Definitions in the literature^{1â€“6} of first-order chromatic aberration treat the optical system in question as homocentric and as having refracting elements that are stigmatic. There appear to be no published definitions of longitudinal and transverse chromatic aberration in systems that are heterocentric and astigmatic. The lack of definitions would seem unfortunate in view of the fact that heterocentricity and astigmatism are features of the typical eye. It might also suggest that such definitions may not be easy to come by. This note has the limited objective of proposing definitions and developing the linear optics of longitudinal and transverse chromatic aberration of systems that may be heterocentric and astigmatic. The definitions are natural generalizations of familiar definitions in Gaussian optics. They hold for systems in general and apply to the eye in particular, and they allow one to explore the effects of changes to the eye including those that accompany accommodation and refractive surgery for example.

There is inconsistency in the optometric literature over the use of the term chromatic aberration, particularly, perhaps, in the more clinically oriented literature. This does not facilitate communication within the discipline and between optometry and other disciplines. Greater care needs to be taken over terminology; usage should be as consistent as possible with that of the broader scientific community, and distinct concepts should be assigned distinct names. (We take up these points at the end of this note.) In keeping with these thoughts we take our point of departure to be a definition of chromatic aberration used commonly in the literature of both general optics and optometry.^{1â€“3}

#### HOMOCENTRIC SYSTEMS WITH STIGMATIC ELEMENTS

Fig. 1 illustrates definitions^{1â€“3} of longitudinal and transverse chromatic aberration. The definitions are in terms of Gaussian optics. System S consists of refracting elements invariant under rotation about, and centered on, a common axis Z, the optical axis of S. None of its refracting surfaces is shown. S has entrance plane T_{0} and exit plane T, both transverse to axis Z. The indices of refraction are n_{0} and n upstream and downstream, respectively, of S. Object point O has longitudinal position z_{O} and transverse position y_{O}. Fig. 1 is drawn with y_{O} > 0 and z_{O}< 0. The location of the image point I depends on the frequency *Î½* of light involved. Consider two particular frequencies *Î½*_{r} and *Î½*_{b}. It will be convenient to refer to the light as red and blue, respectively. The red and blue images of O are represented in Fig. 1 as I_{r} and I_{b}, respectively, with longitudinal z_{r} and z_{b} and transverse y_{r} and y_{b} positions, all being positive in the figure. We need to distinguish the incident n_{0r} and n_{0b} and emergent n_{r} and n_{b} indices corresponding to frequencies *Î½*_{r} and *Î½*_{b}. By definition

and

are *the longitudinal (or axial) and transverse (or lateral) chromatic aberrations,* respectively.

We note that longitudinal and transverse chromatic aberrations do not depend on the properties of system S alone; they depend on the properties of system S and the property of object point O. Most properties of a system depend on the context or environment of the system, that is, the indices n_{0} and n. For convenience, we take reference to properties of a system to imply the context as well. The property of object point O is its position in space. In Fig. 1, it is represented by its longitudinal and transverse positions z_{O} and y_{O}. It follows that for a given system S, the chromatic aberration is not unique. There is usually an infinity of longitudinal and transverse chromatic aberrations. The chromatic aberration of the system becomes unique when the location of the object point is specified.

By these definitions, both longitudinal Î”z and transverse Î”y chromatic aberrations are lengths measured orthogonally and represented by scalars. One is tempted, therefore, to regard the two chromatic aberrations as Cartesian components of a vectorial chromatic aberration and to insert an arrow from I_{r} to I_{b} to represent it in Fig. 1. We could then treat chromatic aberration holistically and not have to write separate equations for the two components. We shall find below, however, that the two aspects are fundamentally different and cannot meaningfully be combined in this way.

#### HETEROCENTRIC SYSTEMS WITH STIGMATIC ELEMENTS

A two-dimensional drawing, like that of Fig. 1, suffices for representing chromatic aberration in the case of homocentric systems free of astigmatism because optical axis Z, object point O, and images I_{r} and I_{b} all lie in a common plane; that plane becomes the plane of the paper. However, when homocentricity is relaxed, this is usually no longer the case. In general, for heterocentric systems with stigmatic elements, we need a 3-dimensional representation like that attempted in Fig. 2. In Fig. 2, system S contains refracting elements that may be mutually decentered. S may contain prisms and tilted surfaces. Z is no longer an optical axis but merely a longitudinal axis.

