1.

Introduction

When evaluating most optical systems or surfaces, the wavefront or surface topography becomes extremely important. Typically, it is represented by a topographic map that can be obtained mostly by optical interferometric methods, for example, the Twyman–Green interferometer in Fig. 1(a), representing an astigmatic (sphero-cylindrical deformation) which is a topographic map with equal elevation deformation, where each line represents the geometrical locus of points of equal elevation or separation from its reference sphere. The difference in elevation between two consecutive fringes is one wavelength. Figure 1(b) shows a representation of an equal wavefront map for the constant deviation points where the different colors represent different values of the elevation.

Fig. 1

Maps of equal elevation deformation in a wavefront or surface of an optical system. (a) Twyman–Green interferograms and (b) color-coded map. An advantage of the Twyman–Green interferogram is that it provides more quantitative results than the color-coded map, but a disadvantage is that its elevation has a sign uncertainty that the color-coded map does not have. In the Twyman–Green interferogram, the fringe separation is an indication of the elevation slope.

Intuitively, we know that the local curvature tells us how fast the elevation slope changes at some point. In some optical systems, the local curvature instead of the elevation becomes more important since the value of the curvature is an indication of the local convergence or divergence power of the optical surface or the local degree of convergence or divergence in a wavefront. This will be described later in more detail.

The geometrical concept of curvature and the mathematical theory of local curvatures are quite old and they are the main subjects of the books about differential geometry. The history of the concept of curvature has been treated recently by many authors with some detail, e.g., by Margalit¹ and Bardini and Gianella.² The first attempts to formally define the curvature started many centuries ago, beginning with the Greek writers, following with Johannes Kepler (1571 to 1630) who was the first to define the curvature as the inverse of the local radius of curvature. However, the first successful study comes from Newton,³ as described by Coolidge.⁴ He said that the curvature of a curved line is equal to the curvature of the largest circle that is tangent to the curve on its concave side and that the center of the circle is the center of curvature. The study of curvatures is increasing its importance and has recently become a very important concept in optometry and ophthalmology.⁵ It is thus surprising that classical optics books ignore this subject almost completely. Even ophthalmology and optometry books study curvatures at just an introductory level. Mathematicians and specialists in this field are the ones that study this subject in detail and great advances have recently been made. It is the purpose of this article to make a general review of the subject at a level so that opticians, optical engineers, and in general non-specialists in the field can get an introductory description of the main concepts today in wide use. In this review, we have made all mathematical derivations using elementary concepts of calculus, without the use of differential geometry to make it simpler to understand for nonspecialists. The concept of curvature is so important that has applications in many fields, even as something that can be perceived by just touching the surface⁶ and as a tool for neurologic studies.⁷

When evaluating an optical surface, we might be interested in the map of the optical surface or wavefront deformations, frequently called aberrations, as given by an elevation map. A typical example includes the surface of a primary mirror for an astronomical telescope under polishing or figuring. In some other optical surfaces, the important characteristics to be measured are not the optical surface or the wavefront deviation but the local curvatures, as in the case of the human eye corneas. The most commonly evaluated curvatures in optometry and ophthalmology are the sagittal and tangential curvatures and the axis orientation.⁵ In most commercial instruments this is done with approximations, frequently with the assumption that the optical surface or wavefront is nearly rotationally symmetric. More general and precise evaluation methods were not found to be described in any general optics, ophthalmology, or optometry books, perhaps with the only exception of the book by Stavroudis.⁸ Surprisingly, to our knowledge, this information was absent even in differential geometry books.

The subject of surface topography and shape, of surfaces studied in differential geometry books, among them, by Stoker,⁹ Lipschutz,¹⁰ Bronshtein and Semendyayev,¹¹ Kepr,¹² Gray et al.,¹³ and Krauthammer and Ventsel¹⁴ is quite important and useful in many fields, mainly in ophthalmic and optometric optics and in geology.¹⁵ Typically, in these books, the first fundamental and the second fundamental forms are developed, leading to expressions for the principal curvatures but not for the sagittal and tangential curvatures. In optics, these concepts are useful for the study of many properties of the optical surfaces and wavefronts. The basic intrinsic properties of a surface are described by the fundamental forms of surfaces, which are the subject of differential geometry books. Their results can be applied to study the sagittal and tangential curvatures, but frequently some other methods are used.¹⁶

2.

Curvatures in a Plane Curve

To fully describe the concept of local curvature in a surface, let us first begin by considering a one-dimensional function $f(x)$ represented in a plane, as in Fig. 2, where three circles are drawn close to the curve.

