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Putting the Eye Into the Equation

2 CPD in Australia | TBC in New Zealand | 1 May 2019

By Nicola Peaper      

Combining measurements that are easily available (aberrometry, pachymetry and topography) with new techniques, makes it possible to calculate beyond the vertex sphere into the eye. This enables lenses to be individually designed and accurately produced for patients, resulting in greater clarity of vision.


  1. Understand the way a lens is currently calculated,
  2. Understand that the individual eye structure has an impact upon the way a lens design reaches the retina, and
  3. Appreciate that while higher order aberration (HOA) cannot be fully compensated, measuring and compensating for the patient’s individual HOA will produce better patient comfort than using a physiological model with average HOA and pupil size.

The earliest spectacle lenses were made from quartz, probably in the 13th century, but spectacles were not in common use until the invention of the printing press in 1450. Other historical influences on spectacle lens use, and indeed lens design, include the industrial revolution in the 18th and 19th centuries and, most recently, digital grind technology. This method of lens production enables lenses to be produced with virtually any surface, without the limitations of traditional grind tooling. As a result, optimisations and compensations for elements such as frame parameters, eye rotation due to Listings Law, and near astigmatism may be included in a lens design. This in turn has opened up incredible possibilities for clearer central and peripheral vision.

In order to use the technology to its fullest in any age, we need to go back to ophthalmic optics and the understanding of the eye as a complex lens system with several refracting surfaces and changes of refractive indices.

In 1877, Josef Rodenstock said, “Visual impairments… can be remedied very easily if you understand the system of vision in its entirety”.

The aim of lens design is to produce a lens such that, for any direction of gaze, the corresponding point on the lens will focus at the fovea. In progressive lenses this is limited by Minkwitz astigmatism and a compromise must be found for different positions of gaze.

To this end, two major breakthroughs in lens design occurred in the 19th and early 20th centuries. Firstly, the development of reduced and schematic eyes, allows us to predict the path light takes through the eye, based on a reduced number of refracting surfaces and refractive indexes. Secondly, the introduction of the far point sphere, then the vertex sphere which is used as the last reference surface in spectacle lens design.

The most basic reduced eye takes the complex ocular system of refracting surfaces and refractive indices and breaks it down for calculation purposes to a single spherical surface 60D in power (F’e). This separates the air outside from a medium of refractive index of 1.333 (the same as for water) inside the eye (Figure 1). The radius of curvature of the refracting surface is 5.55mm, the centre of which is represented by the nodal point, N. It is usual to assume that the eye is emmetropic, such that an object at infinity will be focused on the retina which is 22.22mm behind the refracting surface.

Figure 1. Example of a reduced eye model reproduced from Emsley, HH., 1979. Visual Optics. 5th edition.1


f’e = refractive index x 1/F’e

f’e = 1.333 x 1/60 = 22.22mm.1

Objects at infinity will be focused sharply on the retina.

The drawback of using this reduced eye is that it is emmetropic and has a length of 22.22mm. Very few of the eyes that we are required to correct meet these specifications! Eye length can vary by up to 10mm across the average and while this is related to script, a specific level of ametropia will still have axial length variations2 (Figure 2).

Figure 2. Distribution of eye length across the population2


Gullstrand, at the turn of the 20th century, established more elaborate eye models consisting of an increased number of surfaces compared to the basic one refracting surface model. This allowed better understanding of how light passes through the eye. However, the eye model was still considered to be emmetropic, with spherical corneal and lenticular surfaces, and it used averages for the distances between surfaces.2 This will mean that for a given amount of ametropia, optimal imaging may not occur on the retina, causing a difference in image clarity. Accordingly, patient experience may vary.

The second breakthrough was the development of the vertex sphere. Consider Figure 3.

Figure 3. Vertex sphere of a lens producing an image at the far point sphere through the eye’s centre of rotation.


