## Introduction

The development of new instrumentation to measure human optical aberrations and the recent refinements in the excimer laser delivery systems have opened a new era in vision correction: patient-customized, wavefront-guided treatment. It has been known for a long time that the normal human eye suffers from many monochromatic aberrations that degrade retinal image quality. Current ophthalmic lenses correct defocus and astigmatism but still leave uncorrected additional aberrations. The pattern of these aberrations varies among individuals and reduces the optical performance of the eye for pupil diameters larger than 3 mm. Well-developed techniques previously proposed and developed in astronomy have been employed to estimate the aberration of the eye. In 1997, using adaptive optics, Liang et al. corrected optical aberrations beyond sphere and cylinder and provided normal eyes with supernormal optical quality ; similar results were obtained by other authors. The use of wavefront technology has since come into focus due to recent rapid advancements in technology to measure the optical properties of the human eye.

The application of wavefront-sensing technology might enable the noninvasive observation of living retinal cone cells, the measurement of central nervous visual function by eliminating higher-order aberrations using adaptative optics, and the implementation of higher-order correction in everyday vision through intraocular lenses, customized contact lenses, or laser refractive surgery. Custom corneal ablation procedures involve the use of wavefront analysis to measure the aberrations of the eye beyond sphere and cylinder and to direct the photoablation on the cornea. Although conventional laser procedures increase higher-order aberrations, wavefront-guided profiles of ablation aim to correct both spherocylindrical ametropia and high-order aberrations to optimize the postoperative patient’s visual function. The limits of ocular performance are determined by the quality of the retinal image and by neural architecture and function. At maximal image quality, visual acuity should reach 20/8, or between 20/12 and 20/5, depending on pupil size, which is more than the usual 20/20 visual acuity. To achieve this “super vision,” two conditions must be present: the eye must be free of optical aberrations and the pupil must be dilated to minimize the effects of diffraction.

The cornea is the principal optical component of the human eye. Modern aberrometers are equipped with a corneal topographer system. Such instruments enable computation of the effect of the anterior and/or posterior corneal contribution to the ocular wavefront and, by subtraction, the effect of internal optics (the crystalline lens, or an intraocular lens in pseudophakic eyes). Clinicians are more familiar with the geometric conception of light propagating in a rectilinear fashion as rays. For a better understanding, this chapter will briefly describe some optical principles related to the field of optical aberration and diffraction that derive from the wave properties of light. After discussing the basics of the wavefront theory, we will show how it can be used to predict the optical performance of the human eye.

## History of Wavefront: The Debate Concerning the Phenomenon of Light

The exact nature of light has been an intriguing subject. From ancient times, many scientists have experimented with light in order to better understand its true nature. Among them, Willibrord Snell first formulated what is expressed today as Snell’s law of refraction to describe the properties of light propagation in optical media:

nisini=nrsinr.

Geometric optics embraces the concept of the light ray and wavefront of Huygens. It has allowed the design of many optical instruments, such as telescopes and microscopes. The earlier Newtonian concept of light particles was refined two centuries later (in 1905) by Albert Einstein with the introduction of the photon as the smallest particle of light that retains the information from the initial source. This important step allowed explanation of some observed physical properties of light and prediction of the properties and feasibility of laser systems. Both ideas of light as a wave and light as a particle remain today as the wave/particle dualism in optics.

Optical aberrations and diffraction are physical events that derive from the wave properties of light. We will first review these properties and then study the principles of wavefront-guided ablations.

## Wavefront Theory

## What Is a Wavefront?

A wave, being a light wave or a sound wave, is defined by its frequency (number of oscillations per unit of time) and its propagation speed. The wavelength of a monochromatic light wave is a function of these two parameters. The visible spectrum corresponds to wavelengths between 400 and 700 nm.

A wavefront propagates like the surface ripples that emanate from the point of impact of a stone tossed into a tank of water ( Fig. 5.3 ). In a homogeneous medium, a monochromatic light source emits wavefronts that propagate at a constant speed in all directions from the source. At a given moment, the points in space located at the same distance from that source are in the same state of the value of the electromagnetic field. The wavefront is the envelope of these points and would be spherical in this case ( Figs. 5.4 and 5.5 ).

## How Does a Wavefront Propagate?

The consecutively emitted wavefronts by a monochromatic light source are separated by equal time intervals. This property allows one to prove the law of refraction of Snell (known as Snell-Descartes in France; Fig. 5.6 ). Refraction occurs when the wavefront meets a different environment and its rate of spread is reduced.

