Abstract
Background: Chromatic differences in power, refractive compensation, position and magnification have been described in the literature for Gaussian eyes.
Aim: This article explores these definitions and defines chromatic properties for eyes that have astigmatic and decentred or tilted elements.
Setting: Linear optics.
Methods: The optical model is linear optics and makes use of the ray transference T.
Results: Results are presented as either 2 × 2 matrices or 2 × 1 vectors. The dependence of the chromatic properties on the position of an object and the limiting aperture is explored and results are presented as independent of, or dependent on, object and aperture position. Apertures undergo both longitudinal and transverse shifts in position. The results are general and apertures may include pupils or pinholes, either surgically inserted inside the eye or held in front of the eye. Numerical examples are provided for Le Grand’s foursurface eye and an arbitrary astigmatic heterocentric foursurface eye.
Conclusion: Apertureindependent chromatic properties include chromatic difference in power and refractive compensation, both given as 2 × 2 matrices. Aperturedependent chromatic properties are all dependent on longitudinal shifts in the plane of the limiting aperture. In addition, chromatic difference in position and inclination depend on both object and transverse aperture position. Chromatic difference in image size or angular spread depends on object position and is independent of transverse aperture position. These four aperturedependent chromatic properties are given as 2 × 1 vectors. Chromatic magnifications are independent of object and transverse aperture position and are given as 2 × 2 matrices.
Introduction
The purpose of this article is to define chromatic properties for an eye with astigmatic and heterocentric elements. In physiological optics, longitudinal chromatic aberration^{1,2,3,4,5,6,7} is defined as chromatic difference in focus (either the chromatic difference in power or the chromatic difference in refractive compensation). Transverse chromatic aberration^{1,2,3,6,7,8,9,10,11} is defined as either chromatic difference in position or chromatic difference in magnification. The definitions for transverse chromatic aberration depend on the position of the nodal point.^{1,2,9,11}
We use these definitions as the basis for generalising to chromatic properties of eyes that may have astigmatic and decentred or tilted refractive elements. Like others,^{1,2,3,8,11} we too find that the pupil plays an important role in transverse chromatic properties of the eye; however, we seek definitions that do not depend on the nodal point. Firstly, in an astigmatic eye, the nodal points expand into nodal structures,^{12} much like a focal point with its twoline foci and a circle of least confusion across the interval of Sturm and, secondly, nodal points depend on the frequency of light.^{13} Nodal points are, therefore, not good structures on which to base definitions.
In astigmatic systems, the images at the retina of a dichromatic object point may comprise elliptically shaped red and blue blur patches, even when the pupil is circular. The two blur patches may differ in size and shape and may be relatively rotated. We define two properties that are independent of object and pupil position. Chromatic properties that depend on the object and the pupil position are defined in terms of the position, size and orientation of the red and blue images on the retina, and the inclination of the chief rays subtending these images.
The results obtained are general. The chromatic properties that depend on the pupil position can be extended to any system that has an aperture. Because of this, the results can be applied to any system that includes a pinhole aperture such as the Kamra corneal pinhole inlay^{14} and the IC8 pinhole intraocular lens,^{15} both made by AcuFocus, or the Xtrafocus 93L intraocular pinhole implant^{16} made by Morcher. These pinhole surgeries are all indicated for presbyopia and the Xtrafocus is additionally indicated for irregular astigmatism.