It is apparent that the definition for longitudinal chromatic aberration **Î”z** in the case of homocentric systems with stigmatic elements (Eq. 1) can be generalized to heterocentric systems unchanged. This is not the case for transverse chromatic aberration however. Transverse chromatic aberration becomes a two-component vector **Î”y** defined by

where **y**_{r} and **y**_{b} are the transverse position vectors of images I_{r} and I_{b}. Transverse chromatic aberration **Î”y** can be decomposed into horizontal and vertical components in the transverse plane if desired.

One still needs to resist any temptation to lump longitudinal and transverse chromatic aberration into a single concept of chromatic aberration that could be represented by an arrow (not shown) in Fig. 2 from I_{r} to I_{b}.

#### HETEROCENTRIC ASTIGMATIC SYSTEMS

We now relax the requirement that the elements of system S of Fig. 2 be stigmatic; the elements of S may be heterocentric and astigmatic. Each image point, I_{r} and I_{b}, in Fig. 2 becomes blurred in Fig. 3. One can think of each image point as dissociating longitudinally into a pair of orthogonal image lines. The structure becomes that of the familiar interval of Sturm, its nature and location being dependent on the frequency of the light. We need to allow for the fact that the orientations of the image lines usually do not match; that is, the first image line of the red blurred image is usually not parallel to the first image line of the blue blurred image. How now do we define longitudinal and transverse chromatic aberration?

Let us first consider chromatic aberration in a system with astigmatic elements that are centered on Z. Z then is an optical axis. Suppose, further, that object point O lies on Z. Red image I_{r} is then centered on Z, and its associated line segments intersect Z. The same holds for blue image I_{b}. Evidently, there is no transverse chromatic aberration. What chromatic aberration there is longitudinal. But how do we define it? The definition should surely account for the fact that the two blurred images may differ not only in longitudinal position but also in the nature and degree of blur. In other words, longitudinal chromatic aberration would need at least three numbers for its complete quantitative representation.

The images themselves do not suggest an obvious answer. Instead, we shift focus to the pencils of light containing them. In the absence of astigmatism, the red pencil would have reduced vergence L_{r} = n_{r}/z_{r} in exit plane T where z_{r} is the longitudinal position of the image point relative to T. Turning the equation around, we obtain z_{r} = n_{r}/L_{r}. In the presence of astigmatism, the generalization of the scalar reduced vergence L is the matrix reduced vergence **L** introduced by Fick^{7,8} and, independently, by Keating.^{9} **L** is a 2 Ã— 2 symmetric matrix^{10} identical in mathematical character to the dioptric power matrix F described by Fick^{7,8} and Long^{11} if not by others before them. Its entries have the units of reciprocal length and so can be in diopters. **L**_{r} is the reduced vergence at exit plane T of system S of the red astigmatic pencil defined by O, and **L**_{b} is the same but for the blue pencil.

We define

**Z** is symmetric and has the units of length and can be regarded as a generalized position of the blurred image relative to exit plane T. The eigenvalues of **Z** give the longitudinal positions of the image lines, and the eigenvectors define their orientations. (Eigenvalues and eigenvectors are treated in standard texts in linear algebra and have been applied in this context in several articles.^{9â€“11}) (For the moment, we assume that **L**^{âˆ’1} exists and return to the issue of nonexistence later.) Let the eigenvalues of **Z** be z_{âˆ’} and z_{+} where z_{âˆ’} â‰¤ z_{+} and let the corresponding normalized eigenvectors be **v**_{âˆ’} and **v**_{+}. Then, the first image line has longitudinal position z_{âˆ’} relative to exit plane T and is parallel to the complementary eigenvector v_{+}, and the second image line has longitudinal position z_{+} and is parallel to **v**_{âˆ’}.

Eq. 1 suggests the definition

for the longitudinal chromatic aberration of a homocentric astigmatic system S for object point O on the optical axis. There is no transverse chromatic aberration. **Î”Z** characterizes the longitudinal difference of the two images completely. By this definition, longitudinal chromatic aberration becomes the 2 Ã— 2 symmetric matrix **Î”Z** and, as such, can be characterized by three independent numbers. However, we need to examine it further.

Now reduced vergence (either as scalar L or as matrix **L**) depends on the relative longitudinal positions of object point and refracting elements and is independent of relative transverse positions. Thus, provided relative longitudinal positions are maintained, decentering the object point and elements of the system has no effect on the longitudinal positions and natures of the blurred images including the longitudinal positions and orientations of the image lines. It follows, therefore, that we can relax the requirement that O and the centers of the elements of S be on axis Z and take Eq. 5 to be the definition of the longitudinal chromatic aberration **Î”Z** of a heterocentric astigmatic system for an object point anywhere.