If a wavefront of light propagates in the empty medium, the speed propagation is designated as “c.” If a planar wavefront is refracted by a plano lens, its speeds decrease proportionally to the value of the refractive index of the lens ( Fig. 5.7 ). As the frequency is unchanged, the wavelength is reduced in the lens. When the surface of the plano lens is parallel to the wavefront envelope, no phase shift will appear and the shape of the wavefront will be unchanged as it exits from the lens.

When the surface of the plano lens is not parallel to that of the incident wavefront, the latter will undergo a deviation but no shape modification. Because of the skewed position of the lens as compared to that of the wave, part of it will undergo the reduction in speed while the other part still moves on at unchanged speed. This will cause a change in the position of the whole wavefront ( Fig. 5.8 ).

If a planar wavefront propagates through a planar convex lens, the optical path will be different for the wave entering the lens at a different location (the optical path will be maximal in the center of the lens). The lens introduces a retardation of the phase of the central portion of the wavefront relative to its edges. This will cause the emerging wavefront to converge ( Fig. 5.9 ). Thus given a flat wavefront traveling through a perfect convex lens, the resulting emerging wavefront will be changed to spherical so that all the light rays perpendicular to the wavefront come exactly in one point.

The wavefront distortion can be considered as a phase retardation distribution relative to its most advanced point. After having traveled in a homogeneous medium (constant refractive index), the wavelengths that have the longest path will exit later than those with a shorter path. This difference in optical path can be expressed in microns.

When different colors of light propagate at different speeds in a medium, the refractive index is wavelength dependent. A well-known example is the glass prism that disperses an incident beam of white light at equal angles. Because the various optical media have a different refractive index for each wavelength of light, chromatic aberration in the human eye is the result of the different focus location for different wavelengths. Thus chromatic aberrations correspond to departures from perfect imaging that are owing to dispersion and make their appearance only in polychromatic light. They cause a diminution of the retinal image contrast. However, there is a larger gain when monochromatic aberrations are corrected without correcting chromatic aberration than when polychromatic aberrations are corrected alone. There is currently no practical solution to correct for polychromatic aberrations; we will consider only the field of the correction of monochromatic aberration in the rest of this chapter.

## What Is Diffraction?

Diffraction involves the bending of waves around obstacles. It is generally guided by Huygens’s principle, which states that every point on a wavefront acts as a source of tiny wavelets that move forward with the same speed as the wave; the wavefront at a later instant is the surface that is tangent to the wavelets. The presence of an obstacle induces a distortion in the wavefront propagation ( Figs. 5.10 and 5.11 ). Thus it is impossible to obtain a perfectly spherical wavefront. In the case of the diffraction by an aperture, the narrower the aperture, the greater the effect on the wavefront that has propagated beyond the aperture. Conversely, the larger the entrance pupil in an optical system, the less diffraction will impact the image quality. Diffraction alone causes a minimum blurred image called an *Airy disk* . It represents the “spread” of the incident light caused by pupil diffraction and makes perfect stigmatism practically impossible with any diaphragm optical system. The diffraction phenomenon is wavelength dependent. The longer the wavelength, the narrower the aperture and the larger the light spread. Aberrations in the optical system of the eye counteract the improvements in resolution that are expected according to diffraction theory with increasing pupil size. In the normal eye well corrected for sphere and cylinder, higher-order aberrations that are unmasked by the pupil dilation will start to degrade the image quality more than diffraction for pupil diameters greater than 3 mm.

The diffraction phenomenon, which is consubstantial of the wave properties of light, can be used for the design of multifocal diffractive intraocular optics (IOLs). In such design, diffraction is not initiated by the passage of light through a pupil or its deviation by an obstacle but rather by the controlled spatial modulation of the IOL thickness.

## Diffraction and Fourier Transform

The Fourier transform has become a powerful analytical tool in diverse fields of science. In some cases, the Fourier transform can provide a means of solving unwieldy equations that describe dynamic responses to electricity, heat, or light. In other cases, it can identify the regular contributions to a fluctuating signal, thereby helping to make sense of observations in astronomy, medicine, and chemistry. Light waves can be represented as periodic oscillations of the electromagnetic field. Fourier analysis (or spectral, or harmonic, analysis) indicates that any periodic function can be fairly well approximated by the sum of a series of sinusoidal terms. Given a periodic function (wave) in the space domain, it is possible to break it up into its Fourier components. The Fourier transform accomplishes this by breaking down the original time-based waveform into a series of sinusoidal terms, each with a unique magnitude, frequency, and phase. This process, in effect, converts a waveform in the time domain that is difficult to describe mathematically into a more manageable series of sinusoidal equations.

