On the relation between power and astigmatismnear an umbilic line

Jacob Rubinstein

Department of Mathematics, Technion, Haifa 32000, Israel (koby@math.technion.ac.il)

Received January 11, 2011; revised February 23, 2011; accepted February 23, 2011;posted February 23, 2011 (Doc. ID 140991); published April 5, 2011

The relation between the variation of power and astigmatism in a progressive power optical element is considered.The classical Minkwitz identity is revisited. Then, it is shown how to extend this identity, which applies tothe geometry of surfaces, to optical parameters that are determined by the geometry of reflected or refractedwavefronts. © 2011 Optical Society of America

OCIS codes: 080.4225, 080.2720, 330.7328.

1. INTRODUCTIONProgressive addition lenses (PALs) provide a good solution topeople suffering from presbyopia. The optical power of a PALvaries considerably as the eye scans the lens in different view-ing directions. One of the main drawbacks of PALs is that thevarying power across the lens gives rise to unwanted astigma-tism. Notice that, in optometry, the notion of astigmatism, orcylinder, means the difference between the two principalpowers of the lens (cylinder). Therefore, great effort in thedesign process is invested in reducing this effect. In spiteof many years of experience, and innovative design methods,it is obvious that some level of unwanted astigmatism must bepresent. The immediate explanation for this unwanted astig-matism is a classical theorem in differential geometry, sayingthat only the umbilic surface, i.e., a surface where the princi-pal curvatures equal each other at all points, is a sphere. If weuse for simplicity surface power and surface astigmatism asapproximations of the true optical characteristics of the lens,then the requirement that the power vary across the lens im-plies that the surface of a PAL must deviate substantially froma spherical surface, and, hence, some level of surface astigma-tism must be expected.

A more quantitative result was discovered by Minkwitz [1].To understand his result, we recall that, in a simple PAL de-sign, the power varies from far vision to near vision along acentral vertical curve that we denote here the vertex line.Minkwitz showed that if the vertex line is umbilic, then thelateral derivative of the surface astigmatism at each pointalong the vertex line is twice the vertical derivative of the sur-face power at that point. Therefore, the demand of varyingpower implies a growth of the astigmatism away from the ver-tex line. The goal of this paper is to examine this statement insome detail. In Section 2 we formulate the problem preciselyand clarify some confusion that seems to be present in theliterature regarding Minkwitz’s identity. Minkwitz consideredexplicitly surface power and astigmatism. However, it is wellknown that surface curvatures only approximate the curva-tures of the refracted wavefront. Therefore, we shall presentoptical analogs of the Minkwitz identity. First, we derive, inSection 3, an associated formula for reflective surfaces, and

then, in Section 4, we shall present an optical Minkwitzidentity for a refractive surface.

2. THE SURFACE MINKWITZ IDENTITYTo express the Minkwitz identity in precise mathematicalterms, we should introduce proper coordinate systems. Thus,consider an optical surface z ¼ uðx; yÞ defined over the ðx; yÞplane. The triplet ðx; y; zÞ will be our universal coordinateframe. Other coordinate frames will be defined whenever con-venient. One source of confusion has to do with the meaningof the notion vertex line. As defined above, it is a line on thesurface u. However, for practical reasons, it is convenient totalk of the projection of this line on the ðx; yÞ plane. We shalldenote this projection the central line of the lens. This distinc-tion is important in practice. While Minkwitz derived an iden-tity with respect to the arc length distance along the vertexline, measuring devices, for instance, provide many propertiesof the lens on the two-dimensional projection of it. In suchcases, the optical power variation is seen along the centralline.

The mean curvature Hs and Gaussian curvature Ks of asurface are expressed in terms of the surface graph u by

Hs ¼12∇ ·

�∇u

ð1þ j∇uj2Þ1=2�; Ks ¼

uxxuyy − u2xy

ð1þ j∇uj2Þ2 : ð1Þ

It is more convenient, though, to work in appropriate localcoordinates. Assume that the vertex line is umbilic, and con-sider a point q on this curve. We define a local coordinateframe ðX; Y; ZÞ centered at q such that Z points along the nor-mal to the surface, Y points along the tangent to the vertexline, and X completes Y , Z to a right-hand orthogonal frame.The astigmatism Cs of a surface is defined as the differencebetween the principal curvatures. Minkwitz’s identity canbe expressed in terms of this frame as