Fig. 2

Three circles near a curve. The first circle is intersected by the curve at two points and the second circle is tangent to the curve at one point. There are many possible tangent circles of any size, at any point. The third circle is osculating to the curve at the point $\mathbf{P}$. At a given point in the curve, we can trace only one osculating circle.

The first circle at the left, in Fig. 2 is intersected by the curve at two points. The second circle touches the curve at only one point, where the first derivative (slope) at this point is the same both at the curve and at the circle. At a given point in the curve, we can trace an infinite number of tangent circles of different sizes, on any side of the curve. The third circle touches the curve in a small region, where both, the first and the second derivatives are the same at the center of the region where they touch each other. At a given point in the curve, we can trace only one of these circles. Then, we say that the circle and the curve are osculating (from Latin: osculum = kiss). The curve at this small region has the same radius of curvature as the osculating circle. The unit normal vector is a vector passing through the point in the curve being considered and pointing to the center of curvature of the osculating circle. The curvature is defined by the inverse of the radius of curvature $r$. More formally, we may say that for a small length along with the curve $\mathrm{d}s$, the curvature is given by how fast the slope is changing at the point $\mathbf{P}$ in Fig. 2, then, as defined probably for the first time by Kästner,¹⁷ the curvature along the $x$-direction $c_x$ at the point $\mathbf{P}$ would then be expressed as

This result can also be applied to find the exact value of the local curvature over a curve on a surface $f(x,y)$ in the direction of maximum slope (gradient), where the slope in the perpendicular direction is zero. In the next section, we will describe how to find the local curvature along a curve on the surface $f(x,y)$ in the direction of no slope, perpendicularly to the direction of maximum slope.

3.

Local Curvatures in a Direction Perpendicular to its Maximum Slope

The concept of curvature was extended by Euler,¹⁹ Bernoulli’s doctoral student, to three dimensions. In a more general case we have a surface $f(x,y)$, and a curve on this surface, this expression is strictly valid if the points $\mathbf{P}(x,y)$ where the curvature is to be evaluated has a slope along some direction of the curve but not in the perpendicular direction. We will study the local curvatures over an open surface $f(x,y)$, so that given any point $\mathbf{P}$ with coordinates $(x,y)$ there is only one possible value of the function $f(x,y)$ describing the surface. Let us start by defining some important concepts. A normal plane at a point in a surface is any plane containing the normal to the surface at that point. The intersection of a normal plane with the surface is a curve called a normal section and the curvature of this curve at that point is the normal curvature.

At any point on the surface, we can place a plane tangent at that point. This plane is not necessarily horizontal, which in the general case has a tilt. At the point of tangency on this surface, there is a different slope for different directions, which is equal to the first derivative in the given direction. Thus, at this point, there are two mutually perpendicular directions, one with a zero slope and one with a maximum slope (gradient). It should be pointed out that these zero and maximum slope directions are not necessarily the same as the maximum and minimum curvatures, which are also called the principal curvatures. However, it is interesting to notice two particular cases: (a) If the surface has rotational symmetry, at any off-axis point, the principal curvatures are one in the direction of maximum slope (gradient) and one in the direction of zero slope. In these surfaces, these are the radial and the angular directions. (b) A second interesting case is a surface with symmetry about a straight line in the $x\text{–}y$ plane, e.g., a horizontal cylinder. In this case, the principal curvatures are also one in the direction of maximum slope (gradient) and one in the direction of no slope.

The local curvatures at the point in the surface being considered can be easily calculated in these two special directions, i.e., the direction of maximum slope and the direction of zero slope. For the direction of the maximum slope, there is no slope in the perpendicular direction, and hence the theory in Sec. 2 and the result in Eq. (5) can be applied if the $x$-direction is selected in the direction of maximum slope. Next, we will calculate the local curvature in the direction of zero slope.

Figure 3 illustrates a single-valued function $f(x,y)$, describing a surface on top of a sphere and a cylinder. Let us assume that on this surface we can find two points ${\mathbf{P}}_1$ and ${\mathbf{P}}_2$, with the following characteristics. At the point ${\mathbf{P}}_1$, the local curvatures on this point in the surface $f(x,y)$ are the same in all directions and thus, we can place an osculating sphere at this point. At the point ${\mathbf{P}}_2$ in the surface $f(x,y)$ the curvature along the $x$ coordinate is zero, hence the axis of the cylinder is contained in a plane perpendicular to the $y$ axis. Thus, we can place an osculating cylinder located at this point ${\mathbf{P}}_2$. The unit normal vectors $\mathbf{N}$ are at the points where the local curvatures are to be measured. The unit vectors ${\mathbf{r}}_{\mathrm{M}}$, also at the points ${\mathbf{P}}_1$ and ${\mathbf{P}}_2$, are parallel to the $z$-axis. Thus, the angle between these two vectors $\mathbf{N}$ and ${\mathbf{r}}_{\mathrm{M}}$ is equal to the angle ${\alpha }_g$ formed by the maximum slope (gradient) on the surface.

