Wavefront Analysis





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:


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This formula is discussed in Chapter 3 . In 1690, Christiaan Huygens postulated that light originates from a pulsing source. From the source, pulses of light energy expand into space and create change in the substance that he called “ether,” which was supposed to surround us and be present in any substance. According to Huygens’s theory, light propagation showed similarities with that of fluid and gasses ( Fig. 5.1 ). The concept of light emanating from a pulsing source raised the important point that energy would fluctuate as it propagates. When the distance from the source increases, the energy propagates in a parallel direction ( Fig. 5.2 ). The light emanating from the point source at infinity would then appear to the observer’s eye as a flat wavefront. This is equivalent to the parallel light rays emanating from infinity. Here, the rays represent the direction of propagation of the wavefronts and are mutually perpendicular.


Fig. 5.1


According to the Huygens’ principle, every point on a propagating wavefront serves as the source of spherical secondary wavelets such that the wavefront at some later time is the envelope of these wavelets. The secondary wavelets have the same frequency and speed as that of the wave emanating from the source S when they propagate in the same (isotropic) medium. Fresnel later successfully modified Huygens’s principle, adding the concept of interference.



Fig. 5.2


The principle of Huygens was completed and mathematically formulated by Fresnel at the beginning of the nineteenth century, including the concept of interference. This concept explained that there was no wave reissued “backward” by “light vibration” contiguous sources. In a dense homogeneous medium, there is little or no light scattered sideways or backward. Earth’s atmosphere contains millions of molecules in a cube whose side lengths would be equal to that of a visible light wavelength (e.g., 500 nm). This is related to the size of the molecules in the nanometer range, while the visible light wave is of the order of a micron. Consider that a plane wave propagates in a medium dense and homogeneous, composed of many contiguous atoms, represented here schematically by dots. At the level of atoms, the light is absorbed and reemitted by electrons. This phenomenon is responsible for changes in the phase of the wavefront, which can then appear as slowed in the propagation media (the index of refraction is proportional to the slowdown of the transmitted wave). Perpendicular to the propagation direction, we can form many pairs of atoms separated by half a wavelength (atoms are about a thousand times smaller than the relevant visible wavelength). Given the principle of Huygens, each atom reemits the incident wave in all directions. Thus in a perpendicular direction from the main direction the light spread, there is destructive interference for each of these pairs of atoms, which explains the absence of lateral diffusion of the light. Given the principle of conservation of energy, there cannot be constructive interference in all directions. In a sparse environment (e.g., upper atmosphere), because of the scarcity of the atoms and molecules, the phenomenon of interference disappears. Then, there are random encounters between light and atoms of lateral reemissions, which can show that they are proportional to the fourth power of the frequency of the considered light wave. Blue (higher frequency) is more scattered than red, which explains the color of the sky: our eyes perceive a bluish hue because the blue wavelength is more scattered laterally and toward the ground.


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 ).




Fig. 5.3


(A) When a stone is tossed into a tank of water, the surface ripples emanate from the point of impact and spread out in two-dimensional circular waves. This imagery can be extended to three dimensions, where a small pulsating sphere surrounded by a fluid would generate pressure variations propagating outward as a spherical wave as it expands and contracts. (B) Three-dimensional representation of light propagation showing the wavefronts as concentric spheres that increase in diameter as they expand out into the surrounding space of an idealized point source of light, S . Rays are orthogonal trajectories of the wavefront (one ray is represented as a dotted line emanating from S ).





Fig. 5.4


A propagating wavefront of light can be defined by the locus of the points lying in the same optical path from the source (A, C). When the optical path length is the same for all the rays emitted by a source (A), they interfere constructively to produce a sharp image of the source (B). When the optical path length is different for the emitted rays (C), they arrive at different phases; the system is aberrated and the image suffers from degradation (D).





Fig. 5.5


As a spherical wavefront propagates, its radius increases. A small area of the incoming wavefront, located at a far distance from the source, closely resembles a flat portion of a plane wave: at a given time, all the surfaces on which the disturbance has a constant phase form a set of planes, each perpendicular to the direction of propagation (A). For a “perfect eye” (diffraction limited), the optical paths of all the rays emitted by a single point source are identical: the light oscillates an identical number of times from the source to the fovea. The planar wavefronts are converted into spherical wavefronts centered on the fovea (B).


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.