Because **Î”Z** is 2 Ã— 2 and symmetric, longitudinal chromatic aberration has two orthogonal principal meridians. They are the meridians within which the longitudinal chromatic aberration is a maximum and a minimum. The maximum and minimum values are the *principal longitudinal chromatic aberrations*. They are the eigenvalues of **Î”Z**, and the corresponding principal meridians are the corresponding eigenvectors.

The only effect of relative decentration of object point and system elements is to cause transverse displacement **y**_{r} and **y**_{b} of blurred images I_{r} and I_{b} of object point O in heterocentric astigmatic system S. Then, Eq. 3 defines the transverse chromatic difference **Î”y** of the images. We, therefore, call **Î”y** the transverse chromatic aberration of system S for object point O.

#### CHROMATIC ABERRATION IN GENERAL

Eqs. 3 and 5 represent the generalizations to optical systems in general of the definitions (Eq. 1 and 2) for systems whose refracting elements are all stigmatic and centered on an optical axis. Eq. 3 defines **Î”y** transverse and Eq. 5 **Î”Z** longitudinal chromatic aberration in general; the first is a two-dimensional vector, and the second is a 2 Ã— 2 symmetric matrix. The essential difference in mathematical character between transverse and longitudinal chromatic aberration highlights the fact that the two types of aberration are fundamentally different in nature and cannot meaningfully be combined into a single unified concept of chromatic aberration.

All this holds in particular for systems whose elements are stigmatic and homocentric. However, in a context in which only such systems are under discussion Î”y and Î”Z can be reduced to the scalar quantities Î”y and Î”z and sketched in one plane as in Fig. 1. Then, Î”y is one component of Î”y, the other being zero and perpendicular to the plane of the paper, and Î”z is the scalar coefficient in the scalar matrix Î”Z = IÎ”z, **I** being an identity matrix.

#### QUANTIFYING CHROMATIC ABERRATION

Having defined them, and given the makeup of an optical system, how do we calculate longitudinal and transverse chromatic aberration? Here, we derive general formulae in linear optics. The key is the systemâ€™s ray transference, which is a function of the frequency of light.^{12} For systems that may be heterocentric and astigmatic, the transference is the 5 Ã— 5 matrix^{13,14}

**A**, **B**, **C**, and **D** are 2 Ã— 2 and **e** and **Ï€** are 2 Ã— 1 submatrices. They are the fundamental properties of the system. **o**^{T} is the matrix transpose of the 2 Ã— 1 null matrix **o**. The fifth row of **T** is the trivial (0 0 0 0 1). **e** and **Ï€** account for the effects of tilt and decentration; each is null if the longitudinal axis is an optical axis.^{15}

##### Longitudinal Chromatic Aberration

If the reduced vergence is **L**_{0} at entrance plane T_{0} of system S, then the reduced vergence is^{10,16}

at the exit plane T of S. For an object point O at longitudinal position z_{O} relative to T_{0}

Hence

or

with two special cases,

for z_{O} â†’ âˆž and

for z_{O} = 0. (Eq. 11 represents the back-vertex power of system S.^{17}) Adding subscripts to all the parameters in these equations (except z_{O}) gives expressions for the red and blue reduced vergences **L**_{r} and **L**_{b} at exit plane T. Substitution into Eq. 5 then gives the longitudinal chromatic aberration **Î”Z** for system S and object point O.

##### Transverse Chromatic Aberration

Perhaps surprisingly, the problem of calculating the transverse chromatic aberration is more challenging. We first examine object points at finite distances.

Consider the compound system from the transverse plane of O to the transverse plane containing an image line of a blurred image. Let the longitudinal position of the plane of the image line be z relative to exit plane T of system S. The compound systemâ€™s transference is obtained by multiplying the transferences of the components in reverse order in the usual way.^{14} Its top block row turns out to be

Combining this with the equation for the transverse position at emergence (Eq. 14 of a previous article^{14}), we see that a ray of inclination **aO** at object point O arrives at the transverse plane of the image line with transverse position

