2. THE HUMAN EYE
of n at the back being much smaller (Æn=0.04).
The front of the cornea has the shape of a ball, which is attened to the
periphery and mostly combined with some cylinder. A mathematical description of the shape may be:
cx x + cy y
p
z=
(2.1)
1 + (1 (1 + k)(cxx + cy y )
with z=height cx=curvature in x-direction, cy =curvature in y-direction, k=conic
constant. This model does not regard rotation around the z-axis and tilt.
In general the rear of the cornea has the same shape with a slightly smaller
curvature. The curvature of the cornea can vary by 0.06mm in the course
of each day. This causes a shift in the refraction of about 0.3D. It may also
have an eect on the cylinder.
2
2
2
2.1.3
2
Anterior Chamber
The space between the endothelium of the cornea and anterior surface of the
lens is called the Anterior Chamber.
It measures 12mm in diameter and about 3.6mm in depth. By grows of the
lens during life the depth decreases continually.
The aqueous humor - as medium very clear - is responsible for the intraocular
pressure. It is produced in the ciliary body and is diverted through the
trabecula and the Schlemms canal into the venous system.
2.1.4
Iris
The iris consists of an elastic diaphragm with a central expandable circular
opening, the pupil. The iris has a diameter of 12mm and a thickness of
about 0.6mm. The pupil is the aperture stop of the eye and limits the
passage of light into the eye. Its diameter is shifted by two muscles. The
musculus sphincter pupillae contracts the pupil and the musculus dilatator
pupillae widens it. The diameter can be changed between 2mm and 8mm,
corresponding to approximately 16 times variation in area. The aperture of
an optical system has always great inuence on the optical properties. A
stricture of the pupil has some eects on vision:
8
2.1. ANATOMY OF THE EYE
reducing the brightness
reducing the higher order aberration
increasing the diraction eects
increasing the depths of focus
The higher order aberration and the diraction eects oppose each other with
the change of the pupil size. The optimal vision quality will be achieved at
2mm-3mm depending on the eyes aberration.
The center of the pupil also denes - together with the fovea centralis - the
visual axis of the eye.
2.1.5
Crystalline Lens
The lens of the eye is much more complex than the cornea. It can be seen as
an asymmetric deformable biconvex gradient index lens. It has a diameter
of about 8mm, a thickness of about 4mm and a curvature radius of 10mm
at the front, and 8mm at the back. The refractive index of about n=1.4
decreases from the center to the periphery.
The lens keeps growing during lifetime. Starting with a small core at birth,
new layers of bres keep growing around this core continually. So the lens
can grow up to double of its original size. As a consequence the inner parts
have a reduced metabolism and harden. This leads to a higher refractive
index and a smaller accommodation range. The accommodation of children
can be up to 14D and goes down to about 2D at the age of 50 and less than
0.5D at the age of 70. This causes a shift of the near point, from 7cm for a
child to more than 2m for a seventy year old person in an emmetropic eye.
The accommodation is regulated by the ciliary muscles and by the zonulabres. The zonulabres pull the lens in radial direction. If the ciliary muscles
are relaxed the refraction of the eye is at a minimum. By straining these
muscles the refraction rises.
9
CHAPTER 2. THE HUMAN EYE
2.1.6
Vitreous
The vitreous forms the largest part of the eye. It is the transparent colorless
and gelatinous mass - consisting to 99% of water and 1% of collagen and
hyaluronic acid - between the lens and the retina. The refraction index is
very close to the refraction index of water.
The main importance of the vitreous lies in its high refractive index.
2.1.7
Retina
The retina is the light sensitive part of the eye. It converts the light stimulus
into a signal which can be processed by the brain.
The photosensitive cells can be classied in two groups, the rods and the
cones. These cells are arranged like a mosaic on the retina. The cones support photopic (day, color) vision. The usually smaller rods support scotopic
(twilight) vision, which is in black and white only. In the fovea - the area
receiving the sharpest image - there are no rods at all. Relative the number
of rods increases to the periphery, as the number of cones decreases. The
total number of rods is about 100 million, compared to just 7 million cones.
The size of the cones depends on the position in the eye. Their minimum
lies - with 2.5 m to 4 m - in the fovea.
Looking at the retina from the front two spots attract attention. On the
nasal side there is the optic papilla, the area where the nerves pass out of
the eye into the brain. In this area the retina has no light active cells and
the eye is blind.
The other spot is the macula. It contains the area with the sharpest vision
called the "fovea centralis". This part of the retina is built in a way that
minimizes distortion of the image. There are no rods in the macula and no
blood vessels either. The cones dominating the vision in this area are very
small. The eld of view of the fovea centralis is very small (about 1 degree
or 0.1% of the retina).
Still this area is the only target of the standard visual acuity tests. This area
is night blind due to the absence of rods.
The retina consists of ten layers of cells. The most important are:
10
2.2. THE DIOPTRIC SYSTEM
pigment epithelium.
this layer is responsible for the supply of the retina
light sensitive layer.
It consists of rods and cones. If light hits these cells a chemical reaction
starts.
Layer of neurons called bipolar cells.
These cells transform the chemical signal into an electrical signal.
The ganglion cells.
The innermost layer of neurons.
Layer of nerve ber.
Apart from these layers there are others which establish a horizontal connection between the dierent areas of the retina.
In the retina some rst image processing is taking place. The number of
cells decrease with every layer. So the information of more than 100 million
receptors can be transmitted with about 1 million nerve bres.
The eective place of reection and the place of absorption - corresponding
to the maximum - varies with the wavelength of the light.
2.2 The Dioptric System
From the optical point of view the eye must be seen as a system with four almost spherical surfaces - the anterior and the posterior surfaces of the cornea
and the crystalline lens. The eect grows, as the dierence of the refraction
index between the two materials increases, and the radius of curvature of the
surfaces decreases.
The total refraction of the eye is about 59D, dominated by the refraction of
the front of the cornea.
In contrast to most human made optical systems, the optical axis of the eye
is not identical with the functional axis of the eye. The visual axis is tilted
by about 5degrees against the optical axis (=symmetry axis) which causes
11
CHAPTER 2. THE HUMAN EYE
Cornea
Optical Axis
Aqueous
Visual Axis
Anterior Ocular
Chamber
Lens
Figure 2.2: Optical setup of the human eye
coma. The most important eye model was developed by Gullstrand at the
beginning of the 20th century. It is still in use for many applications, even
the most modern eye models are based on the Gullstrand eye.
The Gullstrand eye has just one lens. The basic values stem from measuring
a very large number of emmetropic eyes, and taking mean values. In this
way he obtained a model for a non accommodating eye with a thin lens with
58.64D 1.48mm behind the cornea and a distance of 17.05mm between lens
and retina.
Apart from the refractive properties of the eye the transparency has another
decisive role for our application. The transparency of the optical components
depends to a great deal on the wavelength. This is shown in Fig.2.3. In the
visible range (550nm to 750nm) as well as in the neighboured near infrared
range (750nm to 900nm) the transparency for the total optical path, from
cornea to retina, is about 75%. The transparency for a wavelength is not
constant throughout life. It decreases with age as shown in g.2.4.
12
transmission
2.2. THE DIOPTRIC SYSTEM
wavelength
Figure 2.3: The transparency of the human eye [Me96]
1) transmission of cornea
2) transmission of lens
3) transmission of the vitreous body
transparency in %
80
70
age:
60
50
30 years
40
70 years
30
20
10
0
300
350
400
450
500
550
600
650
700
750
800
wavelength in nm
Figure 2.4: Change of transparency with age
13
CHAPTER 2. THE HUMAN EYE
refractive index
cornea
aqueous
lens
core lens
place
cornea front
cornea back
lens front
lens back
radius of curvature
cornea front
cornea back
lens front
lens back
total optical system
refractive power
place rst cardinal point
place second cardinal point
place rst focus point
place second focus point
front focal length
back focal length
place of fovea
axial refraction
place of near point
unit no accommodation max accommodation
1.376
1.336
1.386
1.406
1.376
1.336
1.386
1.406
mm
mm
mm
mm
0
0.5
3.6
7.2
0
0.5
3.2
7.2
mm
mm
mm
mm
7.7 - 7.8
6.8
10
6
7.7 - 7.8
6.8
5.33
5.33
D
mm
mm
mm
mm
mm
mm
mm
D
mm
58.64
1.348
1.602
-15.707
24.387
-17.055
22.785
24.0
1.0
|{
70.57
1.722
2.086
-12.397
21.016
-14.169
18.030
24
-9.6
-102
Table 2.1: The optical properties of the eye ([Me96])
14
2.3. STYLES-CRAWFORD EFFECT
air
h
a
n
eye
n´
L´
L
a´
h´
Figure 2.5: The magnication depends from the length of the eye.
2.3 Styles-Crawford Eect
The Styles-Crawford eect lies in an angular dependence of retinal sensitivity.
Rays parallel to retinal receptors, entering the pupil near its center, are
more eective (appear brighter) than oblique rays, entering the pupil near
its margins. This fact reduces the eective pupil size. This phenomenon was
discovered by Styles and Crawford in 1933. In a model this eect goes as a
lter, in which transmission decreases with diameter.
2.4 Magnication
In visual science the linear distance on the retina corresponding to 1degree
of visual angle is called the retinal magnication factor.
The magnication of the optics can be easily computed by applying Snells
law on a simple eye model (g.2.5):
n sin() = n` sin(`)
(2.2)
which gives us - with the small angle approximation and taking the angles
trigonometrical equivalent:
h h`
=
(2.3)
L L`
So we nally get for the magnication m
h` L`
m= =
(2.4)
h
L
Since the length L` grows from a hyperopic to a myopic eye the magnication
also gets larger. For example the magnication factor for a 10D myopic eye
15
CHAPTER 2. THE HUMAN EYE
Figure 2.6: Accommodation of an emmetropic eye
a) the eye is maximally accommodated
b) the eye is not accommodated
is 20% larger than for an emmetropic eye - objects in the same distance seem
to be larger and Vision Charts can be read better.
2.5 Eye Movements
Each eye is moved by six muscles, arranged in couples. Two of the pairs
are responsible for moving the eye up/down and right/left and the third pair
induces a rotation of the eye around the visual axis. The rotation guarantees
that the eye is always horizontal. This is most important for orientation.
Usually all six muscles work together. Only horizontal views, with the head
also being horizontal, can be achieved by using just one pair of muscles.
Horizontal movements - e.g. used for reading - can be realized very fast and
precise.
2.6 Accommodation
Accommodation is the ability of the eye to change the focus to dierent distances. The accommodation is realized by the ability of the eye to change the
shape of the lens (g.2.6). In the normal relaxed state the lens is relatively
at. By tensing a muscle the lens gets rounder in shape and the refraction
of the eye increases. Thus objects closer to the eye will be imaged on the
retina.
16
2.7. MONOCHROMATIC ABERRATIONS
accommodation
width
max.
accommodation
near point
no
accommodation
far point
Figure 2.7: Accommodation of a myopic eye
The \near point\ (g.2.7) is the nearest point for which the sight is sharp.
Likewise the furthermost point is called the "far point". The dierence in D
is called "accommodation width". In a normal young emmetropic eye the far
point is in innity and the near point about 20cm in front of the eye. This
gives an accommodation width of 5D. Due to the hardening of the lens the
accommodation width decreases with age as described above.
2.7 Monochromatic Aberrations
Deviations from the normal abilities of vision are called ametropia. The standard case of ametropia occurs when the image of an innite object - with the
eye relaxed - is not on the retina. The main kinds of ametropia are myopia,
hyperopia and astigmatism. These classical refraction errors are superimposed by higher order aberrations like coma and spherical aberrations.
2.7.1
Myopia and Hyperopia
The cause of ametropia normally lies in a deviation in the length of the eye,
the distance between lens and retina. If this distance is too large the eye
is myopic, if the distance is to small the eye is hyperopic, like can be seen
in gure2.8. A deviation in the curvature of the cornea or in the grade of
refraction may also cause ametropia, being minor in eect.
In a relaxed hyperopic eye the focus point of an object positioned in innity
lies behind the retina.
The focal length can be shortened by accommodating so that the image lies
17
CHAPTER 2. THE HUMAN EYE
PSF
500 µm
50 µm
Hyperopia 2dpt.
Emmetropia
Figure 2.8: Top: Refractive errors of the eye
Bottom: Formation of the PSF
18
500 µm
Myopia -2dpt.
2.7. MONOCHROMATIC ABERRATIONS
minimal
horizontal focus
focusline diameter
vertical
focusline
beam shape
Figure 2.9: Development of astigmatism
on the retina. The nearest point of sharp vision is further away. In an
emmetropic eye with rising age and falling accommodation range, the near
point will rise to some meters.
If the focus point is in front of the retina, the eye is myopic. Even with a
relaxed eye the far point is nite.
By accommodation the area closer than this far point can be seen sharply.
The near point is closer than in a normal eye.
In myopia and hyperopia the image of a point spreads to a blur as can be
seen in gure2.8.
2.7.2
Astigmatism
The third kind of classic refraction failure is astigmatism. Astigmatism lies
in the fact that dierent axes have dierent focal lengths.
The main cause is a kind of barrel shape of the cornea or - in a minor
dimension - of the lens. An average eye has 0.5D of horizontal cylinder
called "regular cylinder". A dierence in the curvature of 0.1mm gives a
cylinder of 0.5D. With cylinder a point-lightsource gives - instead of one
focus point - two focus lines, with a very large focus spot in between, as can
be seen in gure 2.9. If the cylinder is uncorrected, the eye focuses to the
spot with the minimal diameter.
19
CHAPTER 2. THE HUMAN EYE
Figure 2.10: Development of spherical aberration
2.7.3
Higher Order Aberrations
Apart from these deciencies there are higher order aberrations of a wide
range. For their classication and characterization Zernike-polynomials are
in use. The most important kinds of higher order aberrations in the human
eye are spherical aberrations and coma.
Spherical aberrations occur if paraxial beams cross a spherical lens o-axis.
As shown in g.2.10 with a larger distance to the axis the focal length becomes shorter and the focus spot smears.
Coma occurs if a beam crosses a lens o-axis or tilted. If coma exist, the
focal point has a shape like the tail of a comet.
The higher order aberrations mainly occur at the cornea and at the lens.