Fig. 3

A surface $f(x,y)$ with an osculating sphere at the point ${\mathbf{P}}_1$ and an osculating cylinder at the point ${\mathbf{P}}_2$. The curvature at the two points ${\mathbf{P}}_1$ and ${\mathbf{P}}_2$, along the direction of $h$ with no slope is to be measured and the maximum slope is in the direction of $g$.

With a simple geometrical analysis, it is relatively simple to show that in general, for the osculating sphere as well as for the osculating cylinder, the curvature in the direction of no slope, which is the perpendicular direction of maximum slope (called here direction $h$), at a point located along a curve in the plane containing the normal to the surface would be given by the second derivative in this direction, multiplied by $\cos {\alpha }_g$. Thus, if the slope along the curve in the direction $h$ is zero, but different from zero in the perpendicular direction $g$, the curvature $c_h$ is

4.

Curvatures in Different Directions at a Point on a Surface

We have calculated the local curvature at a point in a surface, in two particular mutually perpendicular directions, the direction of maximum slope and the direction of no slope. In any other direction the problem is mathematically more complicated, but the problem has been solved by differential geometry methods, by the development of the first fundamental and the second fundamental form of surfaces. However, the expressions for $c_x$ and $c_h$ [Eqs. (5) and (7)] are enough to obtain a highly accurate and intuitive method for calculating and understanding the local curvatures as described below.

Let us consider a point in a surface $f(x,y)$ and an infinite number of possible directions for the curvature, passing through a point, as illustrated in Fig. 4.

Fig. 4

Different orientations for the zero slope, maximum slope (gradient), the sagittal curvature, the tangential curvature, and the minimum curvature and the maximum curvature (principal curvatures) at a point in a surface.

For this point in this surface, the slopes in the $x$ and $y$ directions are given as

5.

Curvatures in a Surface $f(x,y)$ in Any Direction $\alpha$

Now, to calculate the local curvatures at any point in any desired direction $\alpha$ (see Figs. 4 and 5) we need to know the first and second derivatives at the desired location. Let us assume that we need to calculate the first and second derivatives at the point $(x,y)$ but in the direction $\alpha$ of the rotated coordinates $u$ and $v$, as illustrated in Fig. 5.

Fig. 5

Translation and rotation of coordinates to evaluate the local curvatures at the point $(x,y)$ along the rotated axis $u$ in the direction $\alpha$.

The function representing the surface is $f(x,y)$. The first derivative with respect to the coordinate $u$, in the $\alpha$ direction as

Now, let us now assume that this surface normal is not perpendicular to the $x\text{–}y$ plane. We assume there is a slope tan ${\beta }_u$ along the curve where the curvature is measured, given by the first derivative with respect to the coordinate $u$, in the direction as

The exact formula derived in a rigorous manner from the fundamental forms of the differential geometry. From the book by Stoker⁹ after converting to Cartesian coordinates by a Monge parametrization, the exact curvature expression is quite similar, as follows:

If the expression is converted from Cartesian to polar coordinates, the exact expression for the local curvatures at the point $(\rho ,\theta )$ becomes

6.

Accuracy of the Approximate Formula to Determine Curvatures in any Direction

The determination of the local curvatures in a wavefront in an optical system is of great importance. It was shown by Kneisly²¹ that they can, in principle, be determined by studying the propagation of light beams in an optical system by means of ray tracing. Following our treatment, we have pointed out that the expression 20 provides only an approximate value for the curvatures in directions different from those of the gradient direction and perpendicularly to the gradient.

Exact results can be obtained only using the results from the fundamental forms of surfaces, as studied in differential geometry, in Eqs. (20) or (21). However, the result is quite accurate. An expression for the curvature error can be calculated by taking the difference between the approximate expression obtained here and the exact expression as follows:

We can see that the maximum error is a function only of the magnitude of the gradient, that is, of the maximum slope at the point where the curvature is evaluated. If we add this value of $\mathrm{\Delta }{c}_{\theta }$ to the approximate expression, the small remaining error is compensated, thus obtaining an exact result. As we can observe, the error amplitude depends only on the magnitude of the gradient, or maximum slope of the tangent plane at the point where the curvature is evaluated. We might observe that the error is zero in the directions of maximum and minimum slope (gradient direction) and in the direction of zero slope (perpendicularly to the gradient). The maximum error is $\pm 45\mathrm{deg}$ with respect to the gradient direction.