Fig. 5.6


In the left media, the wavefront moves at speed v i . In the right media, with a higher refractive indice, the propagation speed (v r ) is reduced. The wavefront forms an angle θ i with normal to the surface and is deflected with an angle θ r after refraction. Given the principle of Huygen’s, the time taken by light to go from C to B is the same as for getting from A to D . This time equals distance divided by the speed, and one may write: CB/v i = AD/v r . According to the geometry of the figure and observing that θ i is equal to the angle (CAB) and θ i is equal to the angle (ABD) , one can express that the distance CB = AB sin( θ i ) and that the distance AD = AB sin( θ r ). The first equation becomes AB sin( θ i )/V i = AB sin( θ r )/v r . We can simplify by AB and multiply by a constant c (speed of light in a vacuum) and get that sin( θ i ) c/v i = sin( θ r ) c/v r . The ratio between the speed of light in a vacuum and light in the medium is equal to the index of refraction of the medium: n i = c/v i = 1 if the incident medium is air and n r = c/v r . Finally, we obtain that sin( θ i ) = sin( θ r ) n r .


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.




Fig. 5.7


Representation of a beam of light traveling in a vacuum at a speed c and impinging on a glass interface at a null angle. The glass atoms scatter light; the transmitted wave propagates with an effective speed, less than c. In addition, the wavelength of light decreases in the glass plate, but the oscillation of the wave (frequency) remains constant. When the wave emerges from the glass, its speed is c again.


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 ).




Fig. 5.8


When a beam of light impinges on a glass interface of indice n t at a non-null angle, the transmitted wavefront is slower than the incident electromagnetic wave because the atoms in the region of the surface of the transmitting medium reradiate the wavelets at a slower speed. These wavelets combine constructively to form a refracted beam that is bent as it crosses the boundary. The fact that the incident rays are bent is called refraction . The path actually taken by light in going from point A to point B is the shortest optical path length: OPL = n i × OA + n r OB. Differential calculus leads to the expression: n i × sin(θ i ) = n r × sin(θ r ).


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.




Fig. 5.9


When a portion of wavefront (WF) passes through a material of nonuniform thickness, it is distorted. Because the thickness varies, it causes the rays having the same OPL to bend and take on a spherical shape beyond the lens. In this example, the lens acts as a refracting device that converts a beam of plane waves into converging spherical waves. This assertion is equivalent to the geometrical optics assertion that when a parallel bundle of rays passes through a converging lens, the point to which it converges is a focal point of the lens (insert) .


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.




Fig. 5.10


The diffraction causes the deviation of light from rectilinear propagation, which is not caused by refraction or reflection. Diffraction occurs when the wavelength is large compared to the aperture (A); the waves then spread out at large angles into the region beyond the obstruction. According to the Huygens–Fresnel principle, every unobstructed point of a wavefront serves as a source of spherical secondary wavelets. Thus the multiple wavelets emitted from the aperture interfere constructively or destructively beyond the aperture (B). When the aperture is very small, the parallel beam is reduced to a wave that propagates in all directions (A). The larger the aperture, the less diffraction will take place (B).





Fig. 5.11


As opposed to the geometrical optics, where light rays propagate in rectilinear fashion (A), physical optics deals with light waves emanating from a source (B). Because of the diffraction caused by the edges of the aperture, the transmitted wavefront is slightly distorted beyond the aperture (*). This causes the irradiance produced by any optical system with one or multiple diaphragms to take the form of a blurred spot over a finite area (B). This patch of light in the image plane is called the point-spread function (PSF) . Diffraction thus destroys stigmatism (C). Schematic representation of the irradiance produced by the optical system free of aberrations, which corresponds to the diffraction figure of the input source. When no aberrations are present, an Airy pattern is formed in the image plane.




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).




Fig. 5.12


Any periodic signal (full line) can be broken down into fundamental harmonics selectively weighted (dotted lines) . Conversely, the addition of the weighted fundamentals allows reconstruction of the original signal. This is the basis of Fourier analysis.


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 ).




Fig. 5.13


Spherical aberration of a lens. Rays striking the surface at a greater distance above the axis are focused nearer the vertex. Those rays are stopped when the pupil is narrow. When the pupil is large, the marginal rays are bent too much and focus in front of the paraxial rays. The distance between the axial intersection of a ray and the paraxial focus is known as the longitudinal spherical aberration. Spherical aberration shifts the light out of the central disk to the surrounding rings. If a screen is placed at the focal plane of such a lens, the image of a point source will appear as a bright central spot on the axis surrounded by a symmetrical halo delineated by the cone of marginal rays. The envelope of the refracted rays is called a caustic .