∂Csð0; YÞ∂X

¼ 2∂Hsð0; YÞ

∂Y: ð2Þ

However, this formula is not always convenient to use. Asnoted above, the data are often given on the two-dimensional

734 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 Jacob Rubinstein

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universal plane ðx; yÞ, In addition, to use the ðX; Y; ZÞ frame,we must know, in addition to the optics of the lens, also theactual surface u. Therefore, we write Eq. (2) in the universalcoordinates:

∂Csð0; yÞ∂x

¼ 2

ð1þ h02ðyÞÞ1=2∂Hsð0; yÞ

∂y; ð3Þ

where the vertex line is defined by uð0; yÞ ¼ hðyÞ.An interesting evaluation of the Minkwitz identity [Eq. (2)]

was recently published by Sheedy et al. [2]. The authors askedwhether commercial PALs satisfy this identity. To examinethis question, they measured a few commercial PALs and com-pared their behavior at the corridor near the central line withthe prediction of Eq. (2). They found that only one of thebrands they examined was close to the expected result. Itseems that Eq. (3) would be more appropriate to use insteadof Eq. (2) since the measuring device only provides the opticaldata in the universal frame. In particular, the device they useddoes not give the actual surface topography, and thus Eq. (2)is surely not applicable then.

In addition to the experimental part of their paper, Sheedyet al. argue that an Alvarez surface satisfies the Minkwitz iden-tity. The Alvarez surface [3] has the form

uðx; yÞ ¼ Aðy3=3þ yx2Þ: ð4Þ

Here A is a coefficient that determines the relative effect of thecubic terms. The original idea of Alvarez was that a verticalshift of the surface by δ induces to first order a change ofδAðx2 þ y2Þ, and thus, a power change of δA. However, itwas realized by a number of authors (see, e.g., [4]) that evena stationary Alvarez surface is a useful model for a PAL, sincethe surface power is given paraxially by Ps ¼ 2Ay. Sheedyet al. perform a reverse calculation. They assume that the ver-tex line is umbilic and that the surface power change is linearin y. Under these two assumptions they derived Eq. (4). How-ever, their derivation only holds in the local frame ðX; Y; ZÞand not in the universal frame ðx; y; zÞ. In fact, a direct com-putation shows that the vertex line x ¼ 0 in the Alvarez sur-face is not umbilic at all. This can be seen by applying theclassical formula for the mean curvatureHs and Gaussian cur-vature Ks, given in Eq. (1) to the Alvarez surface in Eq. (4).The difference between the principal curvatures C ¼ jκ1 − κ2jis given by

C ¼ 2ðH2 − KÞ1=2: ð5Þ

We use Eq. (5) to compute the surface astigmatism Cs of theAlvarez surface, and obtain, along the vertex line,

Cs ¼ 2A3y5=ð1þ A2y4Þ3=2: ð6Þ

For example, if we consider a 10mm portion of the centralline, and use A ¼ 10−3, the deviation from umbilicity is as highas 0:2D. Since the vertex line x ¼ 0 in the Alvarez surface isnot surface umbilic, the Minkwitz identity is not relevant for it.

3. AN IDENTITY FOR A REFLECTINGSURFACEThe Minkwitz identity of Eq. (2) [or (3)] refers to surface quan-tities. However, surface power and surface astigmatism arethe same as their optical counterparts only in the paraxial ap-proximation. It is well known (see, e.g., [5]) that the opticalpower and optical astigmatism of a PAL are substantially dif-ferent than the associated surface properties. Therefore, wenow examine whether the Minkwitz identity applies also tothe optical properties of the optical element. Clearly, a con-sideration of optical properties depends not only on the sur-face u, but also on the incident light. Therefore, to form acanonical problem, we consider the reflection or refractionof a parallel wavefront incident from the positive z directionon a given surface uðx; yÞ. We shall say that a point q on thesurface u is optically umbilic if the wavefront surface afterreflection or refraction by this surface is umbilic at q.

We first examine the case of a reflecting surface. We shallestablish here two results. First, we prove the surprising factthat, while the vertex line of an Alvarez surface is not surfaceumblic, it is optically umbilic if the same surface serves as amirror. We proceed to derive an optical version of theMinkwitz identity.

