Aperture referral in dioptric systems with stigmatic elements

A previous paper develops the general theory of aperture referral in linear optics and shows how several ostensibly distinct concepts, including the blur patch on the retina, the effective corneal patch, the projective field and the field of view, are now unified as particular applications of the general theory. The theory allows for astigmatism and heterocentricity. Symplecticity and the generality of the approach, however, make it difficult to gain insight and mean that the material is not accessible to readers unfamiliar with matrices and linear algebra. The purpose of this paper is to examine what is, perhaps, the most important special case, that in which astigmatism is ignored. Symplecticity and, hence, the mathematics become greatly simplified. The mathematics reduces largely to elementary vector algebra and, in some places, simple scalar algebra and yet retains the mathematical form of the general approach. As a result the paper allows insight into and provides a stepping stone to the general theory. Under referral an aperture undergoes simple scalar magnification and transverse translation. The paper pays particular attention to referral to transverse planes in the neighbourhood of a focal point where the magnification may be positive, zero or negative. Circular apertures are treated as special cases of elliptical apertures and the meaning of referred apertures of negative radius is explained briefly. (S Afr Optom 2012 71(1) 3-11)


Introduction
The blur patch on the retina, the effective patch on the cornea, or on any refracting surface, the projective field of a retinal point and the field of view of an optical instrument, all important in vision, are usually thought of as distinct concepts that need to be treated separately.A recent paper 1 shows, however, that each is a special case of a general phenomenon described there as aperture referral.The concepts represent the action of an aperture, often the pupil of the eye, taking place as it were at some other axial position in the system.The paper 1 uses linear optics to develop a general theory of aperture referral for general dioptric systems whose refracting elements may be heterocentric and astigmatic.
Allowance for astigmatism in the general theory means that symplecticity takes its full form.Consequently the mathematics is complicated, qualitative insight is difficult to obtain and many readers find the material inaccessible.Here we treat the special case in which all refracting elements are stigmatic.In other words we exclude astigmatism.The exclusion of astigmatism greatly simplifies the mathematics but retains much of the essential mathematical structure.In effect, because of symplecticity, the transference of an optical system simplifies from 20 numbers related by six equations to only eight related by only one equation.(Results concerning simplecticity in the context of linear optics are summarized elsewhere 2 .)What is a problem in linear algebra becomes, more or less, a problem in the much more familiar vector 157 algebra.It becomes much easier to visualize what is happening and the special theory provides a stepping stone to the general theory.Thus it gives insight into the general theory that one cannot get in any other way.The purpose of this paper, then, is to examine aperture referral in dioptric systems containing only stigmatic refracting elements.The surfaces may be decentred or tilted.We shall also allow apertures to be elliptical and decentred.Circular apertures are treated as a special case.