Figure 6 shows three polar representations of the approximate and the exact curvatures in all directions for a point in a spherical surface and a radius of curvature equal to 7.722 mm, the average radius of the human cornea.

Fig. 6

(a)–(c) Three polar representations of the local curvatures in all directions at three-point in a sphere with different distances from the axis. At these points the gradient or maximum slope is different. Each figure has three different polar plots, one of them, with blue dots, is a perfect circle, calculated with the exact expression for the curvatures (local curvature equal to $0.13{\mathrm{mm}}^{-1}$). Another, with red dots, is calculated with the approximate expression. It is quite close to the circle and touches over the line along the gradient and along a line perpendicular to the gradient. The third polar plot is for the local curvatures with green dots, calculated just with the numerator of the expression for the curvatures. All three polar plots become identical to a circle (zero error) when the maximum slope is zero (at the optical axis).

7.

Euler Curvature Formula

A point in a surface contains an infinite number of normal planes in all possible directions. These normal planes at a point in a surface in most surfaces and points have different values for different directions, except at points with perfectly spherical or flat shapes.

The expression for the curvature in any direction $\theta$ can be expressed in the following form, known as the Euler’s curvature formula,^12,13 graphically represented by a closed figure frequently resembling an ellipse, but sometimes it is more like a nut.

The Euler formula representing the polar distribution of the curvature for different angular directions is illustrated in Fig. 7, with an axis orientation $\psi =0\mathrm{deg}$ and different values of the ratio of the cylindrical curvature to the spherical curvature.

Fig. 7

Polar plots of the Euler formula for an axis orientation $\psi$ equal to 0 deg and different values of the ratio of the cylindrical curvature to the spherical curvature. The color indicates the magnitude of the curvature.

In any kind of surface, except in a sphere, the curvature at any point is not necessarily constant, but variable with the direction, in a nearly ellipsoidal manner. In other words, a plot of the curvature as a function of the angle describes a closed path, as in Fig. 8, with an inclination $\psi$ of the cylinder axis, given by Eq. (24).

Fig. 8

Variation in the value of the normal curvature $\mathrm{c}\alpha$ for all possible directions. The maximum and minimum curvatures, also called principal curvatures ${\kappa }_1$ and ${\kappa }_2$ are orthogonal to each other.

There is a maximum value of the curvature ${\kappa }_1$ in one direction and a minimum value ${\kappa }_2$ in an orthogonal direction. These are the two principal curvatures. The principal curvatures are the maximum and minimum local curvatures, which are always perpendicular to each other. If the principal curvatures at a point in an optical surface are ${\kappa }_1$ at an angle $\alpha =\psi \pm \pi$ and ${\kappa }_2$ at an angle $\alpha =\psi \pm \pi +\pi /2$ the curvature along a direction $\theta$ can also be written in terms of the principal curvatures, as illustrated in Fig. 8.

Analytically, this curvature can be represented by the Euler curvature formula which is illustrated in Fig. 9(a) as

Fig. 9

Curvature variation with the angle at a point in a surface. The total curvature can be expressed as a sum of a spherical and an astigmatic curvature. The spherical component can have: (a) a curvature equal to the average curvature, (b) a curvature equal to its minimum value, and (c) a curvature equal to its maximum value.

The Euler expression for local curvatures can be written in several other different equivalent manners, e.g.

This can be written as

8.

Mean, Gaussian, and Cylindrical Curvatures

Efforts had been made to find parameters that define as simple and as clear as possible the shape of a surface in a small region.²³ Besides the previously described curvatures, in differential geometry, the mean curvature $c_{\text{mean}}$ and the Gaussian curvature $c_{\text{gauss}}$ have been defined as the arithmetic average and the product, respectively, of the two principal curvatures, as follows:

Using Euler [Eq. (31)], we may find the mean curvature as the arithmetic average of the two principal curvatures or in a more general manner as the arithmetic average of any two curvatures in orthogonal directions, as follows:

Individually, these two curvatures hide any information about the difference between the two principal curvatures, frequently called astigmatism, and just specify the arithmetic and the geometric average of the two principal curvatures, representing the curvatures of two intermediate reference spheres, tangent at the point under consideration. The mean curvature is represented by the radius of a circle in Fig. 10.

Fig. 10

Polar representation of the Gaussian and mean curvatures.

