# Optical Forums > Ophthalmic Optics >  Free-form lens: oblique astigmatism and quantitative performance

## hkn73

Hi! This is my first post here, and from browsing other posts, I can tell that I don't know much about lens design. I apologize if this is a stupid question.

A spherical lens works best if one looks through the optical center; looking at an angle through the sides of the lens will result in oblique astigmatism. For a spherical lens, the astigmatism is the same whether the eye looks left/right or up/down. As I understand, a free-form lens design can reduce this oblique astigmatism in part of the field of view, for example on the left and right.

However, from what I understand of the optics, I can't see how this could be achieved without making the astigmatism even worse somewhere else in the field of view, at the same angle from the optical center. However, I don't read this from the marketing literature, e.g. Figure 7 in the Zeiss SV whitepaper. Although, Figures 3 and 5 in this whitepaper do actually suggest that the wider view is at the expense of the up/down field of clear vision.

Do I understand this correctly?

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## Darryl Meister

No, you do not need to make astigmatism worse somewhere else in the field of view in order to correct oblique astigmatism at some point on the lens. In a single-vision lens, it is possible to virtually eliminate oblique astigmatism for most wearers with a complex enough lens design, although a small amount of mean power error or "curvature of the field" will still remain, which is unavoidable in any lens.

In fact, even with a spherical base curve, you can completely eliminate oblique astigmatism by simply selecting a "best form" base (front) curve for prescription sphere powers, if the lens has no tilt and no prescription cylinder power. (At least for prescriptions below +7.00 D.) Oblique astigmatism in a lens with no tilt and no prescription cylinder power is produced in a rotationally symmetrical manner about the center of the lens, as you noted. Standard "best form" and aspheric designs are also rotationally symmetrical, so they can eliminate this oblique astigmatism.

Lenses with prescription cylinder power (but no tilt) will produce oblique astigmatism that varies between the principal meridians of the lens, so that the meridian with sphere power will produce different oblique astigmatism than the meridian with cylinder power. Oblique astigmatism has "plane" symmetry in this case. An atoric design is required to eliminate the oblique astigmatism for lenses with prescription cylinder power but no tilt.

Tilting a lens, on the other hand, will introduce an asymmetrical form of oblique astigmatism, which is neither rotationally symmetrical nor symmetrical about the principal meridian planes. Conventional "best form," aspheric, and atoric lens designs will not eliminate this oblique astigmatism completely, although you can reduce it somewhat by compensating the prescription for lens tilt in addition to using the correct design. Because free-form surfaces do not need to be rotationally symmetrical, however, these surfaces can be made to eliminate the asymmetrical oblique astigmatism of a tilted lens.

For instance, Figure 7, in the white paper, shows a contour plot of the actual, ray-traced oblique astigmatism of Zeiss Individual Single Vision. This is the oblique astigmatism as perceived by the actual wearer with the lens in its position of wear. The oblique astigmatism over the entire lens is less than 0.25 DC. Yet, the conventional single vision lens produces over 1.00 DC of oblique astigmatism through certain regions of the lens.

Best regards,
Darryl

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## hkn73

Thank you for your reply, Darryl. I was - incorrectly - assuming that oblique astigmatism (OA) was a fundamental property of a singlet rotation-symmetric lens (singlet as in not a camera objective). I assumed that OA was eliminated by superimposing some kind of saddle shape on top of the spherical lens surface, i.e., z=a(y^2-x^2) with z representing the additional lens thickness.




> In fact, even with a spherical base curve, you can completely eliminate  oblique astigmatism by simply selecting a "best form" base (front) curve  for prescription sphere powers, if the lens has no tilt and no  prescription cylinder power.


OK, it seems that I misinterpreted figure 2 ("blurred vision through periphery of lens"); if I understand you correctly, this is only the case because the front curve is manufactured in steps of about 4D. Now that I read the whitepaper again, I see that it is actually written in the first paragraph.

Is this also what is indicated by Fig. 4 in the whitepaper? Since you are the author of that whitepaper, could you assign numbers to the color scale? The green-red colors just indicate "low power error" to "high power error", but I would like to know what figure of merit is actually plotted there. I'd assume it's something like "maximum spherical plus astigmatic power error within XYZ degrees field of view, for fitting with zero tilt angle".




> For instance, Figure 7, in the white paper, shows a contour plot of the  actual, ray-traced oblique astigmatism of Zeiss Individual Single  Vision. This is the oblique astigmatism as perceived by the actual  wearer with the lens in its position of wear.


Apparently I misinterpreted the figure. The scale on the side, is that in degrees (+/-25° from the center)? And is there no spherical power error as well?

I also wonder how big the advantage of reducing power/astigmatism aberrations as long as there still is chromatic aberration. For example polymer n=1.67 lens material has Abbe number V=31, which means that the refractive index varies between roughly 1.66 and 1.68 over the visible-light spectrum. I'm not sure how to translate this into a quantity that can be compared to astigmatic power, but it's clear to me that chromatic aberration leads to significant additional "astigmatic" blurring in the periphery of the lens.

Anyway, can I summarize the advantages of a free-form lens as follows?

base curve is exactly matched to lens power, reduces oblique astigmatism (not really a result of free-form manufacturing, but rather of on-demand digital manufacturing);elimination of aberrations in lenses with significant cylindrical powers (>1.25 DC if I guess from Fig 4);elimination of aberrations in frames with significant tilt (21° sum of panto+tilt looks pretty bad with up to 1 DC OA).