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.




Fig. 5.14


Wavefront (A, C, E, G) and point-spread functions (B, D, F, H) as a function of the pupil diameter for a typical uncorrected slightly simple hyperopic astigmatic eye (OPD scan, Nidek). Note the increase of the wavefront error RMS value with pupil dilation, in great part due to the increase in high-order aberrations (A, C, E, G). The point spread function (PSF) represents how a single object is imaged by the optical system; at 3 mm, it resembles a diffraction-limited PSF (B). When the pupil dilates, high-order optical aberrations unmask and inspection of the corresponding respective PSFs reveals asymmetric enlargement (D, F, H). WF, Wavefront.




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.




Fig. 5.15


For an eye limited only by diffraction and chromatic aberration, the image of an optotype falling on the photoreceptors requires one photoreceptor line per dark and light bar to be detected. In the fovea, the cones are equally spaced and the distance between two adjacent cone areas is about 3 µm. If the size of the letter is decreased, undersampling by the foveal cones would occur and the pattern would not be correctly detected. In the represented conditions, the visual acuity of this eye would be approximately 20/10.


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 ).




Fig. 5.16


In an eye with no optical aberrations (A), the point-spread function (PSF) corresponds to an Airy disk pattern. If the source is made with two components, such as the bars of a Snellen E letter, two juxtaposed Airy patterns will result. When these patterns overlap, a certain amount of ambiguity exists in deciding when the two systems are individually discernible or to be resolved. Lord Rayleigh’s criterion states that the sources are resolved when the center of one Airy disk falls on the minimum of the other Airy disk pattern (A, yellow and purple PSF). This condition is achieved in the top part of the figure. The retinal image is sharper, and the area under the modulation transfer function (MTF) curve is maximal (A). In the presence of optical aberrations (B), the PSF is broader and the alignment is less precise, resulting in a blurred Snellen E letter and reduced area under the MTF curve.



Fig. 5.17


For a given spatial frequency (defined by the number of light and dark bars per degree of visual field), the perceived image has a lower contrast owing to the presence of diffraction and possible optical aberrations after passing through the eye optical system. The contrast of the observed vertical sinusoidal grating can reduce to its threshold, that is, the value to which the subject is not able to discern its orientation.


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 ).




Fig. 5.3


(A) When a stone is tossed into a tank of water, the surface ripples emanate from the point of impact and spread out in two-dimensional circular waves. This imagery can be extended to three dimensions, where a small pulsating sphere surrounded by a fluid would generate pressure variations propagating outward as a spherical wave as it expands and contracts. (B) Three-dimensional representation of light propagation showing the wavefronts as concentric spheres that increase in diameter as they expand out into the surrounding space of an idealized point source of light, S . Rays are orthogonal trajectories of the wavefront (one ray is represented as a dotted line emanating from S ).





Fig. 5.4


A propagating wavefront of light can be defined by the locus of the points lying in the same optical path from the source (A, C). When the optical path length is the same for all the rays emitted by a source (A), they interfere constructively to produce a sharp image of the source (B). When the optical path length is different for the emitted rays (C), they arrive at different phases; the system is aberrated and the image suffers from degradation (D).





Fig. 5.5


As a spherical wavefront propagates, its radius increases. A small area of the incoming wavefront, located at a far distance from the source, closely resembles a flat portion of a plane wave: at a given time, all the surfaces on which the disturbance has a constant phase form a set of planes, each perpendicular to the direction of propagation (A). For a “perfect eye” (diffraction limited), the optical paths of all the rays emitted by a single point source are identical: the light oscillates an identical number of times from the source to the fovea. The planar wavefronts are converted into spherical wavefronts centered on the fovea (B).




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.




