Key Features
- •
Aberration theory is strictly confined to geometrical optics and ignores any aspects of physical optics such as diffraction and interference.
- •
Originally, wave aberrations were applied only to symmetrical optical systems, and each aberration was represented by a unique polynomial.
- •
Today other expansions, particularly the Zernike polynomials, are most commonly used, and usually each polynomial is a combination of pure aberrations.
- •
In any case, after defocus and regular astigmatism, the most important aberrations are coma and spherical aberration.
Introduction
From a clinical perspective, aberration theory is simply an extended treatment of refractive error. Only hyperopia, myopia, and regular astigmatism are correctable by spectacles, so in the past these were the only aberrations of clinical interest. The existence of other refractive errors was certainly recognized, but because they were not correctable, clinicians had no need to study them in depth. Instead, the many aberrations not correctable by spectacles were relegated to a single broad category—irregular astigmatism. It is probably fair to say that even today many clinicians believe irregular astigmatism exists only in pathological cases and is not present in normal eyes.
For centuries, designers of optical instruments (e.g., telescopes, microscopes) had methods of correcting aberrations that could not be implemented clinically. Consequently, engineers have carefully characterized the individual aberrations that clinicians refer to collectively as irregular astigmatism. With the advent of keratorefractive surgery came the intriguing possibility of correcting at least some refractive errors not correctable by spectacles. Keratorefractive surgery also brought the disconcerting complication of increased irregular astigmatism in some cases. For the first time it was necessary for clinicians to explore irregular astigmatism more thoroughly. Fortunately, rather than reinvent the wheel, ophthalmologists can turn to the already well-developed aberration theory used by lens designers.
Ray Aberrations
Aberrations can be described by two different but closely related methods: ray and wave aberrations. When a ray is precisely traced through an optical system using the laws of geometrical optics (e.g., Snell’s law) it is found that usually the ray does not exactly intersect the ideal image point. The displacement of the actual ray from the image point is the ray aberration. When a pencil of rays (by definition all originating from a single object point) are traced, the result is a spot diagram ( Fig. 2.6.1 ).
For a specific and familiar example, consider the spot diagram for the axial object point at infinity of a theoretical eye with a 4 mm pupil and −1.00 D of myopia and no other aberrations. A spot diagram can be calculated anywhere within an optical system, and typically several are calculated in the vicinity of the theoretical image plane. The set of through-focus spot diagrams show that a stigmatic focus occurs about one-third of a millimeter anterior to the retina. The size of the spot diagram increases as the pupil enlarges, but the location of the stigmatic focus (and, therefore, the power of the corrective spectacle lens) does not change with pupil size, so the aberration is said to be pupil independent. Likewise, regular astigmatism is also pupil independent, which is why it is unnecessary to check pupil size to prescribe spectacles. The aberration known to lens designers as defocus encompasses both myopia and hyperopia.
In 1857 Ludwig von Seidel described five basic monochromatic ray aberrations of rotationally symmetrical optical systems. When studied individually, each aberration can be easily identified by its characteristic spot diagram. However, virtually all optical systems have many different aberrations. The spot diagrams of actual optical systems are a complex combination of several aberrations. It is difficult to identify and quantify the many individual aberrations from spot diagrams of actual optical systems.
In the early 1950s HH Hopkins developed a different, wavefront-based approach that simplified the definition, identification, and quantification of aberrations. The two approaches, ray and wavefront, are not mutually exclusive but rather complementary.
The Wavefront Approach to Aberrations
Aberration theory falls within the discipline of geometrical optics, so diffraction and other wave phenomena are completely ignored. In aberration theory, light propagation obeys the basic laws of reflection, refraction, and rectilinear propagation. All three laws are encompassed by Fermat’s principle, which states that between any two points light travels the fastest path.
Ideally, a pencil of rays filling the entrance pupil of an optical system emerges from the exit pupil as a pencil converging to a perfect image point, i.e., a stigmatic focus. The reference sphere is an imaginary spherical surface centered on the ideal (paraxial) image point and intersecting the center of the exit pupil. According to Fermat’s principle, to achieve a stigmatic focus, all light radiating from the object point at a particular instant must arrive simultaneously at the image point, or equivalently, must cross the reference sphere simultaneously.