The close to spherical shape in the center of the front of the cornea will make
up most of the spherical abberation. If a beam tilted by 5 degrees enters
the eye on the visual axis, it will also cause coma, which is even added by
a displacement of the center of the cornea to the visual axis. This can be
assumed the most important aberrations caused by the cornea.
The eect of the lens on the higher order aberrations is not quite easy to
see. It causes spherical aberration depending on the state of accommodation, coma and also triangular astigmatism, probably induced by the 3 pairs
of muscles.
If the size of the pupil is large, the aberrations obtain relevance.
20
2.8. CHROMATIC ABERRATIONS
total refraction D
62
61
Theory values
experimental values
60
59
400
500
600
700
800
wavelength (nm)
Figure 2.11: Dependence of the total refraction of the eye from the wavelength
2.8 Chromatic Aberrations
In addition to the monochromatic aberrations the eye also suers from wavelength dependent aberrations. The chromatic aberrations occur in two ways.
The rst one relies on the fact that the refractive index of a material is dependent on the wavelength. The refractive index of the eye compares to that
of water. The second way is generated by the fact that light of dierent
wavelength is absorbed (and reected) in dierent layers of the retina.
Chromatic aberrations can be divided into two kinds:
Longitudinal Chromatic Aberration (LCA)
Transversal Chromatic Aberration (TCA)
2.8.1
LCA
The shift of focal length with wavelength is called LCA. The eect is well
known in the visible range. The dierence in total refraction in an emmetropic eye between 400nm and 600nm goes beyond 1.5D. The dependence
21
CHAPTER 2. THE HUMAN EYE
is shown in g.2.11. The theoretical values are based on Tucker [Tu74] . The
experimental means are based on the values of 20 measured eyes [Li97].
The theoretical model-eye Tucker applied is quite simple. It has just one refractive surface. The refraction index changes in the same way, as the index
of water does. In the visible range the theoretical results come very close
to the experimental values. For the near infrared there are no experimental
results for the LCA. So the assumption is that the models for visible light
are valid also in this range.
2.8.2
TCA
TCA appears if polychromatic light enters the eye at an angle. The difference in the refraction index makes dierences for the refraction of beams
dependent on their color. So the position of the image is shifted transversally
for oblique beams of dierent wavelength. This aects the magnication of
the eye as well as the position of the image. The inuence of the shift in the
reection depth on the TCA is negligible.
The inuence of the TCA on our HSS measurement may be neglected as the
measurements are on-axis.
22
Chapter 3
Basics of Wavefront Sensing
This chapter gives an introduction into Wavefront Sensing. It describes the
basic properties of wavefronts and informs about dierent ways of measuring
and describing them. Furthermore the connections of wavefronts to ophthalmological parameters are pointed out.
Beams from a pointsource are all in phase in the pointsource itself. If you
have dierent beams originating from one pointsource at the same time their
endpoints at any later time will generate a sphere (g.3.1). All points on
the surface of this sphere are in phase again. A surface like this is called a
wavefront: Wavefronts are phasefronts of light.
The direction of the propagation of light inside a medium is always orthogonal to the local surface of the wavefront.
The quality of an optical system (for our use) can be measured by its ability
to keep beams, originating from a pointsource, spherical. This is essential
for focussing them back to one point by another perfect optical system.
The dierence in the optical path between a wavefront surface and the best
tted sphero-cylindrical surface is called the higher order wavefront error.
The sphero-cylindrical surface is always used as reference here.
Detectors only respond to brightness levels and not to the phase of light. So
wavefront sensors register wavefronts in a more indirect way. They transform
optical path dierences (OPD) to dierences in light levels.
Wavefront Sensors dier in kind: Direct wavefront sensors measure the wavefront itself (as in Radial Shear Interferometry). Indirect wavefront sensors
23
CHAPTER 3. BASICS OF WAVEFRONT SENSING
Figure 3.1: Denition of a Wavefront
measure the dierential wavefront either in the pupil plane (e.g. knife edge
test) or in the image plane (e.g. Hartmann-Shack Sensor).
3.1 Wavefront Sensors in Ophthalmology
As to applications for wavefront sensor systems, several uses can be thought
of, especially for measuring cornea topography or the space-resolved refraction of the eye. In this section will be described dierent setups for the
measurement of the space-resolved refraction.
Three types of this kind are in development: the Thinbeam Raytracing Aberrometer (g. 3.2), the Tscherning Aberrometer (g.3.3) and the HartmannShack Method (g.3.4).
The Raytracing Aberrometer uses the thin-beam principle of optical ray
tracing. It rapidly res a sequence of very small light beams into the eye.
Through a beamsplitter a very fast PSD (Positioning Sensing Detector) measures for every beam the position where it hits the retina. This is a great
advantage of this kind of sensor. Each measurement gives the values for one
single beam. Even in case of strong aberration it is guaranteed that the
beams can be discriminated by time. About 64 beams are distributed over
the pupil size at random in a very short time (about 2ms). On the basis
of this the total refractive power and higher order aberrations can be deter24
3.1. WAVEFRONT SENSORS IN OPHTHALMOLOGY
Scanning
Positioning
Sensing
Detector
PSD
n=1
n=2
n=3
n=4
n=... x
n=64
y
Figure 3.2: Thinbeam Ray-Tracing Aberrometer
25
CHAPTER 3. BASICS OF WAVEFRONT SENSING
oberroscope lens
retinal
image
mask of apertures
ophthalmoscope lens
CCD
resulting image
Figure 3.3: Tscherning Aberrometer
retinal
spot
collimated beam
lens array
CCD
Figure 3.4: Hartmann-Shack Method
26
3.2. PRINCIPLE OF A HARTMANN-SHACK SENSOR
Figure 3.5: Hartmann test for testing the quality of lenses. With a Hartmann
aperture in front of the lens pictures will be taken at points si and se (in front
and behind the focus point) and the results will be compared.
mined. The possibility to vary the pattern of the entrance points enables the
operator to concentrate on special areas of the pupil.
The Tscherning Aberrometer bases directly on the Hartmann-Test described
in the next chapter. A collimated laser beam irradiates a mask of about 160
holes. The beams formed by the pattern will be imaged on the retina. A lens
in front of the eye focuses each point 1mm to 3mm in front of the retina,
so a grid with a diameter of 1mm forms on the retina. A ccd camera takes
pictures of this grid from the outside of the eye. By the distortion of the grid
the wavefront can be calculated. In contrast to the Hartmann-Test, only one
image outside the focal plane is used. That amount of aberration that can
be measured in this way depends on the distance between the apertures and
that of the focal plane from the retina.
The Hartmann-Shack Sensor will be described in detail in the next chapter.
3.2 Principle of a Hartmann-Shack Sensor
The conception of the Hartmann-Shack wavefront sensor comes from astronomy. In 1900 Johannes Hartmann introduced a new method for specifying
the quality of large telescopes, called Hartmann test (g.3.5): An array of
apertures (HB=Hartmann Blende) are placed in front of a lens. Light of a
collimated beam passes through the lens. It is focussed with some aberra27
CHAPTER 3. BASICS OF WAVEFRONT SENSING
aberrated
beam
collimated
beam
optical
system
lens arrray CCD-chip
Figure 3.6: Idea of a Hartmann-Shack Sensor
tion. Photographic plates are positioned in front and behind the focus. Every
aperture gives an image on each of the plates. By the total pattern taken
on the plates, every image-point can be associated with one aperture. The
focus position and the total aberration of the lens is calculated by taking the
distance of the images from the optical axis and the positions of the plates.
70 years later Shack and Platt introduced an advanced kind of Hartmann
sensor called Hartmann-Shack Sensor ([Pl71]). They proposed to use a lens
array in the image plane followed by a photographic plate in the focal plane
of the lens array (g.3.6). Later the photo-plate was replaced by a ccd-chip.
The improvement of this setup is remarkable. The number of planes for
measurement is reduced to one. This fact makes real-time measurements
possible, using a ccd-chip. The optical path of the system is not involved.
So it is possible to do measurements while the instrument is being used.
The new setup makes measurement of the wavefront more precise and much
faster.
Figure3.7 shows the main idea of the Hartmann-Shack sensor on a single
lens. A collimated beam hits a single lens. The beam is focussed by the lens
in the focal plane. If the beam hits the lens parallel to axis, the focus point
is on-axis. If the beam is tilted by an angle , the focus lies o-axis by
d = f tan()
(3.1)
with f = focal length of the lens. The tilt of the focus is the clue to the
mean slope of the wavefront on the area of the lens. The extent, to which
28
3.2. PRINCIPLE OF A HARTMANN-SHACK SENSOR
f
a
d
h
Figure 3.7: Functionality of a Hartmann-Shack Sensor demonstrated on a
single lens
the slope varies on the diameter of the lens, should not be too large.
A Hartmann-Shack sensor uses a whole array of lenses instead of a single
lens. The lenses divide the beam into sub-beams. Each sub-beam is focussed
by a single lens on the ccd-chip. The position of the focus depends on the
mean slope of the wavefront on every microlens (g.3.8).
The result is the mean derivation in x- and y-axis for every lens position:
ÆW (xn ; ym ) xn;m
P (x ; y ) =
=
(3.2)
n
m
Æx
f
ÆW (xn ; ym ) yn;m
= f
Æy
= mean wavefront at the microlens (n; m),
Q(xn ; ym ) =
(3.3)
and x, y
with W (xn; ym)
the horizontal tilt of the axis.
This kind of measurement is limited to more or less continuous and dierentiable wavefronts as seen in gure 3.9. Limitations of the HSS are shown in
the diagram on the right. At the top the variation of the tilt between two
microlenses is so strong that the focus points change places. Below, a leap
in the wavefront between microlenses is shown. This leap has no inuence
on the result, so it cannot be measured. At the bottom lens the curvature
29
CHAPTER 3. BASICS OF WAVEFRONT SENSING
Figure 3.8: Image on the ccd-chip. The green crosses show the optical axes of
the microlenses, the white points are the focus points of an uneven wavefront
Figure 3.9: Limitations of the HSS
Left: A smooth wavefront reaches a HSS.
Right: A strongly aberrated wavefront reaches the HSS.
30
3.2. PRINCIPLE OF A HARTMANN-SHACK SENSOR
of the wavefront is too large for having a focus point at all.
Not only the position of the spots varies, their shape does so as well. The
curvature of the wavefront surface on the area of each single lens has to be
small. There are three properties of the HSS, which account for the dynamic
range and the resolution: the pitch of the microlenses, the focal length of
the microlenses (actually the distance between lens and ccd-chip) and the
resolution of the ccd-chip. Their eects will be described in the next three
subsections.
3.2.1
Shape of the Microspots
If the wavefront is not disturbed too much, each point can be seen as diraction limited. So we get an Airy Disc. The diameter depends on the size of
the aperture and the focal length of every microlens and the wavelength of
light:
780nm 60m
(3.4)
s = f = 30mm
h
400m
with s = spotdiameter, = wavelength and h = pitch of the lens array.
With 60 m the size of the spot is less than a sixth of the distance to the
next spot.
The shape of the spot gains relevance when stronger aberrations occur. For
the determination of the focus-position the software uses a center of gravity
algorithm. So a non-symmetric change in the shape of the spot could inuence the result of the spotnding and hereby the shape of the wavefront.
If stronger aberrations occur two eects have to be taken into account for
calculating the spot shape.
Each HS spot is the image of the spot on the retina, imaged through the
optical path through the individual micro lens. So for simulating the shape
of the spots we need two PSF`s:
Firstly the PSF we get from the beam coming into the eye including the
precompensation of sphere and cylinder.
Secondly the PSF of the optical path out of the eye and through the machine
to the HSS.
The nal shape of the spot is now the convolution of both PSF`s.
The rst PSF is identical for all microspots, since the same lightsource on
31
CHAPTER 3. BASICS OF WAVEFRONT SENSING
d
a
r
f
Figure 3.10: Dynamic range of a Hartmann-Shack Sensor.
the retina is used. The second PSF is dierent for every microlens, as every
single lens is part of a dierent optical path through the pupil. The quality
of the rst PSF will normally be inferior as the beam diameter is larger here.
The second PSF will be close to diraction limited. If there is no strong local
perturbation, the shape of all microspots will be very similar to the shape of
the focal point on the retina.
A strong aberration of this spot will cause a shift in nding the center of
gravity. This deviation is not that serious, because it aects all points in the
same way. So it just changes the total tilt of the wavefront - which is not
used anyway.
3.2.2
Dynamic Range
The dynamic range of a HSS species the range of aberrations that can be
measured. As shown in g.3.9 the change in wavefront tilt may get so large
that two focal points may overlap or even change places. To distinguish the
focal points of dierent lenses we have to make sure that every focal point
lies within the area of its own lens. This can be done by a fourier lter in
the optical path.
Fig.3.10 shows, in which way the dynamic range is limited. The displacement
32
3.2. PRINCIPLE OF A HARTMANN-SHACK SENSOR
2 points; Sphere (2nd order))
1point: Tilt (1st order))
3 points; Coma (3rd order))
Figure 3.11: The maximum detectable wavefront complexity depends on the
number of measured points.
of the focus point plus the spot size has to be smaller than the radius of the
lens array. This leads to a maximum angle of:
max tan r
d=2
f
(3.5)
This gives us a maximum wavefront tilt of about 0:3Æ. So the maximum
measurable sphere would be 1.8D at a 6mm pupil.
3.2.3
Resolution
There are two kinds of resolution: there is a minimum for the detectable
wavefront slope and a maximum for the complexity of the wavefront.
Like the dynamic range the minimum wavefront slope depends on the focal
length and the radius of the microlenses. The resolution of the ccd-chip matters at this point too.
For the minimum detectable wavefront slope the accuracy of the focus position is crucial. If the spot quality is high the position can be determined
by t routines by about a hundredth of the pixel-size of the ccd-chip. This
33
CHAPTER 3. BASICS OF WAVEFRONT SENSING
gives a minimum detectable tilt of
0:12m 4 10
ccd
min = res f
30000m
6
(3.6)
So the minimum detectable angle is about 2:310 Æ . This corresponds to a
change of the phase of 0.002 m over a microlens and a minimum detectable
defocus of 0.002D.
The maximum detectable complexity depends on the number of detected
points over the pupil-radius. As gure 3.11 shows the number of radial detected points equals the maximum of radial Zernike-orders that can be used.