The Gaussian curvature is the square of the geometric average of the two principal curvatures and has units of $1/{\mathrm{mm}}^2$ instead of 1/mm as the other curvatures. It is the area of the rectangle in the upper part of Fig. 10. A sphere has a constant curvature over the whole surface. Some other surfaces may have a constant Gaussian curvature over the whole surface, e.g., Fig. 11 illustrates three surfaces with different constant Gaussian curvatures, inside and outside of the surface. The first surface has negative value, the second one zero values and the third one positive values. The measurement of the Gaussian curvature by optical means has been mathematically studied by Roitman.²⁵

Fig. 11

Three surfaces with different values of the Gaussian curvature. The three surfaces have the same constant value of the Gaussian curvature at all points, inside and outside of the surface. (a) Negative, (b) zero, and (c) positive.

If both principal curvatures have the same sign, the Gaussian curvature is positive and that point at the surface is said to be an elliptic point. If both principal curvatures are equal, that point at the surface is said to be an umbilical point and it is locally spherical. The name comes from the Latin umbilicus, meaning navel. If the two principal curvatures have different signs, the Gaussian curvature is negative and that point at the surface is said to be a hyperbolic or saddle point. If one of the principal curvatures is zero, the Gaussian curvature is zero and that point at the surface is said to be a parabolic point. This concept of Gaussian curvature has been used in optical design studies.^25,26

In general, a surface has different values of the mean and the Gaussian curvatures at different points.

The local powers in diopters $D_s$ and $D_t$ are just the curvatures $c_s$ and $c_t$, multiplied by the index of refraction $n=1.3375-1(n-1)$ as follows:

The cylindrical curvature, from Eqs. (29) and (38), is given as

9.

Calculating Astigmatic Parameters with Three Measurements

The curvatures along the gradient and perpendicularly to it do not provide all the information about the curvature variation with normal plane orientation. A third parameter is needed to obtain the Euler equation and thus the curvatures in all directions. This information also allows us to retrieve the cylinder orientation $\psi$.

In general, to calculate axis orientation, $\psi$ we take a minimum of three measurements of the curvature in three different directions, as in phase-shifting techniques used in optical testing.²³ If we set ${\theta }_1=\varphi$, ${\theta }_2=\varphi +45\mathrm{deg}$, and ${\theta }_3=\varphi +90\mathrm{deg}$ and two of these three measurements will be the curvatures along the gradient and along the perpendicular to the gradient. The third measurement is at 45 deg between them. Thus, using the Euler equation we have

10.

Local Curvatures in Optical Surfaces

In lens design and evaluation programs, the optical surface shape²⁸ (sagittal distribution) is defined as

For an optical surface with a circular boundary or a wavefront from an optical system with a circular optical pupil, two curvatures are frequently important. One is in the radial direction, the tangential or radial curvature also called instantaneous curvature. The other curvature in a perpendicular direction, the sagittal or azimuthal curvature $c_s$, is also called axial curvature by optometrists and ophthalmologists (see Fig. 12).

Fig. 12

Tangential or instantaneous and sagittal or axial curvatures in a circular pupil.

Using the general expression in polar coordinates, Eq. (22) for local curvatures, the tangential (also called radial or instantaneous) curvatures can be found by setting $\alpha =\theta$, obtaining

Let us now study with more detail these curvatures for surfaces with nearly rotational symmetry. For these surfaces, both curvatures, tangential and sagittal, are constant for all values of $\theta$ and a given value of $\rho$. In other words, the sagittal and the rotational maps are rotationally invariant. Now, by observing Fig. 13, we see that the axial (or sagittal) curvature $c_s=1/r_s$ is the curvature at a point on the intersection of the optical surface with a plane passing through the local center of curvature, which is perpendicular to the plane containing the optical axis (a tangential plane). Thus, when the optical surface has rotational symmetry, the sagittal (or axial) radius of curvature can be calculated by tracing a ray passing through the point $\mathbf{P}$ and measuring its distance to the intersection with the optical axis. This is the reason for the name “axial curvature.” If the optical surface does not have rotational symmetry, the ray passing through point P does not cross the optical axis. This effect is called skew ray error in the optometric literature.²⁹

Fig. 13

An optical surface with rotational symmetry, showing the osculating sphere and also a sphere touching the aspherical surface along a ring passing through the point $\mathbf{P}$.

When the surface has rotational symmetry about the optical axis, these tangential and sagittal curvatures can be calculated with simpler formulas. In Fig. 13, we have a sphere tangent to a surface with rotational symmetry about the optical axis. The point of tangency is at the point $\mathbf{P}$ and along a circle containing the point $\mathbf{P}$, concentric with the optical surface. Since the sphere and the surface are tangent along this circle, a line being perpendicular to the optical surface is also a radius for the sphere with the axial curvature $1/r_s$

Barbero³⁰ has described the concepts of geodesic curvature and geodesic torsion as a metric of the difference between a rotationally symmetric and a nonrotationally symmetric surface. When nonrotational symmetric aberrations are present and this expression is used, important errors appear, mainly at the periphery of the corneal surface.

