Abstract
D.Phil.
That the eye is essentially a first-order optical instrument is evidenced by the
success Gaussian optics has met with in optometry and ophthalmology. An unfortunate
consequence of this approach is that a brief review of the literature on the topic of
intraocular lens power calculation gives one the impression that the character of such a
lens is described fully by its dioptric power. This is not so. Indeed, the idea that a thin
refracting interface can somehow embody the optical character of the thick intraocular
lens can, and in many ways has, limited the scope of intraocular lens power formula.
The purpose of this dissertation is to apply the methods of linear algebra to the
investigation of the first-order optical character of the stigmatic and astigmatic
pseudophakic eye. This work attempts to lay a solid foundation for the study of the
pseudophakic eye in the context of first-order astigmatic optics. While the majority of
concepts and results of this dissertation are directly applicable to the study of the
pseudophakic eye, an attempt has been made to ensure that the methods outlined in this
work may be applied to the study of optical systems in the broader context of first-order
optics.
Central to this work are the members of the non-abelian symplectic group Sp(2n)
under the operation of conventional matrix multiplication. The elements are evendimensional,
non-singular symplectic matrices with unit determinant which are referred
to here as ray transferences. These matrices act on the members of even-dimensional
vector spaces so as to preserve a particular skew-symmetric, non-degenerate bilinear
pairing referred to as the symplectic form. The laws that govern the operation of these
matrices, the three symplectic relations, flow naturally from the structure of the
symplectic group. From the ray transference four 2„e 2 fundamental properties of an
optical system may be defined, the dilation A , the disjugacy B , the divergence C and
the divarication D. A number of additional optical properties can be derived from the
fundamental properties. Examples of derived properties include the dioptric power F ,
the negative of the divergence C and refractive state 0 F .
The ray transference is used here in the derivation of a set of new intraocular lens
formulae for the pseudophakic eye. These formulae are entirely general, working equally
well in both stigmatic and astigmatic pseudophakic eyes in which additional (possibly
astigmatic) intraocular devices may already be present. Formulae for both distant and
near objects are provided.
The constraints under which the divergence of a thick (possibly bitoric)
intraocular lens is conserved despite changes in the lens are investigated. Furthermore,
the constraints under which the refractive state of the pseudophakic is conserved in spite
of changes in the thick intraocular lens are investigated. We find that there exist an
infinite number of thick intraocular lenses that will produce a given refractive outcome,
say emmetropia, in the pseudophakic eye.
The basic theory of matrix differentiation with respect to a scalar variable is
utilized in the study of the changes in the optical character of the pseudophakic eye
following axial translation of a variety of intraocular lens systems. A novel method of
representing the changes in the stigmatic and antistigmatic properties of refraction on
account of axial translation and rotation of a toric intraocular lens in the astigmatic eye is
presented and numerical examples are provided. The analysis permits the calculation of
the ideal axial lens position and orientation in the astigmatic pseudophakic eye. Such
methods will prove increasingly important in refractive data analysis, particularly in light
of the development of continuously adjustable intraocular devices.
Prof. W.F. Harris