  • The far point of an eye is the point at which the relaxed eye focuses a distant object. In the emmetropic eye it is on the retina, a hyperopic eye behind the retina,  and a myopic eye in front of the retina. If, as the eye rotates, these points are plotted, they form a spherical surface, known as the far point sphere.
  • The vertex sphere is a reference curve with its centre at the centre of rotation of the eye. It doesn’t have power in itself, but is used as a reference from which deviation of curvature can be determined. To do this we use the index of the lens material to calculate the power deviation from that of the existing lens surfaces. From a pure ray tracing perspective (ie disregarding any aberration caused by the eye itself), at a point 25mm out from the OC of a 1.498 index, +4.00 spherical lens, the mean power error is about 0.44D. The oblique astigmatic error is about 0.98D as the sagittal power is +3.95, and the tangential power is +4.94. The powers simultaneously reduce in one plane and increase in the other.
  • The distance between the vertex sphere and the far point sphere, measured through the eye’s centre of rotation, is constant and equal to the back vertex focal length of the lens.3

Using these developments, the last point of lens design is the vertex sphere, producing a lens that focuses on the far point sphere through the centre of rotation of the eye. Lenses calculated as such will optimally correct an eye with the dimensions of the reduced eye used in the calculation.

Lens Design With Wavefront Technology

The process of lens design now uses wavefront technology to design a lens. This follows the path and deformation of a wavefront at any given time. A lens will cause a deformation of the wave front. An aberration free lens will turn a parallel light wave into a convergent spherical wave, which converges at the focal point.

Rodenstock DNEye technology, introduced in 2012, uses individual data in the following way:

The design model of the lens specifies the object distance for each point on the lens. In a progressive lens this includes from distance to near along the corridor. The initial step is to calculate the optimum sphero-cylindrical power for this object distance for each point on the lens as a set value. This not only includes the subjective refraction, both distance and near if available, but also measured aberrations (lower and higher order) and pupil diameter. For near, changes caused by accommodation to both cyl power and axis (near astigmatism) and HOA can be included in this sphero-cylindrical power. Additionally, other compensations, such as the torsional component of the eyes when converging for near vision and the near effective astigmatism due to the modified object distance, can be considered at this stage.

With these results the anterior surface is initially defined and a starting surface for the posterior surface is chosen. Now the passage of the wavefront of the object viewed through the lens is simulated at each point of the lens. This takes into account the propagation of light to the anterior surface of the spectacle lens, refraction at the anterior surface, propagation through the spectacle lens, refraction at the posterior surface and finally the propagation up to the vertex sphere.

The resulting wavefront is compared to the set values for the respective point of the lens. Then the posterior surface is modified on the basis of the comparative results of all points and taking into account physiological criteria. This procedure is repeated until the desired imaging properties are attained. (Figure 4)

Figure 4. Using wavefront technology to calculate a lens.


If we look at the example of a +4.00D lens, which has 1.00D of aberration in the periphery; the wavefront generated in the periphery of the lens, as measured at the vertex sphere, will differ significantly to the intended wavefront which should perfectly focus on the retina (Figure 5). It is this difference that lens design attempts to address.

Figure 5. Current approach with vertex sphere and far-point sphere. Wavefronts of an ideal lens (green) and a lens with a refractive error (pink) are shown. The image in the eye is not considered at all. Refractive error is understood as the wavefront aberration at the vertex sphere, in this example +1.00 D.


Impact of Aberrations

When we consider all of the recent strides forward in lens design and technology it is surprising that we still rely on a decades old model for the final design point of a lens. We can build back vertex distance into a design, but we are using an estimation of eye length that could cause a variance of several mm. We are ignoring aberration caused by the eye’s refracting surfaces that can impact the clarity of the retinal image. The wavefronts relevant for imaging on the retina are actually the ones inside the eye. Direct computation shows that the wavefront generated by the lens (pink in Figure. 6), once refracted through the lens of the eye, in this example, deviates in reality by +2.05 D from the wavefront of the ideal lens. This deviation can differ with eyes of the same ametropia.