The Fourier spectrum can be represented by displaying the frequency along one axis and the magnitude (or amplitude) along a second axis. Plotting the amplitude of each sinusoidal term versus its frequency creates a power spectrum, which is the response of the original waveform in the frequency domain. Inversely, the original periodic function can be synthesized by putting the proper spectral components together. Fourier series are generally the sums of many waves of many frequencies.

To illustrate this concept, we can take as an example the decomposition of a sound or any periodic signal in its different harmonics ( Fig. 5.12 ). The ear formulates a transform by converting sound—the waves of pressure traveling over time and through the atmosphere—into a spectrum, a description of the sound as a series of volumes at distinct pitches. The brain then turns this information into perceived sound. The sound can be reconstructed with fidelity by adding the harmonics that were present in the initial decomposition. This may also allow studying the effect of the removal (filtering) of a particular harmonic (i.e., remove a particular optical aberration).

When a plane parallel beam of monochromatic light is incident upon a small aperture, the diffraction pattern observed at a very large distance from the aperture along the optical axis will contain a very good approximation of the Fourier transform of the aperture function. These conditions (large distance between the aperture and the plane of observation of the diffractive pattern) are termed of *Fraunhofer* type (Fraunhofer diffraction). The Fourier transform can also be displayed in the focal plane of a lens following the diffractive aperture.

If light is generated by a monochromatic source, such as a laser, the light waves that are generated are derived from the same source and exhibit a fixed relationship between their phases. This kind of light is said to be coherent and interference will be an important factor to consider. In daily life, light waves are emitted by effective independent sources (sun, lightbulbs, and so on). Even if these sources were monochromatic, the relations between the waves converging to the image plane would vary randomly. The quantity determining the net effect of these random superpositions is the average light irradiance. Thus at optical wavelength, the only detectable optical signal is the irradiance that is proportional to the square of the Fourier transform of the optical disturbance within the aperture (Fraunhofer irradiance), which corresponds to the point spread function for incoherent imaging. Simply put, in examining the optical properties of the human eye, the application of these concepts allows prediction of how the light emanating from a single point source will be imaged on the retina by combining the effects of both diffraction and ocular aberrations. This is achieved mathematically by computing the square of the Fourier transform of the ocular wavefront within the exit pupil. The inspection of the retinal point spread function is of clinical importance, as the light spread pattern may provide insights to some visual disturbances such as halos and monocular diplopia.

## Ocular Aberrations

Paraxial optics, or first-order optics, rely on the assumption that the height of incident light rays from the optical axis is small and that the considered optical systems are free of aberrations. In such idealized conditions, spherical surfaces yield perfect imagery. Real-life optical systems such as the human eye are not perfect; the description of their optical properties falls out of the paraxial domain. The departures of the idealized conditions of paraxial optics are known as higher-order aberrations.

Two main types of aberration can be distinguished: chromatic aberrations (which arise from the fact that the refractive index is actually a function of frequency or color) and monochromatic aberrations. The latter fall into subgroupings, such as spherical aberration and coma. The monochromatic optical aberrations of optical systems increase as the incident ray height increases ( Fig. 5.13 ).

The normal emmetropic eye is free of aberration when its pupil diameter is less than 2.5 mm. At that pupil diameter size, the diffraction by the edges of the pupil is the only factor that governs the size of the retinal image of a point source. When the pupil diameter increases, the quality of the retinal image decreases owing to the increase in optical aberrations ( Fig. 5.14 ). However, for an eye that would be free of optical aberration, the quality of the retinal image would increase when the pupil dilates owing to the reduction of the effect of diffraction. Such an eye would be said to be diffraction limited.

A signal must be sampled with a frequency at least twice the frequency of the signal itself. The retinal surface is tiled with photoreceptors of discrete areas ( Fig. 5.15 ). This imposes an upper limit to the resolution capacity of the human eye, which is called the *Nyquist limit* . The sampling frequency of the foveal cone mosaic is about 120 c/deg. The Nyquist limit, or maximum detectable frequency without error, is thus half the sampling frequency. Therefore the foveal cones offer a maximal sampling rate of about 60 c/deg (equivalent to a 20/10 line on a letter acuity chart). When spatial frequencies that exceed this limit are formed on the retina, they cannot be correctly interpreted and the image is said to be aliased. In the human eye, these aliases form irregular shapes.

Ocular aberrations are usually quantified in terms of a wavefront aberration that is expressed in microns. They result in an increased spread of the light emanating from an incoherent light point source imaged by a fixating patient on the fovea. Depending on the amount of this spreading, a reduction in contrast sensitivity and visual acuity can result ( Figs. 5.16 and 5.17 ).