In the case of surfaces with symmetry of revolution, these are the tangential and the sagittal curvatures. For surfaces without symmetry of revolution, like the sphero-cylindrical surfaces or astigmatic corneas, in general, the principal curvatures are not in the tangential and sagittal directions. Thus, these principal curvatures are not the same as the averages of the tangential and sagittal curvatures, unless the surface or wavefront deformations are rotationally symmetric.

In surfaces without rotational symmetry, the two curvatures, sagittal and tangential, are completely independent of each other. Under these conditions, it is not possible to derive one type of curvature from the other. There is a lot of confusion in the literature about this topic. However, for surfaces with rotational symmetry, these two curvatures are related to each other as described by Klein and Mandell,³¹ Tang et al.,³² and Schwiegerling,⁵ page 45)

When the surfaces do not have rotational symmetry, the two principal curvatures are not the same as the tangential and sagittal curvatures, at any point in the optical surface, which is the case when: (a) the surface has a cylindrical or sphero-cylindrical shape, (b) when the aberration surface has nonrotationally symmetric deformations.

In general, the principal curvatures cannot be calculated from the sagittal and tangential curvatures only. An extra parameter must be determined at all points in the optical surface, e.g., the orientation of the principal curvatures (cylinder axis) of the difference of their magnitudes (cylinder magnitude), or a curvature in another direction. This means that the sagittal and tangential maps do not provide the whole information about the shape of the surface deformations nor about the curvatures.

Examples of maps for tangential and sagittal curvatures, obtained by computer generation for an astigmatic (sphero-cylindrical) wavefront are in Fig. 14.

Fig. 14

An example of tangential map of curvatures (a) and a sagittal map of curvatures (b) for a sphero-cylindrical optical surface or wavefront. The color indicates the sign, red for positive values and green for negative values.

10.1.

Sagittal and Tangential Curvatures for Conic Surfaces

Conic surfaces are a particular case of surfaces with rotational symmetry. For conic surfaces, the sagittal, azimuthal, or axial curvature is given by³³

Eq. (58)

$$c_s=\frac{c}{{[1-K{c}^2{S}^2]}^{1/2}},$$

and the tangential, radial, or instantaneous curvature is given as

Eq. (59)

$$c_t=\frac{c}{{[1-K{c}^2{S}^2]}^{3/2}}.$$

As in any rotationally symmetric surface, the two curvatures, sagittal and tangential, are related to each other. In this case, they are related as

Eq. (60)

$$c_s^3=c^2{c}_t,$$

where $c$ is the vertex curvature, i.e., the curvature at the intersection of the optical surface with its optical axis. The local radius of curvature is equal to the radius of curvature $r$ plus the aberration of the normal, represented by $\mathrm{\Delta }r$, as illustrated in Fig. 15. Since the slope of the line going from the point $\mathbf{P}$ to the local center of curvature is equal to the first derivative or slope of the aspherical surface, the aberration of the normal can be obtained as

Eq. (61)

$$\mathrm{\Delta }r=\frac{S}{\left(\frac{dz}{dS}\right)}+z-r,$$

which, for conic surfaces becomes

Eq. (62)

$$\mathrm{\Delta }r=-Kz.$$

The anterior surface of the cornea is frequently assumed to be ellipsoidal, with or without rotational symmetry.^34,35 With this fact in mind, Harris,³⁶ and later Bektas,³⁷ studied the curvature of general ellipsoids using the fundamental form of surfaces. A detailed study of the shape of the cornea and its local curvatures has also been reported by Griffits et al.^38,39

Fig. 15

A conic surface with its osculating sphere, illustrating the transverse aberration.

10.2.

Curvatures in Convex or Concave Optical Surfaces

A particular case of special interest is the measurement of local curvatures in a strong spherical convex or concave with relatively small aberrations. It is also important to consider what happens when an unaberrated convergent or divergent wavefront is reflected or refracted in an aberrated optical sphere to measure this optical surface. In the case or reflection, the reflected wavefront will have the same aberrations as the incident wavefront, multiplied by two, and in the case of refraction, the refracted wavefront also will have the same aberrations but multiplied by ($n-1$) which is close to 0.5. A consequence is that the local curvatures at all points over the wavefront the local curvatures after reflection or refraction are scaled by the same factor 2 or ($n-1$).

