1. Introduction

Freeform optics is a recent and revolutionary field of optical design, where the goal is no longer to find the best combinations of positioning and curvature parameters of known surfaces, such as conics, but to directly calculate the surfaces that maximize the desired optical properties, regardless of the mathematical function that describes them. Freeform surfaces are generally devoid of any rotational or translational symmetry axis, and there are several approaches to obtaining their shapes [1], usually involving the optimization of an initial surface and/or the use of ordinary or partial [2,3] differential equations, such as the Wassermann-Wolf equation [4], the variational calculus [5] for the minimization of aberrations, the use of principles such as optical path length invariance [6–8] for perfect focusing, and coupling input and output variables [9,10]. These methods have different advantages and computing time depending on their application and may allow the generation of multiple freeform surfaces. It is not unusual that an algorithm developed to solve three-dimensional problems is derived from a solution in two dimensions, as was the case for the Wasserman-Wolf and simultaneous multiple surface methods [1,11], and so, in this work, we developed a quick algorithm for generating aspheric lenses, where a monochromatic set of rays need to go (or come back) collimated in at least one step of their trajectory, by coupling input and output angles in a two-dimensional approach. We expect that our method can be further generalized to three dimensions with a vectorial formalism to generate freeforms, but it can already be applied to systems where there is translational symmetry along the axis perpendicular to the analyzed plane. Iterative construction algorithms have already been developed for illumination systems [12] and later generalized to three dimensions to generate freeform mirrors [13]. For lenses, a known method is the Construction-Iteration process, which begins with an initial lens and is iteratively optimized [14]. However, our method exploits the nature of collimating lenses to obtain solutions with a simple geometric construct, requiring no optimization processes or solving differential equations.

2. Mathematical framework

In two dimensions, it is possible to treat light rays as linear functions $r(x)$ and the surfaces of lenses and mirrors as functions $f(x)$ in a cartesian space, then compute their interactions using algebra and calculus. For rotationally symmetric systems, it is very convenient to set the $x$ axis as the optical axis. Let ($x_0$, $y_0$) be the initial position and $\theta _{\text {in}}$ the inclination of an input ray in relation to the $x$ axis, then the function that describes the path of light is:

To compute reflection and refraction analytically, one must first obtain the coordinates of the incidence point $I = (x_i,y_i)$, that is, where the ray and the surface meet. The analytical solution can be found by equalling both functions and solving for $x$, as long as the resulting incidence equation is not transcendental, in which numerical approaches must be used.

In this framework, all angles must be measured with respect to the $x$ axis. Since derivatives provide the slope of functions at a given point, the tangent and normal angles $\theta _T$ and $\theta _N$ of the surfaces are obtained directly from them:

By tracing a hypothetical ray towards a generic surface, it is possible to derive through trigonometry that the reflection and refraction laws in this frame of reference are given by:

The focus $F(x)$ can be defined as the value of $x$ for which the rays intersect the optical axis after the interaction, that is, the solution of the equation $r_{\text {out}}(F) = 0$.

An advantage of this "differential optics" is that the properties of the surfaces do not need to be known in order to carry out the calculations. Furthermore, the surfaces themselves can be treated as unknowns, turning the equations presented above into differential equations subject to impositions, for example, that all rays focalize at a single point ($\partial F/\partial y_0 = 0$ or $F =$ constant) or that $F$ is a specific function of height. From this, it is possible to obtain the result that the “perfect mirror” is parabolic and the “perfect lens” for a given index of refraction is a hyperboloid or an ellipsoid, without ever having studied about conic sections. For most systems, however, the math quickly becomes cluttered and analytical solutions do not exist. Despite this, numerical solutions to these problems can still be computed and are naturally aspherical.

Notice that in Eq. (3b) all the information about the surface is contained in $\theta _N$, while the input and output angles are arbitrary. The surface angle can be seen as an unknown, which must be solved for the angular relation to be satisfied. Attempting to do that yields a trigonometric equation:

 

Fig. 1. Color map produced by every combination of input and output angles and the respective relative surface normal, with each color assigned to a relative index of refraction. On the right half, a few curves with common values are highlighted in black.

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By including the corrections above in Eq. (5) and inverting the relation presented in Eq. (2) to return $\theta _N$ to its derivative form, one could obtain a differential equation for the surface $f(x)$

Suppose $n = 1/2$, $\theta _{\text {in}} = 0$ and $\theta _{\text {out}} = 30^{\circ }$. These values yield a constant slope of $1-\sqrt {3}$, so the solution is a line inclined at approximately $-41.94^{\circ }$ in relation to the $x$ axis. The input and output angles in relation to this surface’s normal are approximately $23.79^{\circ }$ and $53.79^{\circ }$, and they satisfy Snell’s law. Although this case seems trivial, it confirms the validity of Eq. (5) as an “inverse” to Snell’s law, where the surface angle is the unknown.