There have been only a few studies on the second- and higher-order aberrations of the eye in the peripheral visual field. These studies show that optical aberration increases rapidly away from the fixation axis. We will focus on the aberrations impairing foveal vision.

## What Is a Wavefront?

A wave, being a light wave or a sound wave, is defined by its frequency (number of oscillations per unit of time) and its propagation speed. The wavelength of a monochromatic light wave is a function of these two parameters. The visible spectrum corresponds to wavelengths between 400 and 700 nm.

A wavefront propagates like the surface ripples that emanate from the point of impact of a stone tossed into a tank of water ( Fig. 5.3 ). In a homogeneous medium, a monochromatic light source emits wavefronts that propagate at a constant speed in all directions from the source. At a given moment, the points in space located at the same distance from that source are in the same state of the value of the electromagnetic field. The wavefront is the envelope of these points and would be spherical in this case ( Figs. 5.4 and 5.5 ).

## How Does a Wavefront Propagate?

The consecutively emitted wavefronts by a monochromatic light source are separated by equal time intervals. This property allows one to prove the law of refraction of Snell (known as Snell-Descartes in France; Fig. 5.6 ). Refraction occurs when the wavefront meets a different environment and its rate of spread is reduced.

If a wavefront of light propagates in the empty medium, the speed propagation is designated as “c.” If a planar wavefront is refracted by a plano lens, its speeds decrease proportionally to the value of the refractive index of the lens ( Fig. 5.7 ). As the frequency is unchanged, the wavelength is reduced in the lens. When the surface of the plano lens is parallel to the wavefront envelope, no phase shift will appear and the shape of the wavefront will be unchanged as it exits from the lens.

When the surface of the plano lens is not parallel to that of the incident wavefront, the latter will undergo a deviation but no shape modification. Because of the skewed position of the lens as compared to that of the wave, part of it will undergo the reduction in speed while the other part still moves on at unchanged speed. This will cause a change in the position of the whole wavefront ( Fig. 5.8 ).

If a planar wavefront propagates through a planar convex lens, the optical path will be different for the wave entering the lens at a different location (the optical path will be maximal in the center of the lens). The lens introduces a retardation of the phase of the central portion of the wavefront relative to its edges. This will cause the emerging wavefront to converge ( Fig. 5.9 ). Thus given a flat wavefront traveling through a perfect convex lens, the resulting emerging wavefront will be changed to spherical so that all the light rays perpendicular to the wavefront come exactly in one point.

The wavefront distortion can be considered as a phase retardation distribution relative to its most advanced point. After having traveled in a homogeneous medium (constant refractive index), the wavelengths that have the longest path will exit later than those with a shorter path. This difference in optical path can be expressed in microns.

When different colors of light propagate at different speeds in a medium, the refractive index is wavelength dependent. A well-known example is the glass prism that disperses an incident beam of white light at equal angles. Because the various optical media have a different refractive index for each wavelength of light, chromatic aberration in the human eye is the result of the different focus location for different wavelengths. Thus chromatic aberrations correspond to departures from perfect imaging that are owing to dispersion and make their appearance only in polychromatic light. They cause a diminution of the retinal image contrast. However, there is a larger gain when monochromatic aberrations are corrected without correcting chromatic aberration than when polychromatic aberrations are corrected alone. There is currently no practical solution to correct for polychromatic aberrations; we will consider only the field of the correction of monochromatic aberration in the rest of this chapter.

## What Is Diffraction?

Diffraction involves the bending of waves around obstacles. It is generally guided by Huygens’s principle, which states that every point on a wavefront acts as a source of tiny wavelets that move forward with the same speed as the wave; the wavefront at a later instant is the surface that is tangent to the wavelets. The presence of an obstacle induces a distortion in the wavefront propagation ( Figs. 5.10 and 5.11 ). Thus it is impossible to obtain a perfectly spherical wavefront. In the case of the diffraction by an aperture, the narrower the aperture, the greater the effect on the wavefront that has propagated beyond the aperture. Conversely, the larger the entrance pupil in an optical system, the less diffraction will impact the image quality. Diffraction alone causes a minimum blurred image called an *Airy disk* . It represents the “spread” of the incident light caused by pupil diffraction and makes perfect stigmatism practically impossible with any diaphragm optical system. The diffraction phenomenon is wavelength dependent. The longer the wavelength, the narrower the aperture and the larger the light spread. Aberrations in the optical system of the eye counteract the improvements in resolution that are expected according to diffraction theory with increasing pupil size. In the normal eye well corrected for sphere and cylinder, higher-order aberrations that are unmasked by the pupil dilation will start to degrade the image quality more than diffraction for pupil diameters greater than 3 mm.