Method
The method relies on linear optics, including the ray transference:
a 5 × 5 matrix,^{17,18} where A the dilation, B the disjugacy, C the divergence and D the divarication are each 2 × 2 submatrices, e the transverse translation and π the deflection are each 2 × 1 submatrices. Together A, B, C, D, e and π are the six fundamental paraxial optical properties of a system.^{17} The bottom row is trivial; o is a 2 × 1 null matrix and superscript T represents the matrix transpose. (A matrix is represented in bold and a scalar in italics, for example, y and y.) The elements of the system may be astigmatic (which affect A, B, C and D) and heterocentric (which affect e and π). A, D and π are unitless; B and e are in units of length, for example, metres; and C is in units of inverse length, for example, dioptres. The transference completely defines the effect the system has on rays traversing it.^{19}
The ray state is defined by the 5 × 1 matrix^{17,18,20}
where y the transverse position and α the reduced inclination, given by α = na, are both 2 × 1 submatrices and n is the refractive index of the surrounding medium.^{20} a is the inclination. The ray state at incidence γ_{0} onto the system is changed to γ at emergence through the relationship^{17,18}
The transference depends on the frequency of light^{21,22} through the refractive indices of the media. Frequency is independent of the medium and provides expressions that are closer to being linear and therefore is preferable to vacuum wavelength.^{22,23} We make use of formulae for the refractive indices of the media based on Le Grand’s ^{24} eye given by Villegas et al.^{25} to obtain frequencydependent transferences and reduced inclination.
Power F = −C and cornealplane refractive compensation F_{0} = B^{−1}A, both 2 × 2 matrices, are obtained from T.^{17} Chromatic difference in power^{22} is defined as δF = F^{b} − F^{r} where superscripts b and r represent two frequencies. For convenience, we shall call them blue and red. Other chromatic differences are defined similarly; they are obtained from transferences corresponding to the red and blue frequencies. The chromatic difference in refractive compensation^{22} is . These two chromatic properties depend on neither object position nor limiting aperture and we refer to them as apertureindependent chromatic properties. Examples are provided in Table 1.
TABLE 1: Apertureindependent chromatic properties. 
Harris^{17} describes the magnification, blur and ray state at the retina for an astigmatic heterocentric eye. The eye naturally partitions into anterior and posterior portions at the plane of the pupil, a limiting aperture. If a pinhole is surgically implanted in the eye, then it partitions the eye into anterior and posterior subsystems at the plane of the pinhole. In this situation, we make the reasonable assumption that the pinhole is the limiting aperture. The anterior subsystem has transference T_{A} and the posterior subsystem T_{B}. Hence, the eye has transference^{17,18} T_{E} = T_{B} T_{A}. Subscripts A, B and E are used to represent the anterior and posterior subsystems and the system of the eye, respectively.
To define chromatic differences at the retina, we are interested not in reduced inclination α but in (unreduced) inclination a of rays. Accordingly, we adjust and summarise Harris’s^{17} results to obtain the coefficient matrix:
a 5 × 5 matrix with the same blockmatrix structure and units as the transference. W_{E} is the distance image blur coefficient, X_{E} is the distance image size coefficient, g_{E} is the distance image displacement, Y_{E} is the directional spread coefficient, Z_{E} is the distant directional coefficient and h_{E} is the distant directional displacement.^{17} Each of the coefficients is made up of the fundamental properties of the eye and the anterior and posterior subsystems. Light from a distant object arrives at the cornea with inclination a_{K}, traverses the pupil or pinhole with position y_{P} and reaches the retina with transverse position y_{R} and inclination a_{R}. We define modified ray state . Then V_{E}v = r_{R} where . Like the transference, V_{E} depends on the frequency of light.
We define chromatic difference in image position δy_{R} as the position of the blue relative to the red retinal image and chromatic difference in inclination of the chief rays at the retina δa_{R}, as shown in Figure 1. These are summarised as the chromatic difference in physical ray states at the retina . Substituting from Equation 1, we obtain δV_{E} v = δr_{R} or
where is the chromatic difference in coefficient matrices. Examples are provided in Table 2. By definition,^{1,2,3,9,10} we are following the chief ray and by implication ignoring blur and the size of the aperture. All chromatic differences are defined as from red to blue, that is, from a low to a high frequency and energy.

FIGURE 1: A general eye with corneal T_{K}, pupillary T_{P} and retinal transverse planes T_{R}. A pencil of rays enters the eye at T_{K} from a distant object with inclination a_{K} and we select only the red and blue chief rays traversing the centre of the (decentred) pupil y_{p} at T_{P}. The chromatic difference in image position δy_{R} is the vector from the position of the red ray to the blue ray at the plane of the retina T_{R}. Similarly, the chromatic difference in inclination of the chief rays at the retina δa_{R} is measured from the red ray to the blue ray . All ray positions, inclinations and pupil decentration are exaggerated for clarity. 