In particular for the ray parallel to longitudinal axis Z at O

We write this as

The first image line goes through the point given by Eq. 16 with z = z_{âˆ’} and is parallel to **v**_{+}. Hence, we can write a parametric equation for the first image line as

for all real scalars k_{âˆ’}. Interchanging the plus and minus signs gives the equation of the second image line. But **v**_{+} is orthogonal to **v**_{âˆ’} so we can write the equation for the second image line as

where

. Subtracting Eq. 17 from Eq. 18 we obtain

where

Now **y**_{+} âˆ’ **y**_{-} is the longitudinal projection of a vector from a point on the first-image line to a point on the second. We make this vector parallel to Z, i.e., **y**_{+} âˆ’ **y**_{-} = **o**. Then, in terms of the components of **v**_{+} and **c**, Eq. 19 becomes

Multiplying out, rearranging, and reassembling into matrices we obtain

Because **v**_{+} is a unit vector, the 2 Ã— 2 matrix on the left has unit determinant. Hence

from which we obtain

in particular. Substituting from Eq. 24 into Eq. 17 and rearranging one finds that the transverse position of the image is

where **V** is the matrix

For a distant object point we take the compound system to be system S and the homogeneous gap between S and an image line and apply a similar method to that used above for an object point at a finite distance. We find that, for a distant object point O, the transverse position of the image turns out to be

where **aO** is the inclination of the rays from O.

Eqs. 25 to 27 can be written for the red and blue blurred images. Eq. 27 then gives **y**_{b} and **y**_{r} for distant object points and Eq. 25 gives them otherwise. Hence, from Eq. 3, we obtain the transverse chromatic aberration

for an object point at a finite distance and

for a distant object point.

The calculation fails when the reduced vergence **L** of either the blue or red light is singular, that is, when an image line is at infinity. However, such cases seem of little practical interest and we consider them no further.

##### Systems with Stigmatic Elements

In particular, if every element of the system is stigmatic, then **A**, **B**, **C**, and **D** are all scalar matrices; that is, **A = I**A where A is a scalar, and similarly for the other three 2 Ã— 2 fundamental properties. The reduced vergence at emergence is also a scalar matrix, **L = I**L, and so is **Z** (Eq. 4), **Z = I**Z. The eigenvalues of **Z** are not distinct: z_{âˆ’} = z_{+} = Z is simply the longitudinal position of the image point relative to exit plane T. The longitudinal chromatic aberration is **Î”Z = IÎ”Z** where **Î”Z** is the longitudinal position of the blue image point relative to the longitudinal position of the red image point. Eq. 26 reduces to **V = I**Z/n and, finally, Eqs. 28 and 29 become

and

##### Summary of the Routine for Calculating Longitudinal and Transverse Chromatic Aberration

Suppose we know the transferences of a system S for blue and red light. We can then calculate the longitudinal and transverse chromatic aberrations of the system for a finite object point O with longitudinal position z_{O} and transverse position **y**_{O}. We proceed as follows. We use Eqs. 9 or 10 to determine the reduced vergence **L** of blue light from O leaving S. Eq. 4 then gives the generalized longitudinal position **Z** of the blue image. We repeat for the red image. The longitudinal chromatic aberration is then **Î”Z** given by Eq. 5. For **Z,** for blue light, we obtain the eigenvalues z_{âˆ’} and z_{+} and the corresponding normalized eigenvectors **v**_{âˆ’} and **v**_{+}. Eq. 26 gives **V**. Hence one determines **A** + **VC** and **e** + **VÏ€** for the blue light. This is repeated for red light. **Î”(A + VC)** is calculated by subtraction (blue minus red) and similarly for **Î”(e + VÏ€)**. Finally, the transverse chromatic aberration **Î”y** is given by Eq. 28.

If object point O is distant then we need the inclination **aO**. The calculation is the same as for a finite object point except that the vergence **L** is obtained via Eq. 11, (**B + VD**)n_{0} replaces **A + VC**, and Eq. 29 is used instead of Eq. 28.

The Appendix illustrates the calculations for a heterocentric astigmatic model eye with four refracting surfaces (available at http://links.lww.com/OPX/A107).

#### CONCLUSIONS

For systems with homocentric stigmatic refracting elements definitions of chromatic aberration differ from author to author. Several authors have remarked on the inconsistency and confusion.^{18â€“20} We believe there is a need for authors to take greater care to define terms in general but particularly in the context of chromatic aberration.

Here, we have offered definitions that are natural generalizations of the familiar concepts^{1â€“6} in Gaussian optics; they hold for the special case of systems with homocentric stigmatic elements, and they hold for systems, like the eye, with elements that are heterocentric and astigmatic. We have also derived expressions for longitudinal and transverse chromatic aberration in terms of the fundamental properties of the optical system.