The diffraction phenomenon, which is consubstantial of the wave properties of light, can be used for the design of multifocal diffractive intraocular optics (IOLs). In such design, diffraction is not initiated by the passage of light through a pupil or its deviation by an obstacle but rather by the controlled spatial modulation of the IOL thickness.

## Diffraction and Fourier Transform

The Fourier transform has become a powerful analytical tool in diverse fields of science. In some cases, the Fourier transform can provide a means of solving unwieldy equations that describe dynamic responses to electricity, heat, or light. In other cases, it can identify the regular contributions to a fluctuating signal, thereby helping to make sense of observations in astronomy, medicine, and chemistry. Light waves can be represented as periodic oscillations of the electromagnetic field. Fourier analysis (or spectral, or harmonic, analysis) indicates that any periodic function can be fairly well approximated by the sum of a series of sinusoidal terms. Given a periodic function (wave) in the space domain, it is possible to break it up into its Fourier components. The Fourier transform accomplishes this by breaking down the original time-based waveform into a series of sinusoidal terms, each with a unique magnitude, frequency, and phase. This process, in effect, converts a waveform in the time domain that is difficult to describe mathematically into a more manageable series of sinusoidal equations.

The Fourier spectrum can be represented by displaying the frequency along one axis and the magnitude (or amplitude) along a second axis. Plotting the amplitude of each sinusoidal term versus its frequency creates a power spectrum, which is the response of the original waveform in the frequency domain. Inversely, the original periodic function can be synthesized by putting the proper spectral components together. Fourier series are generally the sums of many waves of many frequencies.

To illustrate this concept, we can take as an example the decomposition of a sound or any periodic signal in its different harmonics ( Fig. 5.12 ). The ear formulates a transform by converting sound—the waves of pressure traveling over time and through the atmosphere—into a spectrum, a description of the sound as a series of volumes at distinct pitches. The brain then turns this information into perceived sound. The sound can be reconstructed with fidelity by adding the harmonics that were present in the initial decomposition. This may also allow studying the effect of the removal (filtering) of a particular harmonic (i.e., remove a particular optical aberration).

When a plane parallel beam of monochromatic light is incident upon a small aperture, the diffraction pattern observed at a very large distance from the aperture along the optical axis will contain a very good approximation of the Fourier transform of the aperture function. These conditions (large distance between the aperture and the plane of observation of the diffractive pattern) are termed of *Fraunhofer* type (Fraunhofer diffraction). The Fourier transform can also be displayed in the focal plane of a lens following the diffractive aperture.

If light is generated by a monochromatic source, such as a laser, the light waves that are generated are derived from the same source and exhibit a fixed relationship between their phases. This kind of light is said to be coherent and interference will be an important factor to consider. In daily life, light waves are emitted by effective independent sources (sun, lightbulbs, and so on). Even if these sources were monochromatic, the relations between the waves converging to the image plane would vary randomly. The quantity determining the net effect of these random superpositions is the average light irradiance. Thus at optical wavelength, the only detectable optical signal is the irradiance that is proportional to the square of the Fourier transform of the optical disturbance within the aperture (Fraunhofer irradiance), which corresponds to the point spread function for incoherent imaging. Simply put, in examining the optical properties of the human eye, the application of these concepts allows prediction of how the light emanating from a single point source will be imaged on the retina by combining the effects of both diffraction and ocular aberrations. This is achieved mathematically by computing the square of the Fourier transform of the ocular wavefront within the exit pupil. The inspection of the retinal point spread function is of clinical importance, as the light spread pattern may provide insights to some visual disturbances such as halos and monocular diplopia.

## Ocular Aberrations

Paraxial optics, or first-order optics, rely on the assumption that the height of incident light rays from the optical axis is small and that the considered optical systems are free of aberrations. In such idealized conditions, spherical surfaces yield perfect imagery. Real-life optical systems such as the human eye are not perfect; the description of their optical properties falls out of the paraxial domain. The departures of the idealized conditions of paraxial optics are known as higher-order aberrations.

Two main types of aberration can be distinguished: chromatic aberrations (which arise from the fact that the refractive index is actually a function of frequency or color) and monochromatic aberrations. The latter fall into subgroupings, such as spherical aberration and coma. The monochromatic optical aberrations of optical systems increase as the incident ray height increases ( Fig. 5.13 ).