We get 15 Hartmann-Shack points over a 6mm pupil, so we could calculate
Zernike coeÆcients up to the 15th order. In fact we limit our calculations to
the 6th order, due to computation time and necessary resolution.
For a two-dimensional pupil the maximum order of Zernike polynomials is
xed by the degree of freedom of the Hartmann-Shack points. Every point
has two degrees of freedom, every Zernike polynomial one. For describing 6
orders of Zernike polynomials (=28polynomials) we need at least 14 focus
points. That corresponds to a pupil size of about 2.8mm.
(
4)
3.3 Zernike Polynomials
The wavefront error is described as a surface over the exit pupil. To describe
the surface we use a function W (x; y), which attributes a wavefront height for
every position in the pupil P (x; y). This height is the optical path dierence
between the reference sphere and the wavefront.
In ophthalmological optics the use of Zernike polynomials is dominating in
the description of optical aberrations. Zernike polynomials were introduced
in 1934 by F. Zernike as a convenient tool for representing wavefront aberrations over a circular pupil. A great advantage of these polynomials is the
fact, that their relations to the classical aberrations are very simple. The
polynomials have (among other things) the following properties:
They are orthogonal over the circle with unit radius
They are complete
34
3.3. ZERNIKE POLYNOMIALS
jlj 0 1
0 1
1
2
3
4
n
2
2 1
3
2
4
6 + 1
6
4
3 2
2
3
2
4 3
4
3
2
4
Table 3.1: Radial Polynomials Rnjlj(), for jlj 4, n 4
The precision, an aberration can be described with by Zernike polynomials,
depends on the order of the polynomials being used, and has no minor bound.
The extension to higher order terms does not aect the coeÆcients of the
lower order ones - at least in theory, on the assumption that the base of the
t is an innite set of points distributed uniformly. In case a wavefront is
represented as a linear combination of Zernike polynomials, the variance of
the whole term is equal to the sum of the variances of the single terms.
The polynomial can be expressed as a product of two functions, one depending on the radial coordinate only, the other representing the dependence on
the angular coordinate. The total polynomial can be described as follows:
Znl = Rnl ()eil
(3.7)
with n = degree of the polynomial, l = angular dependence parameter,
= normalized radial distance and = angle with the axis x. The numbers n (> l) and l are either both even or both odd.
Tabular3.1 gives the radial polynomials up to 4th order. Tabular3.2 shows
the full Zernike polynomials with their classical equivalent in non-complex
presentation. Most classical aberrations can be represented by just one component, only nonrotationally symmetric aberrations like coma and astigmatism are decomposed into two components.
35
CHAPTER 3. BASICS OF WAVEFRONT SENSING
n
l
no.
Zernike
Monomial
aberration name
Polynomial
Representation
0
0
0
1
1
Piston
1
1
1
sin x
Tilt about y axis
-1
2
cos y
Tilt about x axis
2
3
2 sin 2
2xy
2
0
-2
3
3
1
-1
3
4
4
2
4
2
5
2
7
8
sin 3
(3
(3
11
0
12
y
sin 4
4y
2
3 ) sin 2
62 + 1
Triangular astigmatism on x axis
2
2y + 3y + 3x
4
64
x
3
cos 3
(4
Cylinder with axis at 0/90 degree
3
3
3
2
3x
3
x
4x
6xy + 8y
6y 2
1
5
(4
-4
14
4 cos 4
5
15
5 sin 5
3
16
(55
2
3
Triangular astigmatism on y axis
y
x + 8x3 y
y4
y + 6x
17
(105
4x4
6x2 y 2 + x4
5xy 4
43 ) sin 3
Third order spherical aberration
4
2
10x3 y 2 + x5
12xy 2 + 4x3 + 15xy 4
+10x3 y 2
1
Third order Coma along y axis
3y + 3x + 4y 4
3 ) cos 2
13
y
6x2 + 6y 4 +
+12x
2
-2
Third order Coma along x axis
y
2 2
4
3
2x + 3xy + 3x
2) cos )
4
x
2
123 + 3) sin 3x
12xy 2
5x5
12x3
+10xy 4 + 20x3 y 2 + 10x5
1
18
(105
123 + 3) cos 12y 3
3y
12x2 y + 10y 5
+20x2 y 3 + 10x4 y
-3
19
(55
43 ) sin 3
4y 3 + 12x2 y + 5y 5
10x2 y 3 + 15x4 y
-5
20
5 cos 5
y5
10x2 y 3 + 5x4 y
Table 3.2: First 5 orders of Zernike polynomials with classical description
36
45 degree
Defocus
2
2
3
10
2
3xy
2) sin )
3
9
2
y
3
3
2
1 + 2y + 2x
1
cos 2
6
Cylinder with axis at
2
Z13
Z10
j
r
sin(j)
Z6
Z3
Z11
Z7
Z1
Z0
Z4
Z12
Z2
Z8
Z5
cos(j)
Z9
Z14
3.3. ZERNIKE POLYNOMIALS
Figure 3.12: Chart of Zernike polynomials up to 4th order
37
CHAPTER 3. BASICS OF WAVEFRONT SENSING
3.4 Fourier Optics
This chapter gives a short overview of the use of Fourier methods in describing optics.
The Fourier transformation makes it possible to change over from space domain into frequency domain. Many optical processes - especially imaging
with limited pupil size - can be handled much easier if the considerations and
the calculations are done in the frequency domain. This counts especially for
diraction eects, which cannot be described with ray-tracing anymore.
3.4.1
Fourier Transformation
In the space domain an object is described by an intensity function f (x; y).
In the frequency domain the same object can be represented by use of the
2-dimensional Fourier Transformation as F (; ) with and representing
the frequencies in x- and y-direction.
For the 2-dimensional case the transformation will be performed by
F (; ) =
Z 1Z 1
1
1
f (x; y )e
dxdy
(3.8)
F (; )ei2(x+y) dd
(3.9)
i2(x+y)
The way back will be performed by:
f (x; y ) =
3.4.2
Z 1Z 1
1
1
Optical Imaging in Fourier Representation
f (x; y ) and g (X; Y ) are the complex entrance and exit functions of a linear
system. The optical impact can be described by a linear operator L:
g (X; Y ) = L[f (x; y )]
(3.10)
with the use of the principle of superposition we get:
g (X; Y ) = L
=
Z 1 Z 1
Z 11Z 11
1
1
38
f (x0 ; y 0)Æ (x
x0 )Æ (y
f (x0 ; y 0)L[Æ (x x0 )Æ (y
y 0 )dx0 dy 0
y 0)]dx0 dy 0
(3.11)
(3.12)
3.4. FOURIER OPTICS
object
optic
I0(x,y)
PSF(x,y)
convolution
image
=
I(x,y)
i
Fourier-Transformation
IS0(x,h)
multiplication
=
OTF(i x,h)
IS(i x,h)
Figure 3.13: Fourier Optic in incoherent imaging
The application of the linear operator L on the Æ-function gives us the so
called \impulse answer\.
The further application of the linear operator depends on the kind of illumination used. If the object is illuminated by coherent light, the optical
system is linear in phase, otherwise it is linear in intensity. In the following
we suppose the light to be incoherent.
Equation 3.12 shows a convolution in the space domain. A convolution on
one side of a fourier-transformation is equal to a multiplication on the other
side. This gives a multiplication in the frequency domain.
G(; ) = OT F (; )F (; )
G(; ) and F (; ) are the fourier transformed of f(x,y) and g(x,y).
(3.13)
Optical
imaging can be seen as a ltering in the frequency domain.
Figure3.13 illustrates this with I = Intensity, OT Fi = incoherent optical
transfer function and P SF = point spread function. The amplitude ratio
is called modulation-transfer function (MTF).
The relation between the pupil function and the optical transfer function is
of further interest. The incoherent optical transfer function is the autocorre39
CHAPTER 3. BASICS OF WAVEFRONT SENSING
lation of the pupil function:
= jF T fP (x; y)gj
(3.14)
This correlation makes it possible to obtain the PSF directly from the pupil
function and therewith from the Zernike polynomials.
P SF
2
3.5 Propagating Wavefronts
Every wavefront changes its shape while propagating in space. Only an innite widespread at wavefront would keep unchanged. So the shape of a
wavefront will be dierent in any two places.
A wavefront originating from a pointsource stays spherical in shape, but the
curvature varies with the reciprocal distance from the source. In a suÆcient
distance, however, the wavefront can be seen as at. The dierences between
a real wavefront surface and a theoretical at surface make the total wavefront error.
In most cases it is not possible to have the wavefront sensor at exactly that
place, we want the information about. So the wavefront has to be imaged in
a denite way from the plane, we want to have values for, to a plane we can
actually take values in by our sensor. The easiest way to realize this, is by
using conjugate planes of a lens system.
A Badal system - consisting of two lenses - modies the wavefront in just
that accurately dened way. Defocus and cylinder can be eliminated, so the
remaining higher order wavefront error will be recorded. The elimination of
the lower order aberrations is a suitable means of optimizing the HartmannShack sensor in dierent respects, aiming at either a large active range or
a high accuracy. The active range the higher order aberrations need is far
lower. So we optimize the HSS for a high accuracy and correct the sphere
and cylinder ahead of the wavefront measurement.
A wavefront on one side of the optical system is - apart from small errors
introduced by the system itself - similar to the wavefront in the conjugate
plane.
An alteration in the shape may result from lens-failures or diraction, the nite size of the object or the surfaces being of further inuence. In particular
40
3.5. PROPAGATING WAVEFRONTS
Figure 3.14: Wavefront propagating from a plane to its conjugate plane
a small aperture in the optical path will cause a attening of the wavefront
as it works as a Fourier lter.
For the pre-compensation there are three kinds of correction in use: Shifting
the telescope for correcting the sphere, introducing cylinder lenses for correcting the cylinder and variations of the OPD in a conjugate plane by using
an active mirror for correcting higher order aberrations.
3.5.1
Correcting Aberrations in the Conjugate Plane
Correcting aberrations works by a step-by-step-principle, using a series of
conjugate planes, the rst one being the object plane and the last one the
measurement plane of the sensor. In the conjugate planes in-between the
optical path can be varied for every longitudinal position in a controlled way.
This can be done by an active mirror, a liquid crystal device, or simply by
taking inuence on the optical path-length by inserting a non uniform glass.
3.5.2
Using a Telescope for Correcting Sphere
The Sphere Correction will be realized by shifting the distance of the lenses in
a Badal system. This shift has no inuence on the position of the conjugate
planes in respect to the lenses as you can see in g.3.14. Shifting the length
of the telescope has inuence on the curvature of the wavefront only, all other
deviations staying unchanged. The total sphere is given by D = D + D
1
41
2
CHAPTER 3. BASICS OF WAVEFRONT SENSING
Figure 3.15: A Fourier Transform Lens
dD1 D2 .
sphere of
With the zero position of d =
D=
1
D1
+ D this leads to a change of
1
2
dD D
(3.15)
with D = effective sphere in D, D = sphere of first lens in D, D =
sphere of second lens in D and d = shift between the lenses in m. The
change of corrected sphere is thus proportional to the shift between the lenses
with a paraxial system.
1
1
3.5.3
2
2
Correcting Cylinder
The cylinder pre-correction is realized by a pair of cylindrical lenses, positioned in the optical path of the system. The ideal position would be in a
conjugate plane of the pupil.
3.5.4
Spatial Filtering
The wavefront system has to eliminate the reections from surfaces other
than the retina, especially those from the cornea. For this reason an aperture
is introduced into the last telescope. The size of the aperture is crucial. If it is
too large, unwanted light comes through, if it is too small, spatial frequencies
of the wavefront will be cut o.
The rst lens of the telescope works as a fourier transformer, the aperture in
the Fourier plane as a lter for high frequencies.
42
3.6. SINGLE PASS MEASUREMENT
In a perfect optic the crossing of the aperture will be limited to that part of
the beam, of which the tilt satises the following conditions:
tan fr
(3.16)
The spatial information for smaller frequencies gets lost. With f=80mm and
r=0,5mm should be smaller than 0.00625 which corresponds to a sphere
of 2.1D (PTV=18,7 m) at a 6mm beam.
Higher order wavefront distortions have a stronger maximal tilt with the
same amplitude. So they get cut o with an even smaller amplitude. For
third order spherical aberration the cut-o amplitude goes down - compared
to a ideal sphere - by the factor 1/3, for fourth order terms the factor is
about 1/4. In fact it is the total local tilt in a wavefront that matters, not
the tilt per polynomial.
3.6 Single Pass Measurement
The Hartmann-Shack method works on the assumption that the focus-point
on the retina is a point-lightsource. This is correct, if the reection from the
retina is diuse and the light loses all its phase information of the way into
the eye. Several retina models (e.g [Ar95], [Di00]) attribute the fact that the
retina is diuse to the roughness of the surface made up by the individual
cones.
The single-pass or the double-pass property (corresponding to diuse or specular reex) may prevail. This can be tested by a simple setup. A measurement of a strongly aberrated eye - with a non symmetrical kind of aberration
- is performed. If the double-pass property dominates the wavefront should
be symmetric. In case of specular reex and if incoming and outgoing beams
have the same diameter the result should be an autocorrelation of the Single
Pass. The described setup was simulated with ZEMAX, an optical design
program. The result was compared to a real measurement on an articial
eye with diuse and specular kind of reection, and to the measurement on
a real eye. For the diuse reector SPECTRALON was used, as specular
reector a mirror. The results are shown in gure3.16.
43
CHAPTER 3. BASICS OF WAVEFRONT SENSING
ZEMAX: Simulation
single pass
diffuse reflector
double pass
artificial eye:
specular reflector
human eye:
Figure 3.16: Simulation (by ZEMAX) and Measurement: Double-Pass vs.
Single-Pass
44
3.7. DESCRIBING OPTICAL IMAGING QUALITY
The results of the single pass setup are seen on the left, those for the doublepass on the right. The ZEMAX-simulation is given at the top. The results
for the articial eye are shown below. Those for a real eye with a well known
coma in front of the eye are given at the bottom.
Diaz-Santana points out the independence of phase information and intensity information. While the phase information of the rst pass gets lost, the
intensity information remains unchanged. This can be seen in the shape of
the microspots, which vary with the size of the incoming beam.
3.7 Describing Optical Imaging Quality
For the description of the performance of an optical system there are several
parameters in use. Some of them are applied to the human eye as well. A
short overview of some scales used in ophthalmology will be given in this
section.
