Journal of Refractive Surgery

Letters to the Editor

# Matrix Optics

Richard Sheard, MA, FRCOphth

#### Abstract

I congratulate Dr Haigis for his article on matrix-optical representation of intraocular lens (IOL) power formulae,1 which appeared in the February 2009 issue of the Journal of Refractive Surgery. Analysis of a complex optical system such as the eye is rather cumbersome by conventional vergence methods, and the general vergence formula used in all IOL power formulae is rather complicated and prone to errors in implementation. Matrix representation of optical systems is a well-recognized and powerful technique that is well-suited to computer modeling but, thus far, has been applied to a rather limited range of optical problems in ophthalmology.

In his model, Haigis represents optical rays as a vector of the form

This model is rather counter-intuitive because light rays travel in a straight line in a given medium irrespective of its refractive index, yet the ray vector and translation matrix are dependent upon the refractive index. Similarly, light is refracted at an interface between two media of differing refractive indices, yet in Haigis’ model the refraction matrix

Ultimately, the model works as expected, but to determine the IOL power, it is necessary to calculate the inverse of two matrices. This is potentially problematic for two reasons. First, not all matrices have an inverse (so-called singular matrices), and second, the general calculation for inversion of a matrix may be computationally difficult.

It is possible to propose an alternative model that has none of these potential drawbacks. Using the same symbol notation as above, the ray vector is defined as

Note that the ray vector and translation matrix are refractive index independent. The refraction matrix contains two refractive indices, n0 being the refractive index of the medium transmitting the incident ray and n1 that of the medium transmitting the emergent ray.

Adopting the same notation from Haigis’ figure, the system matrix is

This has the useful property that the position of the back focal plane of the optical system, t, relative to its right-most element is given by

For our model of the eye, the back focal plane must coincide with the right-most element, and therefore t = 0. From this it follows that −a = 0, thus allowing any unknown quantity to be directly calculated from the system matrix without the need for matrix inversion.

Haigis suggests that the transformations required to determine the refractive power of the IOL are easy. I disagree, particularly for those unfamiliar or out of practice with matrix arithmetic, as many ophthalmologists are likely to be. I therefore append solutions for IOL power, spectacle power, and effective lens position using a simpler approach ().

In his discussion, Haigis cites a paper by Basu,2 which claimed to have found an error in the programming of the Hoffer Q formula used by the IOLMaster (Carl Zeiss Meditec, Jena, Germany). Subsequent correspondence on the matter raised concerns about Basu’s programming of the formula,3–5 but the exact cause of the discrepancy has never been published. In fact, in addition to the incorrect inclusion of limits on the anterior chamber depth3 and his confused programming methods,4 Basu fails to add 0.05 to the effective lens position for the IOL power calculation (see, for example, Eq in Haigis’ current paper1) and erroneously “corrects” the keratometric power to a refractive index of 1.336 rather than 1.3375 (as in Table 1 of Haigis’ article1). The IOLMaster’s Hoffer Q implementation is accurate5 and Basu’s own errors account exactly for the discrepancy he reported.

Richard Sheard, MA, FRCOphth
Sheffield, United Kingdom

To calculate the IOL power, fL, we define the system matrix as

Using the back focal plane property of…

### To the Editor:

I congratulate Dr Haigis for his article on matrix-optical representation of intraocular lens (IOL) power formulae,1 which appeared in the February 2009 issue of the Journal of Refractive Surgery. Analysis of a complex optical system such as the eye is rather cumbersome by conventional vergence methods, and the general vergence formula used in all IOL power formulae is rather complicated and prone to errors in implementation. Matrix representation of optical systems is a well-recognized and powerful technique that is well-suited to computer modeling but, thus far, has been applied to a rather limited range of optical problems in ophthalmology.

In his model, Haigis represents optical rays as a vector of the form

where n is the refractive index of the medium through which the ray propagates, θ is the angle of the ray in relation to the optical axis of the system, and h is the height of the ray above the optical axis at a particular plane. A translation matrix models the propagation of a ray between two optical planes and takes the form where d is the distance between the planes and n is the refractive index of the medium.

This model is rather counter-intuitive because light rays travel in a straight line in a given medium irrespective of its refractive index, yet the ray vector and translation matrix are dependent upon the refractive index. Similarly, light is refracted at an interface between two media of differing refractive indices, yet in Haigis’ model the refraction matrix

(where f is the power of the interface) is independent of refractive index.