It is important to notice that if a spherical or conic optical surface is aberrated, those aberrations could be defined and measured in a direction parallel to the optical axis or perpendicularly to a close reference sphere. Thus, we define the absolute aberrations as those in a direction parallel to the optical axis and the relative aberrations as those measured perpendicularly to the close reference sphere. Figure 16 illustrates these concepts of the absolute distance $z_a$ and the relative distance $z_r$. They are related as

Eq. (63)

$$z_r=z_a\cos \theta ,$$

where $\theta$ is the angle between their normal to the sphere and the normal to the aberrated sphere. If the aberrated wavefront is almost flat, the reference sphere is a plane and $\theta$ is zero.

Fig. 16

Illustration of the absolute distance, measured with respect to the reference sphere, in a direction parallel to the optical axis and the relative distance, measured with respect to the reference sphere and in a direction towards the center of the sphere.

The importance of this distinction can be clearly seen when we change the curvature of the reference sphere of a convex (divergent) wavefront by means of a collimating lens. By use of the Fermat principle, we can see that if the relative sagitta in the incident nearly spherical wavefront becomes equal to the absolute sagitta in the collimated wavefront. In Fig. 17, $z_{r1}$ is the relative aberration in the divergent wavefront and it is equal to the absolute aberration $z_{a2}$ in the collimated wavefront.

Fig. 17

Flattening of the curved, aberrated wavefront by means of a collimator.

An interesting conclusion is that if we measure or evaluate the local curvatures in the convex or concave optical surface in Fig. 16 or the convergent wavefront in Fig. 17, the slope is large and thus we must use the exact Eq. (21) to calculate the local curvatures. On the other hand, if we need to calculate the relative local curvatures of the curvatures in the collimated wavefront in Fig. 17, we can use the simplified formula, in Eq. (20), or if the aberrations are not extremely large, even just the numerator in this expression, which can be written as

Eq. (64)

$$c_{\alpha }=\frac{1}{2}(\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2})+{[\frac{1}{4}{(\frac{{\partial }^{2}f}{\partial x^2}-\frac{{\partial }^{2}f}{\partial y^2})}^2+{\left(\frac{{\partial }^{2}f}{\partial x\partial y}\right)}^2]}^{1/2}\cos 2(\alpha -\psi ),$$

which is valid for any almost flat (collimated) wavefront. Then, the local curvature along the $x$ and $y$ axes are given as

Eq. (65)

$$c_x=\frac{{\partial }^{2}f}{\partial x^2}\text{and}{c}_y=\frac{{\partial }^{2}f}{\partial y^2}.$$

The principal curvatures ${\kappa }_1$ and ${\kappa }_2$ can be obtained from Eqs. (21) and (23), as follows:

Eq. (66)

$${\kappa }_1,{\kappa }_2=(\frac{{\partial }^{2}f}{\partial x^2}+\frac{{\partial }^{2}f}{\partial y^2})\pm {[{(\frac{{\partial }^{2}f}{\partial x^2}-\frac{{\partial }^{2}f}{\partial y^2})}^2+4{\left(\frac{{\partial }^{2}f}{\partial x\partial y}\right)}^2]}^{1/2},$$

with an axis orientation given by Eq. (24).

11.

Wavefront Sensing

All preceding treatment of local curvatures assumes that the analytic representation of the surface is known, otherwise, the surface must be measured and mathematically represented to calculate its local curvatures. Optical surfaces can be directly measured optically or mechanically in many ways, typically by profilometry, as described by Schmit et al.⁴⁰ In a more common method, optical surfaces and general optical components are evaluated by measuring the wavefront deformations in the reflected and/or refracted light beam in these systems. This procedure is normally called wavefront sensing, traditionally employed to test all types of optical systems²⁸ and also used to examine the eye,^41,42 mainly since the advent of adaptive optics. In many of these methods, it should be taken into account that nearly always a perfectly nonaberrated wavefront when reflected or refracted in even a perfect optical surface or system becomes aberrated, as described in optical design books.⁴³

11.2.

Interferometric Wavefront Sensing

The devices that can measure directly the wavefront deformations are the interferometers, which produce interferograms, representing curves of equal level, that can be processed to retrieve the slope and the local curvatures.⁴⁴ Two typical examples are the Fizeau⁴⁵ the Twyman–Green^46,47 interferometers (see Fig. 18).

Fig. 18

Twyman–Green interferometer.

