Below we shall make use of two orthogonal matrices, the rotation matrix
and the reflection matrix
To avoid ambiguity of sign later, we impose the restriction −90° < θ ≤ 90° in the case of R θ and −45° < θ ≤ 45° in the case of −Rθ, although it will be convenient occasionally to relax these constraints. As we shall see, the restrictions do not imply a loss of generality. R θ represents rotation through angle θ, and −Rθ represents reflection in a line at angle θ. When we come to use these matrices below, they will represent operations on the image relative to the object. A positive angle θ is an angle measured counterclockwise when one is looking in the positive sense along the longitudinal axis Z. We note, for later reference, that
Any matrix of the form
with unit determinant is a rotation matrix, and any matrix of the form
with determinant −1 is a reflection matrix.18 Under the restrictions on θ imposed above, p ≥ 0.
STIGMATIC AND ASTIGMATIC SYSTEMS
Consider a point object located in transverse plane O in Fig. 2. Suppose the point object maps to a point image through system S. We choose transverse plane I such that it contains the point image. Planes O and I then become object and image planes, respectively. It follows (see the paragraph immediately before Equation 14 that compound system SC is conjugate, and hence its disjugacy is B C = O. Thus, from the top, middle block of Equation 18 we have
We now make the following definitions:
- DEFINITION: A system is called stigmatic if to every point object there corresponds a point image through the system.
- DEFINITION: A system that is not stigmatic is called astigmatic.
For an astigmatic system, there is at least one point object that does not map to a point image. There may be point objects that do map to point images.
For completeness, we must allow object and image planes to be “at infinity” or at any finite distance, negative or positive. Thus, we take ζ0 and ζ in Equation 27 as points on the extended real number line, i.e., the familiar number line plus ∞. That being so, we need to be on the look out for potential indeterminate forms involving ∞ in addition to those involving 0. This is especially so because of the importance in optometry of the case of ζ0 = ∞. ζ0 ≥ 0 implies a real object point; for ζ0 = 0 the object point is on the entrance plane of system S, and for ζ0 < 0 the object is virtual. Similarly for ζ ≥ 0 the image is real; for ζ = 0 the image is on the exit plane of S; and for ζ < 0 the image is virtual. However, these names are of no consequence for our purposes; we shall not need to refer to them again. As mentioned above, systems that function as if they were gaps of negative thickness are realizable. Although compound system SC may be infinite in length, system S itself is finite as are all the entries in its transference, including A, B, C, and D in Equation 27 in particular.
Let a, b, c, and d represent real numbers and R either a particular rotation matrix R θ with 90° < θ ≤ 90° or a particular reflection matrix −Rθ with 45° < θ ≤ 45°. We state and then prove the following theorem:
THEOREM: A system is stigmatic if and only if its transference is of the form
PROOF: The theorem takes the form of a biconditional. Following normal practice, we prove one conditional of the pair and then the other.
For one of the conditionals, we assume that the transference is of the form in Equation 28. We are then required to prove that for every ζ0 there exists a ζ that satisfies Equation 27. From the third symplectic equation (Equation 11) we have
which, because of Equation 23, shows that
The latter is the symplectic equation of two-dimensional linear optics.11,12 Substitution into Equation 27 leads to
and, hence, because of Equation 24,
Solving we obtain
This shows that, with two potential exceptions, there is a ζ that corresponds to every ζ0. The two potential exceptions occur when ζ0 = ∞ and when the denominator on the right side of Equation 33 is zero. In the first case, we set ζ = ζ∞; Equation 33 becomes
Because of Equation 30, c and a cannot both be zero, and when c = 0 ζ∞ = ∞. Thus, ζ∞ is always defined. For a zero denominator, again because of Equation 30, Equation 33 reduces to ζ = 1/(0 × c), which is always ∞. Thus, for every ζ0 there is a ζ, which means that for every point object there is a point image. By definition then the system is stigmatic. We have proved one of the two conditionals.
The converse conditional is more problematic. We assume that for every ζ0 there is a ζ, or in other words that Equation 27 is satisfied for all ζ0. We have then to prove that Equation 28 is a consequence.