Ultimately, the model works as expected, but to determine the IOL power, it is necessary to calculate the inverse of two matrices. This is potentially problematic for two reasons. First, not all matrices have an inverse (so-called singular matrices), and second, the general calculation for inversion of a matrix may be computationally difficult.

It is possible to propose an alternative model that has none of these potential drawbacks. Using the same symbol notation as above, the ray vector is defined as

and the translation and refraction matrices as

Note that the ray vector and translation matrix are refractive index independent. The refraction matrix contains two refractive indices, n0 being the refractive index of the medium transmitting the incident ray and n1 that of the medium transmitting the emergent ray.

Adopting the same notation from Haigis’ figure, the system matrix is

where S is a 2×2 matrix of the form

This has the useful property that the position of the back focal plane of the optical system, t, relative to its right-most element is given by

For our model of the eye, the back focal plane must coincide with the right-most element, and therefore t = 0. From this it follows that −a = 0, thus allowing any unknown quantity to be directly calculated from the system matrix without the need for matrix inversion.

Haigis suggests that the transformations required to determine the refractive power of the IOL are easy. I disagree, particularly for those unfamiliar or out of practice with matrix arithmetic, as many ophthalmologists are likely to be. I therefore append solutions for IOL power, spectacle power, and effective lens position using a simpler approach ().

In his discussion, Haigis cites a paper by Basu,2 which claimed to have found an error in the programming of the Hoffer Q formula used by the IOLMaster (Carl Zeiss Meditec, Jena, Germany). Subsequent correspondence on the matter raised concerns about Basu’s programming of the formula,3–5 but the exact cause of the discrepancy has never been published. In fact, in addition to the incorrect inclusion of limits on the anterior chamber depth3 and his confused programming methods,4 Basu fails to add 0.05 to the effective lens position for the IOL power calculation (see, for example, Eq in Haigis’ current paper1) and erroneously “corrects” the keratometric power to a refractive index of 1.336 rather than 1.3375 (as in Table 1 of Haigis’ article1). The IOLMaster’s Hoffer Q implementation is accurate5 and Basu’s own errors account exactly for the discrepancy he reported.

Richard Sheard, MA, FRCOphth
Sheffield, United Kingdom

### References

1. Haigis W. Matrix-optical representation of currently used intraocular lens power formulas. J Refract Surg. 2009;25:229–234.
2. Basu S. Comparison of IOL power calculations by the IOLMaster vs theoretical calculations. Eye. 2006;20:90–97. doi:10.1038/sj.eye.6701800 [CrossRef]
3. Hoffer KJ. Errors in self-programming the Hoffer Q formula. Eye. 2007;21:429. doi:10.1038/sj.eye.6702559 [CrossRef]
4. Haigis W. IOLMaster vs theoretical calculation in eye. Eye. 2007;21:430–431. doi:10.1038/sj.eye.6702562 [CrossRef]
5. Dreher C. Reply to Basu. Eye. 2007;21:1451. doi:10.1038/sj.eye.6702971 [CrossRef]
Appendix
Calculation of IOL Power

To calculate the IOL power, fL, we define the system matrix as

Using the back focal plane property of the system matrix it can be shown that

where tVij and xij represent the elements from the i-th row and j-th column of the matrices TV and X.

Calculation of Expected Spectacle Power

To calculate the expected spectacle power, fS, we define the system matrix as

and it follows from the back focal plane property that where yij represents the element from the i-th row and j-th column of matrix Y.

Calculation of Effective Lens Position

Calculation of the effective lens position, d, is more involved, because d appears in two of the translation matrices in this optical system, TAC and TV. The system matrix is defined as

It can be shown that if

where L is the axial length of the eye and rLij and zij represent the elements from the i-th row and j-th column of the matrices RL and Z. The quadratic formula may then be used to determine d.

I thank Mr Sheard for his interest in my article1 and his comments and suggestions.

In his letter, he addresses three topics:

1. He agrees that the chosen matrix representation in my article works, but criticizes that for this purpose inverse matrices must be calculated, which may be computationally difficult or mathematically problematic.

2. As an alternative, he proposes a slightly different matrix representation, which he claims to be free of the above drawbacks.