The normal emmetropic eye is free of aberration when its pupil diameter is less than 2.5 mm. At that pupil diameter size, the diffraction by the edges of the pupil is the only factor that governs the size of the retinal image of a point source. When the pupil diameter increases, the quality of the retinal image decreases owing to the increase in optical aberrations ( Fig. 5.14 ). However, for an eye that would be free of optical aberration, the quality of the retinal image would increase when the pupil dilates owing to the reduction of the effect of diffraction. Such an eye would be said to be diffraction limited.

A signal must be sampled with a frequency at least twice the frequency of the signal itself. The retinal surface is tiled with photoreceptors of discrete areas ( Fig. 5.15 ). This imposes an upper limit to the resolution capacity of the human eye, which is called the *Nyquist limit* . The sampling frequency of the foveal cone mosaic is about 120 c/deg. The Nyquist limit, or maximum detectable frequency without error, is thus half the sampling frequency. Therefore the foveal cones offer a maximal sampling rate of about 60 c/deg (equivalent to a 20/10 line on a letter acuity chart). When spatial frequencies that exceed this limit are formed on the retina, they cannot be correctly interpreted and the image is said to be aliased. In the human eye, these aliases form irregular shapes.

Ocular aberrations are usually quantified in terms of a wavefront aberration that is expressed in microns. They result in an increased spread of the light emanating from an incoherent light point source imaged by a fixating patient on the fovea. Depending on the amount of this spreading, a reduction in contrast sensitivity and visual acuity can result ( Figs. 5.16 and 5.17 ).

There have been only a few studies on the second- and higher-order aberrations of the eye in the peripheral visual field. These studies show that optical aberration increases rapidly away from the fixation axis. We will focus on the aberrations impairing foveal vision.

## Wavefront Measurement

Analysis of human ocular optical aberration relies on the pioneering work of Hartmann and Tscherning in the nineteenth century. Hartmann described the principles of “outgoing” objective wavefront analysis. After reflexion of an incident coherent monochromatic light wave on the fovea, the outgoing wavefront is captured outside of the eye on a charge coupled device (CCD) matrix. The analyzed wavefront corresponds to the conjugated effects of all the ocular media (vitreous, crystalline lens, cornea, and tear film). Conversely, Tscherning wavefront measuring machines allow the study of wavefront distortion through the analysis of the image of a distorted projected mire on the retina (ingoing reflective aberrometry). A similar principle is used today by sequential laser ray tracing systems. Automated skiascopy using infrared light projected through a rotative slit scanning can also be used to study the optical path differences and obtain a wavefront reconstruction. Other devices soliciting subjective patient participation, such as spatially resolved refractometry (ingoing aberrometry), are not used for clinical examinations.

All of these systems share a common principle: the wavefront analysis is performed through the study of the distortion of an emitted signal. Because of their widespread use, the wavefront reconstruction using the Hartmann–Shack system will be presented here.

## Outgoing Reflective Aberrometry Using Hartmann–Shack Wavefront Analyzers

These machines are based on the Scheiner disk principle, named after a seventeenth-century philosopher and astronomer. This ingenious apparatus allowed detection of the blur caused by the optical aberrations of the eye ( Fig. 5.18 ). This technique was refined by the consecutive work of Hartmann and Shack.

The principal steps leading to wavefront detection and analysis are the following ( Figs. 5.19–5.23 ):

- •

emission of an incident light ray centered on the fovea

- •

detection of the reflected wavefront out of the eye using a microlenslet array

- •

focalization of the wavefront on a CCD device by each microlenslet (the wavefront is broken down on different contiguous portions)

- •

The location of the spot corresponding to the portion of the refracted wavefront is compared to the reference location (that corresponds to a flat/nonaberrated wavefront).

- •

The average slope of each wavefront portion is calculated.

- •

Mathematical integration calculus allows reconstruction of the three-dimensional shape of the wavefront envelope using Zernike polynomials. These polynomials are selectively weighted depending on their respective contribution to wavefront distortion. The number of Zernike polynomials and the number of microarray lenslets limit the accuracy of the wavefront detection.

For an ideal eye, emmetropic and free of any monochromatic aberration (diffraction-limited eye), the emerging wavefront is flat. There is no deviation from the expected location of the spots imaged by the microarray lenslets because each refracted portion of the wavefront is flat and parallel to the lenslet. If the wavefront was measured “in the eye,” it would theoretically be spherical and centered on the fovea. When aberrations are present, they cause a geometric shift of the spots away from their reference position. The amount of deviation is directly related to the slope of the wavefront.