Transferences of systems with stigmatic elements
An optical system S has transference ( 5 × 5 ) (1) where A (the dilation), B (the disjugacy), C (the divergence) and D (the divarication) are the four 2 × 2 fundamental properties of S, e (the transverse translation) and π (the deflectance) are the two 2 ×1 fundamental properties and o T is the matrix transpose of the 2 ×1 null matrix o. e and π account for decentration or tilt of refracting surfaces in the system.Together the six fundamental properties represent 20 numbers.The top-left submatrix of 2 × 2 fundamental properties is symplectic; results for such matrices are summarized elsewhere 2 .For more information on the transference the reader is referred to previous papers 1,3 .
If the longitudinal axis coincides with the optical axis 4 of the system e and π are null and one can eliminate the fifth row and column and reduce the transference to 4 × 4 .
By a stigmatic refracting element we mean either a refracting surface that is not astigmatic (it is invariant under rotation about a normal) or a gap of material that is homogeneous and isotropic.We make the following assertion: every dioptric system all of whose elements are stigmatic has transference (2) where I is the 2 × 2 identity matrix.A, B, C and D are scalars; we shall call them the four scalar fundamental properties of the system.Thus for a system with only stigmatic refracting elements the dilation A is a scalar matrix IA.We shall also call A the dilation.The same holds for the other 2 × 2 fundamental properties.Because of symplecticity the scalar fundamental properties are related by (3) A, B, C, D, e and π make up the eight numbers referred to in the Introduction above and Equation 3represents the single equation that relates them.
The assertion made above is obviously true for refracting elements: a stigmatic refracting surface has A=D=1, B=0 and C equal to the negative of the (spherical) power F and a homogeneous gap has A=D=1, C=0 and B equal to the reduced length of the gap.Consider now any compound system made up of refracting elements.Its transference can be obtained by multiplying the transferences of the elements.But multiplication of any two matrices of the form in Equation 1 results in a matrix of the same form as can be readily confirmed.This proves the assertion.
We note in passing that the set of all transferences of the form of Equation 2 satisfies all four axioms of a group in mathematics: closure under multiplication as we have just seen; associativity (because (T 1 T 2 )T 3 =T 1 (T 2 T 3 )); existence of an identity (the 5 × 5 identity matrix, which is the transference of a homogeneous gap of zero width and of a flat refracting surface orthogonal to the longitudinal axis); and existence of an inverse T −1 corresponding to every T in the set.We may call this set the group of transferences of systems with stigmatic elements.(The theory of groups is treated in several good texts [5][6][7] .Groups are the generic as it were of which transferences are a specific.It is important, the author believes, to recognize such relationships because it sets what one is doing within a mathematical context and counteracts isolationist tendencies.It helps to increase communication with other disciplines and the possibility of being able to draw on methods developed elsewhere.It also emphasizes that we are not developing new mathematics but merely applying it.) The key to aperture referral is a quantity that is constant along a ray through a system.There are in fact two invariants along a ray, a topic to which we now turn.For the situation represented by Figure 1(a) the positional invariant is listed in Table 1 and the inclinational invariant in Table 2.
By a similar argument one can determine the two invariants for each of the other three situations represented in Figure 1.They are also listed in Tables 1  and 2. In Figure 1(c) there is an image point I with transverse position y I on the exit plane T I of system S while in (d) I is distant and rays to it all have reduced inclination α O I .For each of these cases the positional and inclinational invariants are listed in Tables 1 and 2 respectively.
In what follows we make use only of the positional invariants.Reference to inclinational invariants has been introduced largely for mathematical completeness.We note, however, that the right-hand sides of Equations 4 and 5 can be equated.This gives a relationship between reduced inclination α O and transverse position y of a ray at any longitudinal position which can be solved for one in terms of the other.The same can be done for all the systems represented in Figure 1.
We turn now to the relationship of the transverse position of a ray in one transverse plane to its transverse position in another.Positional transformation from one transverse plane to another Consider Figure 1(a) again.Let us re-label transverse plane T as T 1 .The system from T O to T 1 is now system S 1 .System S 1 has transference T 1 , scalar fundamental properties A 1 , B 1 , C 1 and D 1 and 2 ×1 fundamental properties e 1 and π 1 .Adding subscript 1 to all the variables on the right-hand side of Equation 4 we obtain the positional invariant at transverse plane T 1 .
We can also choose a second transverse plane T 2 in Figure 1(a) in which everything in the previous paragraph holds if we replace subscript 1 by 2. The positional invariants at T 1 and T 2 can be equated because they are both equal to the incident reduced inclination α O : Solving for the emergent position we obtain which we can write where Equations 8 to 10 are Equations 11 to 13 of the previous paper 1 with astigmatism ignored.The important difference is that X y, 1→2 has been replaced by X y, 1→2 .The linear component 8 of the affine magnification 9 associated with the referral now reduces to the much more familiar scalar magnification.Equation 8 tells us that, from transverse plane T 1 to transverse plane T 2 , the transverse position y 1 of a ray gets multiplied by the scalar X y, 1→2 and has a fixed vector d y, 1→2 added to it.In other words if we follow a ray from one transverse plane to another we find that its transverse position relative to the longitudinal axis Z undergoes magnification by the factor X y, 1→2 followed by transverse shift d y, 1→2 .We call X y, 1→2 the positional magnification and d y, 1→2 the common transverse translation from T 1 to T 2 although we shall often use abbreviated terminology.The positional magnification (Equation 9) and the common transverse translation (Equation 10) are listed in Table 1 for the case of object O on the entrance plane.
Just as there is transformation of position there is also transformation of reduced inclination of a ray from one transverse position to another which we now consider briefly in passing.