In most cases, image rays do not emerge from the exit pupil as a pencil converging to a single point but rather converge to a small irregularly shaped region. In other words, the focus is not stigmatic. It always is possible to draw a surface through the center of the exit pupil that all rays cross simultaneously, and when the focus is not stigmatic, that surface—called the actual wavefront—is not spherical. The difference between the reference sphere and actual wavefront is the wave aberration ( Fig. 2.6.2 ). The wavefront and reference sphere are both surfaces, and the difference between these two surfaces, the wave aberration, is likewise a surface. To give an analogy, consider the Earth’s surface as the actual wavefront and sea level as the reference sphere. Height above (or below) sea level is analogous to the wave aberration.
The word wavefront is perhaps an unfortunate choice, suggesting that it is somehow related to the wave nature of light. However, as noted, aberration theory is based on geometrical optics that by definition ignores the wave properties of light. A pencil of object rays entering the optical system upon emerging from the exit pupil always “reaches” the wavefront simultaneously, so the wavefront is a surface of equal time. Arguably, the term isochrone (equal time) is more descriptive, but wavefront is too entrenched to change.
It is a common misconception that there is only one set of aberrations for each optical system. In fact, there is a different set of aberrations for every object point. This discussion began with a single pencil of object rays by definition from a single object point. Each object point produces its own pencil, and each object pencil has its own unique set of aberrations, although nearby object points have very similar aberrations.
Stigmatic focus means image rays converge to a perfect point either at the ideal image point or elsewhere. Because of diffraction, the focus can never be stigmatic, but geometrical optics ignores diffraction. Using only refraction, reflection, and rectilinear propagation, focus can, in theory, be stigmatic. Some use the term “astigmatism” to refer to any focus that is not stigmatic, but the word astigmatism also has several other more specific meanings. To avoid confusion in this chapter, “nonstigmatic” refers to any focus that is not stigmatic. The term astigmatism is reserved for regular astigmatism or astigmatism of oblique incidence. When the focus is nonstigmatic, rays do not converge to a perfect point even at the best possible focus. As in defocus, aberrations can still exist even when the image is stigmatic. If rays converge to a perfect point but the image point is not at the position predicted by the vergence equation, then an aberration is present.
Returning to the example of a theoretical eye with −1.00 D of myopia, the reference sphere is centered on the ideal image point. As shown earlier, in myopia, the image is stigmatic but anterior to the ideal image point. The wavefront emerging from the exit pupil therefore also is spherical but centered on a point anterior (i.e., to the left) of the ideal image point. The difference between the actual wavefront and reference sphere is the wave aberration surface ( Fig. 2.6.3 ). It is worth noting that the maximum separation between the reference sphere and the wavefront is a mere 1.5 µm, which is at the edge of the pupil. For midperipheral and central rays the wave aberration is less.
When wave aberrations are displayed on a graph, the optical axis is rotated 90° from horizontal to vertical. The wave aberration produced by myopia has a bowl shape. All myopias, whether large or small, have the same characteristic bowl shape. The amount of myopia changes the bowl’s depth and steepness of its edges but does not change its basic bowl shape. The wave aberration for hyperopia also is bowl shaped but the bowl is inverted (i.e., spills water). As both refractive errors have the same fundamental shape, lens designers classify them as the single aberration, defocus just as clinicians use “sphere” to specify either myopic or hyperopic correction when writing prescriptions. However, positive defocus is myopia and negative defocus is hyperopia, which may be counterintuitive to clinicians. The difference arises because lens designers classify defocus by the shape of the wave aberration, whereas clinicians think in terms of the power of the correcting lens.
Spherical Aberration
In spherical aberration (SA), rays near the center of the lens focus at or close to the ideal image point, whereas more peripheral rays focus in a different location. In positive SA, peripheral rays focus in front of the ideal image point, and the more peripheral the ray the more anterior its focus ( Fig. 2.6.4 ). In negative SA, peripheral rays focus behind the ideal image point and, again, the more peripheral the rays, the more posterior their focus. The wave aberration representing SA also has a bowl shape. In fact, SA looks more like a bowl than does defocus. Compared with defocus, the wave aberration of SA has a flatter broader center and the edges become much steeper than the bowl representing defocus.