## Wavefront Study With Retinal Imagery

The deformed signal is measured at the retinal level.

## Tscherning Analyzing System

A coarse array of light rays is obtained from the filtration of a 532-nm laser radiation through a perforated mask. Each beam has a diameter of 0.5 mm. The rays are projected on the retina on a surface of approximately 1 mm ^{2} ( Fig. 5.24 ). The image is then imaged on a CCD camera through the 0.9-mm central area of the cornea that is assumed to be free of optical aberrations. The retinal spot pattern is analyzed and compared to the theoretical distribution of an aberration-free eye. The displacement of the retina spots from their aberration-free position is used to calculate the wavefront envelope.

## Retinal Ray Tracing

The procedure is analogous to that of the Tscherning procedure, but the spots are distributed sequentially to avoid reconstruction errors, especially for highly aberrated eyes ( Fig. 5.25 ).

## Ingoing Adjustable Refractometry

The rays are emitted through different precise locations in the entrance pupil, similar to the principle of the Scheiner disk. If aberrations are present, double images will be perceived by the patient. The angle of deviation necessary to superimpose the image on the retina is proportional to the local wavefront distortion.

## Double-Pass Aberrometry (Slit Skiascopy/OPD Scan Device)

This apparatus is based on retinoscopic principles. A slit of light is scanned into the eye along different meridians over the full pupil. The timing and scan rate of the reflected light is analyzed by an array of photo detectors to determine the wave aberrations along these meridians. This technology enables reconstruction of the wavefront in eyes with high ametropias.

## Accuracy and Repeatability of Wavefront Measurements

Several studies address the repeatability of static wavefront measurements. The repeatability decreases with pupil misalignment errors, short-term variation in the actual ocular aberrations, tear film rupture, and small drifts in the measuring equipment.

The accuracy of wavefront measurement may be variable depending on the type of wavefront sensor used. The spots associated with the lenslet array in a Shack–Hartmann sensor can overlap when a patient has a very distorted wavefront. This can be addressed by increasing the dynamic range of the system. High resolution is also important to accurately analyze an eye that has fine structure aberrations.

Ray-tracing instruments may be sensitive to saccadic eye movements, especially if they have a long scan time. The spots being analyzed by ray-tracing instruments are being imaged by the eye and the instruments must make some assumptions regarding the shape of the retina.

## Outgoing Reflective Aberrometry Using Hartmann–Shack Wavefront Analyzers

These machines are based on the Scheiner disk principle, named after a seventeenth-century philosopher and astronomer. This ingenious apparatus allowed detection of the blur caused by the optical aberrations of the eye ( Fig. 5.18 ). This technique was refined by the consecutive work of Hartmann and Shack.

The principal steps leading to wavefront detection and analysis are the following ( Figs. 5.19–5.23 ):

- •

emission of an incident light ray centered on the fovea

- •

detection of the reflected wavefront out of the eye using a microlenslet array

- •

focalization of the wavefront on a CCD device by each microlenslet (the wavefront is broken down on different contiguous portions)

- •

The location of the spot corresponding to the portion of the refracted wavefront is compared to the reference location (that corresponds to a flat/nonaberrated wavefront).

- •

The average slope of each wavefront portion is calculated.

- •

Mathematical integration calculus allows reconstruction of the three-dimensional shape of the wavefront envelope using Zernike polynomials. These polynomials are selectively weighted depending on their respective contribution to wavefront distortion. The number of Zernike polynomials and the number of microarray lenslets limit the accuracy of the wavefront detection.

For an ideal eye, emmetropic and free of any monochromatic aberration (diffraction-limited eye), the emerging wavefront is flat. There is no deviation from the expected location of the spots imaged by the microarray lenslets because each refracted portion of the wavefront is flat and parallel to the lenslet. If the wavefront was measured “in the eye,” it would theoretically be spherical and centered on the fovea. When aberrations are present, they cause a geometric shift of the spots away from their reference position. The amount of deviation is directly related to the slope of the wavefront.

## Wavefront Study With Retinal Imagery

The deformed signal is measured at the retinal level.

## Tscherning Analyzing System

A coarse array of light rays is obtained from the filtration of a 532-nm laser radiation through a perforated mask. Each beam has a diameter of 0.5 mm. The rays are projected on the retina on a surface of approximately 1 mm ^{2} ( Fig. 5.24 ). The image is then imaged on a CCD camera through the 0.9-mm central area of the cornea that is assumed to be free of optical aberrations. The retinal spot pattern is analyzed and compared to the theoretical distribution of an aberration-free eye. The displacement of the retina spots from their aberration-free position is used to calculate the wavefront envelope.