Interferometers have an extremely large sensitivity to small deformations. As a result, the atmospheric turbulence, vibrations, and small movements make the system quite unstable. Interferometers that cancel out these vibrations and rapid movements had been designed,⁴⁸ making them more appropriate for ophthalmologic and optometric studies. This Twyman–Green interferometer uses the phase-shifting technique with four-phase steps taken simultaneously. To achieve this simultaneity, every pixel of the CCD image detector where the interferograms are imaged is formed by four pixels. Each cell has a polarizing layer on top, with different orientations for the linear polarization in each of them, as shown in Fig. 19. Since the light beam arriving in this image detector is circularly polarized, we have four different phase shifts. In all interferometer setups, the cornea or the retina under examination should be conjugate to the observation plane with the light detector.

Fig. 19

Image detector for a vibration and turbulence stabilized Twyman–Green. The light beam arriving at this image detector is circularly polarized to have four different phase shifts.

11.3.

Foucault or Schlieren Wavefront Sensing

In the Foucault test,⁴⁹ also known as Schlieren, when used to detect the flow of fluids or air turbulence, illustrated in Fig. 20, a knife edge is inserted perpendicularly to the point of convergence of the light coming back from a concave mirror, to block the ray lights that have a transverse aberration larger than the distance from center of the image to the edge of the knife edge. An imaging lens or an observing eye is located after the knife, to form an image of the mirror at an observing plane. The image in the observing plane has darker zones where the light rays are blocked. An image that subjectively illustrates the surface with the irregularities is clearly shown. This test measures the transverse aberrations, which are the same, except for a constant, to the surface slopes in a direction perpendicular to the knife edge. This test can be made quantitative.

Fig. 20

Foucault test of a concave mirror.

The imaging lens in contact with the knife, should form at the observation plane an image of exit pupil of the system or the optical surface under test.

11.4.

Pyramidal Prism Wavefront Sensing

An important modification of the Foucault test is obtained by substituting the knife edge with a pyramidal prism to simultaneously produce four Foucault images, with the edges of the four prisms forming the pyramid, as if four knives were introduced instead of just one.^50–55 This makes the test bidimensional, measuring the transverse aberration in the $x$- and $y$-direction. Also, the pyramidal prism has oscillations in both directions to measure the transverse aberrations over the whole aperture.

After obtaining the maps for the transverse aberrations $T{A}_x$ and $T{A}_y$, the wavefront deviations are obtained. With this information, the local curvatures can also be retrieved. Let us assume that the illumination (irradiance) values on the images $A$, $B$, $C$, and $D$ are $I_A(x,y)$, $I_B(x,y)$, $I_C(x,y)$, $I_D(x,y)$. The total amount of light on each of these images is directly proportional to the light passing through the corresponding quadrant in the pyramidal prism. Then, it is easy to show that the irradiance of these images can be added in pairs, obtaining four Foucault images, as if they were produced by four different knife orientations. If we add all four Foucault images, a uniformly illuminated image is obtained (see Fig. 21).

Fig. 21

Pyramidal prism wavefront sensor.

For any position of the pyramidal prism oscillation, we can measure two signals, $S_x(x,y)$ and $S_y(x,y)$, defined as

Eq. (67)

$$\begin{gathered} S_x(x,y)=\frac{(I_B(x,y)+I_D(x,y))-(I_A(x,y)+I_C(x,y))}{I_A(x,y)+I_B(x,y)+I_C(x,y)+I_D(x,y)}, \\ {S}_y(x,y)=\frac{(I_C(x,y)+I_D(x,y))-(I_A(x,y)+I_B(x,y))}{I_A(x,y)+I_B(x,y)+I_C(x,y)+I_D(x,y)}, \end{gathered}$$

where each of these irradiance values for every image pixel, are between zero and one. Then, the values of these signals are directly proportional to the difference between the two complementary irradiance values for two knife orientations in opposite directions, producing values between $-1$ and $+1$, being zero when the two complementary irradiances are equal.

These signals are directly proportional to the slopes in the wavefront. If we represent by $\delta V_x$ and $\delta V_y$ the amplitudes of the oscillations of the pyramidal prism in the $x$- and $y$-directions, respectively, the wavefront slopes, from Eq. (67), can be shown to be

Eq. (69)

$$\begin{gathered} \frac{\partial W(x,y)}{\partial x}=\frac{r_w}{\delta Vx}{S}_x(x,y), \\ \frac{\partial W(x,y)}{\partial y}=\frac{r_w}{\delta Vy}{S}_y(x,y), \end{gathered}$$

where $r_w$ is the distance from the wavefront pupil to the pyramidal prism. The next step is to use Eq. (62) to find the transverse aberration values.

The imaging lens in contact with the pyramidal prism should form at the observation plane an image of exit pupil of the system, the pupil of the observed eye or the optical surface under test.