3.7.1
Root Mean Square
The RMS of the wavefront is a very simple criterion. It is nothing but the
integrated root mean square of the dierences between the wavefront surface
and the mean value of the surface. The complex phenomenon of aberration is
packed into a single number. This makes it so convenient in ophthalmology.
The RMS can be calculated directly from the Zernike polynomials.
For the calculation of the RMS we refer to Zernike polynomials of second
order minimum. The zero order is not measured at all. The rst order gives
information about the tilt only, which is connected to the position of the eye.
It does not supply any information about the characteristics of the eye itself.
The Zernike polynomials are orthogonal and the zero order term is set to
zero. So the mean value of the wavefront surface is zero, too. The RMS is
thus simply the mean squared value of the wavefront over the pupil.
v
u R r R 2
u
W (; )2 dd
RMS = t 0 0R 1 R 2
0
0
dd
s Z Z
1
= 1
0
0
45
2
W (; )2 dd
(3.17)
CHAPTER 3. BASICS OF WAVEFRONT SENSING
In taking mean values of the Zernike polynomials the integral can be replaced
by a sum of the weighted coeÆcients. For a real pupil size the integration
will be from 0 to r. v
uR r R u
W (; ) d d
(3.18)
RMS = t
R R
2
0
=
=
=
v
u
u
t
v
u
u
t
v
u
u
t
0
2
r
0
1
r2
1
r2
1
r2
2
0
Z rZ
0
order
X
i=0
order
X
i=0
d d
order
X
2
0
ci
i=0
Z
2
2
0
Z
1
0
!2
ci Zi (; )
d d
Zi2 (; ) d d
(3.19)
(3.20)
(3.21)
c2i Z 0 2i
with Zi0 = weighting coefficient for each Zernike. It depends from the
radial and angular order.
1
n(n + 1) n l
with
i=
(3.22)
Zi0 =
(2 Æl ) (n + 1)
2 + 2 +1
With 3.21 the RMS can be calculated simply as a root of the sum of coeÆcients. This makes calculations with the RMS very easy.
The Peak To Valley (PTV) is closely connected to the RMS. While the PTV
depends - heavily - on just two extreme values, the RMS is a kind of mean
value received from the complete set of data points. This makes the RMS
much more stable against deviations.
3.7.2
Optical Aberration Index
The Optical Aberration Index (OAI) is dened as
OAI = 1 e RMS
(3.23)
The OAI has values between zero and one. Zero stands for an optical system
that is perfect and 1 for innite aberrations. The OAI is very sensitive in
the typical range for higher order aberrations. It was introduced as an even
simpler scale for the optical quality of an eye.
(
46
)
3.7. DESCRIBING OPTICAL IMAGING QUALITY
3.7.3
Modulation Transfer Function
A typical target for testing the quality of an optical system consists of a
series of alternating black and white bars of equal width with a contrast of
1. These targets are connected to a vision chart with Snellen E`s, as used in
ophthalmology. The Modulation Transfer Function (MTF) gives the contrast
of the image (as percentage of the contrast of the object) in dependence of
max Imin . The MTF
the frequency. The contrast is dened by: Contrast = IImax
Imin
may be compared to the Aerial Image Modulation (AIM) curve. This curve
shows the smallest amount of modulation a sensor like a ccd-camera or the
retina is able to detect. The AIM is a function of the frequency used as
well. As the MTF normally goes down with frequency increasing, the AIM
increases with frequency. The point of intersection gives the resolution.
For a diraction limited optic the MTF can be calculated by
2
M
(3.24)
MT F ( ) = i = ( cos sin)
+
with
Mo
(3.25)
= arccos 2NA
= frequency in cycles
mm , NA=numerical aperture and = wavelength
3.7.4
Point Spread Function
The point response of an optic should still be a point. Even if the optic is
perfect the response is a pattern - due to the diraction. In a real system
the aberrations widen the image up to a spot. The spot is represented by a
2-dimensional distribution. This is described by the Point Spread Function
(PSF).
If the aberrations are smaller than 0.25 (Rayleigh criterion) the diraction
pattern provides a good description of the PSF.
Up to about 2 it is appropriate to consider the manner in which the aberration aects the diraction pattern. For larger wavefront aberrations illumination described by raytracing is suÆcient for description.
The aberrations of eyes are in this transition zone in most cases.
47
CHAPTER 3. BASICS OF WAVEFRONT SENSING
wavefront
RMS: 0.23 µm
OAI: 1-e
(-0.23)
=0.24
1
0.8
0.6
diffraction limited (6mm)
MTF
MTF:
eye (6mm)
0.4
0.2
0
0
20
40
60
80
100
120
cycles/degree
PSF:
Figure 3.17: Dierent representations of the image quality of one eye
48
Chapter 4
Setup
This chapter gives a description of the device as a whole. First the purpose
and the general idea of the optical setup are dealt with. The elements and
their roles in the system make up the main part. Finally measurements at
test optics are presented.
4.1 Specication of the System
Specications originate in the scheduled application in ophthalmology: The
measurement of higher order aberrations in a strongly aberrated human eye
as tool for targeting refractive surgery in the cornea.
The most important specications are the eective and the active range, the
pupil size and the accuracy of the system:
eective range:
The sphere must be measured in a range from +8D to -12D
The astigmatism must be measured in a range from 0D to -5D
pupil size:
An eye must be evaluable in a pupil size up to 6mm.
Accuracy:
The amount of sphere and cylinder must be measured with an accuracy
49
CHAPTER 4. SETUP
of 0.1D, the axis with an accuracy of 2degrees.
The reconstructed wavefront must have an accuracy of 0.1 m
The range is accustomed to the typical range of ametropia. The maximum
pupil size is set by the typical size of a large pupil. A minimal evaluable
pupil size of 3mm derives from the minimum number of data points.
The accuracy should be in the order of the short-time uctuations of the
aberrations of the eye.
4.2 The Optical Setup
First an overview of the setup will be given then the details will be described
in the subsections.
Fig.4.1 gives the setup of the device. There are four main parts, two for
the measurement itself and two for the accurate positioning of the eye. To
prevent interferences between the parts each part uses a beam with a special
wavelength or polarization optimized for the particular use.
The rst part (blue) is used for the operators surveillance of the patient`s
eye position. The eye is illuminated by two LEDs at 900nm. The anterior
near-infrared ccd-camera gets a direct image of the illuminated eye through
a dichroic beamsplitter. This picture enables the operator to align the device
to the pupil in lateral and axial position.
The second part is a target provided for the patient (green dots). The green
target ( 550nm) leads the patient's line of sight to the right angle. A
further task of the target is to relax the accommodation. The target is a very
small picture in the focal plane of a lens. The beam originating from this
lens is coupled into the optical axis of the device by a dichroic beamsplitter.
Then the beam crosses both precompensation units. So the image arriving
at the retina is already sphero-cylindrically corrected.
Using both parts in combination - the surveillance unit and the target - makes
sure that the real line of sight of the eye will be measured.
The third part (red) is the illumination beam (red-dashed, 780nm). At the
top you see the laser diode. The collimated polarized beam originating from
the diode is coupled into the optical axis by a polarizing beamsplitter-cube.
50
4.2. THE OPTICAL SETUP
precompensation:
sphere
mirror
laser
eye
quarterwaveplate
precompensation:
cylinder
quarterwaveplate
active mirror
polarizing
beamsplittercubes
dichroic
beamsplitter
CCD
dichroic
beamsplittercube
target
mirror
CCD
lens-array
aperture
pupilplane
active
mirrorplane
Lenses
pinhole
defocuscorrection
Cylindercorrection
Figure 4.1: Setup of the measurement device
51
HSS
CHAPTER 4. SETUP
After the precompensation of the sphero-cylindrical aberrations of the eye
the beam crosses a quarterwave-plate before entering the eye. The eye collimates the beam onto the retina. The spot we get here is - depending on
the higher order aberrations - close to being diraction limited. The diuse
reection of this spot is used as lightsource for the measurement.
The fourth part is the sensor beam itself (also 780nm). The intensity of
the entering beam exceeds the intensity of the beam coming back from the
retina by the factor 1000. So reections from optical surfaces have to be kept
small.
The quarterwave-plate changes the polarization of the light from linear to
circular polarization. Crossing the quarterwave plate once again on the way
back the polarization rotates by 90degrees compared to the illumination
beam. So the way back can be distinguished from the way into the eye.
For directing the beam there are two telescopes (g.4.1, bottom). The rst
one images the wavefront at the pupil plane onto the plane of the active mirror, the second one takes the image over to the HSS plane. For the rst telescope the length - and thereby the optical power - can be shifted by moving
the retro mirror. So it is used for compensating the beam for defocus. After
the second lens of the anterior telescope there is a cylinder compensation
unit consisting of two rotatable cylinder lenses with focal lengths identical
in amount and diverse in sign. The beam reaching the mirror behind these
elements has only higher order aberrations and some residual cylinder and
sphere.
The second telescope is xed. In the focal point of the rst lens there is an
aperture used to lter out the reections from the lenses and the cornea. It
also limits the dynamic range of the wavefront sensor. In the focal plane
of the second telescope there is the HSS measuring the residual aberrations.
This plane is once more conjugate to the pupil plane again. The outgoing
and the incoming beam are discriminated by dierent linear polarization and
by their direction.
52
4.3. THE OBSERVATION UNIT
4.3 The Observation Unit
The observation unit provides a life image of the eye and permits the operator
to adjust the device precisely.
It consists of a camera with one lens imaging the retina to the ccd-chip. The
camera is directed onto the eye through a dichroic mirror directly on the
optical axis of the machine. The aperture is very large in order to have the
depth of focus very small. The operator watches the image on the monitor.
He gets information about any transversal or axial displacement of the eye.
A transversal displacement makes the image of the eye move o center, thus
it can be recognized on the screen directly. The interest in this kind of
displacement is minor, as it can also be measured by the HSS, and a shift
can be considered in the results. The inuence of a lateral shift on the
measurement wouldn't be large anyway. An axial displacement results in the
image on the observation camera being blurred. This displacement cannot
be reconstructed, therefore the eect must be kept small.
4.3.1
Determination of the Axial Position of the Eye
Fig.4.2 shows the importance of a precise adjustment of the the z-position
of the eye. The transversal position is not that important as it can be reconstructed with the HSS-picture.
A wrong estimation in the z-position of zmm in an eye with sphere S causes
a failure in the determination of the sphere of
S = S +1 z
(4.1)
S
If the eye is 1mm o, at a 10D eye the result will be wrong by 0.1D. So the
depths of focus should be smaller than 2mm to meet the specications.
It is not possible to reconstruct the z-position of the eye, so it has to be set
precisely with the monitoring camera.
The monitoring camera is designed with a very low depth of focus. This
causes a blur of the image if the eye is out of the perfect position. An image
can be considered as blurred if the image of a point-source gets so large that
it will be imaged on two pixels. A single pixel of our camera has a size of
1
53
CHAPTER 4. SETUP
0.6
error in D
0.1 mm
0.3 mm
0.4
1 mm
5 mm
acceptable error
0.2
0
-12 -10
-8
-6
-4
-2
0
2
4
6
8
10
12
existing refraction in D
Figure 4.2: Correlation between the precision in z-position and the following
error in sphere. In the y-axis is shown the error in D with the existing
sphere on the x-axis. The dierent curves show the dierent facilities in the
determination of the eye position.
object plane
pupil
image plane
size of cones=u´
a
av
ah
depth of focus
Figure 4.3: Denition of the depth of focus
54
4.4. THE TARGET AND VISION-CHART UNIT
1 arcmin=20/20
Foggen
Figure 4.4: Target and Vision Chart Unit:
Left: Target
Right: Vision Chart
about 11 m. As seen in g.4.3 the depth of focus can be calculated by
af 0 2u0k(a + f 0 )
af = ah av 0
(4.2)
f
(u0k(a + f ))
with af = depth of focus, a = optimal object distance, f ` = image focal
distance, k = numerical aperture, u` = blur size. If the blur size is set to
a more realistic 0.1mm (=5pixel) the depth of focus in this setup is about
0.3mm.
2
4
2
4.4 The Target and Vision-Chart Unit
The Target Unit presents a stimulus to the proband. This has two main
functions:
Providing a relaxed accommodation status of the eye.
Doing Vision Tests in the machine.
55
CHAPTER 4. SETUP
200x240 px
thickness of mirror
mirror deflection
Ux,y
36µm
40µm
Ux+1,y
Figure 4.5: Setup of the micromirror
The optic of the vision chart is limited to a lens with a photolithographic
monochrome drawing in its focal plane. The drawings can be swapped by a
revolving mechanism. It is back-lighted with an alternatively green or red
LED.
The lens works like a simple magnifying glass. The object is projected to
innity, so that the proband relaxes his accommodation to obtain a sharp
image. To make sure that the eye is relaxed the distance between lens and
object can be increased. So the patient gets a blurred image that is shifted
1.5D to the hyperope even in a relaxed eye. This shifting is called fogging,
since the object cannot be seen clearly anymore.
The target is fogged during the measurement. It shows a schematic of a
tunnel leading the eye to the far point.
After the measurement the lens moves back to its original position. Now
Snellen lines of dierent sizes can be presented. The structure size on the
target reaches from 8 m to 25 m. These sizes correspond to a vision range
from 20/10 to 20/32 (0.5arcmin to 1.6arcmin).
56
4.5. THE ACTIVE MIRROR
outgoing beam
incoming beam
maximal shift=l/2
micro mirrors
l/2 phaseshift
Figure 4.6: Functionality of the micromirror
4.5 The Active Mirror
An active matrix mirror (Aktive Senk-Spiegel Matrix, AKSM) is used in the
device (see g.4.5). It is an array of 200x240 micromirrors ((40 m x 40 m
each). Each one of the mirrors can be lowered up to 400 m independently.
The mirrors can only be lowered without the facility of tilting. With this
technique wavefronts can be corrected up to the double height of deection
- more than one wavelength. By using the 2- method (g.4.6) the range of
the wavefront deformations to be corrected can be enlarged by far.
The 2- method makes use of the phase properties of light. A sag of 2-
between two neighboring mirrors has no eect on the direction of the light
and can be subtracted without any eect on the wavefront. So the range
of movement needed for the correction of any wavefront-deformation can
be reduced to =2. In fact the use of the mirror is limited to light of one
wavelength when using the 2- method.