TABLE 2: Chromatic properties dependent on object (a_{K}) and aperture (y_{P}) position (Equation 2). 
When a pinhole is placed in front of the eye, the anterior transference becomes an identity matrix and Equation 2 simplifies. In this situation, only δX_{E} and δZ_{E} are dependent on z, the position of the pinhole in front of the eye. An example is provided in Table 2c. A pinhole placed in front of the eye holds more interest for clinical and experimental settings; some phenomena are exaggerated and, so, may lead to more insight.
∆a_{K} is the angular spread subtended by the two extreme points of a distant object. Red and blue images of sizes and are formed on the retina. We use δ to represent a chromatic difference and ∆ to represent size or angular spread between rays of the same frequency. The chromatic difference in image size at the retina is δ(∆y_{R}). Similarly, δ(∆a_{R}) is the difference in angular spread between the red and blue chief rays arriving at the retina. δ(∆y_{R}) and δ(∆a_{R}) are both linear in ∆a_{K}. We obtain the summary δV_{E}∆v = δ(∆r_{R}) where ∆y_{P} = o, ignoring any blur component. Consequently, aperture size and shape are irrelevant. Hence,
where • represents entries that are nullified. This is illustrated in Figure 2. δ(∆y_{R}) and δ(∆a_{R}) are independent of the transverse position of the aperture y_{p} but are dependent on the longitudinal position of the aperture (which are contained in δX_{E} and δZ_{E}) and the object’s angular size ∆a_{K}.

FIGURE 2: A general eye with corneal T_{K}, pupillary T_{P} and retinal transverse planes T_{R}. A pencil of rays enters the eye at T_{K} from a distant object with angular size ∆a_{K}. We trace only the red and blue rays traversing the same position in the pupil y_{P} in T_{P}. The chromatic difference in image size δ(∆y_{R}) is the vector from the position of the red image of size to the blue image of size in T_{R} and shown in the inset. Similarly, the chromatic difference in angular spread at the retina is δ(∆a_{R}). All ray positions, inclinations, object and image sizes, and pupil decentration are exaggerated for clarity. 

We define chromatic image size magnification M_{yR} by means of . From Equation 3 and manipulating we obtain
and are shown in Figure 2. Similarly, we define the retinal chromatic angular spread magnification as . Hence,
M_{yR} and M_{aR} are 2 × 2 magnification matrices whose eigenstructure reveals the magnitude and direction of minimum and maximum magnification. Examples are given in Table 3.
Results – numerical examples
For a Gaussian eye, such as a model eye, all the results reduce to scalars. We present numerical examples for two eyes; Le Grand’s foursurface eye^{24} and an arbitrary foursurface astigmatic heterocentric eye for a chromatic difference from 430 THz to 750 THz (vacuum wavelengths 697.2 nm and 399.7 nm). The parameters of this eye are provided in an appendix for the two frequencies, along with the transferences and coefficient matrices for the red and blue frequencies. The apertureindependent properties are given in Table 1. Le Grand’s eye ^{24} is emmetropic at a reference frequency of 509 THz (vacuum wavelength 589.0 nm) and the magnitude δF_{0} is as expected.^{1,4,8,26,27}
For the astigmatic eye δF may be an asymmetric matrix with nonorthogonal principal meridians. The results highlight the fact that the eye’s power and refractive compensation are not simply related. δF and δF_{0} both yield astigmatic results; hence, the apertureindependent chromatic properties vary with the ametropia of the eye.
