Cardinal points and generalizations.

In the presence of astigmatism a focal point typically becomes the well-known interval of Sturm with its pair of axially-separated orthogonal line singularities. The same is true of nodal points except that the issues are more complicated: a nodal point may become a nodal interval with a pair of nodal line singularities, but they are not generally orthogonal, and it is possible for there to be only one line singularity or even none at all. The effect of astigmatism on principal points is the motivation behind this paper. The three classes of cardinal points are defined in the literature in a disjointed fashion. Here a unified approach is adopted, phrased in terms of rays and linear optics, in which focal, nodal and principal points are defined as particular cases of a large class of special structures. The special structures arising in the presence of astigmatism turn out to be described by mathematical expressions of the same form as those that describe nodal structures. As a consequence everything that holds for nodal points, lines and other structures now extends to all other special points as well, including principal points and the lesser-known anti-principal and anti-nodal points. Thus the paper unifies Gauss's and Listing's concepts of cardinal points within a large class of special structures and generalizes them to allow for refracting elements which may be astigmatic and relatively decentred. A numerical example illustrates the calculation of cardinal structures in a model eye with astigmatic and heterocentric refracting surfaces.

Nodes and nodal points and lines in eyes and other optical systems.

The typical stigmatic optical system has two nodal points: an incident nodal point and an emergent nodal point. A ray through the incident nodal point emerges from the system through the emergent nodal point with its direction unchanged. In the presence of astigmatism nodal points are not possible in most cases. Instead there are structures, called nodes in this paper, of which nodal points are special cases. Because of astigmatism most eyes do not have nodal points a fact with obvious implications for concepts, such as the visual axis, which are based on nodal points. In order to gain insight into the issues this paper develops a general theory of nodes which holds for optical systems in general, including eyes, and makes particular allowance for astigmatism and relative decentration of refracting elements in the system. Key concepts are the incident and emergent nodal characteristics of the optical system. They are represented by 2 x 2 matrices whose eigenstructures define the nature and longitudinal position of the nodes. If a system's nodal characteristic is a scalar matrix then the node is a nodal point. Otherwise there are several possibilities: Firstly, a node may take the form of a single nodal line. Second, a node may consist of two separated nodal lines reminiscent of the familiar interval of Sturm although the nodal lines are not necessarily orthogonal. Third, a node may have no obvious nodal line or point. In the second and third of these classes one can define mid-nodal ellipses. Astigmatic systems exist with nodal points and stigmatic systems exist with no nodal points. The nodal centre may serve as an approximation for a nodal point if the node is not a point. Examples in the Appendix, including a model eye, illustrate the several possibilities.

Visual axes in eyes that may be astigmatic and have decentred elements.

The visual axis of the eye has been defined in terms of nodal points. However, astigmatic systems usually do not have nodal points. The purpose of this note is to offer a modified definition of visual axes that is in terms of nodal rays instead of nodal points and to show how to locate them from knowledge of the structure of the eye. A pair of visual axes (internal and external) is defined for each eye. The visual axes then become well defined in linear optics for eyes whether or not they are astigmatic or have decentred elements. The vectorial angle between the visual axes and the optical axis defines the visuo-optical angle of the eye.

Flipped optical systems.

In the standard refraction routine, a Jackson cross-cylinder is reversed in front of the eye by the process of flipping it about its handle. Reverse telescopes have applications in low vision. In ophthalmoscopy, the practitioner views a patient's retina using light that traverses the eye in a direction opposite to that involved in vision. These represent examples of flipped systems that are involved in optometry. The purpose of this paper is to examine what happens to the optical character of an arbitrary optical system when it is flipped in this manner. The analysis is in terms of the ray transference of the system and, because the transference completely characterises the first-order optics of a system, is complete within the limitations of linear optics. It allows for elements in the system to be astigmatic and decentred. An expression is derived for the transference of the flipped system in terms of the transference of the system itself. Expressions are also obtained for the fundamental first-order optical properties and the dioptric power of the flipped system. Three numerical examples are given in the Appendix.

Subjective refraction: the mechanism underlying the routine.

The routine of subjective refraction is usually understood, explained and taught in terms of the relative positions of line or point foci and the retina. This paper argues that such an approach makes unnecessary and sometimes invalid assumptions about what is actually happening inside the eye. The only assumption necessary in fact is that the subject is able to guide the refractionist to (or close to) the optimum power for refractive compensation. The routine works even in eyes in which the interval of Sturm does not behave as supposed; it would work, in fact, regardless of the structure of the eye. The idealized subjective refraction routine consists of two steps: the first finds the best sphere (the stigmatic component) and the second finds the remaining Jackson cross-cylinder (the antistigmatic component). The model makes use of the concept of symmetric dioptric power space. The second part of the refraction routine can be performed with Jackson cross-cylinders alone. However, it is usually taught and practiced using spheres, cylinders and Jackson cross-cylinders in a procedure that is not easy to understand and learn. Recognizing that this part of the routine is equivalent to one involving Jackson cross-cylinders only allows one to teach and understand the procedure more naturally and easily.

Technical note: dimensionless ray transference.