Fig. 5.6


In the left media, the wavefront moves at speed v i . In the right media, with a higher refractive indice, the propagation speed (v r ) is reduced. The wavefront forms an angle θ i with normal to the surface and is deflected with an angle θ r after refraction. Given the principle of Huygen’s, the time taken by light to go from C to B is the same as for getting from A to D . This time equals distance divided by the speed, and one may write: CB/v i = AD/v r . According to the geometry of the figure and observing that θ i is equal to the angle (CAB) and θ i is equal to the angle (ABD) , one can express that the distance CB = AB sin( θ i ) and that the distance AD = AB sin( θ r ). The first equation becomes AB sin( θ i )/V i = AB sin( θ r )/v r . We can simplify by AB and multiply by a constant c (speed of light in a vacuum) and get that sin( θ i ) c/v i = sin( θ r ) c/v r . The ratio between the speed of light in a vacuum and light in the medium is equal to the index of refraction of the medium: n i = c/v i = 1 if the incident medium is air and n r = c/v r . Finally, we obtain that sin( θ i ) = sin( θ r ) n r .


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.




Fig. 5.7


Representation of a beam of light traveling in a vacuum at a speed c and impinging on a glass interface at a null angle. The glass atoms scatter light; the transmitted wave propagates with an effective speed, less than c. In addition, the wavelength of light decreases in the glass plate, but the oscillation of the wave (frequency) remains constant. When the wave emerges from the glass, its speed is c again.


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 ).




Fig. 5.8


When a beam of light impinges on a glass interface of indice n t at a non-null angle, the transmitted wavefront is slower than the incident electromagnetic wave because the atoms in the region of the surface of the transmitting medium reradiate the wavelets at a slower speed. These wavelets combine constructively to form a refracted beam that is bent as it crosses the boundary. The fact that the incident rays are bent is called refraction . The path actually taken by light in going from point A to point B is the shortest optical path length: OPL = n i × OA + n r OB. Differential calculus leads to the expression: n i × sin(θ i ) = n r × sin(θ r ).


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.




Fig. 5.9


When a portion of wavefront (WF) passes through a material of nonuniform thickness, it is distorted. Because the thickness varies, it causes the rays having the same OPL to bend and take on a spherical shape beyond the lens. In this example, the lens acts as a refracting device that converts a beam of plane waves into converging spherical waves. This assertion is equivalent to the geometrical optics assertion that when a parallel bundle of rays passes through a converging lens, the point to which it converges is a focal point of the lens (insert) .


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.




Fig. 5.10


The diffraction causes the deviation of light from rectilinear propagation, which is not caused by refraction or reflection. Diffraction occurs when the wavelength is large compared to the aperture (A); the waves then spread out at large angles into the region beyond the obstruction. According to the Huygens–Fresnel principle, every unobstructed point of a wavefront serves as a source of spherical secondary wavelets. Thus the multiple wavelets emitted from the aperture interfere constructively or destructively beyond the aperture (B). When the aperture is very small, the parallel beam is reduced to a wave that propagates in all directions (A). The larger the aperture, the less diffraction will take place (B).





Fig. 5.11


As opposed to the geometrical optics, where light rays propagate in rectilinear fashion (A), physical optics deals with light waves emanating from a source (B). Because of the diffraction caused by the edges of the aperture, the transmitted wavefront is slightly distorted beyond the aperture (*). This causes the irradiance produced by any optical system with one or multiple diaphragms to take the form of a blurred spot over a finite area (B). This patch of light in the image plane is called the point-spread function (PSF) . Diffraction thus destroys stigmatism (C). Schematic representation of the irradiance produced by the optical system free of aberrations, which corresponds to the diffraction figure of the input source. When no aberrations are present, an Airy pattern is formed in the image plane.




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).




Fig. 5.12


Any periodic signal (full line) can be broken down into fundamental harmonics selectively weighted (dotted lines) . Conversely, the addition of the weighted fundamentals allows reconstruction of the original signal. This is the basis of Fourier analysis.


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 ).




Fig. 5.13


Spherical aberration of a lens. Rays striking the surface at a greater distance above the axis are focused nearer the vertex. Those rays are stopped when the pupil is narrow. When the pupil is large, the marginal rays are bent too much and focus in front of the paraxial rays. The distance between the axial intersection of a ray and the paraxial focus is known as the longitudinal spherical aberration. Spherical aberration shifts the light out of the central disk to the surrounding rings. If a screen is placed at the focal plane of such a lens, the image of a point source will appear as a bright central spot on the axis surrounded by a symmetrical halo delineated by the cone of marginal rays. The envelope of the refracted rays is called a caustic .


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.