## Retinal Ray Tracing

The procedure is analogous to that of the Tscherning procedure, but the spots are distributed sequentially to avoid reconstruction errors, especially for highly aberrated eyes ( Fig. 5.25 ).

## Ingoing Adjustable Refractometry

The rays are emitted through different precise locations in the entrance pupil, similar to the principle of the Scheiner disk. If aberrations are present, double images will be perceived by the patient. The angle of deviation necessary to superimpose the image on the retina is proportional to the local wavefront distortion.

## Tscherning Analyzing System

A coarse array of light rays is obtained from the filtration of a 532-nm laser radiation through a perforated mask. Each beam has a diameter of 0.5 mm. The rays are projected on the retina on a surface of approximately 1 mm ^{2} ( Fig. 5.24 ). The image is then imaged on a CCD camera through the 0.9-mm central area of the cornea that is assumed to be free of optical aberrations. The retinal spot pattern is analyzed and compared to the theoretical distribution of an aberration-free eye. The displacement of the retina spots from their aberration-free position is used to calculate the wavefront envelope.

## Retinal Ray Tracing

The procedure is analogous to that of the Tscherning procedure, but the spots are distributed sequentially to avoid reconstruction errors, especially for highly aberrated eyes ( Fig. 5.25 ).

## Ingoing Adjustable Refractometry

The rays are emitted through different precise locations in the entrance pupil, similar to the principle of the Scheiner disk. If aberrations are present, double images will be perceived by the patient. The angle of deviation necessary to superimpose the image on the retina is proportional to the local wavefront distortion.

## Double-Pass Aberrometry (Slit Skiascopy/OPD Scan Device)

This apparatus is based on retinoscopic principles. A slit of light is scanned into the eye along different meridians over the full pupil. The timing and scan rate of the reflected light is analyzed by an array of photo detectors to determine the wave aberrations along these meridians. This technology enables reconstruction of the wavefront in eyes with high ametropias.

## Accuracy and Repeatability of Wavefront Measurements

Several studies address the repeatability of static wavefront measurements. The repeatability decreases with pupil misalignment errors, short-term variation in the actual ocular aberrations, tear film rupture, and small drifts in the measuring equipment.

The accuracy of wavefront measurement may be variable depending on the type of wavefront sensor used. The spots associated with the lenslet array in a Shack–Hartmann sensor can overlap when a patient has a very distorted wavefront. This can be addressed by increasing the dynamic range of the system. High resolution is also important to accurately analyze an eye that has fine structure aberrations.

Ray-tracing instruments may be sensitive to saccadic eye movements, especially if they have a long scan time. The spots being analyzed by ray-tracing instruments are being imaged by the eye and the instruments must make some assumptions regarding the shape of the retina.

## Wavefront Analysis and Map Interpretation

## Principles of Wavefront Reconstruction

The total wavefront is converted in the sum of elementary aberrations that are selectively weighted. The preferred surface fitting method to characterize the wavefront envelope characteristics at this time uses Zernike polynomials. Reconstruction of the wavefront using Zernike polynomials allows the extraction of useful information. This mathematical expansion has been used extensively in optics and astronomy to decompose the optical aberrations of an optical system into well-described aberrations. These aberrations include sphere and cylinder, but Zernike analysis also allows extraction of higher-order aberrations, such as coma and spherical aberration, that is, Zernike terms above the third order.

This concept derives from the Fourier decomposition but, rather than using simple sine/cosine functions, relies on the use of Zernike functions. The attentive reader will notice the implementation of sine/cosine functions (whose frequency corresponds to the azimuthal frequency) in every nonrotationally symmetric Zernike polynomial expressed in polar form ( Table 5.1 ).

## Use of Zernike Polynomials in Wavefront Sensing

In the field of adaptive optics, Zernike polynomials are particularly useful for wavefront decomposition. These functions are usually represented in a pyramid ( Fig. 5.26 ). They are normalized and expressed on the unit pupil disk, and the human ocular pupil is also circular. These functions are defined in a Cartesian conventional system centered on the center of the ocular entrance pupil ( Fig. 5.27 ). The first Zernike polynomials have a physical practical interpretation because they correspond to classical optical aberrations. Each Zernike function can be expressed in polar coordinates (r, theta), where r is the distance from the pupil center and theta its azimuthal angle, as the product of a polynomial function in r and a cosine or sine function in theta ( Figs. 5.28 and 5.29 ). The degree n of a polynomial is defined as the highest value of the r ^{n} term of the polynomial function.