Optical axes of catadioptric systems including visual , Purkinje and other nonvisual systems of a het-erocentric astigmatic eye

For a dioptric system with elements which may be heterocentric and astigmatic an optical axis has been defined to be a straight line along which a ray both enters and emerges from the system. Previous work shows that the dioptric system may or may not have an optical axis and that, if it does have one, then that optical axis may or may not be unique. Formulae were derived for the locations of any optical axes. The purpose of this paper is to extend those results to allow for reflecting surfaces in the system in addition to refracting elements. Thus the paper locates any optical axes in catadioptric systems (including dioptric systems as a special case). The reflecting surfaces may be astigmatic and decentred or tilted. The theory is illustrated by means of numerical examples. The locations of the optical axes are calculated for seven optical systems associated with a particular heterocentric astigmatic model eye. The optical systems are the visual system, the four Purkinje systems and two other nonvisual systems of the eye. The Purkinje systems each have an infinity of optical axes whereas the other nonvisual systems, and the visual system, each have a unique optical axis. (S Afr Optom 2010 69(3) 152-160)


Introduction
For a general dioptric system, that is, a dioptric system with elements that may be heterocentric and astigmatic, an optical axis is, by definition, a straight line along which a ray traversing the system both enters and leaves. 1 Given the length and transference of the system, and the indices of refraction of the media immediately before and after it, one can determine whether the system has an optical axis; and if the system does have an optical axis one can determine whether that optical axis is unique or not and find the location or locations of all of them in the system 1, 2 .
The purpose of this paper is to generalize the definition and the results to catadioptric systems.In other words the purpose is to define optical axes for systems that may also contain reflecting surfaces which may be astigmatic and tilted or decentred.The optical model used here is linear optics.

Catadioptric systems
Let us imagine a ray traversing a general optical system.Tracing the ray through the system we observe that it undergoes a sequence of steps involving refraction and reflection.Let µ be the number of re- The South African Optometrist ISSN 0378-9411 153 within the system.What it is doing there is of no consequence.Within the system segments of the ray that defines an optical axis may or may not lie on the optical axis.
Figure 1 shows the entrance T 0 and exit T planes of an arbitrary even-catadioptric system.The system itself is not shown.If there are no reflections, that is, if the system is dioptric ( 0 = µ ) then the system lies between T 0 and T. If there are reflections ( 0 > µ ) the system may extend upstream from T 0 and downstream from T. A ray R 0 R traverses the system in Figure 1.Only the incident R 0 and emergent R segments of the ray are shown.In the figure R 0 and R happen to lie on straight line O.By definition, then, O is an optical axis of the system.
Figure 2 shows a ray traversing an odd-catadioptric system.Again segments R 0 and R lie on the same straight line which then, by definition, is an optical axis of the system.In contrast to the case of evencatadioptric systems the segments are oppositely directed.
This definition of optical axis depends on the concept of the ray 4, 5 ; it is not a definition limited to linear optics.In order to examine questions of existence and location of optical axes we now invoke linear optics and so introduce the limitations associated therewith.For a treatment of linear optics and its limitations and relation to other optical models the reader is referred elsewhere 6 .
flections.We partition the set of all systems into those for which µ is an odd number and those for which µ is even number including 0; we call the former oddcatadioptric and the latter even-catadioptric 3 .Dioptric systems are even-catadioptric because zero is an even number.
The analysis below holds for catadioptric systems in general.The theory will be illustrated numerically by application to seven optical systems associated with the eye and described recently 3 .Three of the systems are even-catadioptric (the visual system with 0 = µ and two nonvisual systems with 2 = µ ) and four odd-catadioptric (the Purkinje systems which have 1 = µ ).The transferences of these systems have been calculated 3 .Here we draw on these previous results to locate optical axes in those systems.

Definition of an optical axis
The definition given before 1 for an optical axis in a dioptric system we simply generalize unchanged to catadioptric systems.Thus we have the following: DEFINITION If the incident and emergent segments of a ray traversing a catadioptric system lie on the same straight line then that line is an optical axis of the system.

Nothing is said in this definition about the ray
Figure 1 An optical axis O and entrance T0 and exit T planes of an even-catadioptric system.A ray with incident segment R0 on O has emergent segment R also on O, with the two segments pointing in the same direction.The system itself is not shown; it may extend upstream from T0 and downstream from T.