Expressions for the mean curvature Hm and Gaussian cur-vature Km of a reflected wavefront in terms of the reflectorsurface uðx; yÞwere derived in [6]. The authors obtained there

Hm ¼ Δu

1þ j∇uj2 ; Km ¼ 4Ks: ð7Þ

Note that we work here in the universal frame, and that theseformula are exact (and not a paraxial approximation). We ap-ply the expressions in Eqs. (7) to compute the geometry of thewavefront reflected by an Alvarez mirror on the vertex linex ¼ 0:

Hm ¼ 4Ay=ð1þ A2y4Þ; Km ¼ 16A2y2=ð1þ A2y4Þ2: ð8Þ

Equations (8), together with Eq. (5), show that the vertex lineof the Alvarez mirror is indeed optically umbilic. Does theMinkwitz identity hold there? The answer is negative. Againa direct computation using Eqs. (7) and (4) gives for the as-tigmatism Cm near the vertex line and for Hm on the vertexline

Cm ¼ 8Ax=ð1þ A2y4Þ; Hm ¼ 4Ay=ð1þ A2y4Þ: ð9Þ

Clearly, the expressions for Cs and Hs do not satisfy Eq. (3).However, it turns out that one can derive an optical analog

of the Minkwitz identity for a reflector. To see this, let the ver-tex line of the reflector by given by uð0; yÞ ¼ hðyÞ for someprofile h. We assume that the vertex line is optically umbilicand that the surface u is symmetric about the line x ¼ 0.Recalling Eq. (5) and using Eqs. (7), we see that the conditionCmð0; yÞ ¼ 0 implies that, on the vertex line,

∂2u

∂x2¼ ∂2u

∂y2;

∂2u∂x∂y

¼ 0: ð10Þ

Equations (10), together with the symmetry assumption∂u∂x ð0; yÞ ¼ 0 and the profile h of the vertex line, imply that,near x ¼ 0, the reflector surface u has the form

Jacob Rubinstein Vol. 28, No. 5 / May 2011 / J. Opt. Soc. Am. A 735

uðx; yÞ ¼ hðyÞ þ 12h00ðyÞx2 þ Oðx4Þ: ð11Þ

Substituting this formula for u into Eqs. (7) gives to leadingorder

Cm ¼ 4h000ðyÞx1þ ðh0ðyÞÞ2 ; Hmð0; yÞ ¼

2h00ðyÞ1þ ðh0ðyÞÞ2 : ð12Þ

We therefore obtain the optical Minkwitz identity for areflector:

∂

∂xðð1þ ðh0ðyÞÞ2ÞCmÞ ¼ 2

∂

∂yðð1þ ðh0ðyÞÞ2ÞHmÞ: ð13Þ

It is easy to verify that the Alvarez reflector indeed satisfiesthe identity in Eq. (13).

We pointed out earlier that the only umbilic surface is asphere. Similarly, Eqs. (10) imply that the only optically um-bilic reflector (for an object on the optical axis) is a parabo-loid. These two statements are closely related. To see this,consider the wavefront some distance after reflection. Denotethe wavefront surface by Sw. On each point on Sw, constructthe normal, which defines the ray direction, and move along ituntil the point qwhere it intersects the mirror surface. Now, atq, the wavefront is umbilic. Moreover, from the law of propa-gation of principal curvatures, it is clear the they remain thesame along the ray all the way to Sw. However, as this argu-ment works for all points on the wavefront Sw, it follows thatSw is an umbilic surface, and thus it is spherical. This impliesthat the wavefront converges to a focal point. It is well knownthat paraboloids are the only reflectors that focus a parallelwavefront onto a single point. By exactly the same argument,an optically umbilic refractive surface must be a Cartesianoval.

4. AN IDENTITY FOR A REFRACTINGSURFACEThe case of a refracting surface is a little more complicatedthan that of a reflecting surface. Our starting point is again[6]. The authors derive there a formula for the mean curvatureof a refracted parallel wavefront by a surface u. To write downthis formula, we need some notation. Let the refraction in-dices before and after refraction be n1 and n2, respectively.Define μ ¼ n1=n2, and β ¼ 1 − μ2. Next we define a new coor-dinate frame. We use the universal z direction, which, in thiscase, is the direction of the normal to the incident wavefront.Let ν be the normal vector to the surface u at some point q onit. We choose the η coordinate to lie in the refraction planedefined by the z direction and the normal ν, normal to z inthis plane. Finally we define ξ to complete z and η to aright-hand orthogonal system. Note that, by symmetry, theframe ðξ; η; zÞ identifies at the vertex line with the universalframe ðx; y; zÞ. Working in the local ðξ; η; zÞ frame, we expressthe mean curvature of the refracted wavefront Hr as [6]