Inclinational transformation from one transverse plane to another
Starting with Equation 5 instead of Equation 4 and applying a similar argument one finds that the reduced inclination changes from transverse plane T 1 to transverse plane T 2 in a similar manner.It turns out that α O 1 is magnified by scalar X α, 1→2 and has a fixed vector d α, 1→2 added to it.Expressions for inclinational magnification X α, 1→2 and fixed shift d α, 1→2 for O on the entrance plane are listed in Table 2.
So far we have examined only the case represented by Figure 1 as given in Table 1 and with expressions for X α, 1→2 and d α, 1→2 as given in Table 2.
Apertures may be of any shape.We turn now to apertures that are elliptical (including circular) in particular. is a radial vector of the ellipse.Ellipse E has equation r T Mr = 1 (13) where M is positive definite, that is, both its eigenvalues are greater than zero.(This is the same as Equation 19 of the previous paper 1 .)M represents the shape, size and orientation of E; we refer to it simply as the geometry of E. M 1/2 is the principal square root of M and

Elliptical apertures
the positive definite generalized radius of E. Equation 13 can be written R p can be obtained from the spectral theorem 10 of linear algebra: An example is treated in the Appendix of the previous paper 1 .
Figure 2 An ellipse E in a transverse plane T. a is half the major diameter and b half the minor diameter.v a and v b are unit vectors along the major and minor diameters respectively.Z is the longitudinal axis and is orthogonal to the plane of the paper.
Relative to Z the centre of the ellipse has position vector y 0 .y is the position vector of a point on the ellipse.r = y − y 0 is the position vector of the point relative to the centre of the ellipse; it is a radial vector of the ellipse.
Suppose in transverse plane T there is a thin opaque plate with an elliptical aperture A with centre y 0 and positive definite generalized radius R p .We define A to be all the points in T for which r T Mr ≤ 1 .
(17) Equality implies points on the margin.
An aperture may be referred to another transverse plane by an object or image point.The question is: What is the geometry and location of the referred ap-erture?

Aperture referral
Consider now an elliptical aperture A 1 in transverse plane T 1 .Its centre is at y 1 0 , its geometry is M 1 and its positive definite generalized radius is R p1 .A radial vector r 1 has equation Suppose aperture A 1 is referred to another transverse plane T 2 by an object or image point to become referred aperture A 2 .Every position vector y 1 in A 1 undergoes affine magnification to y 2 in A 2 according to Equation 8.In particular y 1 0 becomes Applying Equations 8 and 20 to Equation 12 we see that radial vector r 1 of A 1 undergoes scalar magnification according to Solving for r 1 and substituting into Equation 18we obtain where Equation 22 shows that referred aperture A 2 is also elliptical; its centre is at y 2 0 given by Equation 20 and its geometry is given by Equation 23.In fact Equation 21 shows that referral magnifies aperture A 1 by the scalar X y, 1→2 ; thus referral changes the size but not the shape or orientation of the ellipse.
Substituting for r 1 in Equation 19 we find that is a generalized radius of A 2 .The positive definite generalized radius is The South African Optometrist ISSN 0378-9411 Circles are simpler to handle than ellipses and the pupil of the eye and most other apertures of relevance in optometry are close to circular.Accordingly we now examine that special case.