When SA is present, the location of the best focus is no longer at the ideal image point predicted by the vergence equation ( U + P = V ). Positive SA shifts the best focus toward myopia, whereas negative SA shifts the focus toward hyperopia. However, even at the best position the focus is not stigmatic.
SA is strongly pupil dependent, because the position of best focus changes with pupil size. In positive SA, myopia increases as the pupil enlarges, which explains the uncommon clinical condition known as night myopia. In patients with a large amount of positive SA the increase in myopia under low light conditions is typically about −0.50 D and produces symptoms of blurred vision and halos. The condition can be treated by a second pair of spectacles or clip-ons with an extra −0.50 D over the daytime correction for night use.
SA can be reduced by aspherical refracting surfaces that flatten peripherally, which is typical of most corneas. Indeed, the cornea can flatten so much that it overcompensates, producing negative SA. Theoretically, large amounts of negative SA would cause a hyperopic shift in low light conditions, but this is rarely if ever seen clinically. The Stiles–Crawford effect (see later) also reduces the effect of SA on vision. The fact that refractive error usually does not fluctuate with pupil size in most eyes suggests that SA is adequately compensated over the typical range of pupil sizes.
It has been claimed that the crystalline lens compensates for corneal SA, and when the crystalline lens is removed, it should be replaced by an SA-compensating implant. However, studies that have measured SA in the cornea and crystalline lens separately have not found any tendency for the crystalline lens to compensate for corneal SA. Also, corneal SA varies from person to person, so an implant that corrects a fixed amount of SA is unlikely to benefit all patients. The patient’s corneal SA should be measured preoperatively to properly compensate corneal SA with an implant, and an implant should be selected not only by its power but also by the amount of SA corrected, which is not current clinical practice. Raytracing data on model eyes show that the amount of SA corrected by an implant varies substantially if the anterior chamber depth changes by tenths of millimeters or if the implant is tilted or decentered. Refractive error rarely fluctuates with pupil size, l suggesting SA is not a problem in patients with conventional implants. SA increases depth of focus, so reducing it is not always desirable.
Coma
Coma causes a pencil of rays to “flare out” from the image point in a fashion reminiscent of a comet’s tail ( Figs. 2.6.5 and 2.6.6 ). The shape of the wave aberration resembles a lounge chair.
Coma is increased when multiple optical elements do not share the same optical axis. Howland showed that after best spectacle correction, coma accounts for most of the residual aberration in otherwise normal eyes, which is not surprising because none of the ocular media share a common axis.
Clinically, coma rarely produces recognizable symptoms. However, coma can be symptomatic if there is a large amount present, which can occur (along with other higher-order aberrations) in off-axis keratorefractive surgery, dislocated or tilted crystalline lens or implant, or when the corneal vertex of a full-thickness graft is displaced by an unusual amount.
Distortion
Ideally, the image is geometrically similar to the object, which is to say that the image is a scale model of the object with all image distances proportional to corresponding object distances. Distortion is absent when the transverse magnification is constant over the entire image plane.
In pincushion distortion the transverse magnification increases as distance from the optical axis increases ( Fig. 2.6.7 ). In barrel distortion magnification decreases as distance from the axis increases. Distortion is similar to defocus in that rays still focus stigmatically but in the wrong place. In defocus, rays focus stigmatically but in front or behind the ideal image point. In distortion, rays also focus stigmatically and in the same plane as the ideal image but too far (or too close) to the optical axis.
Pincushion distortion occurs in practically all spectacle-corrected, aphakic patients and is occasionally seen in high myopes after clear lens extraction. Originally, one of the major reasons for the use of intraocular lens implants was to overcome the distortion suffered by spectacle-corrected aphakes. Once common, distortion is now rare since the introduction of intraocular lenses, although there has been a minor resurgence in patients undergoing clear lens extraction.