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## Darryl Meister

> I assumed that OA was eliminated by superimposing some kind of saddle shape on top of the spherical lens surface, i.e., z=a(y^2-x^2) with z representing the additional lens thickness.


You could conceivably create an _atoric_ lens design by applying a saddle shape (such as a hyperbolic paraboloid) to one of the lens surfaces, or some other surface height function that varies from meridian to meridian, in order to eliminate oblique astigmatism in a non-tilted lens.




> OK, it seems that I misinterpreted figure 2 ("blurred vision through periphery of lens")


Figure 2 is a conceptual drawing of how optical aberrations may be introduced by a spectacle lens. In particular, the oblique astigmatism of a "flat" lens has been shown. If you are using the ideal aspheric or best form lens design, the middle illustration would no longer show blurred peripheral vision, at least for spherical prescription powers. Of course, if a best form lens had been used in the illustration, the lens shown would have also been considerably steeper and thicker.




> Is this also what is indicated by Fig. 4 in the whitepaper? Since you are the author of that whitepaper, could you assign numbers to the color scale?


This is, again, a conceptual illustration meant to convey the fact that lens designers can really only fully correct optical aberrations for a single prescription power with a conventional base curve. As for assigning numbers or showing a figure of merit, it really doesn't matter: Regardless of your figure of merit, you can only minimize that figure for one spherical prescription with a conventional base curve. ZEISS lens designers typically minimize astigmatism, while also keeping an eye on excess mean power errors though.




> The scale on the side, is that in degrees (+/-25° from the center)? And is there no spherical power error as well?


This is a contour plot of ray-traced astigmatism over a 50 mm lens (up to +/-25 mm from the center). You can deduce the spherical power error at the center of each lens by taking the spherical equivalent of the prescription indicated at the center.




> I also wonder how big the advantage of reducing power/astigmatism aberrations as long as there still is chromatic aberration.


Reducing power and astigmatic errors becomes even _more_ important with chromatic aberration, since these aberrations will exacerbate the effects of chromatic aberration, contributing further to the degradation in vision quality away from the center of the lens. Chromatic and monochromatic aberrations don't simply "overlap" each other; instead, the separate images formed by chromatic aberration are then individually aberrated by monochromatic aberrations.

It is probably also worth mentioning that, in addition to greater chromatic aberration, high-index materials also produce more residual monochromatic aberrations, due to a more curved image (Petzval's) surface. This also makes proper lens design that much more crucial.




> I'm not sure how to translate this into a quantity that can be compared to astigmatic power


The focal error associated with axial (or longitudinal) chromatic aberration is simply equal to the ratio of the power of the lens to the Abbe value of the lens material (F / v). Lateral (or transverse) chromatic aberration is generally more bothersome to the wearer, however, since this results in the characteristic "color fringing" around objects seen through the periphery of the lens.




> base curve is exactly matched to lens power, reduces oblique astigmatism (not really a result of free-form manufacturing, but rather of on-demand digital manufacturing);


With free-form and even aspheric lenses, you are no longer constrained by base curve selection. You can choose a base curve that produces a flatter, thinner lens or a base curve that more closely matches the frame shape (for wrap lenses, for example), and then apply asphericity, atoricity, or a more complex design to minimize the optical aberrations that would normally result from the departure of the base curve from the "best form" front curve.

Otherwise, your summary is correct.

Best regards,
Darryl

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## hkn73

Thank you again for your elaborate answer.




> You could conceivably create an atoric lens design by applying a saddle shape [...] in order to eliminate oblique astigmatism in a non-tilted lens.


Huh? But that would eliminate oblique astigmatism (OA) only along certain meridians and make it worse on others; the lack of rotation symmetry prohibits that you get proper correction of OA everywhere. That was my original point when I started this topic.




> [Figure 4] is, again, a conceptual illustration meant to convey the fact that lens designers can really only fully correct optical aberrations for a single prescription power with a conventional base curve. As for assigning numbers or showing a figure of merit, it really doesn't matter:


Well, I understand that the definition of the FoM does not matter to illustrate the concept of "fully correct optical aberrations", but such data would be very helpful for advising a patient as to whether in their particular case (frame and prescription) the added value of a free-form lens is worth the premium price. Right now, we have to live with the nonquantitative data in the whitepaper or phrases such as

Up to 50% wider fields of view with superior visual acuity for true edge-to-edge clarity. (Zeiss)recommended for high ametropes - myopes with sphere measurements >3.00D, hyperopes with sphere measurements >2.00D and astigmats with cylinder >1.00D ... visual fields up to 30% wider (BBGR).




> You can deduce the spherical power error at the center of each lens by taking the spherical equivalent of the prescription indicated at the center.


I meant to ask: is there no spherical power error at the edge of the lens in a spherical design?




> Chromatic and monochromatic aberrations don't simply "overlap" each other; instead, the separate images formed by chromatic aberration are then individually


Yes, I understand, but both monochromatic OA and chromatic aberration (CA) lead to a smearing of the image. If the smearing due to CA is much bigger than due to OA, then this will negate the benefit of a better lens design.




> Lateral (or transverse) chromatic aberration is generally more bothersome to the wearer, however, since this results in the characteristic "color fringing" around objects seen through the periphery of the lens.


