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Presbyopia Correction, Differential Geometry, and Free Boundary PDEs

Sergio Barbero
María del Mar González

Communicated by Notices Associate Editor Daniela De Silva

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We survey some mathematical tools involved in the design of multifocal surfaces which are used in presbyopia correction. After giving a brief background in geometrical optics, we consider the Willmore functional in differential geometry, summarizing some classical results, and then explaining its significance in this particular optical application.

1. A Basic Introduction to Geometrical Optics

Modeling precisely how light propagates in a physical medium has been, and still is, one of the great challenges of mathematical physics. Fortunately, in many cases of practical interest, light propagation can just be modeled by means of energy transported along rays; this is the crux of the so-called geometrical optics theory. Beyond geometrical optics, one has the wave optics model, a more sophisticated and accurate theory of light propagation which we will not consider here. For the interested reader, a classical text in optics is BW65.

In general, rays are space curves in the Euclidean space, which obey a stationary action principle; namely, the trajectories are stationary solutions of path length integrals weighted by the refractive index of the propagating medium (this refractive index is related to the microscopic electromagnetic properties of the material of which the medium is made). Note that rays are straight lines when the refractive index is constant.

Image formation

Within the geometrical optics framework, given a bundle of light rays generated by a punctual source, there exists a normal surface to all the rays containing a fixed point, even if the bundle of rays has suffered multiple refractions or reflections (Malus–Dupin theorem). This surface is called the wavefront. Therefore, light propagation can be modeled by propagating either rays or wavefronts. Rays emanating from a point source, within a homogeneous refractive index medium, form a spherical wavefront which can be transformed after refraction or reflection by an optical element.

However, in practice, real objects are not point objects. We owe Ibn al-Haytham (c. 965–c. 1041), latinized as Alhacen, the idea that an extended object can be represented by the union of multiple points and thus, from every point of the surface of an object, light is emitted in all directions in the form of rays. These rays can, for instance after crossing a lens, converge onto another point, which is called focus. Following Alhacen, we say that a set of focal points is the image of an object if rays emerging from each point of the latter converge onto points of the former. Notice that the focus is the only geometrical light concept we actually see or measure; rays and wavefronts are merely convenient mathematical tools.

Then, an imaging system is a set of lenses, mirrors, or other types of optical elements that transform rays, and hence wavefronts, onto converging rays/wavefronts that focus on a light detector—such as a camera sensor or the retina of the human eye—forming a set of images of the object’s points. Ideally, in order to have a “perfect image” of an extended object, each spherical wavefront generated by a point of the extended object should be transformed, by the imaging system, into an outgoing spherical wavefront. Such a system is called an absolute instrument. However, as already noted by James Clerk Maxwell in 1858 Max58, there are some limitations to the existence of absolute instruments and, indeed, forming an image of an extended object with depth into a planar light detector is a specially difficult task.

Optical power

In any case, under the so-called paraxial approximation, i.e., when the sine of the angle is approximated by the angle itself in the refraction law, the imaging system’s capacity of transforming wavefronts depends on the radii of curvature of both the incoming () and the outgoing () wavefronts by means of the vergence equation:

being the optical power of the imaging system, and and the refractive indexes at the incoming and outgoing mediums. In consequence, if we have object points located at different distances from the imaging system, its optical power must change accordingly.

The optical power of the imaging system depends on the geometrical properties of the optical element surfaces and the refractive indexes of the mediums at each side (denoted by ). For instance, in the simple case of a single reflective/refractive spherical optical surface , its optical power is , where is the radius of curvature. Another simple example is given by traditional zoom camera systems, which change the optical power through axial distance shifts between two lenses. Our eyes, however, change the optical power, when looking at near distances–for instance, when reading a book—by means of bending the crystalline lens with the help of the ciliary muscle, thus modifying (see the scheme of a human eye in Figure 1.)

Figure 1.

Scheme showing the optical elements of the human eye. Two lenses: the cornea and crystalline lens form images on the retina. The human eye sees nearby objects by means of increasing the optical power of the crystalline lens.