Figure 6. View of the actual imaging in the individual eye. The size of the circle of least confusion on the retina is decisive for perception. Accordingly, after the last refracting surface, the deviation of the actual wavefront (pink) from the ideal wavefront (green) is +2.05D.


If it were possible to consider the aberration caused at the corneal and lenticular surfaces in the design of a lens, then the design would take into account not just the propagation of light up to the vertex sphere, but beyond that, through both the cornea and crystalline lens up to the retina. In this way a wavefront would be produced that conforms to the individual eye. Consider the impact this would have on individual lens design. The point of a lens design is to reduce aberrations and to arrange them across the lens surface into areas to reduce visual impairment during specific visual tasks.

If the final point of calculation is the vertex sphere, then the fields of vision imaged onto the retina can only be assumed. So, patients with the same degree of ametropia but with different axial lengths, lens and cornea powers may have completely different visual experiences. If the calculation is performed as far as the retina, then a different lens surface will be calculated for every patient, with each lens surface producing the same image clarity at the retina. In other words, every patient will receive the experience intended by the lens designer (Figure 7).

Figure 7 A. Even in patients with the same degree of ametropia, the image clarity may be adversely affected by individual eye structures.

Figure 7 B. Not all patients will receive the experience intended by the lens designer.


To make this possible it would be necessary to know:

  1. How much of the lower order aberrations (LOAs) and higher order aberrations (HOAs) of the eye are due to the cornea, and how much are due to the crystalline lens, 
  2. The anterior chamber depth, and
  3. The length of the eye.

An aberrometer will measure the total deviated wavefront for the eye. If this is used in conjunction with topography, then the corneal contribution is known, and the lenticular contribution can be calculated.

The anterior depth can be measured by pachymetry.

The eye length would need to be calculated from the known variables.

With Rodenstock’s DNEye PRO, it is now possible to use these measurements and calculations to allow the individual eye to be included in the optimisation of a spectacle lens.

When considering visual acuity, we need to consider both LOAs and HOAs. We are completely familiar with LOAs as sphere, cyl and axis. HOAs, such as spherical aberration, coma and trefoil will distort a wavefront of light.

The visual effect of these is to reduce contrast and cause blur, halos and glare, especially in low light levels.

The general symptoms of a patient experiencing HOAs include:

  • Ghosting, shadowing or monocular diplopia of the Snellen letter chart, which may indicate the presence of coma.
  • Uncertainty of cyl axis which may be indicative of trefoil.
  • Indecisive duochrome and problems night driving, which may indicate spherical aberration.

It is also important to note that the belief that young patients do not exhibit HOAs is incorrect. A study of young emmetropes (average age 21.73 years, with ammetropia of -0.50 to +1.25) showed 46% had 0.336um Root Mean Square (RMS), which accounts for approximately 0.25D. The figures were similar for young myopes (average age 25 with ammetropia of -0.50 to -10.00D).4

The HOAs of the individual eye can be measured with an aberrometer using a sensor, such as the Shack Hartmann sensor, to analyse a wavefront produced by a ray of near infra-red light reflected back by the retina. The reflected light passes through the vitreous and crystalline lens and is then limited by the pupil to pass through the cornea. This gives all aberrations, lower and higher order, in detail.

The perfect emmetropic eye with no HOAs will produce a wavefront that is smooth and parallel (Figure 8). In reality the wavefront will be distorted (Figure 9).

Figure 8. Wavefront of a perfect eye.

Figure 9. Wavefront from an eye with both LOA and HOA.


While the effect of LOAs tends to be unaffected by pupil size, the effect of HOAs increases as pupil size increases. This is due in part to spherical aberration from the lens periphery. Accommodation can also effect the degree of HOAs present. Because of this, it is important to consider the aberrometry of the eye under photopic and mesopic conditions and under the influence of accommodation. This allows the  assessment of the effects of different pupil size and the degree of lenticular accommodation.