The ray transference matrix completely characterises the first-order optical nature of an optical system including the eye. It is in terms of the transference that quantitative analyses (for example, calculation of an average eye) can be performed. However, the fact that the entries of the transference do not have the same physical dimensions precludes the calculation of the usual scalar value (a Frobenius norm for example) for a change or difference between two optical systems. The purpose of this note is to show how to use the wavelength of the light as a natural unit of length to define a dimensionless transference and so make it possible to calculate a meaningful norm. In most practical applications, some components of the dimensionless transference may dominate unreasonably in the resulting norm in which case suitably weighted norms may be more appropriate. In an appendix, some of the issues are illustrated by application to a lens.

Technical note: accounting for anatomical symmetry in the first-order optical character of left and right eyes.

In quantitative analyses of the optical character of eyes (and related systems) it is sometimes necessary to deal with left and right eyes in the same context. In accounting for anatomical symmetry (mirror symmetry in the mid-sagittal plane) one treats a cylinder axis at 20 degrees , say, in a left eye as equivalent to an axis at 160 degrees in a right eye. But this is only one aspect of the linear optical character of an eye. The purpose of this note is to show how to account for anatomical symmetry in the linear optical character of eyes in general. In particular the note shows how to modify the optical properties of left (or right) eyes so that anatomical symmetry is accounted for in quantitative analyses in contexts in which both left and right eyes are involved.

Effect of tilt on the tilted power vector of a thin lens.

PURPOSE: The purpose of this study is to derive the equation for the effect of tilt on the tilted power vector of a possibly astigmatic thin lens. METHODS: The analysis makes use of the equation for tilted power in terms of the dioptric power matrix. RESULTS: A simple equation is obtained for the tilted power vector of a thin lens in terms of the untilted power vector of the lens. A solution of the equation provides a means for compensating spectacle lenses for tilt. Numerical examples are presented in the (available online at www.optvissci.com). CONCLUSIONS: The equation gives insight and is useful for researchers and clinicians working in terms of power expressed as power vectors.

Curvature of ellipsoids and other surfaces.

From differential geometry one obtains an expression for the curvature in any direction at a point on a surface. The general theory is outlined. The theory is then specialised for surfaces that are represented parametrically as height over a transverse plane. The general ellipsoid is treated in detail as a special case. A quadratic equation gives the principal directions at the point and, hence, the principal curvatures associated with them. Equations are obtained for ellipsoids in general that are generalisations of Bennett's equations for sagittal and tangential curvature of ellipsoids of revolution. Equations are also presented for the locations of umbilic points on the ellipsoid.

The exponential-mean-log-transference as a possible representation of the optical character of an average eye.

Considering a set of eyes, how does one define an average whose optical character represents an average of the optical characters of the eyes in the set? The recent proposal of the exponential-mean-log-transference was based on a conjecture. The purpose of this note is to provide justification by proving the conjecture and defining the conditions under which it fails.

Sequential tilt and tilted power of thin lenses.

PURPOSE: The purposes of this article were, for any sequence of tilts applied to a thin lens, to calculate an equivalent turn and single tilt, to show how to use the equivalent turn and tilt to calculate the tilted power of the lens, and, given the desired tilted power, to calculate the power of the untilted lens necessary to compensate for the effects of tilt. The untilted lens may be stigmatic (spherical) or astigmatic (spherocylindrical). METHODS: The analysis makes use of rotation matrices to represent rotation in space and previous work in third-order optics on oblique central refraction. RESULTS: Equations are presented for calculating the combination of turn and tilt that is equivalent to any sequence of tilts. They are specialized for the particular case of combinations of faceform and pantoscopic tilts and allow the decomposition of an arbitrary tilt into a combination of turn and pantoscopic and faceform tilts. The equations also lead to a procedure for calculating or compensating for the tilted power of a sequentially tilted thin lens. CONCLUSIONS: Previous work on the effect of tilt on thin lenses has been generalized to handle combinations of arbitrary tilts.

Error cells for spherical powers in symmetric dioptric power space.

PURPOSE: : The purpose of this article is to analyze the geometry and examine the implications of the error cells of purely spherical powers in symmetric dioptric power space. METHODS: : In the context of spherocylindrical data spherical data typically implies a cylindrical component that is less than some particular amount (often 0.125 D) in magnitude. This error or uncertainty in cylinder is over and above the error in sphere itself. The two components of error are used to define the error cells in symmetric dioptric power space. RESULTS: : Error cells of spherical powers are constructed and presented as stereopairs. They are also shown in relation to error cells of powers in general. CONCLUSIONS: : An understanding of error cells can help the researcher avoid pitfalls in the analysis of spherocylindrical data. Perhaps surprisingly, the error cells of spherical powers are not invariant under spherocylindrical transposition.

Reduction of artefact in scatter plots of spherocylindrical data.

Round-off of spherocylindrical powers, to multiples of 0.25 D (for example) in the case of sphere and cylinder, and 1 or 5 degrees in the case of axis, represents a type of distortion of the data. The result can be artefacts in graphical representations, which can mislead the researcher. Lines and clusters can appear, some caused by moiré effects, which have no deeper significance. Furthermore artefacts can obscure meaningful information in the data including bimodality and other forms of departure from normality. A process called unrounding is described which largely eliminates these artefacts; each rounded power is replaced by a power chosen randomly from the powers that make up what is called the error cell of the rounded power.