4.6 The Measurement Unit
This section gives a description of the main components of the process of
wavefront measuring.
57
CHAPTER 4. SETUP
4.6.1
Light Source
As light source for the sample beam we use an infrared laser diode. It is
optimized to operate close to the lasing threshold. So compared to a standard
laser the coherence level is lower, the quality of the wavefront of the beam
staying high. The high coherence level leads to a speckle-eect described
below. A polarizer in front of the laser supplies a polarization ratio greater
than 99%.
The light of the diode has a wavelength of 780nm at a power of 60 W. The
power arriving at the eye is reduced to less than 50 W by the polarizer and
as well as by several mirrors and lenses in the beam path.
4.6.2
Hartmann-Shack Sensor
The Hartmann-Shack Sensor consists of a microlens array and a ccd-camera.
The microlens array is made of gradient index lenses. Each of the lenses has
a diameter of 400 m and a focal length of 30mm at a wavelength of 780nm.
The camera has a 2/3\ ccd-chip positioned in the focal plane of the microlenses. It has 737(H)x575(V) pixel with a size of 11.6 m(H)x11.2 m(V)
() total size: 6.5mmx8.6mm). The signal to noise (S/N) ratio of the
camera is very low (59dB), the sensitivity is very high (min. illumination:
0.05lx).
4.7 Software
This section is about the use of the software, the Graphical User Interface
(GUI) and the options for data export and data analysis.
The software bases on routines developed at the Institut fur Angewandte
Physik by Michael Schottner, Frank Muller and Stefan Wuhl. The version
actually used was designed by Frank Muller and Stefan Wuhl at 20/10 Perfect Vision.
Fig. 4.7 shows the GUI of \WavePrint\, the software predominantly used. At
the top left the Patients Data Input is shown. By pressing \GO\ a second
window opens (g.4.8) giving the examiner the Hartmann-Shack pattern.
58
4.7. SOFTWARE
Figure 4.7: The Graphical User Interface
Figure 4.8: The manual control during the measurement
59
CHAPTER 4. SETUP
This window enables the user either to start the autofocus for the precompensation of sphere, cylinder and axis or to choose a setting manually.
The \Fog\ button starts the fogging of the target. This shifts the target
1.5D to the hyperope and prevents the patient from accommodation.
As soon as the picture of the iris in the PupilCam and the picture of the
Hartmann-Shack array are both sharp the measurement can be started. This
is done by pressing the \Acquire\ button.
The results are shown in the bottom part of the window. On the left you
see the total wavefront map. Next to it on the right you see the higher order
aberration map. It shows the wavefront formed by third to sixth order of
Zernike polynomials. At the bottom right the refraction data like sphere,
astigmatism and axis are given. By checking the vertex distance box the
values will be converted to the plane 16mm in front of the cornea - the plane
where glasses would be positioned. This makes them comparable to the manifest refraction normally used by ophthalmologists. The vertex distance is
calculated the same way as the failure in the sphere, by the determination of
the axial position of the eye.
The programm allows the export of sphere, cylinder, axis, pupil size and the
rst 27 Zernike coeÆcients, as well as the total RMS from second to 6th
order. This export function opens a way to further investigations.
4.8 Precompensation of Lower Order Aberrations
The principle of the Hartmann-Shack Sensor gives restrictions to the measuringrange. On the one hand the range is restricted by the optical properties of
the eye - especially if the higher order aberrations are very large - on the
other hand it is restricted by the setup of the WaveScan in multiple ways.
The task of the precompensation unit is not restricted to the compensation
of the beam coming out of the eye for the lower orders. It also has to supply
a good light-source for the sensor beam by keeping the focus-spot on the
retina very small. The larger the light source, the poorer the results at the
HSS-measurement will be. Every point of the HSS-pattern is a convolution
60
4.8. PRECOMPENSATION OF LOWER ORDER ABERRATIONS
of the point-lightsource on the retina, with the PSF of the particular part
of the total optics passed (eye plus WaveScan). The sphere and the cylinder
are corrected (sphere from -12D to +12D,_ cylinder from (-6D to 6D)). Only
higher order terms can restrict the measurement range.
The restrictions in the measuring range are twofold. One is caused by the
aperture of the second telescope, the other one by the principle of the HSS
itself.
The rst lens in our telescope works as a Fourier-Transformer. It causes a
Fourier-Transformation of the incoming beam on the focal plane. An aperture placed in this plane works as a high-frequency lter for the beam. The
smaller the aperture gets the lower the frequencies ltered out are.
Moreover the active range is limited by the spot distance on the HSS-camera
as described in chapter 3.
4.8.1
Pre-Correction of Sphere
The Precompensation of the sphere is done by a Badal system. It consists of
two similar lenses. The distance between them can be varied by the use of a
mirror system.
The eye is positioned in the focal plane of the rst lens. The HSS is xed
in the focal plain of the second lens. These positions do not depend on the
distance between the lenses. The Badal system images the wavefront of the
eye-plane to the HSS-plane, modifying the sphere by changing the distance
between the lenses. For thin lenses the relation between the distance of the
lenses and the sphere is given by
Sphere = f2
d
f2
(4.3)
For d=2f the change in sphere is zero. The relation between the correction
and the distance of the lenses is linear. The compensation becomes far more
complicated if the eye has not just sphere, but also astigmatism and higher
order aberrations.
61
CHAPTER 4. SETUP
4.8.2
Pre-Correction of Astigmatism
To precompensate the cylinder of the eye there are dierent setups. Most
widely-used in autorefractometer is a system with two crossing cylinder
lenses. All setups of this kind have two main problems in common:
a) The distance between the cylinder-lenses is small, but not zero. So the
correction is not perfect. Higher order aberrations occur, as well as a beam
distortion.
b) The cylinder-lenses commercially available normally induce a sphere. So
you cannot simply take two identical lenses. You can either take cylinderlenses dierent in sign or add a spherical lens close to the cylinder-unit.
4.8.3
Calculating Sphero-Cylindrical Lenses
A short introduction into a mathematical formalism for sphero-cylindrical
lenses will be given. It is based on an article by Thibos, Wheeler and Horner
[Th94].
Thibos formalism uses a vector method. The main advantage - compared
to the convention used in Ophthalmology - lies in the fact that the vector
components are independent. So lenses can be added and the eect of a lens
on a wavefront can be determined simply by a vector addition. The length
of the vector is connected to the size of the blur on the retina. The RMS can
also be taken directly from the vector. So the relation between the vector
and the Visual Acuity is very close. Thibos supposes that the length of the
vector is an even better characteristic of the Visual Acuity than the RMS.
The optical power of a refracting surface is dened by:
P
= (n0
n)
(4.4)
with = curvature and n0 and n the refractive indices of the media in front
of and behind the media separated by the surface.
() = x cos2 ( ) + y sin2 (
= x + (x y ) cos2(
62
)
)
(4.5)
(4.6)
4.8. PRECOMPENSATION OF LOWER ORDER ABERRATIONS
If you substitute
S = x (n n0 )
C = (x y )(n n0 )
(4.7)
(4.8)
you get
P () = S + C cos2 ( )
= S + C + C cos(2( ))
(4.9)
(4.10)
2 2
The last equation shows how the \refracting power\ of an arbitrary surface
changes with the meridian. The denition of \refracting power\ used here is
slightly dierent from the denition technically used, but it is very convenient
as approximation.
Starting with this equation a Fourier approach can be used. M = S + C form
the constant term and J cos(2( )), with J = C the harmonic term.
This leads directly to
P () = M + J cos(2( ))
(4.11)
This equation can be converted from polar into rectangular form. This results
in:
P () = M + J cos(2) + J sin(2)
(4.12)
After this transformation any sphero-cylindrical lens can be represented by
the 3 independent values M, J , J . Combined they can be written as a
power vector. In this convention lenses can be handled as vectors which
makes all calculations very simple.
2
2
0
0
4.8.4
45
45
The Use of Power Vectors
The optometric convention for describing sphero-cylindrical lenses are in polar form:
Refraction = Sphere (in D) Cylinder (in D) axis (inÆ)
(4.13)
This notation is very useful for an optometrist as it tells him how to design
a lens.
63
CHAPTER 4. SETUP
However there are various disadvantages as well. Higher sphere may even improve vision in cases where the sign of the cylinder is opposite. So problems
arise in connecting the Visual Acuity to the refraction data and in calculating
the eect of several cylinder lenses lined up. Doing statistics with the values
used by optometrists brings further problems.
The power vector method gives an answer to just these problems. A spherocylindrical lens is represented by a triple of values, the mean sphere, a crosscylinder at 0degree, and a cross cylinder at 45degree. These values can be
interpreted as coordinates of a vector representation. You can add two lenses
by simply adding the two vectors. The conversion from medical notation to
vector notation is given by:
= S + C2
C
J = cos(2 )
2
C
J = sin(2 )
2
The transformation back is given by:
q
S=M
J +J
q
C =2 J +J
1
=
2 tan( J )
M
0
45
2
0
2
45
2
0
2
45
45
J0
(4.14)
(4.15)
(4.16)
(4.17)
(4.18)
(4.19)
4.9 Speckles
If the light is collimated the spots seem to have steady granular patterns.
These patterns are called speckles. Speckles always occur if coherent light
is scattered from a stationary rough surface. The phase of the light eld is
shifted in space randomly.
The size of the speckles can be estimated as shown in g.4.9. There is no
detailed information about the surface, so it is not possible to know where
an interference will occur. If we know a place with maximum interference
we can estimate the distance to the place of minimum interference from the
64
4.9. SPECKLES
m
ea
l
tb
ren
he
co
ds
a
D
a
a
d
l
l
rough
surface
Figure 4.9: Development of speckles
opening angle and the wavelength . The smallest Speckle diameter ds is
given by the maximum angle:
= l
(4.20)
2 sin(=2) D
with = wavelength, = apex angle, l = length and D = diameter of the
scattering surface. For smaller angles the distance rises. The roughness of
ds =
the surface does not have any inuence on the size of the speckles, but it has
an eect on their contrast.
In our setup there are two locations which have an eect on the size of
speckles: Firstly the eye with the size of the light-spot as dimension of the
scattering surface (objective speckles) and secondly the HSS with the microlens diameter as dominating size (subjective speckles). So the speckle
sizes dier with kind, the rst eect leading to up to about 800 m, the second one to 60 m. While the rst kind of speckles will shade an area of some
focal points, the second kind destroys the structure of a single focal point.
For handling the speckles - reduction or elimination - there are various methods. Having the coherence length of the laser far below the roughness of the
object reduces the intensity of the speckles. In a dierent approach the
phases are changed very rapidly. Though speckles occur, they do not aect
the measurement: Since their locations shift rapidly a time integrated picture is free of any speckles.
A third possibility is be to reduce the speckles to a size that does not aect
65
CHAPTER 4. SETUP
12
0.1
10
0.08
8
0.06
0.04
4
2
0.02
0
0
-2
-0.02
-4
difference in D
measured sphere in D
6
-0.04
-6
-0.06
-8
-0.08
-10
-12
-0.1
-12
-10
-8
-6
-4
-2
0
2
4
6
8
10
12
adjusted sphere in D
Figure 4.10: Measurement of best corrected sphere: Dierence between ZEMAX results and measured results
our measurement.
In our device we use a laser with a small coherence length and a time integration of about 25ms to reduce speckles.
4.10 Test Measurements on Articial Eyes
The precision of the device has been tested for dierent kinds of the performance. Various articial eyes were designed for this purpose.
4.10.1
Testing Sphero-Cylindrical Measurements
The articial eye designed for testing sphere is very simple. The optic consists of a single lens with a reector behind. By shifting the distance between
lens and retina the sphere can be adjusted. The correlation between sphere
and distance was simulated with ZEMAX. As retina a SPECTRALON plate
which has a high diuse reection is used.
66
4.10. TEST MEASUREMENTS ON ARTIFICIAL EYES
0.1
0
0.08
0.06
0.04
0.02
-2
0
-0.02
-3
difference in D
measured Cylinder in D
-1
-0.04
-0.06
-4
-0.08
-5
-0.1
-4
-3
-2
-1
0
adjusted Cylinder in D
Figure 4.11: Measurement of best corrected cylinder: Dierence between
simulated results (ZEMAX) and measured results
0.8
mismeasured Sphere in D
0.6
0.4
0.2
0
-3
-2
-1
0
1
2
3
-0.2
-0.4
-0.6
-0.8
residual Sphere in D
Figure 4.12: Measurement of sphere without precompensation: Dierence
between ZEMAX results and measured results
67
CHAPTER 4. SETUP
0.3
measured Cylinder (D)
0.25
0.2
0.15
0.1
0.05
0
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
residual Cylinder (D)
Figure 4.13: Measurement of cylinder without precompensation: Dierence
between ZEMAX results and measured results
In a rst test the device measures the sphere with an optimal precompensation. The information given by this test is more about the acuity of the
precompensation than about the Hartmann-Shack Sensor itself.
The test for cylinder is quite similar. The articial eye in this case consists of
an achromatic lens and a second cylindrical lens in front of a reector. The
distance between the achromatic lens and the reector is kept constant. The
cylinder can be changed by moving the cylindrical lens.
The results are shown in g.4.10 and g.4.11. For both sphere and cylinder
the accuracy is higher than 0.04D throughout measuring range.
The Hartmann-Shack Sensor itself was object of the second test. The eyes
used were the same. The sphere slider and the cylinder precompensation
were not moved at all. This is also a measurement for the active range of the
device.
The precision of the measurement of sphere is very high (dierence 0.05D)
for an active range up to 1D and the precision is acceptable (dierence
0.1D) for an active range up to 1.5D (g.4.12). Outside this range the
failure rises rapidly. This is due to the pinhole in the second telescope which
68
4.10. TEST MEASUREMENTS ON ARTIFICIAL EYES
1
value in µm
0
-1
1 µm Coma
2 µm Coma
-2
3 µm Coma
4 µm Coma
-3
-4
-5
z6 z7 z8 z9 z10 z11 z12 z13 z14 z15 z16 z17 z18 z19 z20 z21 z22 z23 z24 z25 z26 z27
Zernike (no.)
Figure 4.14: Measurement of dierent Coma plates: Each Graph shows mean
values from 4 measurements.
prevents light with higher angles from arriving at the sensor.
These failures can be avoided by disregard HSS-values larger than a set limit.