Agreed. I sat down and wrote down some equations (based on simple flat-lens equations), to get an idea of the magnitude of the lateral chromatic aberration.

A power error Ep will lead to a smearing of the image on the retina over an angle (in radians) α = d Ep, where d is the pupil diameter. Alternatively, one can write E = α/d as the equivalent power error for a mechanism that leads to an angular smearing angle α.

The chromatic power variation is, as you say, Ec = P / V for a lens with nominal power P and Abbe number V. For example, a -6 D lens with Abbe number V=32 would have Ec = 0.19 D. As you already hinted at, this longitudinal power error is rather small. However, a ray of white light hitting the lens at a distance r from the optical center will be refracted over a range of angles due to wavelength-dependence of the chromatic aberration. The angular spread is then α_c = r Ec. We can translate this angular spread into an effective chromatic error 



For example, at r=25 mm, d=5 mm, V=32, and P=6 D, we would have Ece = 0.94 D.

I would say that once the OA is reduced to less than about 0.5 D over +/-25 mm field of view, the added benefit of further reduction of OA becomes rather limited due to the color fringing being the limiting factor in image sharpness, at least for this particular example. Unfortunately, for lack of quantitative data on the (monochromatic) aberrations of various lens offerings, such evaluations cannot be made.

One could argue that the effective OA power errors should be divided by two, since the eye will likely accommodate such that the focus error is the same (but with opposite sign) in the radial and azimuthal angles. I mean: if the OA at some point in the field of view is +0.5 D x 180°, it is in practice more like a simultaneous +0.25 D x 180° and -0.25 D x 90°. That would make the effect of lateral CA even more dominant over OA errors, since such a compensation mechanism does not exist for lateral CA.

I apologize that I am probably using a notation that is probably uncommon for professional ophthalmic lens designers, but I hope that I got my point across.

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## Darryl Meister

> Huh? But that would eliminate oblique astigmatism (OA) only along certain meridians and make it worse on others; the lack of rotation symmetry prohibits that you get proper correction of OA everywhere. That was my original point when I started this topic


As I indicated in my earlier post, you do not need to make oblique astigmatism worse in other meridians, if you use a sophisticated enough lens design. Also, I do not understand your use of "rotation symmetry" here. If you are designing an atoric lens, by definition it should not have rotational symmetry. Atoric lenses are not rotationally symmetrical, and generally reduce or eliminate oblique astigmatism through both principal meridians.




> but such data would be very helpful for advising a patient as to whether in their particular case (frame and prescription) the added value of a free-form lens is worth the premium price. Right now, we have to live with the nonquantitative data in the whitepaper or phrases such as


Unfortunately, the vast number of possible prescription and fitting combinations make it impractical provide extensive details, particularly when it comes to lenses that must be individually calculated. Further, most eyecare professionals would find this overwhelming. And even then, some consensus must be reached regarding the details to supply and test methods to use. Nevertheless, I am confident that Carl Zeiss Vision makes more qualititative and quantitative data available for free-form lenses than any other lens manufacturer.

But, more importantly, what you are suggesting is against how we position customized products (at least at CZV). We do not position lenses like Zeiss Individual as "the perfect lens for the wearer with an Rx of +4.50 DS -3.00 DC x 023." Instead, we position it as the best possible lens for everyone, delivering optimal optics for any prescription, regardless of whether wearers with a Plano Rx and a low add power may notice less improvement compared to more "difficult" prescriptions. We do not want lenses like Zeiss Individual prescribed only to the handful of wearers you may see with a 3.00 DC cylinder power.

That said, I could probably write a small program to ray-trace a number of possible prescription combinations in order to provide you with some quantitative numbers for different base curves, assuming that I have time later this week. But keep in mind that, since you are asking about the optical errors of "standard" lens, anyone can calculate the optical aberrations of a theoretical single vision spectacle lens with some arbitrary base curve. And your local laboratory is probably the single biggest variable in determining the specifics of the lens design in this case, including the typical base curve that they would pull for a given Rx power.




> I meant to ask: is there no spherical power error at the edge of the lens in a spherical design


Yes, there are generally spherical power errors at the edge of just about any lens design, although these errors will be minimized with a sophisticated enough surface design. But, again, it is impossible to eliminate all optical aberrations completely due to curvature of the field.




> [I would say that once the OA is reduced to less than about 0.5 D over +/-25 mm field of view, the added benefit of further reduction of OA becomes rather limited due to the color fringing being the limiting factor in image sharpness, at least for this particular example. Unfortunately, for lack of quantitative data on the (monochromatic) aberrations of various lens offerings, such evaluations cannot be made


Generally, chromatic aberration will influence vision less than monochromatic aberrations. First, the errors are generally smaller. Second, the sensitivity of the eye to different wavelengths drops off toward the blue and red ends of the spectrum. In fact, the eye suffers from nearly 1.0 D of chromatic aberration that we are generally unaware of. "Color fringing" is unique in that it doesn't just blur vision, it results in an obvious visual artifact.

I'd have to give your "effective chromatic error" some thought before commenting on your proposed result.