The chromatic properties dependent on object and aperture position are given in Table 2 and Table 3. The chromatic differences in position and inclination are given for Le Grand’s eye for three different aperture positions in Table 2a to Table 2c. Included are examples with the Kamra corneal pinhole inlay,^{14} implanted at a depth of 200 µm into the stroma and a pinhole held in front of the eye. Handheld pinholes and Kamra corneal inlays are more susceptible to transverse displacement than the natural pupil, which in turn affects δy_{R} and δa_{R}. When a_{K} = 5^{°}, δy_{R} is zero at a position of y_{P} = −0.60 mm and δa_{R} is zero at y_{P} = −0.56 mm which represents nasal displacement in the left eye. This agrees approximately with the findings of Tabernero and Artal.^{28} (They used wavelengths of 470 nm, 510 nm, 550 nm, 610 nm and 650 nm. Results will depend on the individual eye, the value used for angle κ and the formulae used for the wavelengthdependent refractive indices.) A comparison of Table 2a and Table 2c shows that δX_{E} and δZ_{E} increase in magnitude as the aperture plane moves anterior in Le Grand’s eye. Hence, once a corneal inlay is positioned, offaxial object points increase the magnitude of δy_{R} and δa_{R}.
The results given in Table 2d for δV_{E} show that each of δW_{E}, δX_{E}, δY_{E} and δZ_{E} are asymmetric. All of the coefficient matrices given in Table 2 can be applied to Equation 3 to obtain the δ(∆y_{R}) and δ(∆a_{R}). From the asymmetric entries of δV_{E} in Table 2d, we conclude that the red and blue images of size and may differ not only in size, but also in orientation.
Table 3a gives the chromatic magnifications that compare favourably with experimental results for chromatic difference in magnification.^{1,6,11} In Table 3b, M_{yR} and M_{aR} increase as the plane of the limiting aperture is moved to the cornealplane, as in the case of a Kamra corneal inlay. The values in Table 3c show that chromatic magnification in an astigmatic eye may differ along different meridians. The unequal diagonal entries and nonnull offdiagonal entries imply that the chromatic magnification is not equal in all directions. The eigenstructure illustrates how the red and blue images are magnified by maximum and minimum amounts. The asymmetry of the chromatic magnification matrices indicates that the red and blue images are rotated with respect to each other. This is confirmed by the nonorthogonal eigenvectors. Although the misalignment of the eigenmeridians is small, this gives us insight into the effect of astigmatism.
Discussion
Traditional definitions for chromatic difference in focus, position and magnification are extended here to include eyes that have astigmatic and decentred or tilted refracting elements. We consistently use differences from red to blue (from low to high frequency and energy) and to avoid definitions that depend on nodal points because these are frequencydependent in real eyes and not necessarily points in the presence of astigmatism.
Definitions for transverse chromatic aberration in the literature^{1,2,3, 8,9,10,11} depend on the position of the rays reaching the retina. Angular measurements are taken from the nodal point and are not true ray inclinations. Within the limitations of firstorder optics, we have used actual ray inclinations. The results for the illustrated astigmatic eye highlight the fact that chromatic properties are dependent on the ametropia of the eye.
Two apertureindependent chromatic properties are defined for eyes that include astigmatic elements, namely chromatic difference in power δF and refractive compensation δF_{0}. These depend, among other things, on the ametropia of the eye and δF may be asymmetric. In an astigmatic eye, δF and δF_{0} may both be astigmatic powers. The principal meridians may be nonorthogonal for chromatic difference in power because the eye is a thicklens system.
The aperturedependent chromatic properties are all dependent on the longitudinal position of the aperture. In addition, chromatic difference in image position δy_{R} and inclination δa_{R} at the retina depend on y_{P} and a_{K}. The positional differences between the red and blue rays are not limited to the Gaussian plane but are given as vectors across the transverse plane of the retina. Similarly, the inclinational differences represent the difference in inclination from red to blue chief rays and do not depend on the position of the nodal point or nodal structure. A transverse shift in aperture position y_{P}, such as a misaligned corneal inlay or pinhole, will affect δy_{R} and δa_{R}. This has importance for surgically inserted apertures, particularly corneal pinhole inlays which are sensitive to misalignment.^{28}
The chromatic difference in image size δ(∆y_{R}) and angular spread δ(∆a_{R}) at the retina are independent of y_{P}, but are dependent on the size of the object ∆a_{K}. The relationships are linear in ∆a_{K}. The red and blue images at the retina may differ both in size and orientation, represented by the vectorial difference. A shift in longitudinal position of the aperture, for example, from the iridial plane to the cornealplane, increases the magnitude and exaggerates any relative rotation of red and blue images. However, of greater significance is chromatic magnification.