The result at high cylinder is close to this (g.4.13). The cylinder precompensation was not used. Half of the sphere was corrected by the sphere
slider, so the cross-cylinder was isolated for measurement. The result was
very much the same as at the sphere measurement. In a range up to 2D of
cylinder the precision was very high (dierence 0.03D). Up to a range of
2.5D the result was acceptable (dierence 0.1D). Outside this range the
result was poor due to the same eect of a wavefront tilt at the pinhole.
4.10.2
Testing Higher Order Aberration Measurements
For the performance of the equipment at higher order aberrations a special
kind of articial eye was produced: PMMA discs were taken and treated
with a STAR S3 laser from VISX, an excimer laser system for refractive eye
surgery, described below. Its variable spot scanning system permits removing
any quantity of material wanted and altering the anterior prole of the plastic
discs.
69
CHAPTER 4. SETUP
absolute value in µm
0.5
0
-0.5
-1
z6 z7 z8 z9 z10 z11 z12 z13 z14 z15 z16 z17 z18 z19 z20 z21 z22 z23 z24 z25 z26 z27
Zernike (no.)
0,5 µm sphere
0,75 µm sphere
1 µm sphere
Figure 4.15: Measurement of dierent plates with 3rd order spherical aberration: Each Graph shows mean values from 4 measurements.
Seven discs of this kind were produced: Three discs with a coma-like shape
for generating coma of an amount of 2 m to 6 m peak to valley (PTV)
in steps of 2 m, three discs with third order spherical aberration (0.5 m to
1 m PTV in steps of 0.5 m) and one disc with the aberrations of an average
left eye.
These discs were placed in the measurement plane of the sensor having the
SPECTRALON-plate as reector in a denite position behind them. In the
test setup there are no other lenses which could cause additional higher order
aberrations.
In g.4.14 the results for the the coma plates are shown. The plates were
introduced in such a way that the coma is orientated along y-axis (Z8). The
Zernike terms are all below 0.3 m except the Z7-term. The existence of a
higher coeÆcient of Z7 can be attributed to a failure in the alignment of the
plate. Fig.4.16 shows the attempt to simulate the aberrations of a normal
eye with a plastic disc. You can see that the centering of the ablation could
be a problem.
In g.4.15 the results of the higher order spherical plates are shown. Here
70
4.10. TEST MEASUREMENTS ON ARTIFICIAL EYES
RMS:
0.25 µm
measured wavefront corrected wavefront
+1µm
+0.5 µm
0 µm
-0.5 µm
RMS:
0.09 µm
residual wavefront
-1µm
Figure 4.16: Wavefront of an average eye (average of 140 measurements of
left eyes): The picture top left shows the average eye, the picture top right
shows the measured result of the plate and bottom left shows the dierence.
71
CHAPTER 4. SETUP
phaseplate
dichroic
beamsplitter
cube
diffuser
camera
Figure 4.17: Test device for the active mirror: Light entering from the left
through the phase plate is divided by the dichroic beamsplitter cube. The
aberrations get measured in the right arm. The bottom arm is used to record
an image of the target.
are larger eects to the coma terms Z7 and Z8 which may be evoked by a
non-central ablation. While the results for the two plates with the larger
ablations were precise, those for the third plate proved to be to small.
Fig.4.16 shows the attempt to simulate the aberrations of a normal eye with
a plastic disc. On top left the average wavefront of a left eye is shown. Top
right can be seen the measurement of the plate manufactured with these
data. On the bottom the dierence between these two aberrations is shown.
It can be seen that the rotation of the ablation was not done precisely.
4.10.3
Performance Test of the Active Mirror
For an objective test of the active mirror a test device was constructed
(g.4.17). This device enables us to measure a phase-plate and look through
it into the machine at the same time. By the camera at the test device we
get an image of the target. For the measurements presented here, a target
was used with 1` apex angle corresponding to a VA of 1.0.
72
4.10. TEST MEASUREMENTS ON ARTIFICIAL EYES
Figure 4.18: Active mirror correcting cylinder
73
CHAPTER 4. SETUP
Figure 4.19: Active mirror correcting coma
74
4.10. TEST MEASUREMENTS ON ARTIFICIAL EYES
Fig.4.18 shows the correction of a small cylinder of about 0.3D. In the top
left can be seen the wavefront without using the active mirror, below that
picture the image of the vision chart received by the camera. Activating
the mirror leads to a large enhancement of both, wavefront and image. The
wavefront error can be minimized to 0.033 m, which is less than a tenth of
the earlier error. Also the image gets much sharper. While the left is on the
border of the detectable, the right image looks quite sharp.
In g.4.19 can be seen the eect on a strong coma. Without using the
mirror the RMS is larger than 1 m and the image of the VA-chart is not
detectable.
By activating the mirror the wavefront error can be minimized to about
0.15 m, which is about a seventh of the uncorrected wavefront error. Since
the PTV change in the wavefront is close to 7000nm and the wavelength of
the vision chart is close to 550nm more than 10 2- jumps were used for correcting this wavefront. This shows how well the principle of this mirror works.
75
CHAPTER 4. SETUP
76
Chapter 5
The Hartmann-Shack Sensor at
the Human Eye
Dierent measurements at human eyes were made. First the capability of
measuring the refraction of the eye was tested. Then typical patterns and
changes of higher order aberrations were studied.
At the end of this chapter two studies will be presented.
The rst study was about perfect vision. 70 eyes were measured in order
to compare the higher order aberrations with their Best Spherical Corrected
Visual Acuity (BSCVA). The assumption was that some special wavefront
shape is connected to high visual acuity. A small wavefront error should be
better than a larger one.
In the second study the device was tested for use in ophthalmology, in special
for planning LASIK or PRK treatment in the human eye. For 38 patients the
eyes were measured before operation. In every patient one eye was treated
the classical way, the other one wavefront guided. The results were compared.
Furthermore one device was installed in the Praxisaugenklinik in Heidelberg
to take values for comparing the results.
5.1 Measurements at the Human Eye
In the last chapter the accuracy in measuring test eyes was analyzed and it
proved to be very high. But there is a big dierence between measuring a
77
CHAPTER 5. THE HARTMANN-SHACK SENSOR AT THE HUMAN EYE
6
4
HSS
Measured Sphere in D
2
Autorefraktometer
Linear (HSS)
0
Linear (Autorefraktometer)
-2
y = 0.9919x + 0.0645
-4
y = 0.9697x - 0.2833
-6
-8
-10
-10
-8
-6
-4
-2
0
2
4
6
Manifest Sphere in D
Figure 5.1: Comparison of a manifest measurement of sphere with an autorefractor and by wavefront method.
1.5
100
90
80
ratio in percent
difference in D
1.25
1
0.75
0.5
70
60
50
40
30
20
0.25
10
0
0
-10
-5
0
Manifest Refraction in D
5
0
0.25
0.5
0.75
HSS
1.25
Autorefraktor
Figure 5.2: HSS vs. Autorefractometer: Sphere
Left: Absolute error vs. sphere
Right: Deviation of objective to subjective measurement
78
1
difference in sphere in D
5.1. MEASUREMENTS AT THE HUMAN EYE
xed unmoveable articial eye and a real human eye.
The main problems in measuring a real eye are caused by accommodation,
movement and rotation of the eye and the fact, that the variety of interferences in the optics is much wider in a natural eye.
Moreover there is the fact that what we measure is not really the classical
refraction: a mean value for the radius of curvature of the wavefront for just
one pupil size. On our results we can calculate the average refraction for
dierent pupil sizes, but these sizes may be dierent from the size used for
the subjective refraction.
5.1.1
Comparing the Sphero-Cylindrical Refraction
A rst requirement for the sensor is that its refraction values (sphere, cylinder
and axis) stay in close agreement with the values measured by an ophthalmologist. Common autorefractometers aim at having more than 80% of the
results within 0.5D of the manifest refraction. As a rst test we compared
the wavefront refraction to the manifest refraction. In cooperation with the
Praxis-Augenklinik in Heidelberg we took measurements on 132 eyes with
the wavefront system. Parallel to this an ophthalmologist determined the
manifest refraction and furthermore to this measured the refraction with an
autorefractometer. These results were used to test the capability in measuring eyes of untrained people in a typical environment.
In measuring the manifest refraction the ophthalmologist puts a lens in the
visual axis in front of a patients eye and asks him if his visual performance
rises or falls. By repeating this with other lenses in steps of quarter diopters
the best correction lens is found. The procedure is the same with cylinder
lenses. This method depends strongly on the patient`s help, so it is very
subjective.
Fig.5.1 shows the results. The x-axis gives the manifest refraction, the y-axis
the objective refraction. The manifest refraction and the autorefractometer
refraction are given in quarter diopter steps, the HSS values in steps of a
hundredth of a diopter.
The blue dots show the results of the HSS measurement, the red dots those
of the autorefractometer. The black dotted line gives the 0.5D tolerance of
79
CHAPTER 5. THE HARTMANN-SHACK SENSOR AT THE HUMAN EYE
100
90
80
ratio in percent
70
60
50
40
30
20
10
0
0
0.25
0.5
0.75
difference in sphere in D
1
HSS
1.25
Autorefraktor
Figure 5.3: Comparison of measuring the cylinder with a wavefront system
and in a subjective way.
the manifest refraction. The correlation of the HSS-refraction measurement
- 0.993 - is far better than that of the autorefractometer (AR) (0.982): The
spreading of the results is much smaller. Furthermore the regression of the
HSS is much closer with a regression coeÆcient of 0.99 (AR:0.97) and an
axis interception of 0.06 (AR:-0.28).
As can be seen in the right diagram in g.5.2 more than 50% of the results
are closer than 0.25D to the manifest refraction and about 80% are within a
range of 0.5D. No measurement was more than 1D o. The agreement of the
results does not depend strongly on the total sphere, as can be seen on the
left. So the z-positioning works well. Fig.5.3 shows the results of the quality
in measuring cylinder. In this examination the lengths of the power vectors
were used. Their values - given in this gure - correspond to the amount of
the cylinder lens that would correct the aberration.
Accommodation has no strong eect on measuring the cylinder. So it can
be expected to match even better. Here only 40% of the results are closer
than 0.25D, more than 85% of the results lie within a range of 0.5D of the
manifest refraction. The results show: For the specied task the setup is
80
5.1. MEASUREMENTS AT THE HUMAN EYE
0.5
3rd
4th
5th
6th
amount in µm
0.4
0.3
0.2
0.1
z9
z1
0
z1
1
z1
2
z1
3
z1
4
z1
5
z1
6
z1
7
z1
8
z1
9
z2
0
z2
1
z2
2
z2
3
z2
4
z2
5
z2
6
z2
7
z8
z7
z6
0
Figure 5.4: Reproducibility of 10 measurements: Zernike polynomials
very well suited.
At cylinder, too the result of the HSS is more reliable than that of the autorefractometer. This is, what the result of the autorefractometer tells us.
5.1.2
Reproducibility of the Results
For checking the reproducibility of the measurements on 84 eyes measurements were repeated 10 consecutive times. In g.5.4 you see the average
amount of Zernike coeÆcients in m, the error beams give the mean standard deviation. In third and 4th order the error beams are smaller than the
amount, in 5th and 6th order they are larger in most cases. This is not necessarily due to changes in the eye. It may as well be ascribed to failures in the
centering of the pupil with a strong impact on the higher order aberrations.
The reproducibility of the RMS is shown in g.5.5. The standard deviation of the total higher order RMS is about 0.03 m at a total amount of
81
CHAPTER 5. THE HARTMANN-SHACK SENSOR AT THE HUMAN EYE
amount in µm
0.30
0.20
0.10
0.00
RMS 3
RMS 4
RMS 5
RMS 6
total HO-RMS
order of RMS
Figure 5.5: Reproducibility of 10 measurements: RMS
0.25 m. It is remarkable that the mean standard deviation of the 3rd order
RMS exceeds that of the total RMS.
5.2 Standard Deviation of Sphere and Cylinder
Fig.5.6 shows the reproducibility of the measurements of sphere and cylinder. Sphere can be measured with a reproducibility below 0.13D, cylinder
with less than 0.1D. As shown on the left the axis can be measured with
a reproducibility of 3 degrees for cylinders larger than 0.5D. Unfortunately
there are no measurements for the autorefractometer to compare.
82
5.3. CHANGE OF HIGHER ORDER ABERRATION
40
deviation in degree
mean standard deviation in D
(10 measurements)
0.25
0.20
0.15
0.10
0.05
30
20
10
0.00
0
Defocus
Cylinder
(vector)
-3.0
-2.0
Cylinder in D
-1.0
0.0
Figure 5.6: Reproducibility of values: RMS
5.3 Change of Higher Order Aberration
As to the use of the wavefront sensor for planning refractive surgery it is
important to know how much the wavefront error of the eye - for dierent
reasons - changes. The change of the lens due to accommodation and the
growth in life-time will have large inuence. The change of the shape of the
cornea during the day should be taken into account too.
5.3.1
Age
Long-time measurements on the same eye were not possible. So changes in
the mean wavefront errors with age in a set of dierent eyes were analyzed
instead. With respect to a single eye this procedure can only give an idea
of the changes with age, probably leading to an underestimation of the real
eect.
The results are shown in g.5.7. In particular the mean third order spherical
aberration shows a strong change with age. From age 20 to 70 the mean value
of spherical aberration shifts about 0.5 m. The changes of the coeÆcients
of Z9 and Z12 seem to be a physiological eect that can be attributed to
the growth of the lens.
With a RMS of 0.28 m the amount of the change is about the same order
as the total wavefront error.
The other aberrations will change with age too, but their mean values do not
83
CHAPTER 5. THE HARTMANN-SHACK SENSOR AT THE HUMAN EYE
1.0
R = 0.47
value in µm
0.5
0.0
Z9
Z12
-0.5
Linear (Z 9)
Linear (Z12)
R = 0.3
-1.0
-1.5
20
25
30
35
40
45
50
55
60
65
70
age in years
Figure 5.7: Change of the HOA with age
change that much.
5.3.2
Accommodation
Modifying the shape of the lens - for accommodation of the eye - is a major
invasion in the optical properties. An inuence of the state of accommodation on the HOA is presumed.
The wavefront device is highly qualied for measuring just this. Without
using the fogging-feature the eye accommodates with the sphere precompensation. By shifting the sphere-slider from innity to the near point of the eye
the wavefront error for the whole accommodation range can be measured.