> One could argue that the effective OA power errors should be divided by two, since the eye will likely accommodate such that the focus error is the same (but with opposite sign) in the radial and azimuthal angle


Yes, the cylinder power error is left unchanged, although the blur circle on the retina will be half the size of a spherical error of equal value, if accommodation can move the circle of least confusion to the retina, which is only possible with minus mean power errors, and if the accommodative response is fast enough. In lens design, it is not uncommon to use an aberration criterion or metric that takes this into account, such as RMS power error.

Best regards,
Darryl

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## MakeOptics

> I also wonder how big the advantage of reducing power/astigmatism  aberrations as long as there still is chromatic aberration. For example  polymer n=1.67 lens material has Abbe number V=31, which means that the  refractive index varies between roughly 1.66 and 1.68 over the  visible-light spectrum. I'm not sure how to translate this into a  quantity that can be compared to astigmatic power, but it's clear to me  that chromatic aberration leads to significant additional "astigmatic"  blurring in the periphery of the lens.


You are right, the monochromatic aberration can and do have an additional chromatic component when we consider them across the light spectrum.  To quantify that you would have to change the index in your various equations and ray traces to match the particular spectrum you would be looking for.  This would become tedious so lenses are designed around the reference wavelength.  Also improving chromatic aberration occurs with proper material choice, which is in the dispensers domain of lens design.  I would consider proper material selection to be the dispensers side of the design process.

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## Darryl Meister

For simple monochromatic defocus combined with axial chromatic aberration, the focal power should simply add to the dispersive power of each wavelength. In the absence of a monochromatic power error, the chromatic interval should straddle the retina (or the plane conjugate to it at the far-point of the eye, anyway), leaving the blue and red ends of the spectrum very nearly equally defocused.

Any monochromatic power error will then shift the entire chromatic interval either forward or backward, causing one color to become more defocused. This is exactly what happens during the duochrome test, for instance, in which the chromatic aberration of the eye is used for subjective spherical refinement of the refraction.

Since the reference wavelength used to determine the vertex power (_F_V) and mean power error (_E_M) lies in the middle of the chromatic interval, we can estimate the dioptric focal error of light at the blue and red ends of the interval using:



where _v_ is the Abbe value of the lens material and _E_C is the dioptric focal error for either blue light (+) or red light (-).

So, for example, a +6.00 D lens with an Abbe value of 30 that produces a mean power error of +0.50 D will have a blue focal error of +0.60 D and a red focal error of +0.40 D. Consequently, the dispersive power of 0.20 D is roughly maintained, although blue light now has a greater focal error than the monochromatic estimate of +0.50 D would at first reveal, whereas red light would have a smaller error.

For oblique astigmatism, this could be applied to the individual tangential and sagittal power errors.

Best regards,
Darryl

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## hkn73

> Also improving chromatic aberration occurs with proper material choice, which is in the dispensers domain of lens design.


I'm a bit unfamiliar with the terminology in the US, but assuming that you're referring to the customer-servicing optician by 'dispenser', then I don't really understand your point. The optician does not participate in the lens design, but selects one (material and possibly free-form options) from a list, with only a rough idea of the trade-offs.

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## hkn73

> But, more importantly, what you are suggesting is against how we position customized products (at least at CZV). [...] we position it as the best possible lens for everyone, delivering optimal optics for any prescription, regardless of whether wearers with a Plano Rx and a low add power may notice less improvement compared to more "difficult" prescriptions. We do not want lenses like Zeiss Individual prescribed only to the handful of wearers you may see with a 3.00 DC cylinder power.


From the commercial point of view of a lens manufacturer I can understand this attitude. However, the dispensing optician must balance two interests: earning money (expensive lenses are good for that) and having satisfied customers who don't feel tricked into spending 800 on a new frame and lenses with hardly any improvement over their previous pair.




> Second, the sensitivity of the eye to different wavelengths drops off toward the blue and red ends of the spectrum. In fact, the eye suffers from nearly 1.0 D of chromatic aberration that we are generally unaware of.


Agreed. On the other hand, the Abbe number is a measure for dispersion in the range 486 to 656 nm, and 486 nm is not particularly far into the edge of the visible spectrum. (see Luminosity curve on Wikipedia) 

From your other post:




> the chromatic interval should straddle the retina (or the plane conjugate to it at the far-point of the eye, anyway), leaving the blue and red ends of the spectrum very nearly equally defocused.[...]


OK, but this is all for axial chromatic aberration, i.e., when the position of gaze is at the optical center of the lens. My equations were an attempt to quantify color fringing in the case that the position of gaze is at the edge of the lens. The idea is that if the circle (or rather, line) of confusion due to color fringing is the same as the line of confusion due to an astigmatic power error, then they are equivalent in terms of detrimental effect to visual acuity.

We can then debate whether various correction factors should be inserted or not; with an astigmatic error, the "line of confusion" can turn into a smaller circle of confusion by accommodation, which gives you a factor 2. For color fringing, the line of confusion cannot be reduced by accommodation, but one could argue that the blue and red cone cells in the retina do not contribute much to image sharpness (both due to focus errors of the eye lens and due to a low density of blue cone cells on the retina). Based on the cone cell response curves (Wikipedia), I'd say that the focus in the wavelength range 500-600 nm dominates for the perception of visual acuity, i.e., a factor 0.59 of the Abbe-based chromatic focus error.

(To the moderator: since we've drifted away from my original question, could the topic title be changed into _Free-form lens: oblique astigmatism and quantitative performance_ ?)