Linear optics
Consider a ray traversing catadioptric system S (Figures 1 and 2 (2) (The symbolism used here is the same as that used in other papers 1-3, 7, 8 .)Transverse positions y 0 and y and inclinations a 0 and a are all relative to longitudinal axis Z.They are 1 2 × matrices with Cartesian coordinates with respect to transverse axes Y 1 and Y 2 .Y 1 and Y 2 are mutually orthogonal; we shall usually think of them as horizontal and vertical respectively with Y 1 pointing to the right and Y 2 pointing upward when one is looking in the direction defined by Z. Incident segment R 0 of the ray is in a medium (outside system S) of index of refraction n 0 and emergent segment R (also outside S) is in a medium of the index n.
A, B, C, D, e and π are the six fundamental firstorder optical properties of system S.We call them the dilation, the disjugacy, the divergence, the divarication, the translation and the deflectance of S respectively.The first four are 2 2 × matrices.The last two are 1 2 × matrices; they account for the effects of tilt, decentration or prism in the system.They are submatrices of the transference T ( 5 5 × ) and arranged as follows where o is the 1 2 × null matrix and o T its matrix transpose.We define the T is an augmented symplectic matrix.(Important consequences of symplecticity are summarized elsewhere. 9) Suppose all the optical elements in system S are centred on axis Z; none is tilted or decentred.A ray incident along Z will traverse and emerge from the system along Z. Thus Z is an optical axis of S. For that ray y 0 and a 0 are both null matrices and so are y and a.It follows from Equations 1 and 2 that e and π must also be ( 1 2 × ) null matrices.In this case then the system has o = δ (5) where o is a 1 4 × null matrix.When the elements of S are not centred on Z, or are tilted, then, usually, δ δ is not null.A ray incident along Z emerges with transverse position e y = and as is clear from Equations 1 and The South African Optometrist ISSN 0378-9411 155 2 respectively.In other words matrix δ δ of system S is a measure of the transverse displacement away from longitudinal axis Z undergone by a ray incident onto S along Z.For want of a name we shall call δ δ the transversion of the system.

Even-catadioptric systems
The distance parallel to Z from T 0 to T of the evencatadioptric system in Figure 1 is z.We shall call z the effective length of the system.z may or may not be the actual length of the optical instrument; because of reflection within it the instrument may extend upstream of T 0 and downstream of T. z may be positive (as in Figure 1), zero or negative.
From Figure 1 one sees that, for the ray that defines optical axis O, a 0 = a.
(6) Under the assumptions of linear optics it also follows that Substituting from these equations into Equations 1 and 2 and rearranging we obtain and Cy 0 ( ) It is convenient to combine Equations 8 and 9 into the single equation (13) This is the same as the result obtained before 1 for dioptric systems ( ).In other words the result for dioptric systems now generalizes to all even-catadioptric systems ( 0 = µ , 2, 4, …).We shall call P the locator and d 0 the location of optical axis O in system S. d 0 is with respect to longitudinal axis Z and at entrance plane T 0 .
Given a particular system S the problem of finding its optical axes reduces to finding solutions d 0 (if any) to Equation 13.Before seeking solutions we consider odd-catadioptric systems.

Odd-catadioptric systems
Consider now the odd-catadioptric system of Figure 2. The ray leaves system S in the opposite direction to its direction at incidence.So, instead of Equation 6, we have Equation 7 applies as before.Substituting into Equations 1 and 2 one obtains Equation 8 as before but instead of Equation 9 one obtains Cy 0 ( ) The result is Equation 13 again except that locator P is now defined by

Optical axis locator
The optical axis locator P for odd-catadioptric systems (Equation 16) differs from the locator for evencatadioptric systems (Equation 11) only in that it has a plus sign instead of a minus sign in the bottom-right block.The two equations can be combined in one as ( Alternatively one could use Equation 11for both even-and odd-catadioptric systems provided that, for odd-catadioptric systems, one adopted the trick of The South African Optometrist ISSN 0378-9411 156 substituting n − for n.In this paper we prefer to use the more clumsy-looking but, perhaps, safer Equation 17.

