The refractive group

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.

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.

Power vectors versus power matrices, and the mathematical nature of dioptric power

Representation of astigmatic dioptric power as a power vector is satisfactory for basic operations such as summing and averaging powers. However, power vectors do not fully characterize the nature of dioptric power and are, therefore, unsatisfactory for representing power in general. The purpose of this note is to make the case that it is the power matrix instead that is proper for the representation of power in general. The mathematical nature of dioptric power is examined.

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: 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.

Generalized Coddington equations found via an operator method

Generalized Coddington equations allow the optical properties of an arbitrarily oriented incoming astigmatic ray bundle to be found following refraction by an arbitrary surface. Generalized Coddington equations are developed using the abstract concept of vergence and refraction operators. After suitable incoming vergence and refraction operators have been formed, these operators are re-expressed in a common coordinate system via similarity transformations created from the series of space rotations necessary to align the coordinate systems. The transformed operators are then added together to produce the vergence operator of the refracted ray bundle. When properly applied, these generalized Coddington equations may be used with complex wavefronts and complex refracting surfaces if local surface curvature properties are known for both where the two intersect. The generalized Coddington equations are given in matrix form so that they may be easily implemented.

Dihedral representations and statistical geometric optics. I. Spherocylindrical lenses

The linear 2-dim irreducible representations of the dihedral groups (Dn) are interpreted as classical linear operators of geometrical optics. It is shown that the 2-dim irreducible representation of D4 is simply the refractive group described by Campbell [Optom. Vision Sci. 74, 381 (1997)]. The dihedral Fourier-inverse mechanism is introduced and shown to provide a systematic connection between the standard refractive data and their vector space representation, as proposed by Thibos et al. [Vision Sci. Appl. 2, 14 (1994)].

Analysis of astigmatism in anterior segment surgery

PURPOSE

To demonstrate techniques for analyzing astigmatism in the context of anterior segment surgery. The techniques are applied to refractive and corneal astigmatism and to astigmatism associated with the surgery.

METHOD

The analysis is based on the 3-dimensional first-order ray optics known as linear optics, in which astigmatism is axial. Anterior segment surgery, whose only direct effect is to change corneal curvature, is optically equivalent to placing a thin lens immediately in front of the (preoperative) eye; the surgery can be described as anterior thin-lens equivalent. In principle, however, surgery may also change the relative axial positions of refracting elements. In such cases, anterior segment surgery is anterior thick-system equivalent and the analysis of astigmatism is considerably more complicated, with astigmatism manifesting in power and in other ways. Fundamentally, the method is based on the ray transference, a 4 x 4 matrix that contains the 4 fundamental optical properties of a system. Each property may contain an antistigmatic component, and it is this component that is analyzed.

RESULTS

The data are not consistent with the anterior thin-lens model of anterior segment surgery but rather with the anterior thick-lens model. Many components of refraction and corneal power, their means, and their surgically induced changes were calculated and are presented in tabular and graphical form. Graphical representations include polar profiles and stereo-pair scatterplots in dioptric power space. Results are also presented for all the fundamental properties of the surgery.

CONCLUSION

Ocular astigmatism can manifest in more ways than just refraction and corneal power. Traditional approaches, which treat astigmatism as cylinder in some sense and only as manifested in power, are unsatisfactory. In particular, current approaches weight astigmatism too heavily relative to stigmatism and to other optical phenomena and are inadequate for thick systems such as the eye. The ray transference and the 4 fundamental properties it contains provide the basis for the only methods that can adequately cope with analysis of astigmatism in general and in surgically induced astigmatism in particular.

Lie algebraic treatment of dioptric power and optical aberrations

The dioptric power of an optical system can be expressed as a four-component dioptric power matrix. We generalize and reformulate the standard matrix approach by utilizing the methods of Lie algebra. This generalization helps one deal with nonlinear problems (such as aberrations) and further extends the standard matrix formulation. Explicit formulas giving the relationship between the incident and the emergent rays are presented. Examples include the general case of thick and thin lenses. The treatment of a graded-index medium is outlined.