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## Darryl Meister

> The optician does not participate in the lens design, but selects one (material and possibly free-form options) from a list, with only a rough idea of the trade-offs


Yes, "dispenser" refers to an ophthalmic dispenser or dispensing optician in the US. For conventional (non-aspheric) lenses, the optician or optometrist really designs the lens anytime he or she specifies the base curve. Of course, for aspheric, atoric, and free-form lenses, there is additional optical manipulation that the optician has little if any direct control over.

Best regards,
Darryl

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## Darryl Meister

> having satisfied customers who don't feel tricked into spending 800 on a new frame and lenses with hardly any improvement over their previous pair


I agree that if you are charging an additional 800 euros for a customized lens, it should be a consideration for your patient, although that still doesn't make quantifying these differences any less problematic or less subjective on the part of consumers, who often have their own notions of value and return on investment in addition to their own subjective sensitivity to blur.




> Agreed. On the other hand, the Abbe number is a measure for dispersion in the range 486 to 656 nm, and 486 nm is not particularly far into the edge of the visible spectrum


True, although I'm not sure that this changes the point at all. The eye tolerates a lot of axial chromatic aberration. You can change the wavelengths at which the Abbe value is measured, but this will not change the actual chromatic aberration at all, just how we measure it.




> confusion due to color fringing is the same as the line of confusion due to an astigmatic power error, then they are equivalent in terms of detrimental effect to visual acuity


I am not entirely certain that I followed the point you were trying to make here, but I do agree that wavelengths near the peak luminous sensitivity of the eye, around 555 nm, will have a greater impact on vision quality and the perception of blur. Ultimately, each wavelength of light will create its own interval of Sturm and circle of least confusion in the presence of chromatic aberration and oblique astigmatism, resulting in a poorer image than either aberration alone, since the astigmatic bundle is essentially stretched out even farther at either end of the color spectrum.




> (To the moderator: since we've drifted away from my original question, could the topic title be changed into _Free-form lens: oblique astigmatism and quantitative performance_ ?


I'll change it.

Best regards,
Darryl

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## MakeOptics

> I'm a bit unfamiliar with the terminology in the US, but assuming that you're referring to the customer-servicing optician by 'dispenser', then I don't really understand your point. The optician does not participate in the lens design, but selects one (material and possibly free-form options) from a list, with only a rough idea of the trade-offs.


You are correct in the sense that the dispenser (customer servicing optician) doesn't traditionally design the lens, however material selection is an integral part of the design and some will even specify base curves and thicknesses, better controlling the optics of the lens an example would be isekonic lenses.  All dispensers are not dim witted, just most.

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## hkn73

> I am not entirely certain that I followed the point you were trying to make here,


I'll try again, this time with a picture to illustrate my point.

Figure A shows an eye of diameter D, pupil diameter d, and a minus lens of power P. If the lens+eye have a monochromatic power error E, then the blur disc (line) on the retina has diameter (length)



Figure B shows an eye with an off-axis gaze, intersecting the lens at a distance r from the optical center. Parallel light rays will be refracted to different angles, depending on their wavelength. Given an Abbe number V, the chromatic blur line has a length



which can be derived from simple flat/thin lens formulas. Now I define a new quantity: the effective color-fringing power error E_ce, such that 

,

i.e. the a monochromatic power error of the same magnitude would result in the same size of the blur disc or line on the retina. Rewriting and substitution results in



which conveniently eliminates the eye diameter D from the equation. For a typical V=32, d=5 mm, P=-6 D, and r=25 mm, we get Ece = 0.94 D. This is why I'm saying that reducing power and astigmatic errors in the periphery of the lens may have a limited added value. Under bright conditions, with the pupil reduced to d=2 mm, it gets even worse (Ece=2.3 D).

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## Darryl Meister

For the initial calculation of the blur ellipse on the retina, you would want to use the reduced thickness of the eye to account for the refraction index, or _n_/_D_. You would also have to assume a relatively short vertex distance to ignore the difference between the ocular refraction and the refraction at the spectacle plane. Of course, none of this matters, if you eventually factor out the eye or work in object space at the far-point plane.

At the far-point plane of the eye, the size of the blur ellipse _e_ in millimeters is given by:



The lateral chromatic aberration _e_C at the far-point plane in millimeters is given by:



However, the blurred image produced by astigmatic defocus in the image plane will generally be a circle or an ellipse, and occasionally a line, whereas the image produced by lateral chromatic aberration is a line or streak, so even for the same "size" error, the astigmatic defocus will cover more area of blur in many cases. And the following points need to be considered:

1. The size of the lateral chromatic aberration "streak" is not equivalent to dioptric defocus, unless one meridian of the astigmatic bundle has zero error, resulting in a line focus in the image plane. And, even then, the decreased sensitivity of the eye for the wavelengths associated with the "streak" will still reduce some of the impact of this aberration.

2. These aberrations interact with each other, they do not simply "overlap," so the presence of one aberration will make the other aberration worse. In fact, you would need to add the size of the power error through the tangential meridian to the size of the error due to lateral chromatic aberration.

3. Lastly, you are calculating your lateral chromatic aberration effect at an extreme angle of view (25 mm from the optical center, approaching a 45 degree angle of view). You would be considering much less chromatic aberration at the typical angles of view utilized in single vision lens design between 30 to 35 degrees.