Singularity of the optical axis locator
Whether or not the optical axis locator P is singular will become an important issue below.We therefore examine conditions under which it is singular.
The symplectic unit matrix 9

Location of the optical axis
We are now in a position to solve Equation 13 for the location d 0 of an optical axis.
If for a particular system the optical axis locator P is nonsingular then we can pre-multiply both sides of Equation 13 by the inverse of P to obtain the unique solution It follows that such a system has a single optical axis, its location being given by Equation 22.
For a system for which the locator P is singular the mathematics is messier.In such cases the inverse Here I is the 4 4 × identity matrix and g is an arbitrary 1 4 × matrix we may call the generator.Choosing a particular g one generates the location d 0 of an optical axis.Choosing all possible generators g one locates all the optical axes of system S. (When the inverse does exist 1 − − = P P , Equation 23is always satisfied and Equation 24 reduces to Equation 22.) In keeping with common mathematical terminology we say that, for a particular optical system, an optical axis exists if Equation 23 is satisfied and exists uniquely if Equation 22 holds.

Routine for locating optical axes
We can now lay out a general procedure for determining whether a catadioptric system has any optical axes and, if it does, for locating all of them.
At the outset we need four items of information about the system: its transference T, its effective .( 19) The South African Optometrist ISSN 0378-9411 157 length z (that is, the distance from its entrance plane to its exit plane T 0 ), the indices of refraction n 0 and n and whether the number µ of reflections in it is odd or even.T can be calculated from a knowledge of the structure of the system as described elsewhere 3 .Software such as Matlab is ideal for handling the matrices.
One begins by calculating the locator P by means of Equation 17 in which A, B, C and D are obtained from T. The transversion δ is also obtained from T (according to Equation 4).
Next one needs to determine if P is singular.In some cases this is obvious as in the case of the thin prism described below and in the case of Purkinje systems; in other cases one has to evaluate the determinant or rank of P. (If the determinant is 0 or the rank is less than 4 then P is singular.) If P turns out to be nonsingular then the system has a unique optical axis; its location d 0 is determined using Equation 22. (In Matlab the inverse is given by ( ) On the other hand if P turns out to be singular then one has to check whether Equation 23 is satisfied.This requires the Moore-Penrose inverse − P (given by ( ) • pinv in Matlab and very tedious to calculate if done by hand).
If Equation 23 is not satisfied then the system has no optical axis.
If Equation 23 is satisfied then the system has at least one optical axis.All of the optical axes can be located using Equation 24.The generator g contains four numbers which take on all possible real values.In particular setting all four numbers in g equal to 0 gives the location of one of the optical axes.Equation 12shows that the top half of the location d 0 is the transverse position y 0 of the optical axis at the entrance plane T 0 and the bottom half is the inclination a 0 of the optical axis, both y 0 and a 0 being relative to longitudinal axis Z.

Numerical results for seven ocular systems of a model eye
Table 1 shows the results calculated for seven ocular systems of the heterocentric astigmatic model eye considered elsewhere 3 .S P0 represents the visual system of the eye, an even-catadioptric system ( 0 = µ ); it is the dioptric system from entrance plane T 0 just in front of the cornea to exit plane T just in front of the retina.S P1 to S P6 are six nonvisual ocular systems.S P1 to S P4 are the four Purkinje systems, odd-catadioptric systems ( 0 = µ ) with reflection off the anterior (S P1 ) and posterior (S P2 ) surfaces of the cornea and the anterior (S P3 ) and posterior (S P4 ) surfaces of the lens of the eye; they are the systems responsible for the first to fourth Purkinje images.Their entrance and exit planes coincide and are immediately in front of the cornea.S P5 and S P6 are two nonvisual ocular systems Tshcerning 12 describes as being responsible for harmful rays reaching the retina.They are evencatadioptric systems ( 2 = µ ) with the same entrance and exit planes as the visual system; there is anterior reflection off the anterior (S P5 ) and posterior (S P6 ) surfaces of the lens of the eye followed by posterior reflection off the anterior surface of the cornea.
The optical axis locator P, listed in Table 1 for each system, is calculated using Equation 17.The top-right block of four entries are in millimetres (mm); the bottom-left entries are in reciprocal millimetres, that is, kilodioptres (kD).For all of the systems 1 0 = n (light enters from air).n is the index of the medium in front of the retina (taken as 1.336 here) in the case of systems S P0 , S P5 and S P6 ; those systems have the same effective length namely For the visual system S P0 , and for nonvisual systems S P5 and S P6 , the locators P are all nonsingular (they have rank 4); thus each has a unique optical axis, its location d 0 (calculated by means of Equation 22) being listed in the last column of Table 1.For 158 Table 1 The locator P for and location d 0 of the optical axes of seven optical systems in a heterocentric astigmatic model eye.
The entries in the top-right 2 2 × block of P and the top two entries of d 0 are in millimetres and entries in the bottom-left block of P are in kilodioptres.