Stigmatic optical systems.

There would appear to be little disagreement on what constitutes an astigmatic system in the case of a thin lens: the cylinder is not zero. A spherical thin lens is stigmatic or not astigmatic. The issue is less clear in the case of a thick system. For example, is an eye stigmatic merely because its refraction is stigmatic (spherical)? In this article, a system is defined to be stigmatic if and only if, through the system, every point object maps to a point image. Every other system is astigmatic. Thus, a system is astigmatic if and only if there exists a point object for which the image is not a point. This article is restricted to linear optics. The optical character of a system is completely determined by the ray transference of the system. The objective here is to find those conditions on the transference for which the system is stigmatic or astigmatic. The result is that, for a stigmatic system, all the 2 x 2 submatrices are scalar multiples of a common orthogonal matrix. For a system to be stigmatic, it is not sufficient that its power be stigmatic. An eye may be astigmatic despite having a stigmatic refraction.

Proper and improper stigmatic optical systems.

Stigmatic optical systems are of two classes: proper and improper stigmatic systems. The purpose of the article is to explore the nature of the two classes. The image may be rotated in the case of proper stigmatic systems and is reflected in a line in the image plane in improper stigmatic systems. Chirality is preserved through proper stigmatic systems but reversed in improper systems. The image also undergoes magnification that is the same in all meridians and, in the case of decentered systems, a transverse translation. Negative magnification implies inversion in a point. The magnification depends on the distance of the object plane from the system, whereas the rotation and reflection do not. The article shows how to identify a system as astigmatic or proper or improper stigmatic from the transference and how to construct the transference of a system that will achieve a particular magnification and rotation or reflection.

Realizability of optical systems of given linear optical character.

The linear optical character of an optical system is represented by a particular type of 5 x 5 matrix. This article shows that the converse is also true, namely, that an optical system can be constructed in principle with linear optical character represented by any matrix of this type. In other words, every matrix of this type is realizable as an optical system. The system may have astigmatic and decentered elements.

The average eye.

For statistical and other purposes one needs to be able to determine an average eye. An average of refractive errors is readily calculated as an average of dioptric power matrices. A refractive error, however, is not so much a property of the eye as a property of the compensating lens in front of the eye. As such, it ignores other aspects of the optical character of the eye. This paper discusses the difficulties of finding a suitable average that fully accounts for the first-order optics of a set of optical systems. It proposes an average based on ray transferences and logarithms and exponentials of matrices. Application to eyes in particular is discussed.

Proximity factor in image size magnification for optical instruments in general.

A general expression is derived for the proximity factor in near image size magnification for an arbitrary instrument in front of an arbitrary eye. The proximity factor is a 2 x 2 matrix. The instrument and eye may be astigmatic and have decentred elements. The image on the retina may be blurred or not. The analysis is exact within the limitations of linear optics. The general results are specialized for the case of a stigmatic instrument and a stigmatic eye. The results are applied to the case of a thick, possibly bitoric, spectacle lens. The Appendix treats two numerical examples.

Near image size magnification for optical instruments in general.

A previous article derives expressions for coefficients and magnifications for an arbitrary optical instrument in front of an arbitrary eye and for near object points. The purpose of this note is to correct an error in the interpretation of a parameter in that article, to show that the expression derived there for near image size magnification holds only under particular conditions, and to modify the expression so that it holds in general.

Image size magnification and power and dilation factors for optical instruments in general.

Traditional treatments of spectacle magnification for distant objects consider only stigmatic spectacle lenses and they compare the retinal image size in a refractively fully compensated eye with the image size in the uncompensated eye. Spectacle magnification is expressed as a product of two factors, the power and shape factors of the lens. The power factor depends on the position of the entrance pupil of the eye. For an eye with an astigmatic cornea, however, the position of the entrance pupil is not well defined. Thus, the traditional approach to spectacle magnification does not generalize properly to allow for astigmatism. Within the constraints of linear optics and subject to the restriction that the eye's iris remains the aperture stop, this paper provides a complete, unified and exact treatment for optical instruments in general. It compares retinal image size in a generalized sense (including image shape and orientation) for any instrument in front of an eye with that of the eye alone irrespective of whether the instrument compensates or not. The approach does not make use of the concept of the entrance pupil at all and it allows for astigmatism and for non-alignment of refracting elements in the instrument and in the eye. The concept of spectacle magnification generalizes to the concept of instrument size magnification. Instrument size magnification can be expressed as the product of two matrix factors one of which can be interpreted as a power factor (as back-vertex power) and the other factor for which the name dilation factor is more appropriate in general. The general treatment is then applied to a number of special cases including afocal instruments, spectacle lenses (including obliquely crossing thick bitoric lenses), contact lenses, stigmatic systems and stigmatic eyes. In the case of spectacle lenses, the dilation factor reduces to the usual shape factor.