Jalie provides a pretty comprehensive discussion of this particular effect, resulting from the combination of lateral chromatic aberration with monochromatic aberration, in _Principles of Ophthalmic Lenses_.

Best regards,
Darryl

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## hkn73

> 


I think in your equation, P is the power of the eye, not the power of the spectacle lens, right? Essentially, P=1/D, where D is the effective diameter of the eye.




> The lateral chromatic aberration _e_C at the far-point plane in millimeters is given by:


I'm not sure what the symbol v (Greek nu) stands for. I'll assume that it is the Abbe number. The power of the spectacle lens is not in this equation, which cannot be correct, since a zero-power spectacle lens will not provide lateral chromatic aberration. Could it be that you tried to write

and accidentally canceled out the two different P values?




> even for the same "size" error, the astigmatic defocus will cover more area of blur in many cases.


Agreed, that's why I mentioned optional factors 2 in my earlier post.




> The size of the lateral chromatic aberration "streak" is not equivalent to dioptric defocus, unless one meridian of the astigmatic bundle has zero error, resulting in a line focus in the image plane.


I'm not sure that I understand your point.




> the decreased sensitivity of the eye for the wavelengths associated with the "streak" will still reduce some of the impact of this aberration.


As I stated earlier, you can multiply e_C by 0.59, based on the full width at half maximum (FWHM) of the response curve of the green cones in the retina. But you would have to do the same thing with the line focus in the case of astigmatism, because it is more intense in the center than at the ends (of the line).




> These aberrations interact with each other, they do not simply "overlap," so the presence of one aberration will make the other aberration worse. In fact, you would need to add the size of the power error through the tangential meridian to the size of the error due to lateral chromatic aberration.


I'm not sure what kind of interaction you are referring to. I'd say that the total blur kernel is the convolution of the power-error blur kernel and the chromatic/fringing blur kernel. (I mean kernel in the mathematical sense)

As a rule of thumb, the sizes of these kernels can be added quadratically, i.e.,

This should be done separately for the sagittal and tangential axes, if I use the correct terminology, and by defining their width as the standard deviation. My point is: if e<e_C/2, then the total error is dominated by the e_C contribution; reducing _e_ further down to zero will only reduce the total error by 10%.




> Lastly, you are calculating your lateral chromatic aberration effect at an extreme angle of view (25 mm from the optical center, approaching a 45 degree angle of view).


OK, then we evaluate it at r=16 mm (33 deg). 

So, using all of the above, if there is an astigmatic error E_a, and the eye accommodates such that the astigmatism results in a blur disc, we have a blur-disc standard deviation (for each axis)

where we have a factor 1/2 to go from blur streak to blur disc and a factor 1/4 to convert diameter into standard deviation. Read "effective diameter of the eye" for D. The lateral chromatic streak has a standard deviation

where the factor 0.59 is to restrict the relevant spectral range and the factor 1/2 is to turn FWHM into standard deviation. Following the same reasoning as before, the effective "chromatic fringing error" is
.
Taking the daytime conditions P=-6 D, V=32, d=3 mm, r=16 mm, I get E_ce=2.4 D. That is pretty bad and I suppose that this will completely overwhelm other astigmatic power aberations.

I'm a bit suspicious about my own calculation because this is not what I see personally (I have a -9 D lens for one eye).




> Jalie provides a pretty comprehensive discussion of this particular effect, resulting from the combination of lateral chromatic aberration with monochromatic aberration, in _Principles of Ophthalmic Lenses_.


Thank you for mentioning it; I ordered this book today, but shipping will take up to two weeks.

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## Darryl Meister

> I'm not sure what the symbol v (Greek nu) stands for


Yes, the Abbe number is traditionally represented with the Greek nu symbol, which looks like a 'v'. I assumed that this is why you were using a 'v' for Abbe value.




> I think in your equation, P is the power of the eye, not the power of the spectacle lens, right?


No, it is the power of the spectacle lens. I have used your variables where applicable. The power of the eye is not 1/_D_. If you assume a "reduced eye" model, it is _n_/_D_, where _n_ is approximately equal to the refractive index of water at 1.333.




> But you would have to do the same thing with the line focus in the case of astigmatism, because it is more intense in the center than at the ends (of the line)


Not for the image of a point with a finite pupil, unless you are also considering spherical aberration or coma or the effects of diffraction, which none of these equations do. But, yes, as you consider finite objects, like letters, it will appear that way.




> The power of the spectacle lens is not in this equation, which cannot be correct, since a zero-power spectacle lens will not provide lateral chromatic aberration


Remember that this is the size of the lateral chromatic aberration streak at the _far-point plane of the eye_, which is a distance directly related to the power of the lens _P_ by 1/_P_ from the spectacle plane. As the power of the lens approaches zero, the far-point approaches infinity and therefore size of the streak as perceived by the eye also collapses to zero (that is, the angle subtended by the streak at the far-point approaches zero).

Another way to think of this is to realize that _P_ would appear in both the numerator (for the prism calculation) and the denominator (for the far-point distance), thus canceling out.




> I'm not sure what kind of interaction you are referring to. I'd say that the total blur kernel is the convolution of


Yes, image convolution is involved. I imagine that you could apply a Fourier transform of the astigmatism point spread function to each color image. It's honestly not something that I've ever given much thought to, but I'm sure that clever ways to work this out have already been developed.