Circular apertures
If a = b > 0 then ellipse E is a circle of radius a.We represent the radius by r p , that is, r p = a.Subscript p (for positive) distinguishes this as the radius of a circle as conventionally defined.We refer to it as the positive radius of the circle in contrast to other radii (without subscript p) which may be positive or negative.Ellipse E has geometry and positive definite generalized radius R p = Ir p .
(28) Its equation simplifies to A circular aperture A 1 with positive radius r p1 and centre at y 1 0 in transverse plane T 1 becomes referred aperture A 2 in T 2 .A 2 is circular with centre given by Equation 20 and radius r 2 = X y, 1→2 r p1 (30) which is positive, zero or negative according as the magnification X y, 1→2 is positive, zero or negative.The positive radius of A 2 is (31) The following simple example provides insight into the significance of negative magnification of an aperture in particular.

Example
Figure 3 represents an example of referral of a circular aperture.The longitudinal axis Z is the optical axis of the system.Circular aperture A 1 of positive radius r p1 is referred to transverse plane T 2 by an object point O on Z.A 2 is the referred aperture.T 2 is a distance z downstream from A 1 and the intervening medium has index n.The transference of system S 1 from object plane T O to the plane of A 1 can be written where we have deleted the fifth row and column (which we can do because both e and π null).The transference of system S 2 from T O to T 2 is  From Equation 9we obtain the scalar magnification X y, 1→2 = 1+ ζD 1 / B 1 .
(34) Also, from Equation 10, because y O = e 2 = e 1 = o.As z increases from zero referred aperture A 2 initially decreases in size in Figure 3; this implies that system S 1 has D 1 and B 1 of opposite signs.From Equations 30 and 34 we see that referred aperture A 2 has radius Solving we find that r 2 reduces to zero when ζ = ζ f where Equation 36 can then be written It is clear that r 2 is positive, zero or negative according as ζ is less than, equal to or greater than ζ f .A 2 has positive radius Note that r 2 decreases linearly with z; the function The South African Optometrist ISSN 0378-9411 is smooth.On the other hand r p2 decreases linearly until z = z f where the derivative is not defined after which it increases linearly.
Figure 3 is drawn for ζ = 2ζ f .In that case the magnification is X y, 1→2 = −1 (from Equations 34 and 37) and r 2 = −r p1 (from Equation 38) and r p1 = r p2 (from Equation 39). Figure 4 shows the view along longitudinal axis Z. Points P 1 , Q 1 and R 1 on the margin of A 1 become points P 2 , Q 2 and R 2 , respectively, on the margin of A 2 ; referral preserves the anticlockwise order of the points.In effect referral causes rotation through 180°.However if points are ignored referred aperture A 2 is identical to aperture A .
Figure 4 View along longitudinal axis Z of aperture A 1 and referred aperture A 2 in Figure 3.A 1 is circular with positive radius r p1 .The scalar magnification associated with the referral is X y, 1→2 = −1 and, hence, A 2 has (negative) radius r 2 = −r p1 and positive radius r p2 = r p1 .Points P 1 , Q 1 and R 1 in anticlockwise order on the margin of A 1 points P 2 , Q 2 and R 2 , respectively, also in anticlockwise order on the margin of A 2 .In effect referral causes rotation through 180°.