The South African Optometrist
ISSN 0378-9411 159 here.One expects the same to be true for any eye.For Purkinje systems the locator P is singular.For the eye examined here there is an infinity of optical axes for each Purkinje system and one expects this also to be true of any eye.We mention in passing that Purkinje is a common misspelling of the name of Czech scientist J. E.

∨ e
Purkyn (1787-1869).It is one of nine different spellings of the name that John has encountered in the literature. 13hen we say here that an optical axis exists and has some particular location we are taking no account of stops or apertures in the system.Furthermore the results are subject to the limitations of the optical model (linear optics) that has been used.The definition of optical axis itself, however, is limited only in so far as the concept of the ray is limited.
example the optical axis of the visual system (S P0 ) intersects the entrance plane just in front of the cornea at the point with horizontal and vertical coordinates 0.0486 mm and 0.3251 mm (to the right and above) longitudinal axis Z respectively.Its inclination has horizontal coordinate 0111 .0 − and vertical coordinate 0608 .0 − (radians) also relative to Z; into the eye the optical axis slopes to the left and down.
The locators P of the Purkinje systems are singular (they each have rank 2 in fact).Equation 23 is satisfied for each of the systems.Hence each of the systems has at least one optical axis.In fact the Purkinje systems have an infinity of optical axes as represented in the last column of Table 1 in the form of Equation 24, the 1 4 × column matrix being δ − −P (the location of one of the optical axes) and the 4 4 × coefficient of generator g being P P I − − .A simple example of a system for which the condition of existence of an optical axis (Equation 23) is not satisfied is a thin prism in air.

Concluding remarks
This paper generalizes the result obtained before 1 for dioptric systems with elements that may be heterocentric and astigmatic to catadioptric systems of which dioptric systems are a special case.The expression for the optical axis locator P for dioptric systems holds in fact for all even-catadioptric systems (Equation 11).For odd-catadioptric systems the locator is given by Equation 16.The two equations combine as the single equation, Equation 17.
An optical system may or may not have an optical axis, and, if it does have one that axis may or may not be unique.A system does have an optical axis if and only if Equation 23 (the condition of existence) is satisfied.The system has only one optical axis if its optical axis locator P is nonsingular (the condition of uniqueness).(Its location is given by Equation 22.)This is the case for the visual system and for nonvisual systems S P5 and S P6 of the eye examined ).In linear optics the emergent transverse position y and the emergent inclination a of the ray are represented in terms of the incident transverse position y 0 and a 0 incident inclination by

Figure 2
Figure 2 An optical axis O and entrance T0 and exit T planes of an odd-catadioptric system.The incident and emergent segments of a ray lie on O but point in opposite directions.Notice that emergent segment R is in a medium of index n in both this figure and Figure 1.
Making use of Equations 17 and 18 one finds that In obtaining Equation 19 use has also been made of symplecticity 9 and, more particularly, of the facts that A T C and B T C are symmetric and that locator P is singular or nonsingular according as P T EP is singular or nonsingular.(This is a consequence of Equations 5 and 6 of a previous paper addition, that the system is such that C and B are symmetric and D = A T .Equation 20 reduces to P T EP = O.(21) Hence, for optical systems of this class, the optical axis locator P is singular.Among the systems of this class are the Purkinje systems.Numerical examples are presented in the appendix to a previous paper 3 (available at http://links.lww.com/OPX/A29).(A paper in preparation provides a general proof.)

PP
does not exist and Equation22does not hold.The system may or may not have an optical axis; it has an optical axis if and only if10, 11    is the Moore-Penrose inverse of P. If Equation 23 holds then all of the system's optical axes are defined and D are obtained from the transferences of the systems presented before 3 .The transversions δ are also obtained from the transferences.
axes of catadioptric systems ... nonvisual systems of a heterocentric astigmatic eye The South African Optometrist ISSN 0378-9411

.
C are null and A and D are identity matrices.Equation 17 reduces to O Hence the left-hand side of Equation 23 is null and, therefore, not equal to the right-hand side unless the prism has null deflectance π .