One minor problem is that the pupil size has changed with accommodation,
so that the analysis had to be done with a pupil size of only 5mm.
The results of this measurement can be seen in g.5.8. The dependence
between accommodation and Z12 is strong for all four eyes. Other Zernike
coeÆcients changed too, especially the coma term which is involved in the
age related change of the wavefront already.
84
5.4. PERFECT VISION STUDY
0.2
shift of Z12 in µm
0.1
0
-0.1
-0.2
-0.3
-0.4
-4
-3.5
-3
-2.5
-2
-1.5
-1
-0.5
0
accommodation in D
UPl
UPr
SSl
SSr
Linear (UPl)
Linear (UPr)
Linear (SSl)
Linear (SSr)
Figure 5.8: Change of the HOA with accomodation on 4 eyes
5.3.3
Daily Fluctuations
The refraction of the eye varies - as pointed out in many articles - during
the day up to 0.5D. For the cylinder values alter due to the pressure of the
eyelid on the cornea during night. The change of cylinder suggests that the
HOA of the eye diers during daytime too.
To verify this assumption, four eyes were measured in intervals of 2hours
during the day. The wavefront maps are shown in g.5.9. No larger changes
occur and - above all - there is no time trend to be seen.
The same is true for the RMS (g.5.10). There are small uctuations but
there is no general trend. The same applies for coma and spherical aberration
terms which had the largest uctuations in the other measurements. For
sphere and cylinder the variation stayed behind expectation.
The change of the shape may possibly be due to temporary eects in the
morning.
5.4 Perfect Vision Study
There are many factors inuencing the visual acuity. In the next chapter
the inuences of the single steps - from optical image-forming to image pro85
CHAPTER 5. THE HARTMANN-SHACK SENSOR AT THE HUMAN EYE
OD 1
OS 1
OD 2
OS 2
10 Uhr
12 Uhr
14 Uhr
16 Uhr
18 Uhr
20 Uhr
22 Uhr
Figure 5.9: Daily uctuations on 4 eyes: Higher Order Aberration Wavefront
Maps
86
5.4. PERFECT VISION STUDY
Figure 5.10: Daily uctuation of the total HOA-RMS on 4 eyes
0
Cylinder in D
-0.1
-0.2
-0.3
-0.4
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
BCVA
Figure 5.11: Inuence of small lower order aberrations on the BCVA
87
CHAPTER 5. THE HARTMANN-SHACK SENSOR AT THE HUMAN EYE
0.4
RMS 3
RMS 4
RMS 5
aberrations in µm
0.3
RMS 6
RMS - total higher order
0.2
Linear (RMS - total higher
order)
Linear (RMS 3)
0.1
Linear (RMS 4)
Linear (RMS 6)
0
1
1.2
1.4
1.6
Linear (RMS 5)
Best Spherical Corrected Visaul Acuity
Figure 5.12: Inuence of higher order aberrations on the BCVA
cessing in the brain - will be discussed. The Hartmann-Shack Sensor gives
us a chance to examine the rst of these steps - the image forming - very
precisely.
In this section the dependence of visual acuity on higher order aberration
will be studied.
The Best Corrected Visual Acuity (BCVA) was measured for 80 eyes, the
result was compared to their higher order aberrations.
The BCVA species the Visual Acuity of an eye with the best possible correction of sphere and cylinder. Ophthalmologists measure the spherocylindrical
correction parameters in steps of quarter diopters only. So the BCVA is affected by residual lower order aberrations, too. The inuence of the sphere
is less important as it can be compensated by accommodation. Provided the
ophthalmologist is absolutely perfect only cylinder up to 0.125D remains.
This cylinder corresponds to a RMS of about 0.18 m. In fact this value will
be even larger. An eighth diopter of sphere would lead to a RMS of 0.25 m.
Fig.5.11 shows the inuence of cylinder on the BCVA. Each single point
stands for the mean of about 18 eyes. The error bars give the 95% condence intervals for the values.
Only cylinder values below 0.5D were considered as larger cylinder had prob88
5.5. EXCIMER STUDY
ably been corrected. An explicit dependence of the BCVA on the cylinder
can be seen. In average eyes with a visual acuity of 1.75 the cylinder value is
only half compared to eyes with a BCVA of 1.15. However, no interdependence between sphere and BCVA could be made out.
The inuence of higher order aberrations is shown in g.5.12. For the BCVA
a dependence occurs, but it stays small. An increase in the BCVA from 1.1
to 1.55 is associated with a decrease of the total higher order aberrations by
about 0.05 m. This relatively small decrease can be ascribed to the strong
inuence of the cylinder.
It would be of great interest to have a closer correlation between higher order aberration and Visual Acuity. The connection can be studied with the
lately integrated cylinder correction unit. As described in the last chapter
this unit can correct cylinder to less than a twentieth of a diopter. With the
implemented vision chart unit it would be possible to correct the lower order
aberrations much better than up to now.
The active mirror enables us to perform a real BCVA test with all kinds of
aberrations removed.
5.5 Excimer Study
The object of our study was: Is there any dierence in the result if in the
planning of the laser-ablation pattern higher order aberrations are taken into
account? 42 patients took part in the study. 37 were treated with LASIK
and 5 with PRK.
What we wanted to show is: The results that excimer-laser refractive surgery
with wavefront-derived ablation targets yield in the treatment of refractive
error and higher order aberrations are clinically acceptable.
Treatment of the wavefront error of the eye should improve - not worsen - the
patients Uncorrected Visual Acuity (UCVA), in particular the Best Spherical
Corrected Visual Acuity (BSCVA).
One eye of the patient was treated with a wavefront-derived pattern, the
other one - serving as control - in the conventional manner. The study is
prospective, single-center, non-randomized and unmasked.
89
CHAPTER 5. THE HARTMANN-SHACK SENSOR AT THE HUMAN EYE
0.4
HO-RMS in m m
0.35
wavefront
0.3
classic
0.25
0.2
Preop
1 month
3 months
6 months
time
Figure 5.13: Excimer Laser Study: time trend of total higher order RMS
0.3
aberrations in µm
0.25
0.2
wavefront
0.15
classic
0.1
0.05
0
RMS 3
RMS 4
RMS 5
RMS 6
Figure 5.14: Excimer Laser Study: Distribution of RMS 6m postop
90
5.5. EXCIMER STUDY
45
40
no. of eyes
35
30
25
w avefront
20
classic
15
10
5
0
2
1.7
1.5
1.3
1.2
1.1
1
0.9
Visual Acuity
Figure 5.15: Excimer Laser Study: Best Spherical Corrected Visual Acuity
(BSCVA)
45
40
no. of eyes
35
30
25
wavefront
20
classic
15
10
5
0
2
1.7
1.5
1.3
1.2
1.1
1
0.9
0.8
0.6
Visual Acuity
Figure 5.16: Excimer Laser Study: Un-Corrected Visual Acuity (UCVA)
91
CHAPTER 5. THE HARTMANN-SHACK SENSOR AT THE HUMAN EYE
amount of coefficient in µm
0.5
0.4
0.3
wavefront
classic
0.2
0.1
0
z6 z7 z8 z9 z10 z11 z12 z13 z14 z15 z16 z17 z18 z19 z20 z21 z22 z23 z24 z25 z26 z27
Zernike coefficients
Figure 5.17: Excimer Laser Study: Comparison of the single Zernike coeÆcients
5.5.1
The Excimer Laser System
For doing the reshaping of the cornea, a Star S3 Laser System from VISX was
used. This is a 193nm argon-uoride excimer laser system with a repetition
rate of 10Hz. Compared to common excimer laser systems two features are
new: An eye-tracker and the Variable Spot Scanning (VSS).
The eye-tracker has 2 infrared cameras using the natural pupil as a landmark
to monitor the x-, y-, and z- movements of the eye at 60Hz. It serves as a
user control for the alignment of the eye.
The VSS permits the STAR laser to ablate complex non symmetric shapes
and - in this way - to correct higher order aberrations. The spot size varies:
During the wavefront treatments it is between 0.65mm to 6.5mm in diameter.
The data les of the wavefront device were transferred to a VISX researcher
for conversion to an ablation plan. The individual treatment tables contain
the necessary information for the treatment, as there are the position of every
pulse, the size of the spot and the dwell.
92
5.5. EXCIMER STUDY
5.5.2
Refractive Surgery Methods
For refractive laser surgery there are two methods most widely in use today:
Photorefractive Keratectomy (PRK) and Laser Assisted In-Situ Keratomileusis (LASIK).
In PRK the excimer procedure is applied directly at the surface of the cornea.
For the laser procedure only the epithelium has to be removed. It needs about
3 days for healing. This healing process may be non uniform and provoke
new aberrations.
LASIK is much more popular. The eye surgeon creates a thin surface ap of
the cornea using a microkeratome. This ap will be opened, so the deeper
layers can be exposed to the Excimer Laser. The invasion into the eye is
more extensive.
Still the eye can be used very soon after closing the ap. This is a great
advantage over the PRK. The problem here lies in the fact that if the repositioning of the ap is not perfect, strong higher order aberrations occur.
So both methods bear a risk of provoking further aberrations.
5.5.3
Study Group
42 patients - ages ranging from 21 to 52 - were tested for their aptitude rst.
The conditions were: Cylinder less than 2D, wavefront error less than 4 m
PTV, maximum sphere 4D. The dierence between manifest and wavefront
refraction had to be smaller than 0.5D. Contact lens wearers had to remove
their soft lenses at least one week prior to measurements (two weeks for rigid
lenses).
10 measurements were taken of each eye with the Hartmann-Shack Sensor.
All eyes were treated with the VISX STAR S3. At one eye the treatment was
based on the results of the wavefront sensor, at the other it relied on preoperative manifest refraction. The eyetracker and VSS were used throughout.
93
CHAPTER 5. THE HARTMANN-SHACK SENSOR AT THE HUMAN EYE
5.5.4
Results
Follow-up examinations took place at 1month, 3months and 6months after
treatment. The examinations consisted of 5 wavefront measurements, manifest refraction and a determination of UCVA and BSCVA.
Fig.5.13 shows the time trend of the total higher order RMS before and after the intervention, separated for the classic method and the one using the
higher order aberrations. The error beams show the 95% condence interval.
We should assume that in wavefront treated eyes the aberrations are smaller.
For denite results the sample-size obviously is too small.
Fig.5.14 shows that the dierences are spread over all orders of Zernike coeÆcients. Obviously the results are better not only in the treatment of spherical
and coma aberration, the advantages cover the more complex kinds of aberrations as well. Here again the number of measurements and the dierences
in eect are too small to obtain assured results.
Fig.5.17 shows the mean absolute values for both methods. The dierences
in the single coeÆcients stay small, the values for the classic method being
larger in most cases.
These results are promising. Nobody is interested in reducing the wavefront
error however, it is the actual visual acuity that we are all interested in. In
fact the visual acuity seems to prot as well, especially at the higher end of
the visual acuity chart.
The dierence should be largest at the BSCVA. Here in wavefront treatment
8 eyes reached a visual acuity of 1.7 and better compared to 3 eyes with
classical treatment.
A similar success was achieved at the UCVA. In this case the advantage disappears for a visual acuity of 1.5 or worse, while at the BSCVA the wavefront
treatment is superior on the whole range. An explanation could lie in the
fact that lower order aberrations are more or less the same for both methods.
94
Chapter 6
Visual Acuity
Visual Acuity is the ability to resolve a spatial pattern separated by a visual
angle. A Visual Acuity of 1 corresponds to a resolution of one minute of arc.
For measurements there are eye charts with optotypes (g.6.1).
Resolving a pattern works in two steps (g.6.4). First the optic of the eye
images the object on the retina. In the second step the neuronal system of
the retina converts the retinal image into a neuronal image in the brain.
This chapter gives some information about the procedure of measuring visual
acuity. From the point of physiology for the Visual Acuity achievable there
is a limit. An estimation will be given.
6.1 Vision Charts
The optotypes implemented in the device are Snellen-type. The shape is that
one of the letter \E\ with height and width identical, the width of the lines
being the same as the distance between. The types are arranged on a chart
in lines of 6 with dierent scales. The chart is put up at a distance of about
two meters. An apex angle of 0.86` corresponds to a Visual Acuity of 1.0
(European notation). Exact values are given below.
The relation between VA and the size of the optotypes is linear. A VA
of 1.0 is testied if a person is able to determine the open sides for all 1.0
optotypes but fails with the smaller ones in the next line. The standard of 1.0
is accustomed to the average VA of a young person. The American notation
95
CHAPTER 6. VISUAL ACUITY
Figure 6.1: Snellen Optotypes
Decimal 20 feet 6 meters logMAR apex angle
2,0
20/10
6/3
-0,3
0.43
1,6 20/12.5 6/3.75
-0,2
0.537
1,25 20/16
6/5
-0.1
0.688
1
20/20
6/6
0
0.86
0,8
20/25 6/7.5
0.10
1.075
0,63 20/32 6/10
0.20
1.43
Table 6.1: Dierent Notations of the Visual Acuity (Snellen)
for VA simply gives the distance at which a normal person can read the same
sign. A vision of 20/10 means that the distance at which a sign can be read
is double that of a normal person. The reference distance is always 20foot
(about 6m) The notations are compared in tabular 6.1. The dierence in
notation does not imply any dierence in the technique of measurement.
6.1.1
Conditions For Visual Acuity Measurements
For performing a Visual Acuity test there are very precise regulations. In
Germany this is given by DIN 58220. The most important rules are listed
96
6.2. FUNDAMENTAL LIMITS TO VISUAL PERFORMANCE
below. The concern is visual acuity in the distance.
The eye chart should have a minimum distance of 4m. The optimal
distance is about 6m ( 20foot).
The size of the test eld should be at least 4Æ ( 10%).
The luminescence of the eye chart must lie between 160cd/m and
320cd/m .
The luminescence outside the test eld must lie between 10% and 25%
of the luminescence inside.
The contrast of the optotypes must be more than 85%.
The distance between the optotypes must be larger than the optotype
itself.
Every line of optotypes must have at least 5 optotypes.
A degree of acuity is achieved if at least 60% of the optotypes are
identied correctly.
2
2
6.2 Fundamental Limits to Visual Performance
For the Visual Acuity man can achieve there are limitations of two kinds.