Chromatic magnification is a generalised ratio of the blue to red image size M_{yR} and angular spread M_{aR}. Chromatic magnification is independent of a_{K} and y_{P}, but is dependent on the longitudinal position of the limiting aperture. The chromatic magnification matrices show that the red and blue images undergo unequal magnification in the two principal meridians and that the red and blue images are rotated with respect to each other. (A scalar matrix implies equal chromatic magnification in all meridians. Unequal diagonal and nonzero offdiagonal entries imply unequal magnification along principal meridians. Unequal offdiagonal entries show that the images are rotated.)
The apertureindependent chromatic properties δF and δF_{0} are obtained directly from the red and blue transferences T_{E}. The aperturedependent chromatic properties are all interrelated through the red and blue coefficient matrices V_{E} (Equation 1). Chromatic magnification is obtained from the entries of V_{E}. Subtracting the red from the blue coefficient matrices gives us the chromatic difference of coefficient matrices δV_{E}, from which we obtain the chromatic difference in position, inclination, image size and angular spread. The relationships of the entries of δV_{E} with aperture position and object position or size are given by Equations 2 and 3. Through T_{E} and V_{E} for red and blue light we obtain simple relationships among the chromatic properties of the eye.
The introduction of surgically implanted pinholes creates new interest in the effects of aperturedependent chromatic properties. The corneal inlay increases the magnitude of chromatic magnification because of its longitudinal positioning and is sensitive to misalignment. The Xtrafocus 93L is intended for treating irregular astigmatism, so the chromatic effects of such a system in an astigmatic eye need to be known.
The values obtained for each of the aperturedependent chromatic properties are small. However, these results give insight not only into the chromatic properties of the eye and the influence of limiting apertures, but also into the effect of astigmatism.
Acknowledgements
The authors would like to express their appreciation for the support from the National Research Foundation of South Africa. This article was presented as a poster at the conference Visual and Physiological Optics 2016, held in Antwerp, Belgium.
Competing interests
The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this article.
Authors’ contributions
This work is based on research performed by TE towards a higher degree under the guidance of WFH.
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Appendix 1
The results given in Table 2d and Table 3c are based on an arbitrary foursurface astigmatic heterocentric eye. The parameters of this eye are given in Table 1A1. The parameters were introduced previously,^{1} however, different frequencies have been chosen for this example. K1 and K2 are the first and second surfaces of the cornea, while L1 and L2 are the first and second surfaces of the crystalline lens. The tilt is given as horizontal and vertical components in radians. For K1, the right side of the cornea will be tilted away from and the top towards an observer looking at the eye. Villegas, Carretero and Fimia^{2} published equations for the four media of Le Grand’s eye and we apply these here. The media are given as cornea (K), aqueous humour (A), lens (L) and vitreous humour (V). Refractive indices for the four media for frequencies of 430 THz and 750 THz are given in Table 1A1.
TABLE 1A1: Principal radii of curvature and tilts of the four surfaces and separations and refractive indices of the four media of the arbitrary astigmatic heterocentric eye used in the numerical example. 
The transferences for red and blue light are:
and
where B and e are in metres and C is in dioptres.
Apertureindependent chromatic properties
For illustrative purposes, the calculation of chromatic difference in power and refractive compensation is:
and
from which it is clear that this is a myopic astigmatic eye.
Aperturedependent chromatic properties
In order to obtain δy_{R}, δa_{R}, δ(∆y_{R}), δ(∆a_{R}), M_{yR} and M_{aR}, we need the chromatic difference in coefficient matrices. The red and blue coefficient matrices are
and
with units given in metres (m) and dioptres (D).
References