Fig. 5.14


Wavefront (A, C, E, G) and point-spread functions (B, D, F, H) as a function of the pupil diameter for a typical uncorrected slightly simple hyperopic astigmatic eye (OPD scan, Nidek). Note the increase of the wavefront error RMS value with pupil dilation, in great part due to the increase in high-order aberrations (A, C, E, G). The point spread function (PSF) represents how a single object is imaged by the optical system; at 3 mm, it resembles a diffraction-limited PSF (B). When the pupil dilates, high-order optical aberrations unmask and inspection of the corresponding respective PSFs reveals asymmetric enlargement (D, F, H). WF, Wavefront.




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.




Fig. 5.15


For an eye limited only by diffraction and chromatic aberration, the image of an optotype falling on the photoreceptors requires one photoreceptor line per dark and light bar to be detected. In the fovea, the cones are equally spaced and the distance between two adjacent cone areas is about 3 µm. If the size of the letter is decreased, undersampling by the foveal cones would occur and the pattern would not be correctly detected. In the represented conditions, the visual acuity of this eye would be approximately 20/10.


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 ).




Fig. 5.16


In an eye with no optical aberrations (A), the point-spread function (PSF) corresponds to an Airy disk pattern. If the source is made with two components, such as the bars of a Snellen E letter, two juxtaposed Airy patterns will result. When these patterns overlap, a certain amount of ambiguity exists in deciding when the two systems are individually discernible or to be resolved. Lord Rayleigh’s criterion states that the sources are resolved when the center of one Airy disk falls on the minimum of the other Airy disk pattern (A, yellow and purple PSF). This condition is achieved in the top part of the figure. The retinal image is sharper, and the area under the modulation transfer function (MTF) curve is maximal (A). In the presence of optical aberrations (B), the PSF is broader and the alignment is less precise, resulting in a blurred Snellen E letter and reduced area under the MTF curve.



Fig. 5.17


For a given spatial frequency (defined by the number of light and dark bars per degree of visual field), the perceived image has a lower contrast owing to the presence of diffraction and possible optical aberrations after passing through the eye optical system. The contrast of the observed vertical sinusoidal grating can reduce to its threshold, that is, the value to which the subject is not able to discern its orientation.


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.




Fig. 5.18


The Hartmann aberrometer derives from the seventeenth-century Scheiner disk. The optical path length is related to the number of times the light wave must oscillate traveling from one point to another. When aberrations are present, the optical path of rays emanating from a single point source will differ to the fovea. The observer will see two images instead of one. When there are no optical aberrations (insert) , the OPL is the same for all the light rays traveling from the object point to the image point.


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.




Fig. 5.19


Reflected light from a point source on the retina will emerge from a perfect eye as a plane wave. This reflected wavefront (WF) is then focused by a lenslet array in a perfect lattice of point images. This focusing is achieved in the plane of the entrance pupil of the eye.



Fig. 5.20


When no optical aberrations are present, each image focused by a given lenslet falls on the optical axis of the lenslet. WF, Wavefront.



Fig. 5.21


By contrast, an aberrated eye reflects a distorted wavefront (WF) . The slope of the wavefront is different in front of each lenslet.



Fig. 5.22


The local slope of the wavefront is different in front of each lenslet. The measure of the displacement of each spot from its corresponding lenslet axis allows the computation of the local slope of the wavefront. A mathematical integration of the slopes leads to the three-dimensional reconstruction of the wavefront envelope.



Fig. 5.23


Examples of CCD video of a Hartmann-Shack wavefront sensor after focusing of the array of lenslets of different altered wavefronts: (A) diffraction-limited eye, (B) defocus, (C) coma, and (D) spherical aberration.


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.




Fig. 5.24


(A) Diagrammatic illustration of the laser dot matrix of a Tscherning aberrometer (wavelight, Tscherning aberrometer). (B) Retinal snapshot after projection of the laser grid on the fovea. The location of the centroids (colored spots) is performed by the software. This dot pattern map is then compared to a reference pattern map of an eye without aberration.




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 ).




Fig. 5.25


Wavefront analysis using laser ray tracing: a laser collimated beam (L) is shone through different locations of the entrance pupil (ingoing aberrometry) after mirror deflections (M) . During the scan of the pupil, the deviation of the position of each ray from its reference position Δ(xy) is registered sequentially on a numerical camera (C) . The reference axis is shown in red. In subjective aberrometry, the subject adjusts the incident angle of light until the retinal spot intersects the reference spot.


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.