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Presbyopia

When we get old we lose the capacity to bend our crystalline lens because of the increase in the stiffness of the crystalline lens. This phenomenon is called presbyopia. Then, as shown in Figure 1, our eyes (formed by two lenses: the cornea and the crystalline lens) cannot achieve the required radius of curvature of the outgoing wavefront associated with near vision. The result is that, at the retina, instead of having an image point we perceive a blurred disk. Of course, it is possible to correct that using reading eyeglasses. However, these are monofocal, meaning that they only provide a single optical power and entail shifting between spectacles for far (in case of being myopic or hypermetropic) and near vision tasks. An alternative to that nuisance is to use multifocal instruments.

2. Multifocality

Multifocality is the property of a surface—which could be a wavefront or an optical element—of providing light intensity distributions, not concentrated around a single image point (monofocality) but, instead, distributed along different foci or an extended region. While a spherical surface produces a single focus, multifocality requires a surface with nonconstant curvature.

Geometrically, at any point of a smooth surface in Euclidean space there are two principal curvatures and , which are the maximal and minimal normal curvatures, respectively. Hence it is reasonable to take the mean curvature as an average measurement of the inverse of (radius of curvature) overall normal planes. Another important quantity is the absolute difference between principal curvatures , known as cylinder or astigmatism, which quantifies, at second order approximation, the blurring of the image point due to that difference in ray trajectories for each normal plane.

A caustic digression

Caustics, the geometrical location of light energy concentration, is an ancient concept in geometrical optics because of the caustic (hence the name) properties of some burning lenses and mirrors known since antiquity. Caustics are ubiquitous and provide us with beautiful patterns (see Figure 2).

Figure 2.

The effect of sunlight on clear sea water.

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We underline here the work of Leonardo Da Vinci who, fascinated with light as he was, drew the caustics generated by a circular mirror (these are known as catacaustics): see Figure 3 for the original drawings and Figure 4 for a real life example.

For a planar curve, Tschirnhausen defined the caustic curve as its evolute or the envelope of the normal family of rays to the original curve. In the case of having a wavefront surface, the analog of the evolute of a plane curve is the so-called focal set; then the caustic can be defined as the geometric locus of centers of principal curvatures of the wavefront KO93. Since there are two principal curvatures, each wavefront generates two caustic sheets. From a Riemannian geometry perspective, caustics are the set where rays (geodesics in the optical Riemannian space) concentrate, i.e., the cut locus, the location where more than one minimizing geodesic arrives.

Figure 3.

Da Vinci’s drawing of a reflected caustic by a spherical mirror.

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Figure 4.

Caustics in a cup of milk.

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Ideally, in a perfect multifocal wavefront, the two caustic sheets should coincide and degenerate into a set of points, or a line, because this provides maximum intensity concentration in the regions of interest. However, the two caustic surfaces only coincide when the associated principal curvatures are equal (), i.e., at umbilical points where the cylinder vanishes.

Minkwitz theorem

It is known that, within a smooth Euclidean surface, umbilicity is only possible either at isolated points or lines. Günter von Minkwitz, a young mathematician working on progressive addition lenses, deduced in 1963 what came to be known as Minkwitz’s theorem Min63, which establishes that if a smooth surface contains an umbilical line, then the cylinder along the orthogonal direction increases twice as quickly as the growth rate of the mean curvature along the umbilical line. The effect of Minkwitz’s theorem is illustrated in Figure 5, which shows the mean curvature and cylinder distribution across a circular domain of a real PAL surface RY11.

Figure 5.

Mean power and cylinder of a real PAL surface.

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Recently, a generalization of Minkwitz’s theorem has been derived that provides the exact magnitude of the cylinder at any arbitrary point as a function of the geodesic curvature and the ratio of change of a principal curvature along a principal line BdMG20.

Presbyopia correction: A multifocal approach

There are two ways of applying multifocality for presbyopia correction:

A.

Multifocal intraocular or contact lenses.

B.

Progressive addition spectacles (PALs, for short).

The former solution (A) is based on generating an outgoing wavefront with a varying mean curvature which, for different viewing distances, provides superimposed images on the retina. Then, the neural system is capable of selecting the correct focus at each moment; this is called the simultaneous vision principle.

Figure 6 illustrates this principle: of all rays (green arrow) emerging from a far object (butterfly) only those passing through the central part of the contact lens, converge after refraction onto the retina. On the other hand, of the rays coming from a near object (book) only those (red arrow) passing through the peripheral part of the contact lens converge onto the retina, after passing through the contact lens and the eye. This is possible because the central and peripheral parts of the contact lens are spheres of different radii of curvature: and respectively. Thus the main question is how to design the transition zone between both spheres while keeping astigmatism under control.