Because, by assumption, Equation 27 holds in general, it holds for ζ0 = 0 in particular. For ζ0 = 0 we represent the corresponding ζ by ζ0. Substituting into Equation 27 we obtain
The only potential indeterminacy is in the term ζCζ0 in Equation 27: it could be that ζ0 = ∞, in which case we have the form ∞ × 0. The problem is overcome by rewriting Equation 27 as
and then setting ζ = ζ0 = ∞. One obtains D = O, a result already covered by Equation 35. Equation 35 then shows that B and D are scalar multiples of a common matrix, M, say. Thus, we can write
where ˜b and ˜d are real numbers. ˜b = 0 when B = O and ˜d = 0 when D = O. Because of the third symplectic equation (Equation 11), M is not null and ˜b and ˜d are not both zero. Hence, because of Equations 36 and 37, Equation 35 reduces to
ζ0 = ∞ when ˜d = 0. We substitute from Equations 36 and 37 into Equation 27 and rearrange to give
We choose any particular ζ0 other than 0 and ∞ and such that ζ is either 0 or ∞. Then Equation 39 shows that A + ζC is a scalar multiple of M. Hence, we can write
where s is a scalar dependent on ζ0. For a particular ζ0, say ζ′0, ζ and s are ζ′ and s′, respectively, and Equation 40 becomes
for another particular ζ0. Subtraction shows that we can write
where ˜c is a real number. Substitution from Equations 36, 37, and 43 into Equation 27 gives
which shows that
for a real number ã.
Substituting from Equations 36, 37, 43, and 45 into the third symplectic equation (Equation 11), we obtain
As for P in Equation 13, we express M in terms of I, J, K, and L (Equation 12). Multiplying we obtain
Comparison with Equation 46 shows that the coefficients of J and K must be zero, i.e.,
There are two distinct ways to satisfy Equations 48 and 49 simultaneously: either
It follows from Equations 12 and 13 that M has one of two forms, namely,
We choose M I and M J such that M I ≥ 0 and M J ≥ 0. (We can always do so with the appropriate choice of sign for ã, ˜b, ˜c, and ˜d.) M R and M S are of the forms in Equations 25 and 26, respectively, except that their determinants are not generally 1 or −1, respectively. Dividing by √|det M| converts M R to a rotation matrix (Equation 19) and M S to a reflection matrix (Equation 20). Thus, Equation 45 can be written
and R is a rotation or a reflection matrix. Similarly, one obtains B = b R (from Equation 36, C = c R (from Equation 43, and D = d R (from Equation 37, in which the R’s are all the same matrix and scalars b, c, and d in turn each replace a on both sides of Equation 55. Hence, Equation 28 holds; we have proved the converse conditional.
Having proved the two component conditionals of the biconditional, we have proved the theorem.
The proof above is rather tedious; possibly, a more elegant one is waiting in the wings.
The restrictions imposed on rotations by −90° < θ ≤ 90° and on reflections by −45° < θ ≤ 45° remove ambiguity in the signs of the scalars a, b, c, and d and the orthogonal matrix R; they make the scalars and the matrix unique.
We shall say a transference has stigmatic form or is stigmatic if it has the form of Equation 28.
It is a corollary of the theorem above that if a system has a transference of stigmatic form, then the transference is necessarily constrained by Equation 30.
We have here defined a system to be stigmatic if and only if, through the system, every object point maps to a point image. Astigmatic systems are defined by means of negation: they are systems that are not stigmatic. In other words, through an astigmatic system there is at least one point object that does not map to a point image through it; there may be a point object that maps to a point image. That a point object maps to a point image is not sufficient to make a system stigmatic.
The mathematics has shown that the condition that a system be stigmatic is that all four 2 × 2 fundamental properties of the system are scalar multiples of a common orthogonal matrix R (Equation 28). Because products of orthogonal matrices are orthogonal, a system compounded of stigmatic systems is stigmatic. The two types of orthogonal 2 × 2 matrix define two disjoint classes of stigmatic systems. They are the topic of an accompanying article.1
The questions posed at the outset have now been answered. In general, an optical device is astigmatic when its four 2 × 2 fundamental properties are not all scalar multiples of the same orthogonal matrix. The same applies to an eye and an eye in combination with an optical instrument. Representing, in effect, a thin system, the refraction of an eye is astigmatic if the cylinder is not zero. However, an eye is more than its refraction and may well be astigmatic even if its refraction is scalar. For a system to be stigmatic, it is not sufficient that its dioptric power be spherical; in fact, if the dioptric power is spherical, the system is astigmatic unless all four 2 × 2 fundamental properties are scalar matrices.
I thank J. Rubinstein, Department of Mathematics, Indiana University, for useful comments on the manuscript. I also thank R. Blendowske of the Department of Optical Technologies and Image Processing, University of Applied Sciences, Darmstadt, and R. D. van Gool, G. E. MacKenzie, H. Abelman, W. Heath, A. Rubin, A. S. Carlson, and W. D. H. Gillan of the Optometric Science Research Group for continuing discussions.
W. F. Harris
Optometric Science Research Group
Department of Optometry
Rand Afrikaans University
P.O. Box 524
Auckland Park, Johannesburg, 2006
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Keywords:© 2004 American Academy of Optometry
stigmatic system; astigmatic system; linear optics; ray transference; dioptric power matrix