Concluding remarks
We have here specialized the results of an earlier paper 1 by ignoring astigmatism.Referral from transverse plane T 1 to transverse plane T 2 results in displacement of a point with position vector y 1 in T 1 to a point with position vector y 2 given by Equation 8.The transformation is a special case of affine magnification 9 ; there is scalar magnification by X y, 1→2 followed by transverse translation d y, 1→2 expressions for which are given for the four situations in Table 1.Thus eliminating astigmatism simplifies the linear magnification 8 X y, 1→2 of the general case 1 to the conceptually much simpler and more familiar scalar magnification X y, 1→2 . 2 × 2 matrices have been eliminated in Equation 8 and in Table 1 (and Table   2); only scalars and vectors remain.The mathematics becomes more accessible for persons less familiar with matrix algebra.Scalar magnification X y, 1→2 may be positive, zero or negative.If it is positive referral magnifies the aperture in the familiar sense; there is change in size but not in shape or orientation.If X y, 1→2 is zero the referred aperture is a point.If X y, 1→2 is negative there is inversion through its centre and magnification by −X y, 1→2 .Inversion effectively rotates the shape through 180° about Z. 8 If such rotation represents a symmetry of the aperture then referral does not change the shape or orientation.Figure 4 illustrates the case of X y, 1→2 = −1 and a circular aperture; the aperture and the referred aperture are identical.The same would be true of an elliptical aperture and an aperture of any other shape with Z an axis of two-fold rotational symmetry.However the same is not true of apertures of other shapes as illustrated in Figure 5; triangular aperture P 1 Q 1 R 1 in transverse plane T 1 undergoes referral and magnification X y, 1→2 = −1 to become referred aperture P 2 Q 2 R 2 in transverse plane T 2 .In contrast to the case in Figure 4 the referred aperture is distinguishable from the aperture.Note that the effect is to turn the shape through 180°.To say that the shape is turned upside down would be misleading.In contrast to the circular aperture in Figure 4 the referred aperture is distinguishable from the aperture.To describe the change as turning the shape upside down is not correct for that would require R 2 and P 2 to be interchanged.
Consider an eye with stigmatic elements.The reti-

Figure 1 Figure 1
Figure 1 represents four dioptric systems S each with a transference T of the form of Equation 2. In each case Z is the longitudinal axis; it defines the positive sense through the system.It represents the general direction in which light travels through S. No component of S is shown apart from the entrance and exit planes labelled T with or without a subscript.In each case the index of refraction is α O = n 0 a O on the negative side of the entrance plane (to the left in Figure 1) and n 1 on the positive side of the exit plane.In (a) and (b) there is an object point O and in (c) and (d) an image point I.
Figure 1(b) represents a ray through system S from 6 a distant object point O.The entrance plane of S is now not T O but T 0 .All the rays from O have the same reduced inclination α .Choosing one of the rays is now equivalent to choosing the transverse position y 0 of the ray at T 0 .

Table 1
Positional invariant along a ray through an object (O) or image (I) point and scalar magnification X y, 1→2 and transverse translation d y, 1→2 of transverse position y from transverse plane T 1 to transverse plane T 2 .Table 2 Inclinational invariant along a ray through an object (O) or image (I) point and scalar magnification X α, 1→2 and transverse displacement d α, 1→2 of reduced inclination α transverse plane T 1 to transverse plane T 2 .S Afr Optom 2012 71(1) 3-11 WF Harris -Aperture referral in dioptric systems with stigmatic elements The South African Optometrist ISSN 0378-9411 (a), namely for referral by an object point O on the entrance plane.The other three situations represented by Figure 1 are handled in a similar fashion.Equations 8 and 11 are obtained again for all three cases but with expressions for X y, 1→2 and d y, 1→2

Figure 2
Figure 2 shows an ellipse E in a transverse plane T. a and b are half the major and minor diameters of E and v a and v b are unit vectors along those diameters.For the present we assume a > b > 0 .The centre of E has position vector y 0 in T relative to the longitudinal axis Z.A point is shown on E with position vector y relative to Z and position vector r relative to the centre.Then 33) a result obtained in the usual way.Here ζ = z / n .

Figure 3
Figure3Object point O on longitudinal axis Z refers circular aperture A 1 of radius r p1 = r 1 > 0 in transverse plane T 1 to transverse plane T 2 at distance z downstream in a homogeneous medium.Referred aperture A 2 is circular with radius r 2 .As z increases r 2 decreases reaching 0 for z = z f and then goes negative.For z = 2z f (as in the figure) r 2 = _ r 1 .The positive radius of A 2 is r p2 = r 2 2 .

Figure 5
Figure5Triangular aperture P 1 Q 1 R 1 in transverse plane T 1 of Figure3and referred aperture P 2 Q 2 R 2 .The positional magnification is X y, 1→2 = −1 .Referral rotates the shape through 180°.In contrast to the circular aperture in Figure4the referred aperture is distinguishable from the aperture.To describe the change as turning the shape upside down is not correct for that would require R 2 and P 2 to be interchanged.