For general systems, which may be heterocentric and astigmatic, we have defined longitudinal chromatic aberration to be the 2 Ã— 2 symmetric matrix **Î”Z** given by Eq. 5. It depends on the longitudinal position z_{O} of the object point O but is independent of the transverse position **yO**. Its eigenvectors are principal meridians of longitudinal chromatic aberration, and its eigenvalues are the principal longitudinal chromatic aberrations along them.

The transverse chromatic aberration **Î”y**, a vector defined by Eq. 3, can be calculated by means of Eqs. 28 or 29. In general, it is an affine function of the objectâ€™s transverse position **yO** (Eq. 28) in the case of objects at finite distances or of its direction, in effect **a**_{O}, (Eq. 29) in the case of distant objects. If the refracting elements of the system are all centered on longitudinal axis Z, then Z is an optical axis, and because **e** and **Ï€** are both null,^{15} the constant term **Î”(e + VÏ€)** in those equations vanishes, and the transverse chromatic aberration becomes linear in **yO** or **a**_{O}.

It may be worth mentioning that the principal meridians of the red and blue pencils, leaving the optical system, need not match. This is why one cannot, in general, simply calculate longitudinal chromatic aberration separately in two orthogonal principal meridians. Nevertheless, preliminary calculations (such as those in the Appendix) suggest that, for many practical purposes, it may well be sufficiently accurate to do so.

If the system in question is composed of stigmatic elements arranged homocentrically, then the definitions here reduce to the familiar definitions^{1â€“6} of chromatic aberration in Gaussian optics. This special case has been treated above. We note, however, that stigmatic systems exist with astigmatic elements.^{21,22} For them, the special case does not apply, although their longitudinal chromatic aberration **Î”Z** is a scalar matrix.

We are not entirely comfortable with the word aberration in the terms longitudinal and transverse chromatic aberration. It suggests an optical concept beyond first order, whereas here, and in most cases in the literature, the concept is one in first-order optics. However, until a more suitable term is suggested, we believe longitudinal and transverse chromatic aberration be reserved for the concepts defined here.

The definitions proposed here are not specific to the eye. The retina, in particular, is not mentioned in the definitions. When applied to the eye, as to any other system, it is important to be unambiguous about how the definitions are being used. First, it should be clear what the system is whose chromatic aberration is being defined; in particular, the entrance and exit planes T_{0}and T should be defined. For the visual optical system of the eye that would most likely have T_{0} immediately in front of the tear film on the cornea and T immediately in front of the retina. Second, the location of longitudinal axis Z should be specified in some way. Third, the location of the object point relative to Z should be given. Finally, the two frequencies v_{r} and v_{b} of the light should be given or understood.

In his or her introduction to optometry, the beginning student often learns to refract with the interval of Sturm and its relation to the retina in mind. What is clear from the analysis here is that there is such an interval for each frequency, that they differ longitudinally and transversely by the longitudinal and transverse chromatic aberration, and that, from a knowledge of the structure of the eye, we are now able to calculate these differences. (A somewhat different perspective on what underlies the routine of refraction is presented elsewhere.^{23})

It would seem that the familiar concepts of longitudinal and transverse chromatic aberration, as defined in Gaussian optics,^{1â€“6} are probably less directly useful in the clinical context which may be why a variety of concepts related to them has been devised for use in practice. Confusion arises, however, because many of these concepts are called by the same names. They should, we believe, be assigned suitable distinguishing designations. Our generalization of the concepts in Gaussian optics to allow for heterocentricity and astigmatism may also be of less direct use in the clinical environment. Nevertheless, it has its place in optometric didactics and in the broader understanding of the optics of vision. Furthermore, the theory provides tools for exploring the effects of changes to the eye that accompany accommodation and refractive surgery for example.

#### APPENDIX

The appendix is available online at http://links.lww.com/OPX/A107.

William F. Harris

Department of Optometry,

University of Johannesburg

APK Campus, PO Box 524,

Auckland Park, Johannesburg 2006

South Africa

e-mail: wharris@uj.ac.za

##### ACKNOWLEDGMENTS

We thank L. N. Thibos and R. D. van Gool for continuing discussions. WFH gratefully acknowledges a grant from the National Research Foundation of South Africa. TE, a graduate student working with him, acknowledges support from the Medical Research Council of South Africa.

Received January 15, 2012; accepted July 2, 2012.