Figure 6.

Scheme showing how a multifocal intraocular or contact lens works (in this case a contact lens). Rays from a far object (green arrow emerging from the butterfly) that pass through the central part of the contact lens, where the radius of curvature is , converge onto the retina. On the other hand, rays from a near object (red arrow emerging from the book) only converge onto the retina if they pass through the peripheral part of the contact lens because, there, the radius of curvature () is different from that of the central part ().

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In PALs, the latter solution (B), a surface is designed containing a spatially varying mean curvature. Contrary to contact and intraocular lenses, in PALs, the eye can move independently of the lens, so through gaze movements it chooses, sequentially on time, each viewing area of the PAL providing the required optical power for: far distance vision (center of the lens) and near-view (low nasal portion). Between these two zones, the power varies progressively. Figure 7 shows this.

Figure 7.

Scheme showing how the eye changes its gazes (top figure) when using different parts of the PAL surface (bottom figure). and denote mean curvatures for far and near vision, respectively.

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An ideal progressive surface is one with the prescribed smooth progressive power (mean curvature) and with zero astigmatism (cylinder) everywhere. But, as we have seen above, some amount of astigmatism is unavoidable unless the surface is a plane or a sphere, which does not provide progressive power. In any case, an archetypical model of a PAL is that of an elephant’s trunk (see Figure 8), where there is an umbilical line (red solid line) and the circles obtained by cutting the trunk with parallel planes (dashed blue lines) exhibit reducing radii of curvature when moving from the upper to the lower part of the trunk.

In general, to obtain designs for both solutions, (A) and (B), one should consider the combined action of the eye and the artificial lenses Rub17. However, since multifocality is usually restricted to a single surface, both in intraocular, contact, and PALs, the mathematical study can be, as a reasonable approximation, restricted to a single surface. Then, one may use the calculus of variations and PDEs to give optimal designs in both settings (A) and (B).

Yet, there is a crucial difference between solutions (A) and (B). In the former case (intraocular and contact lenses), axial rotational symmetry is, in most cases, applied, which can be mathematically exploited. Then, the problem is finding a surface that interpolates between two mean curvature values (for adjusting far and near vision, respectively) and minimizes astigmatism. On the other hand, for PALs one instead introduces some weights in the variational functional to include this spatial dependence. Both approaches will be explained in Section 4.

Figure 8.

An elephant trunk is a primitive type of PAL surface model with an umbilical line.

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3. The Willmore Functional

The research of Thomas J. Willmore (1919–2005) was fundamental in the understanding of the following functional, and this is the reason it bears his name Wil93. Let be a smooth, closed, embedded surface in . We define Willmore energy as

where is the mean curvature of and the surface area differential. The factor in front is just a normalization constant.

One can also consider the related functional

for the Gauss curvature of the surface. Remark that an equivalent expression is given by

which is the relevant formulation for our application to the design of multifocal surfaces as we are interested in minimizing astigmatism (cylinder). Indeed, we can think of the functional as a measure of the deviation of from being a sphere (note that, if is exactly a round sphere, vanishes.)

Taking into account the Gauss–Bonnet formula

which relates the Euler characteristic of the surface to its total Gauss curvature, one sees that

Since the last term in the formula above is a topological invariant, in minimization problems for closed surfaces both and are equivalent.

An important property of the Willmore functional is its conformal invariance (recall that conformal map in is a transformation of space that locally preserves angles.) To prove this statement, one observes that any such conformal transformation can be decomposed into a composition of translations, dilations, and inversions. The precise calculation is due to White Whi73 although the result was known to Blaschke in 1929.

The Willmore functional in elasticity

Describing elasticity in terms of geometry is an old subject. In this regard, we underline the works of Sophie Germain and Siméon D. Poisson in the 19th century, in which they modeled the bending energy of a thin plate as the integral with respect to the surface area of an even, symmetric function of the principal curvatures (note here that the Willmore energy is the simplest possible example).

A classical generalization of the Willmore functional comes from the theory of membranes. Indeed, most of the cell membranes of living organisms are made of a lipid bilayer, modeled as a surface minimizing the Canham–Helfrich bending energy, which is essentially

where and are given constants. This approach explains, for instance, the biconcave discoid shape of red blood cells. (See ZCOY99 for an exhaustive discussion on this topic.)