3. Sheard refers to a paper by Basu,2 which contends that the implementation of the Hoffer-Q formula in the IOLMaster (Carl Zeiss Meditec, Jena, Germany) is erroneous. He comes to the conclusion that Basu’s programming was erroneous and that the IOLMaster implementation is accurate.

Regarding topic 1, the purpose of my article was to apply the formalism of matrix optics to currently used intraocular lens (IOL) power formulas to illustrate their similarities and their differences. In a second article,3 this technique was also applied to other problems within the context of IOL calculation, eg, to the derivation of the expected spectacle power as in the of Sheard’s letter. In each article, matrix optics was applied as a tool—a detailed treatise of the method itself was not intended.

Therefore, in my article, inverse matrices and/or their components, which are intrinsic to matrix algebra, were not converted to yield more simple expressions but were intentionally left in place to illustrate their origin within the calculation scheme.

Sheard is correct in pointing out that inverse matrices may be mathematically problematic. In the current context, however, this is not the case: from the formalism to calculate an inverse matrix as given in the Appendix of my article, it follows directly that for our matrices

because the determinant det (A) = a11a22 – a12a21 = 1 as a direct consequence of the definition of translation and refraction matrices. With this result Eq.(3) for the IOL power in my article1 is easily transformed into with no more need to calculate inverse matrices. This equation for fL is—mutatis mutandis—identical to the one in Sheard’s .

Regarding topic 2, Sheard proposes to represent a ray by

ie, with interchanged components and the use of the angle Ø instead of the product (optical direction cosine). Accordingly, the respective matrices in his approach have interchanged components and the refractive index n appears inside the matrices. The ray definition I applied is commonly used in ophthalmic optics and physics,4–6 whereas Sheard’s can be found in technical optics.7,8

Both approaches are completely equivalent. This holds especially for the system matrix.

For Sheard’s system matrix

the vertex distance tS to the back focal plane is given by whereas for my system matrix (Eq. in the article1) the vertex distance tH to the back focal plane is given by

Evidently, all calculations possible in Sheard’s representation can likewise be carried out in the notation of my article. For example, the IOL power given above or by Sheard can be derived from tS as well as from tH using vergence considerations. Vergences, however, were intentionally not used in my article, which deliberately concentrates on matrix optical methods.

On the other hand, if matrix methods are consequently applied in Sheard’s representation, then a similar equation for the IOL power with inverse matrix components as in my article is obtained. Again, the explicit matrix inversion can easily be performed and an equivalent expression for the IOL power is obtained as above.

This shows once more that—as always in physics—there is more than one path to arrive at specific equations or conclusions. In whatever representation, the matrix optical approach is an elegant way to tackle problems in clinical IOL calculation.

Regarding topic 3, I am pleased to learn that Sheard comes to the same conclusion as already expressed in my paper: that Basu’s program code was erroneous and that the implementation of the Hoffer-Q formula in the IOLMaster is correct.

Wolfgang Haigis, MS, PhD
Wuerzburg, Germany

### References

1. Haigis W. Matrix-optical representation of currently used intra-ocular lens power formulas. J Refract Surg. 2009;25:229–234.
2. Basu S. Comparison of IOL power calculations by the IOLMaster vs theoretical calculations. Eye. 2006;20:90–97. doi:10.1038/sj.eye.6701800 [CrossRef]
3. Haigis W. IOL calculation using paraxial matrix optics. Ophthalmic Physiol Opt. 2009;29:458–463. doi:10.1111/j.1475-1313.2008.00629.x [CrossRef]
4. Welford WT. Optical calculations and optical instruments, an introduction. In: Flügge S, ed. Encyclopedia of Physics. Vol XXIX Optical Instruments. Heidelberg, Germany: Springer; 1967:1–42.
5. Rosenblum WM, Christensen JL. Optical matrix method: optometric applications. Am J Optom Physiol Opt. 1974;51:961–968.
6. Harris WF. Paraxial ray tracing through noncoaxial astigmatic optical systems, and a 5x5 augmented system matrix. Optom Vis Sci. 1994;71:282–285. doi:10.1097/00006324-199404000-00009 [CrossRef]
7. Meschede D. Optics, Light and Lasers. Weinheim, Germany: Wiley-VCH; 2003.
8. Dagg IR. Matrix Optics. 2nd ed. Waterloo, Ontario: Otter Press; 1980.

10.3928/1081597X-20090917-01