> As a rule of thumb, the sizes of these kernels can be added quadratically, i.e.,


I haven't heard of the term "kernel" used in this kind of context, so I cannot comment on it. I do note that your equation is equivalent to the modulus or vector length of an orthonormal set of vectors, although I don't necessarily know that this would apply to _e_ and _e_C.




> This should be done separately for the sagittal and tangential axes, if I use the correct terminology, and by defining their width as the standard deviation.


Yes, this is reasonable.




> I'm not sure that I understand your point


If I have understood _your_ earlier point, you were attempting to equate dioptric defocus to lateral chromatic aberration by determining a defocus error with a diameter equal to the length of the streak produced by lateral chromatic aberration. I do not necessarily agree that this is appropriate for the reasons that I described earlier, such as the comparison of a line image to a more diffused blur circle image and the wavelength sensitivity issue.

If I have simply misinterpreted your point, however, please feel free to disregard this observation.




> My point is: if e<e_C/2, then the total error is dominated by the e_C contribution; reducing _e_ further down to zero will only reduce the total error by 10%


I certainly agree that you can reduce monochromatic aberration to the point that lateral chromatic aberration is much more severe. This statement is actually consistent with the point that I made previously regarding the importance of minimizing monochromatic aberrations in order to reduce the overall impact of chromatic aberration.

Honestly, I'm not sure whether we are actually debating the same point now or not. It has been an interesting conversation topic though.

Best regards,
Darryl

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## hkn73

I'll reply to this particular point; for the rest I have to do some thinking to make sure that I understand your points. 




> the line focus in the case of astigmatism, because it is more intense in the center than at the ends (of the line)
> 			
> 		
> 
> Not for the image of a point with a finite pupil, unless you are also considering spherical aberration or coma or the effects of diffraction, which none of these equations do.


Please see the attached sketch, for the simple case of a pure cylindrical lens with a round aperture.

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## Darryl Meister

Ah, I see. I wasn't really referring to the luminance profile, but I understand now what you meant.

I'd have to mull over how this applies to an astigmatic focus with both a tangential line focus and a sagittal line focus though, since the light intersecting the center of the first line focus actually diverges to join light from the edge of the aperture that forms the ends of the second line focus, but I imagine that it is probably the same either way.

Best regards,
Darryl

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## hkn73

> Yes, the Abbe number is traditionally represented with the Greek nu symbol, which looks like a 'v'. I assumed that this is why you were using a 'v' for Abbe value.


Hmm, Wikipedia and E. Hecht, _Optics_ (2nd ed.) both use V (capital vee) as the Abbe number.




> The power of the eye is not 1/D. If you assume a "reduced eye" model, it is n/D, where n is approximately equal to the refractive index of water at 1.333.


I'm not convinced that this is relevant for the size of the image on the retina. However, both your equations (size of blurring in the conjugate plane of the spectacle lens) and my equations (size of blurring on the retina) result in the same end result in terms of the equivalent power error of the color fringing,

where _a_ is some dimensionless scaling factor. So at least we can agree on this?




> Yes, image convolution is involved. I imagine that you could apply a Fourier transform of the astigmatism point spread function to each color image. It's honestly not something that I've ever given much thought to, but I'm sure that clever ways to work this out have already been developed. ... I haven't heard of the term "kernel" used in this kind of context, so I cannot comment on it. I do note that your equation is equivalent to the modulus or vector length of an orthonormal set of vectors, although I don't necessarily know that this would apply to _e_ and _e_C.


With kernel, I mean actually "convolution kernel", which is the same thing as what you call the point spread function. The latter term is probably better to use in the present context. 

If you have point spread functions f1(x) and f2(x), with standard deviations σ1 and σ2, then the convolution 

has a standard deviation

See for example Standard deviation on Wikipedia; note that in the case of a convolution, the covariance is zero.

This can also be applied to two-dimensional point spread functions g(x,y) if they are separable as g(x,y) = g1(x) g2(y); then the standard deviations in x and y follow the identities as above. I think that this applies to astigmatic point spread functions (ellipses) and chromatic streaks (lines), as long as you identify x and y with the tangential and sagittal planes. It will not hold for aberrations such as coma. For spherical aberrations I'm not sure. Fortunately,  we're discussing neither coma nor spherical aberrations.




> If I have understood _your_ earlier point, you were attempting to equate dioptric defocus to lateral chromatic aberration by determining a defocus error with a diameter equal to the length of the streak produced by lateral chromatic aberration. I do not necessarily agree that this is appropriate for the reasons that I described earlier, such as the comparison of a line image to a more diffused blur circle image and the wavelength sensitivity issue.


Mostly correct, but I also tried to take into account the wavelength sensitivity and I actually compare standard deviations of point spread functions rather than their edge-to-edge dimension. For the wavelength sensitivity, I assume the interval 500--600 nm, rather than 486--655 nm as with the Abbe number definition.

On second thought, I think I should not have converted the astigmatic power error into a spherical power error. I considered an astigmatic error of 1 DC to be equivalent to a spherical power error of 0.5 D. If we wish to compare apples with apples, then we should compare the chromatic fringing error to an astigmatic error. Then we get the astigmatic equivalent chromatic error 





> I certainly agree that you can reduce monochromatic aberration to the point that lateral chromatic aberration is much more severe. This statement is actually consistent with the point that I made previously regarding the importance of minimizing monochromatic aberrations in order to reduce the overall impact of chromatic aberration.