A variational point of view

Going back to the geometric problem, the first question one may ask is how to find the infimum of the Willmore energy over the space of closed surfaces in . Willmore himself showed that the sphere is a minimizer; more precisely,

among all such , with equality if and only if is the round sphere. The proof of this result is quite simple in geometric terms: if we lay the surface on the floor, the point where touches the ground (so the normal vector is vertical) must be elliptic, that is . But now we can repeat this process in any direction. So, for any direction of the normal, we can find a point on the surface such that . This means that the image of the Gauss map covers the whole sphere and, moreover, since the Jacobian of the Gauss map is precisely the Gauss curvature, we have that

To conclude, just note that , with equality iff .

Willmore’s next question was to find a minimizer in a class of fixed topology, for instance, in the class of tori. He proved, in particular, that any torus generated by a small circle moving (perpendicularly) along a closed plane curve fulfills

with equality if the generating curve is a circle and the ratio of the radii is , which corresponds to the parametrization

for . Note that can be conformally mapped to the Clifford torus.

Willmore conjectured in 1965 that this result could be generalized to all surfaces in . However, this conjecture remained open until the seminal paper of Marques and Neves in 2014:

Theorem 3.1 (MN14a).

Every embedded compact surface in with positive genus satisfies 3.3. Up to rigid motions, the equality holds only for stereographic projections of the Clifford torus.

Finally, a classical result of Li–Yau LY82 states that when an immersion has multiplicity (i.e., points in get mapped to the same point by ), then

This settles the Willmore conjecture for all surfaces. For further insight in this topic and additional references, we refer to the beautiful survey MN14b.

Willmore surfaces

The Euler–Lagrange equation for is the fourth order quasi-linear elliptic PDE

where is the Laplacian on . Solutions to this equation are known as Willmore surfaces, even if not necessarily minimizers.

This is a highly nonlinear fourth-order PDE (see GGS10 for an introduction to fourth-order boundary value problems.) An important remark here is that these equations do not usually enjoy many of the nice properties of second-order elliptic problems:

The first issue is that we do not have a priori bounds for the regularity of the solution.

The second point is the lack of a comparison principle that says that any subsolution of the equation stays below any supersolution. For second-order equations with a variational formulation, the classical trick to prove a maximum principle is to look at , (positive and negative parts of , respectively). However, these test functions do not have enough regularity in a fourth-order equation, even in a weak sense.

4. Practical Applications of Willmore-Type Functionals

In the previous section, we considered closed surfaces, that is, without boundary. However, lenses have boundaries! So we should let be a surface with boundary. The natural generalization is, thus, to replace the functional 3.1 by:

where is the geodesic curvature of the boundary curve and the arclength. Some existence results, in the spirit of those of the closed case, are considered by Sch10.

In the following, we will consider two direct applications of the Willmore functional: designing a PAL surface and a multifocal intraocular (or contact) lens.

Modified Willmore functionals in PALs design

The Willmore functional has a relevant industrial application in PAL design. On a PAL surface, one can first prescribe a desired mean curvature distribution within the domain of the surface to be designed, then measure the difference with the true mean curvature, and finally minimize a functional where both the mean curvature difference and the astigmatism are weighted KR99WGS03.

More precisely, let us parameterize the surface as a graph of a function over a domain in . We denote the desired mean curvature distribution by . One tries to minimize

where and are prescribed weights. In fact, in the design of ophthalmic lenses (recall Figure 7), certain regions on the surface are required to have very small astigmatism (where we impose to be large), while in other regions its more desirable that the lens has the correct power (where we set large). Nevertheless, this technique involves prescribing a varying mean curvature on the whole surface, which may not be the optimal solution.

A relevant theoretical problem associated with both the functional 4.2 and Willmore energy, is whether it is possible to find its minimum value on open surfaces, as we did for closed surfaces in Section 3. This would offer a kind of Minkwitz theorem in the large. Knowing a lower bound for the amount of total cylinder achievable by a PAL surface would offer useful insights to an optical designer about how close his/her PAL design is to an ideal one.

A free boundary problem for Willmore surfaces

As mentioned in Section 2, in intraocular and contact lens multifocal applications, we are interested in the following geometric problem: find a Willmore surface that interpolates between two spheres and of different radius and , respectively. Of course, one would like to have a minimizer of Willmore energy, but for now, let us be satisfied with a solution of the Euler–Lagrange equation 3.4.