Field Curvature (FC)
Ideally a flat object perpendicular to the optical axis is imaged as a flat object perpendicular to the axis. When field curvature is present, the object is imaged onto a curved surface (as in an IMAX theater). Field curvature does not become significant until the image is outside the fovea, where retinal acuity is too low to detect the blurring produced by this aberration.
Oblique Astigmatism (OA)
Oblique astigmatism is not to be confused with regular astigmatism. Seidel only considered rotationally symmetrical optical systems, which excludes regular astigmatism. Like field curvature, oblique astigmatism does not become significant until the image is outside the fovea where retinal acuity is too low to detect the blurring produced by this aberration.
Higher-Order Aberrations (HOA)
When the importance of aberration theory began to be appreciated, it was initially thought that understanding in detail the unique characteristics of many individual aberrations would be helpful. Theoretically, an infinite number of aberrations exist, so how many aberrations do clinicians really need to understand thoroughly?
Lens designers have found empirically that 36 aberrations adequately describe the imaging properties of even very complex optical systems. The number of aberrations required for clinical purposes is uncertain but is probably much less. However, lacking clear evidence to the contrary, many authors have adopted the use 36 polynomials based on engineering, not clinical experience.
The order of an aberration is defined rigorously later, but basically an order is a family of aberrations. The clinical triad of myopia, hyperopia, and regular astigmatism all belong to the family of second order wave aberrations. The five Seidel aberrations (coma, SA, FC, distortion, OA) are all fourth order aberrations. As the order increases so does the number of aberrations belonging to the order.
It now appears that besides a general familiarity with aberration theory, a thorough understanding of just a few specific aberrations is sufficient. In addition to defocus and regular astigmatism, the Seidel aberrations of SA and coma and perhaps a few others in specific cases are all that need to be understood in depth.
The situation has returned somewhat, but not completely, to the idea of lumping many aberrations into a single category. Whereas irregular astigmatism refers to any aberration higher than second order, “higher-order aberrations” generally refers to sixth order and above and thus excludes the Seidel aberrations.
In any case, it is clear that typically coma and SA account for most of the eye’s irregular astigmatism, with only small amounts of other aberrations contributing to the total irregular astigmatism. Higher-order aberrations may be increased in pathological cases or after some refractive procedures, particularly keratorefractive surgery.
Chromatic Aberration
Wave aberration theory usually adopts the simplifying assumption that the light traversing the optical system is monochromatic. However, chromatic aberration has a significant effect on the image.
Longitudinal chromatic aberration (LCA) is the difference in image location between short and long wavelength light measured along the optical axis ( Fig. 2.6.8 ). LCA is the basis for the duochrome test. The green half of the chart focuses anterior to the red half, and the power of the correcting sphere is adjusted until both sides appear equally sharp. In most cases there is about 0.50 D of separation between the two images.
Transverse chromatic aberration (TCA) is the difference in image location between short and long wavelength light measured in the image plane ( Fig. 2.6.9 ). Although the chromatic aberration of the eye is significant, color effects are rarely noticeable because of visual processing. Occasionally patients notice colored fringes around the edges of objects or letters when reading; these are usually produced by spectacle lenses made from highly dispersive materials. The problem generally is transient, but if not, can be overcome by using lenses fabricated from less dispersive (i.e., higher Abbe number) materials.
Measurement of Ocular Aberrations
In 1896 Tscherning was the first to observe irregular astigmatism (in his own eyes). A small, distant source is viewed through a grid of pinhole apertures placed in front of an eye that is or has been rendered myopic (perhaps by fogging with, for example, +5.00 D lens). The grid divides the light into parallel pencils that focus in front of the retina then diverge, forming a pattern on the retina that is essentially the physical embodiment of a spot diagram. The grid will not be regularly spaced if aberrations (other than defocus) are present. If one draws the appearance of the grid, ocular aberrations can be estimated. Readers can easily reproduce this experiment for themselves just as Tscherning did.