The differences between the profile of a perfect emmetropic eye and the measured eye can be expressed as Zernike Polynomials (Figure 10). Put very simply, these are used to describe the shape of complex wavefronts.

Figure 10. A map of ocular aberrations can be expressed as Zernike polynomials.


Second order polynomials represent the LOAs of defocus and astigmatism (sphero-cylindrical script). Aberrations up to the fourth order represent 99.8% of all wavefront aberrations.5

As previously stated HOAs have ‘shapes’ which are difficult to quantify in terms of their magnitude due to their geometry. RMS is a useful method to describe the deviation from the expected norm. Therefore, RMS can be used as a measure of the difference between a plain wave front and the wavefront distorted by aberrations (which can be HOA, LOA or some of both). At each point, the distance between the wavefronts is taken and squared. The squares of all points are added up and the square root of the sum is taken. An example of this can be seen in the table in Figure 11. This table is for an eye in mesopic conditions with a large pupil of 4.3mm. Aberrations are shown to level seven.

Figure 11. Table of LOAs and HOAs with each aberration expressed in micrometers.


Second order (LOAs) can be corrected for with a sphero-cylindrical script. When considering the use of an aberrometer to determine script, the results gained by a subjective refraction still play an important role. While the aberrometer gives accurate results it cannot take into account aspects of visual processing in the visual cortex and binocular information available from a subjective examination.

Looking at the example in Figure 11, we are then presented with HOA to the seventh level. These values can be used to correct an eye for HOA with ablation during corneal surgery, or they could perhaps be included in a contact lens that is fairly fixed in position on the eye. They cannot, however, be fully corrected for in a spectacle lens as the eye continually moves over the lens surface and if the HOAs are fully corrected at one point on the lens the adjacent points are affected. As previously stated, HOAs are included in the calculation of the best sphero cylindrical power for each point across the lens in conjunction with the subjective refraction, to improve the imaging properties of the lens/eye system.

To better understand the degree of effect that HOAs may have, they can be expressed as a spherical equivalent. It should be understood that this does not represent how a prescription is optimised.

To translate to a diopter value, the formula is:      

SE = 4√3 RMS 
                 p ²

SE = spherical equivalent

P = pupil radius (mm).

A patient will have significantly less effect from HOAs with a smaller pupil, so compensation for HOAs at near should differ from that at distance, otherwise vision may be impaired.

One of the most powerful demonstrations of the effect HOAs have on vision is to examine a map of point spread function. This shows how the image of a light spot may be formed (Figure 12).

Figure 12. Point spread function. Left. How an image of a spot of light may be formed. Right. T hen acted upon by HOAs.


Compensating for Aberrations

For over a decade, lens manufacturers have been examining the effects of HOAs on a spectacle lens and various designs have been produced based on average pupil size for a given task. More recently, HOAs of the eye have been taken into account, specifically spherical aberration. This has been done by considering the mean spherical aberration, the pupil size (again based on averages) and building compensations into the lens. The problem with using averages is, as with the reduced eye, not all of the population will receive the optimal correction. Patients with smaller pupils or low amounts of HOAs will receive too much compensation and those with larger pupils or high amounts of HOAs will receive too little. 

Figure 13 demonstrates that pupil size has a large impact upon the effect that spherical aberration has on the best spherical correction. Taking average values across the population, it can be seen that an increase in pupil size from 2mm to 6mm leads to a change of around -0.25D to 0.30D. If two standard deviations either side of the mean are taken into account, then the power change can be between -0.99D and +0.43D.

Figure 13. Influence of spherical aberration on refraction depending on pupil size.


While this is a good indicator of the effect of spherical aberration, it does not completely describe the compensation applied to the lens as additional factors need to be taken into account.