Yes. But using the same parameters as before: under daylight conditions (d=3 mm, r=16 mm, V=32, P=-6 D), I get E_aec=1.2 D due to chromatic aberrations in the periphery of the spectacle lens.  So, my position is as follows: with a -6D lens, oblique astigmatism that is smaller than 0.6 D is small enough to be negligible.  I assume that  customized free-form lenses can stay well below that target for most prescriptions. However, it will only be a visible advantage if a standard spherical or toroidal lens for the same prescription generates (much) more than 0.6 D in oblique astigmatism. 

Public quantatitive data is lacking, but I suspect that for moderate cylinders (< 1.5 DC) and small-tilt frames, there is no added value of a customized free-form lens. 

I agree that this reasoning is probably too difficult for the majority of the consumers, and probably a sizeable fraction of the opticians as well (at least those in Netherlands who did not get certified as optometrist). But from an academic or engineering point of view I am interested in these things.

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## Darryl Meister

> Hmm, Wikipedia and E. Hecht, _Optics_ (2nd ed.) both use V (capital vee) as the Abbe number.


Not that it really matters, since you are welcome to use whatever symbol you want for it, but the nu symbol was originally used. Even today, Zeiss and Schott, companies Abbe helped start, still use it for lens material specifications.

The use of the letter "v" probably resulted from the fact that it was more convenient, since it does not require a change in typeface and looks a lot like nu, anyway. But you can still see the nu value used in early optical texts, such as Southall's "Mirrors, Prisms, and Lenses" (1918), which Hecht actually uses as a reference.




> I'm not convinced that this is relevant for the size of the image on the retina. However, both your equations (size of blurring in the conjugate plane of the spectacle lens) and my equations (size of blurring on the retina) result in the same end result in terms of the equivalent power error of the color fringing


The refractive index of the eye is relevant both to the _power_ (or _ocular refraction_) of the eye and to the image size, just as an object in water will appear to have a different size compared to the same object in air due to the "reduced thickness" of water. For instance, in order to determine the retinal image size, you would typically trace a ray either through the entrance/exit pupils or the nodal points, and the locations of both are influenced by the assumed refractive index of the eye.

Your initial calculation of the size of the blur patch _e_ on the _retina_ would not be correct, if indeed you really want to determine this value at some point. The correct expression (see Jalie, when your book arrives) would be:



where _P_' is the ocular refraction, which is also equal to the power of the lens _P_ corrected for the vertex distance, and _K_ is the power of the eye.

However, this does not influence your final calculation, because you (rightly or wrongly) factor out the size (_D_) of your eye later, when you equate your _e_ to _e_C. I never used it in the first place, so it doesn't appear in mine. Our equations do, ultimately, arrive at the same result, perhaps even by luck.




> See for example Standard deviation on Wikipedia; note that in the case of a convolution, the covariance is zero.


That sounds reasonable.




> So, my position is as follows: with a -6D lens, oblique astigmatism that is smaller than 0.6 D is small enough to be negligible.


Well, I believe that most eyecare professionals would feel that 0.6 D of unwanted cylinder power isn't "negligible." But I believe the point that you are trying to make (whether I concur or not, aside) is that at 16 mm from the optical center of a -6.00 D lens with a low Abbe value, there is so much lateral chromatic aberration that it overwhelms monochromatic aberrations below 0.6 D.

Of course, I would argue that the real issue here, knowing that 0.6 D of unwanted astigmatism is well above the depth of focus of the eye, is the lateral chromatic aberration generated by the lens material. So you should really be arguing against the use of high-index lens materials, not optical optimization for monochromatic aberrations.

Further, at least one scientific study has been to done assess the effect of lateral chromatic aberration on visual acuity (see "Effect of chromatic dispersion of a lens on visual acuity" by Meslin and Obrecht) and there is already a well-established relationship between power error and visual acuity. Consequently, it would probably be more clinically meaningful to associate these two acuity functions, rather than to attempt to derive a similar relationship based purely upon mathematical assumptions that may still be in question.




> [Public quantatitive data is lacking, but I suspect that for moderate cylinders (< 1.5 DC) and small-tilt frames, there is no added value of a customized free-form lens.


We will have to agree to disagree regarding this observation, particularly since even low- to moderate-cylinder powers can cause the central viewing zones of a progressive lens to become distorted in shape and reduced in size, as the oblique astigmatism interacts with the surface astigmatism of the progressive lens design.

Best regards,
Darryl

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## Darryl Meister

By the way, if you aren't already using ray tracing software, you might want to download my *Spectacle Optics Program* to calculate some of these aberrations precisely.

Best regards,
Darryl

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## Darryl Meister

Given an RMS monochromatic power error _P_, the decimal visual acuity _A_ is represented approximately by the following function:



And, based upon a linear approximation of Meslin and Obrecht's data, given a lateral chromatic aberration error _C_, the decimal visual acuity _A_ is represented approximately by the following function:



where _F_ is the power of the lens, _d_ is the decentration in centimeters, and _v_ is the Abbe value.

Consequently, you can then equate and rearrange these two equations to solve for either aberration as a function of the other in terms of their effect upon visual acuity, since:



Best regards,
Darryl

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