As a method for measuring aberrations, the Tscherning aberrometer suffers from the obvious limitation that it is subjective, relying on the patient’s ability to observe and accurately locate the displaced pencils. Nonetheless, it is a useful conceptual tool. The subjective limitations of Tscherning’s approach can be overcome by photographing the grid on the retina. However, photography introduces another problem—double pass. To be photographed, each grid spot on the retina acts as a light source, which passes through the eye, suffering its aberrations a second time. The second pass complicates the analysis of aberrations, although the Allegretto Wave Analyzer is based on this approach.
There are several ways to overcome the double pass problem. One approach is to focus a laser (at low power of course) on the retina. Since the laser traverses only a small amount of the pupil, it is essentially unaffected by ocular aberrations; practically speaking, this is a single pass technique. The small area of laser illuminated retina becomes a source from which light emerges from the eye. If the eye is aberration-free, the emerging light will have planar wavefronts, otherwise aberrations distort the wavefront shape.
Using an approach suggested by Shack, the wavefront is sampled using a lenslet array that focuses small areas of the wavefront onto a detector. Each lenslet produces a small dot image on the detector that is displaced depending on the wavefront’s shape. The full wavefront is reconstructed from the displaced position of dots. The Hartman–Shack, as it is called today, is probably the most reliable way of measuring small amounts of higher-order aberration in an otherwise well-corrected eye or optical system. The technique is not as well suited to the measurement of large amounts of low-order aberrations. Recent modifications of the technique improve the performance for low-order aberrations. Many aberrometers utilize some variation of the Hartmann–Shack approach.
Automated retinoscopy is another means of measuring aberrations and is incorporated into the ARK 10000 (Nidek). The Tracey uses the Tscherning approach but performs it sequentially by scanning a single laser beam over the pupil and observing the beam’s deviation on the pupil. Besides being essentially a double pass system, movements during data acquisition are a potential problem.
It is important to realize that just because an instrument claims to measure aberrations does not mean it actually does so. It is very difficult to verify the accuracy of an aberrometer. Ocular aberrations, especially higher-order aberrations, change with the slightest change in tear film thickness, or changes in choroidal thickness that occur during the cardiac cycle, accommodation, hippus, ocular movement, head movement, or fixation change. The active optics photographic system that captures high resolution fundus images requires a bite bar for adequate head stabilization, which no clinical aberrometer utilizes. Too often reproducibility is considered indicative of accuracy, but a stopped watch gives highly reproducible yet useless results. Accuracy cannot be inferred simply because a device gives reproducible results. Given the number of factors that affect ocular aberrations, an instrument that gives highly reproducible results, especially for higher-order aberrations, is suspect. Keep in mind that with all the advances in the clinical measurement of aberrations, there is still no objective measurement of hyperopia, myopia, or regular astigmatism that is consistently superior to subjective refraction. If low-order aberrations cannot be measured satisfactorily by objective methods, good reason exists to suspect that objective measurements of higher-order aberrations are somewhat inaccurate.
Regardless of the method used, all clinical measurements produce a set of discrete data points. The wavefront’s shape between data points is not known. Mathematics, no matter how sophisticated, cannot create information beyond that present in the data. To summarize, the mathematical techniques do not find the actual wavefront, only the most likely wavefront given the data values, regardless of how the data were acquired (e.g., Hartman–Shack, Tscherning). More data points, provided they are accurately measured, and the more Zernike polynomials considered, the more likely the computed wavefront is a reasonable representation of the actual wavefront.
Mathematical Considerations
There is no need to be intimidated by the mathematical aspects of aberration theory. Just as a spectacle lens is a combination of spherical and cylindrical contributions, total aberration is the combination of individual aberrations. The only difference is that in aberration theory several more individual aberrations occur, and they are less familiar to clinicians.
An individual aberration can be specified by a graph or its associated formula. As noted, when displayed graphically, the optical axis is rotated 90° from horizontal to vertical. Zernike polynomials are the formulas representing each individual graph.
Every Zernike polynomial has the same structure consisting of a coefficient that specifies the amount of the aberration (analogous to power, but in units of distance, not diopters), a radial polynomial, and usually a sine or cosine function. For instance,
SecondaryAstigmatismX=Z11(4ρ4−3ρ2)Cos2θ
Stay updated, free articles. Join our Telegram channel
Full access? Get Clinical Tree