One limitation is set by the optics of the eye, the other by the neuronal
structure of the retina and the image processing. The rst limitation can
be calculated without problems, for the second there are some questions still
open.
6.2.1
Optical Limits
The optical limits are given in two ways:
By the size of the pupil (diraction limited) and by aberrations of the optical
system.
The refraction limited resolution is given by the Rayleigh criterion:
97
CHAPTER 6. VISUAL ACUITY
Figure 6.2: Diraction limited resolution as function of pupil size and wavelength
The coincidence of the zero order diraction maximum of one object with
the rst order diraction minimum of another object gives the minimumcondition for separate perception of two objects. This case is given by:
sin Æ 1; 22 b
(6.1)
with Æ = apex angle, = wavelength and b = size of aperture. The refraction index in the eye is about 1.33, so the wavelength in the eye is smaller
by the factor .
As g.6.2 shows the diraction limited resolution depends on the pupil size
and the wavelength to a higher degree. Resolution increases with pupil size.
This would make vision better at twilight.
To get a vision of 20/10 at 650nm - with the assumption of a perfect optic
- you need at least a 4mm pupil as shown in g.6.2
The eect in real optics is opposite: With the pupil size the aberrations of
the eye increase, especially spherical aberrations and cylinder.
Fig.6.3 compares the diraction limited MTF`s to the MTF`s for an average
3
4
98
6.2. FUNDAMENTAL LIMITS TO VISUAL PERFORMANCE
1
amount of the MTF
0.8
0.6
0.4
0.2
0
0
20
40
60
80
100
120
140
cycles/degree
6mm
5mm
4mm
3mm
2mm
6mm real
5mm real
4mm real
3mm real
2mm real
Figure 6.3: Dependence of the MTF on pupil size at a diraction limited eye
and at a real eye (at 550nm)
99
CHAPTER 6. VISUAL ACUITY
eye. The model of an average eye bases on measurements taken from 90 eyes.
To obtain aberrations of higher order the mean absolute values of the Zernike
coeÆcients were used with the sign of their mean values. The x-axis gives
the frequency in cycles/degree, the y-axis the contrast of the image. The
graph results from a simulation with ZEMAX, an optic design programme.
The MTF of the diraction limited eye increases with pupil size. In contrast
the mean real MTF has its minimum for the largest pupil, it increases with
pupils size going down. The maximum lies at about 3mm pupil size. For the
2mm pupil the real MTF is very close to the diraction limited MTF.
Compared to Visual Acuity a spatial frequency of 30cycles/degree corresponds to a Visual Acuity of 1.0 (=20/20).
6.2.2
Retinal Limits
There is a fundamental retinal limitation to visual performance too: The
ability of the photoreceptors to sample the retinal image is restricted. The
sample is always a discrete array.
In a very simple model the condition for two points to be separated by the
foveola is, that their images are recognized by two neurons with a third neuron in between without a signal (Helmholtz 1867). With a neuron distance
of about 2 m the minimum for the visual angle can be calculated by
(6.2)
sin Æ 2d
l
with d = distance between two cones (about 2.5 m) and l = length of the
eye. The length of the eye has to be corrected by the refraction index of
the eye (n=1,3). This allows a resolution of about 0.4`, corresponding to a
visual acuity of about 2.5 (20=8) at the smallest cones in the fovea. Further
improvement of the optics will not improve acuity any more, it only increases
contrast, in particular for larger pupil sizes.
A more precise model takes into account the image processing of the retina
and the more complex conguration of the cones. This - in fact - could
increase the visual performance.
These considerations show that the retinal limitations are close to the 2.0
vision. For all developments in refractive surgery this is the target.
100
6.2. FUNDAMENTAL LIMITS TO VISUAL PERFORMANCE
OPTICAL
object
PSF
- aberrations
- pupil size
NEURONAL
retinal image
discrete
neuronal
filtering
neuronal image
Figure 6.4: Origin of the Retinal Image
101
CHAPTER 6. VISUAL ACUITY
log(contrast)
1.000
0.100
MTF 2 mm Pupille
Foveal Neural Threshold (Campbell
Green)
0.010
0.001
0
10
20
30
40
50
60
cycles/degree
Figure 6.5: Foveal neural threshold
6.3 Predicting Visual Performance
The Visual Performance can be predicted by comparing the MTF of the eye
with the Foveal Neural Threshold, the AIM of the retina. The MTF gives
the remaining contrast of the retinal image at any spatial frequency. The
Foveal Neural Threshold is the contrast necessary for the fovea to record a
signal as function of spatial frequency: If the MTF lies above this threshold
a structure can be detected otherwise the information gets lost. The MTF
decreases with spatial frequency, the foveal neural threshold increases. To
determine the neural threshold of a single eye is a major problem. Fig.6.5
shows the aberration free MTF for a 2mm pupil and a mean foveal neural
threshold identied by Campbell and Green (1965). For dierent eyes this
curve varies considerably.
Opticians and medical doctors are not familiar with the MTF. So a characteristic more suitable is to be found. Comparing the blur caused by higher
order aberrations with the defocus blur gives a value apt to easy interpretation. The eect of higher order aberrations would be accessible in D as
well as in Visual Acuity. A straight way would be using the RMS get for
the blur of defocus and compare it to the RMS received from higher order
aberrations. This value of course makes a gross simplication: A function
102
6.3. PREDICTING VISUAL PERFORMANCE
20
18
16
blur size in µm
14
12
10
8
6
4
2
0
0
1
2
3
4
5
pupil diameter in mm
perfect eye (no sphere or cylinder) with mean aberrations
eye without higher order aberration
eye with 0.25D sphere
eye with 0.25 D astigmatism
Figure 6.6: Sizes of blur circles for dierent kinds of aberrations
103
6
CHAPTER 6. VISUAL ACUITY
describing the contrast sensitivity for every spatial frequency is replaced by
a single gure. However the gure is much easier to handle.
Fig.6.6 shows blur sizes for dierent kinds of aberrations. The blur size here
is dened as the diameter encircling 70% of the energy. For a pupil size
below 2.5mm the blur size is dominated by diraction as can be seen. For
larger pupils the importance of the aberrations grows. For a 6mm pupil the
blur is less than the cone size in the fovea. The blur caused by higher order
aberrations is much larger than the blur of a quarter D of astigmatism or
even of sphere.
104
Chapter 7
Conclusion and Outlook
The precise measurement of the total aberrations of the human eye is the
basis for obtaining good results in refractive surgery. The high potential of
the wavefront device in measuring wavefront errors is demonstrated on test
optics. In application to human eyes the device shows to be good for results
needed in surgery. The precision of the device exceeds the standard set by
the temporary uctuations of the aberrations.
The visual acuity study gives evidence for the importance of high precision
in measuring sphere and astigmatism: In most eyes the higher order aberrations lie below the range of the residual 2nd order aberrations sphere and
astigmatism, not detected due to the resolution in quarter diopter steps.
Some wavefront errors occur in one state of accommodation or just for a
short period in life. As to the importance of detecting and correcting these
errors there is some doubt. Measurements in relation to accommodation and
to age suggest: The correction of very small higher order aberrations of eyes
without any other deciency may be not reasonable. Finding limits is of
great interest here.
The excimer laser study is encouraging in whole. For a precise documentation of the advantage of wavefront guided to the traditional procedure further
studies are needed. Improving the interface between wavefront sensor and
excimer laser will be of great importance here. A major improvement would
lie in using the observation image of the iris for nding the pupil center and
for detecting any rotation of the eye.
105
CHAPTER 7. CONCLUSION AND OUTLOOK
It would be ideal to have this information directly in the eyetracker of the
excimer laser.
Transferring the application to a further laser technology for refractive surgery
could be an important step: With intrastromal femtosecond lasers the cornea
could be reshaped without having the unpredictable eects in the healingprocess. In case the higher order aberrations have changed in time there
could be repetition in the reshaping.
The most recent implemented improvements - the continuously adjustable
cylinder compensation and the adaptive optics - open the gate for real Best
Corrected Visual Acuity tests. The eects of higher order aberrations on the
visual ability and in special on the maximum in the visual acuity achievable
can be studied now.
First in interest is the further development of the wavefront device itself:
Enlarging the diameter of the ccd-camera gives the option for measuring bigger pupil sizes. Correcting the higher order aberrations of the illumination
beam provides a new line of improvement. In addition to an increase in the
precision the eective range for strongly aberrated eyes would be enlarged.
106
List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
Cross section of the eye . . . . . . . . . . . . . . . . . . . . . .
Optical setup of the human eye . . . . . . . . . . . . . . . . .
The transparency of the human eye . . . . . . . . . . . . . . .
Change of transparency with age . . . . . . . . . . . . . . . .
The magnication depends from the length of the eye. . . . .
Accommodation . . . . . . . . . . . . . . . . . . . . . . . . . .
Accommodation of a myopic eye . . . . . . . . . . . . . . . . .
Refractive errors of the eye and there formation . . . . . . . .
Development of astigmatism . . . . . . . . . . . . . . . . . . .
Development of spherical aberration . . . . . . . . . . . . . . .
Dependence of the total refraction of the eye from the wavelength
Denition of a Wavefront . . . . . . . . . . . . . . . . . . . . .
Thinbeam Ray-Tracing Aberrometer . . . . . . . . . . . . . .
Tscherning Aberrometer . . . . . . . . . . . . . . . . . . . . .
Hartmann-Shack Method . . . . . . . . . . . . . . . . . . . . .
Hartmann-Test . . . . . . . . . . . . . . . . . . . . . . . . . .
Idea of a Hartmann-Shack Sensor . . . . . . . . . . . . . . . .
Functionality of a HSS demonstrated on a single lens . . . . .
Image on ccd-chip . . . . . . . . . . . . . . . . . . . . . . . . .
Limitations of the HSS . . . . . . . . . . . . . . . . . . . . . .
Dynamic range of a Hartmann-Shack Sensor. . . . . . . . . . .
Maximal measurable orders of Zernike . . . . . . . . . . . . .
Chart of Zernike polynomials up to 4th order . . . . . . . . .
Fourier Optic in incoherent imaging . . . . . . . . . . . . . . .
Wavefront propagating from a plane to its conjugate plane . .
107
6
12
13
13
15
16
17
18
19
20
21
24
25
26
26
27
28
29
30
30
32
33
37
39
41
LIST OF FIGURES
3.15
3.16
3.17
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
A Fourier Transform Lens . . . . . . . . . . . . . . . . .
Single-Pass measurement . . . . . . . . . . . . . . . . . .
Dierent representations of the image quality of one eye .
Setup of the measurement device . . . . . . . . . . . . .
Eect of precision of z-position . . . . . . . . . . . . . .
Denition of the depth of focus . . . . . . . . . . . . . .
Target and Vision Chart Unit . . . . . . . . . . . . . . .
Setup of the micromirror . . . . . . . . . . . . . . . . . .
Functionality of the micromirror . . . . . . . . . . . . . .
The Graphical User Interface . . . . . . . . . . . . . . .
The manual control during the measurement . . . . . . .
Development of speckles . . . . . . . . . . . . . . . . . .
Best corrected sphere . . . . . . . . . . . . . . . . . . . .
Best corrected cylinder . . . . . . . . . . . . . . . . . . .
Pure HSS sphere measurement . . . . . . . . . . . . . . .
Pure HSS cylinder measurement . . . . . . . . . . . . . .
Measurement of coma plates . . . . . . . . . . . . . . . .
Measurement of spherical plates . . . . . . . . . . . . . .
Wavefront of an average eye . . . . . . . . . . . . . . . .
Test device for the active mirror . . . . . . . . . . . . . .
Active Mirror correcting Cylinder . . . . . . . . . . . . .
Active Mirror correcting coma . . . . . . . . . . . . . . .
Sphere measurement: HSS vs. autorefractor . . . . . . .
HSS vs. Autorefractometer: Sphere . . . . . . . . . . . .
Real Cylinder Measurement . . . . . . . . . . . . . . . .
Reproducibility of Zernike Polynomials . . . . . . . . . .
Reproducibility of 10 measurements: RMS . . . . . . . .
Reproducibility of values: RMS . . . . . . . . . . . . . .
Change of the HOA with age . . . . . . . . . . . . . . .
Change of the HOA with accomodation on 4 eyes . . . .
Daily Fluctuation of the Wavefront . . . . . . . . . . . .
Daily uctuation of the total HOA-RMS . . . . . . . . .
Inuence of small lower order aberrations on the BCVA .
108
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42
44
48
51
54
54
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56
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59
65
66
67
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78
80
81
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84
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86
87
87
LIST OF FIGURES
5.12
5.13
5.14
5.15
5.16
5.17
6.1
6.2
6.3
6.4
6.5
6.6
Inuence of higher order aberrations on the BCVA
Study: time trend of total higher order RMS . . . .
Study: distribution of RMS 6m postop . . . . . . .
Study: BCVA . . . . . . . . . . . . . . . . . . . . .
Study: UCVA . . . . . . . . . . . . . . . . . . . . .
Study: Zernike coeÆcients . . . . . . . . . . . . . .
Snellen Optotypes . . . . . . . . . . . . . . . . . . .
Diraction limited resolution . . . . . . . . . . . . .
MTF of a real eye . . . . . . . . . . . . . . . . . . .
Origin of the Retinal Image . . . . . . . . . . . . .
Foveal neural threshold . . . . . . . . . . . . . . . .
Blur circles for dierent kinds of aberrations . . . .
109
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. 99
. 101
. 102
. 103
LIST OF FIGURES
110
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116
Acknowledgement
My thanks go to ...
Prof. Dr. Josef Bille who gave me the chance to work on such a
challenging project in the eld of ophtomological optics.
Prof. Dr. Brenner for his interest in my work and his willingness to
take on the second referees.
20/10 Perfect Vision: for making this project possible and for giving
me their support.
Dr. med. Volz, Dr. med. Gleibs and Mrs. Pankrath for their cooperation and help in the clinical studies.
Peter Brockhaus who helped me getting started with my dissertation.
Tobias Kuhn, Stefan Wuhl, Bernhard Gress, Michael Schumacher and
my other colleagues for the excellent atmosphere at work.
Frank Muller, Joana Costa, Michael Schottner, Karaneh Razavi, Nina
Korablinova and the other group members for critical discussions and
support.
Sylvia for the wonderful time we spent together when not working on
this project.
my parents who encouraged me. Without their support throughout my
entire study all this would not have been possible.
117
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