Hr ¼12ð1þ βj∇uj2Þ1=2 − μ

1þ j∇uj2�

uηη1þ βj∇uj2 þ uξξ

�: ð14Þ

Using the same analysis as in [6], one can also derive a formulafor the Gaussian curvature of the refracted wavefront:

Kr ¼ðð1þ βj∇uj2Þ1=2 − μÞ2

1þ βj∇uj2uξξuηη − u2

ξηð1þ j∇uj2Þ2 : ð15Þ

We proceed just as Section 3 to compute the local shape of thesurface uðξ; ηÞ near the vertex line. Symmetry implies that, toleading order in ξ, it has the form uðξ; ηÞ ¼ hðηÞ þ gðηÞξ2,where h is the vertex line profile and g is a function that isdetermined by the condition that the vertex line is opticallyumbilic. Substituting the formula for u into the expressionsabove for Hr and Kr and setting H2

r ¼ Kr implies, after a littlealgebra,

uðξ; ηÞ ¼ hðηÞ þ 12

h00

1þ βh02 ξ2: ð16Þ

We substitute this local form of u into formula (14) toobtain Hr at the central line

Hr ¼ð1þ βh02Þ1=2 − μ

1þ h02h00

1þ βh02 : ð17Þ

Similarly, we substitute the local shape function in Eq. (16)into Eqs. (14) and (15) for the mean curvature Hr and Gaus-sian curvature Kr . This is used to compute the astigmatismCr ¼ 2ðH2

r − KrÞ1=2. By expanding the result to first orderin ξ, we obtain

Cr ¼ 2ð1þ βh02Þ1=2 − μ

1þ h021

ð1þ βh02Þ1=2�

h00

1þ βh02�0ξ: ð18Þ

We thus obtain at the central line the following identity:

∂

∂x

�1þ h02

ð1þ βh02Þ1=2 − μð1þ βh02Þ1=2Cr

�

¼ 2∂

∂y

�1þ h02

ð1þ βh02Þ1=2 − μHr

�: ð19Þ

Equation (19) is the Minkwitz identity for a refractingsurface. Note that, if we set n2 ¼ −n1, which correspondsto reflection by a mirror, then μ ¼ −1, and β ¼ 0. Therefore,Eq. (19) reduces to Eq. (13).

5. CONCLUSIONThe Minkwitz identity that relates the variation of power andastigmatism in a PAL was examined. In particular, we derivedtwo optical versions of this identity, one for a reflector andone for a refracting surface. In both cases we found that,for any point on the central umbilic line of the lens, the lateralchange of the astigmatism is roughly twice the vertical varia-tion of the power at that point. The precise statement includesappropriate geometrical factors.

ACKNOWLEDGMENTSI thank Sergio Barbero for bringing [2,4] to my attention. Thiswork is supported in part by the Israel Science Foundation(ISF).

REFERENCES1. G. Minkwitz, “Uber den Flachenastigmatismus Bei Gewissen

Symmetrischen Aspharen,” Opt. Acta 10, 223–227 (1963).

736 J. Opt. Soc. Am. A / Vol. 28, No. 5 / May 2011 Jacob Rubinstein

2. J. E. Sheedy, C. Campbell, E. King-Smith, and J. R. Hayes,“Progressive powered lenses: the Minkwitz theorem,” Opt.Vis. Sci. 82, 916–924 (2005).

3. L. W. Alvarez, “Two-element variable-power spherical lens,”U.S.patent 3,305,294 (21 February 1967).

4. R. Blendowske, E. A. Villegas, and P. Artal, “An analyticalmodel describing aberrations in the progression corridor of

progressive addition lenses,” Opt. Vis. Sci. 83, 666–671(2006).

5. B. Bourdoncle, J. O. Chauveau, and J. L. Mercier, “Traps in dis-playing optical performances of a progressive addition lens,”Appl. Opt. 31, 3586–3593 (1992).

6. J. Rubinstein and G. Wolansky, “A class of elliptic equationsrelated to optical design,” Math. Res. Lett. 9, 537–548 (2002).

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