For practical reasons (that is, having sharp jumps on a lens may introduce undesired visual effects for the wearer), we need to impose that this interpolation solution is smooth up to second derivatives at each of the two boundaries (the two contact lines, let us denote them by and , respectively). This imposes three boundary conditions at each and , which make equation 3.4 overdetermined and, as a consequence, one cannot prescribe and in advance. Remark that we should not expect higher regularity than second derivatives since this is a fourth-order obstacle-type problem. For the interested reader, we refer to a 2020 Notices paper by D. Danielli Dan20 on this topic.

In general, such a is called a free boundary Willmore surface. A free boundary problem is PDE (or ODE) defined on a domain whose boundary is a priori unknown and it has to be determined from the solution of the problem a posteriori. The archetypal example of a free boundary model is that of understanding the interface between ice and water when studying the heat equation (Stefan problem).

Unfortunately, the free boundary Willmore problem has not been solved yet in full generality. A partial answer can be given if one assumes additional symmetries.

The one-dimensional problem

Assuming translation invariance, one can reduce to finding a curve that minimizes the curvature functional for

where is the curvature of and the arclength. Such is called an elastic curve, and satisfies the Euler–Lagrange equation

If the curve is given by the graph of a function , then equation 4.3 reduces to the nonlinear ODE

where

There are various sets of boundary conditions that one may impose. Since we are interested in controlling the curvature at the endpoints, let us consider Navier conditions:

In the symmetric case one knows the following bifurcation picture (see Figure 9)

Theorem 4.1 (DG07).

There exists such that for , the problem 4.4 with boundary conditions 4.5 has precisely two smooth solutions in the class of smooth symmetric functions. If there exists only one such solution, for one only has the trivial solution and no such solutions exist for .

The proof of this result goes back to an observation by Euler in 1952. He found that such a solution must satisfy the conservation law

Figure 9.

Solutions of the Navier boundary value problem from Theorem 4.1 for , , and (left to right). Taken from DG07.

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There are several other pairs of conditions one can impose, for this we refer to the same paper DG07. In particular, the nonsymmetric case may be obtained by scaling, translating, and rotating the coordinate system.

Now, going back at our initial question of interpolating between two circles of different radii by a Willmore curve, it is clear that we need to impose boundary conditions on , , and at both endpoints of the interval, and . A possible strategy is, leaving the contact points and free, to use the solution from Theorem 4.1 in order to also match the first derivatives of at the endpoints.

Willmore surfaces of revolution

In this particular case, the surface is obtained by rotating a curve in the -plane around the axis. We assume that this curve is given by the graph of a function

which, in some cases, can be taken to be symmetric . The surface of revolution is then parametrized by

After some calculation, we arrive at

which reveals the highly nonlinear character of 3.4. Unfortunately, conservation laws such as 4.6 are no longer available in this case.

There are many works that explore rotationally symmetric Willmore surfaces with boundaries, imposing different restrictions at the boundary. We recall some existence results when natural boundary conditions (from the variational point of view) are considered. These are given by

Note that the simplest solution is the circular arc for . In general we have:

Theorem 4.2 (BDF10).

For each and , there exists a positive, smooth, and symmetric function such that the surface generated satisfies 3.4 with conditions 4.7.

Obviously, we are interested in the nonsymmetric case. Unfortunately, there are still no results on the associated free boundary problem, despite its potential applications in the optics field.

Acknowledgments

We would like to thank Jacob Rubinstein and Hans-Christoph Grunau for their valuable comments. S. B. and M. G. are supported by the Spanish Government grant PID2020113596GB-I00.

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Credits

The opening image is courtesy of Fermate via Getty.

Figures 1, 4, 5, 6, 7, and 8 are courtesy of Sergio Barbero.

Figure 2 is courtesy of Brocken Inaglory, CC-BY-SA 3.0.

Figure 3 is courtesy of The British Library, folio 87r Codex Arundel.

Figure 9 is courtesy of Klaus Deckelnick and Hans-Christoph Grunau, Boundary value problems for the one-dimensional Willmore equation, Calc. Var. Partial Differential Equations 30 (2007), no. 3, 293–314, DOI 10.1007/s00526-007-0089-6. Reprinted with permission.

Photo of Sergio Barbero is courtesy of Natalia Barcenilla.

Photo of María del Mar González is courtesy of Jose Pedro Moreno.