From this table it is clear that using models to help calculate the amount of spherical aberration compensation will give good results for the average eye, however in reality this compensation may be too great or too small depending on the amount of HOA present and the patient’s pupil size. It would be far more preferable to measure individual aberrometry under differing levels of illumination and with accommodation, then to build compensation around those results.

New Technology Enables Lenses Compensated For Individual Eye Indeed, technology now exists to measure the individual HOA under photopic, mesopic and accommodative pupil sizes, and to calculate compensation for each visual point on the lens. With Rodenstock’s DNEye PRO, each point can now be based on the object distance, pupil size and relative accommodative condition of the eye. With differing levels of HOAs and the variance of pupil size across the population, it is apparent that the majority of patients will benefit from this technology and calculation.

The greatest effects that the patient wearing a lens individually compensated for HOAs should notice are:

  • A reduction of haloes around lights when driving at night, and
  • An increase in contrast – this has an obvious impact on clarity of vision, especially in low light levels.

It is an exciting time in lens design and manufacturing. By using measurements that are easily available (aberrometry, pachymetry and topography) and new calculation techniques it is now possible to calculate beyond the vertex sphere into the eye. Our understanding of how the individual structures of the eye effect the distribution of aberrations contribute to finding a better optimised sphero / cylindrical correction. This represents a significant break with a century old method of using a reduced eye model and vertex sphere as tools for lens calculation.

For practitioners, instrumentation is now available to tie the refracting and dispensing functions together. This in turn assists when explaining to the patient the differences in spectacle lenses available. For the patient it allows for better understanding of the correction supplied, along with greater clarity of vision.

Thanks to Grant Hannaford, adjunct senior lecturer at the School of Optometry and Vision Science UNSW, Dr Stephan Trumm, senior expert innovations optics and standardisations and Dr. Wolfgang Becken, senior principle research optics at Rodenstock Munich for their help and advice with this article.


Nicola Peaper BSc (Hons) Ophthalmic Optics, Cert 4 TAE, qualified as an optometrist in the UK in 1985 and practiced in private and corporate practice in the UK for 20 years. In 2001/2 she was employed as ophthalmic advisor to Kensington, Chelsea & Westminster Health Authority.

Since moving to Australia in 2005, she has worked in fitting laboratories advising on procedures and quality. In roles as state and national training manager, she has gained extensive experience in presenting the technology behind, and the prescribing and fitting of, ophthalmic lenses.

Nicola Peaper is currently Professional Services Manager for Rodenstock Australia.


1. Emsley, HH., 1979. Visual Optics. 5th edition. Butterworths. P40-41, 343-345.
2. C. W. Oyster: The human eye: structure and function. Sinauer Associates (imprint of Oxford University Press), 1999
3. Jalie M. 2008. Ophthalmic lenses and dispensing. Elsevier Butterworth Heinemann. P30-31.
4. Jinhua Bao,  Rongrong Le , Jiangxiu Wu , Yeyu Shen , Fan Lu , Ji C. He. J Optom.2009;2:51-8 - Vol. 2 Num.1. Higher order wavefront aberrations for populations of young emmetropes and myopes
5. Castejón-Mochón JF, López-Gil N, Benito A, Artal P. Vision Res. 2002 Jun;42(13):1611-7. Ocular wavefront aberration statistics in a normal young population.
Additional sources:
• Dr Stephan Trumm, Katrin Nicke, Daria Evdokimova and Dr Wolfgang Becken. Goodbye Gullstrand! or Why stop at the vertex sphere.
• Daria Evdokimova, Katrin Nicke, Dr Stephan Trumm. New multifunctional measuring device.
• Dr Wolfgang Becken, Christina Butz, Gregor Esser. Lenses of the future – the DNEye optimisation.
• 100% Success with the DNEye SCcanner. A seminar of the Rodenstock Academy with Prof Dr Stephan Degle

' In 1877, Josef Rodenstock said Visual impairments… can be remedied very easily if you understand the system of vision in its entirety '