Fig. 5.18


The Hartmann aberrometer derives from the seventeenth-century Scheiner disk. The optical path length is related to the number of times the light wave must oscillate traveling from one point to another. When aberrations are present, the optical path of rays emanating from a single point source will differ to the fovea. The observer will see two images instead of one. When there are no optical aberrations (insert) , the OPL is the same for all the light rays traveling from the object point to the image point.


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.




Fig. 5.19


Reflected light from a point source on the retina will emerge from a perfect eye as a plane wave. This reflected wavefront (WF) is then focused by a lenslet array in a perfect lattice of point images. This focusing is achieved in the plane of the entrance pupil of the eye.



Fig. 5.20


When no optical aberrations are present, each image focused by a given lenslet falls on the optical axis of the lenslet. WF, Wavefront.



Fig. 5.21


By contrast, an aberrated eye reflects a distorted wavefront (WF) . The slope of the wavefront is different in front of each lenslet.



Fig. 5.22


The local slope of the wavefront is different in front of each lenslet. The measure of the displacement of each spot from its corresponding lenslet axis allows the computation of the local slope of the wavefront. A mathematical integration of the slopes leads to the three-dimensional reconstruction of the wavefront envelope.



Fig. 5.23


Examples of CCD video of a Hartmann-Shack wavefront sensor after focusing of the array of lenslets of different altered wavefronts: (A) diffraction-limited eye, (B) defocus, (C) coma, and (D) spherical aberration.


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.




Fig. 5.24


(A) Diagrammatic illustration of the laser dot matrix of a Tscherning aberrometer (wavelight, Tscherning aberrometer). (B) Retinal snapshot after projection of the laser grid on the fovea. The location of the centroids (colored spots) is performed by the software. This dot pattern map is then compared to a reference pattern map of an eye without aberration.




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 ).




Fig. 5.25


Wavefront analysis using laser ray tracing: a laser collimated beam (L) is shone through different locations of the entrance pupil (ingoing aberrometry) after mirror deflections (M) . During the scan of the pupil, the deviation of the position of each ray from its reference position Δ(xy) is registered sequentially on a numerical camera (C) . The reference axis is shown in red. In subjective aberrometry, the subject adjusts the incident angle of light until the retinal spot intersects the reference spot.


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.




Fig. 5.24


(A) Diagrammatic illustration of the laser dot matrix of a Tscherning aberrometer (wavelight, Tscherning aberrometer). (B) Retinal snapshot after projection of the laser grid on the fovea. The location of the centroids (colored spots) is performed by the software. This dot pattern map is then compared to a reference pattern map of an eye without aberration.






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 ).




Fig. 5.25


Wavefront analysis using laser ray tracing: a laser collimated beam (L) is shone through different locations of the entrance pupil (ingoing aberrometry) after mirror deflections (M) . During the scan of the pupil, the deviation of the position of each ray from its reference position Δ(xy) is registered sequentially on a numerical camera (C) . The reference axis is shown in red. In subjective aberrometry, the subject adjusts the incident angle of light until the retinal spot intersects the reference spot.




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.




Fig. 5.26


Colorscale representation of the first 28 Zernike polynomials (sixth radial order). Rotationally symmetric polynomials (piston, defocus, spherical aberration) are located in the central column. Oriented polynomials (i.e., having no rotational invariance) are disposed by pairs having the same radial order and absolute value but opposite sign azimuthal frequency. The selective weighting of each coefficient of a given pair allows for tuning of the orientation and amplitude of the aberration.

The mean value of each mode (except for the piston) is 0. Each mode as a unit vector. In a two-dimensional (2D) Cartesian domain, one could plot two unit vectors along the X and Y axis, which would be orthogonal and with a norm (length) equal to 1. In a three-dimensional (3D) domain, you could add a third unit vector along the Z axis, which is orthogonal to the X and Y axis. In such 3D space, any vector can be decomposed on a weighted sum of the X, Y, and Z axis unit vectors. In the “Zernike polynomial domain,” one extends such a concept to a space with more than three dimensions. Each of these dimensions has its own unit vector, represented by a particular Zernike mode. Each of these dimensions would be perpendicular (orthogonal) to all the other ones. In such a multidimensional space, wavefront error would correspond to a particular vector, which could be broken down in a sum of weighted unit vectors (i.e., Zernike mode). The weight of each mode corresponds to the root mean square (RMS) coefficient of a Zernike wavefront decomposition.

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Oct 10, 2019 | Posted by in OPHTHALMOLOGY | Comments Off on Wavefront Analysis
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