Now let’s imagine the case where the input angle is left as a constant while the output is a function of $x$: The position of incidence is completely unknown until the ODE (ordinary differential equation) is solved, but we know the function is continuous and must force the rays to be deflected according to the chosen function. Suppose $\theta _{\text {in}} = 0$, and $\theta _{\text {out}}(x) = x, -\pi /2<x<0$. Since the ray hitting the surface at $x=0$ must remain with an angle equal to zero, we know the function must have a vertical slope at that coordinate. In the chosen interval, the output angle must be increasingly negative, thus the solution must be a convergent lens. It is easy to infer that a solution does not exist for $0<n<1$ since the function is concave left and that would result in a divergent lens. All ODEs of this class are solved via direct integration, but the antiderivatives may not have a closed-form expression. Besides, for other functions, the shape of the surface can be hard to predict and require precautions with the sign issues discussed earlier. One could also attempt to set the output angle as a function of $f$, that is, the $y$ coordinate of the incidence. This results in a highly non-linear ODE, and the solvability is even less likely than in the previous case.

3. Assembly methods: direct and midpoint

The remaining possibility is to make the output angle dependent only on the input angle, but that precludes the ODE from being solved, as the differentials do not correspond to the variables on the right side of the equation, and an alternative approach must be made. Luckily, the angular relation of Eq. (5) gives a hint about the way to go. For each set of input-output pairs, a unique set of surface angles can satisfy Snell’s law. We can imagine an infinitesimal piece of a surface’s cross section being assigned to each input ray, and the junction of all of their pieces results in an aspheric curve (for an aspheric surface, the pieces would be infinitesimal planes), which is smooth if the angular distributions are also smooth. We can also assume that there are infinitely many solutions since the shape of the aspheric must change accordingly to its position. With that principle in mind, a piecewise lens can be constructed from a starting vertex point, each piece satisfying the desired angular relation for a ray, and as the amount of pieces tends to infinity, the curve converges to the real solution.

Here, elements of a set will be denoted as a function of their position in the set, so $r_{\text {in}}(1,x)$ is the first element of the set of input rays $r_{\text {in}}$, but also a function of $x$. From Eq. (1) we know rays are described by position and angles, so the sets $x_{\text {in}}$, $y_{\text {in}}$ and $\theta _{\text {in}}$ are defined to make it possible to define the set of rays $r_{\text {in}}$, whose $i$-th element is

 

Fig. 2. Mathematical objects involved in the direct iterative assembly.

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Thus, the next lens vertex is the solution to the linear intercept equation between the current input ray and segment $s(i,v(i+1)) = r_{\text {in}}(i,v(i+1))$. A precaution to be taken is that the aspheric must be placed before the input rays intersect each other (if they do), otherwise the solution will break in that region and some vertices may even intersect each other. This is not necessarily a flaw in the algorithm but an indication that the solution does not exist in that region.

It should be noticed this method makes every ray hit a corner, where the normal is undefined, and only the immediate neighbor rays below them (but not above) will be refracted correctly. A workaround to this issue requires extra steps:

 

Fig. 3. Comparison between direct and midpoint iterative assembly methods for aspherical lens generation.

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Fig. 4. Diagram relating all variables involved in the iterative assembly. The free variables are highlighted in blue, the asterisk representing $v(1)$, which is contained in the set of vertices $v$.

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Although it is not an integration technique, but a geometric construction using intercepting lines, this method seems similar to an Euler integration, as it constructs the solution iteratively from a starting point based on known slopes, but here the step size is determined by the input rays. While it should also accumulate error with each step, the restriction of passing through every beam should drastically reduce deviations. To show this, we used the plano-convex lens with a perfect focus at $F=0$ and vertex at $x = V$, which is described by the following hyperboloid

 

Fig. 5. Comparison between a perfect hyperboloid lens and the aspherical generated by our algorithm.

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It is noticeable in Fig. 5 that the generated solution has a higher concentration of vertices towards the center. That is because there is also a higher density of input rays in that region, which coincides with the region with the most delicate variation in curvature of the aspheric. Other aspherics should have the same order of precision as long as those regions coincide. The cost of that advantage in precision is that every piece of the lens has a different length. If the aspherical lens is fabricated as a piecewise and it is preferable that the intervals have the same size, then some manipulation with the ray input distribution or the resulting data must be done, such as interpolating the vertices. In any case, this method is expected to provide solutions for cases where analytical solutions do not exist.

By making $\theta _{\text {out}} = k \theta _{\text {in}}$, where $k \neq 0$, the resulting lens generates virtual images with a considerable positive spherical aberration and magnification equal to $k$ for central rays. Given that the real output rays are equal to the input bundle but translated and broadened or narrowed, an arbitrary amount of surfaces could be generated to increase or decrease magnification. To assess the image quality of an object with finite size, additional ray tracing compatible with piecewise lenses is required and aberrations are to be expected, since the angles are corrected perfectly only for a single point.

4. Correcting the spherical aberration of mirrors and manipulating their focus

The parabolic mirror has a special property of reflecting rays parallel to its axis of symmetry towards a single point, a perfect focus. Meanwhile, the spherical mirror is easier to fabricate but lacks a perfect focus, instead creating a funnel-like array of rays identified as the spherical aberration. We took advantage of the optical reversibility and the presented method to correct a spherical mirror with a unitary radius whose vertex is located at $(-1,0)$. First, a point source of light was created at ($F = -1/2,0$), then its rays were traced towards the mirror with a uniform angular distribution from approximately $\pi$ to $\pi /2$, generating a second set of rays, which become the set of inputs. The next step was to generate an aspheric that collimates all the light using the relation $\theta _{\text {out}}(\theta _{\text {in}}) = 0$, that is, all rays become parallel to the $x$ axis. The vertex point for the aspheric was defined as $(V_x,V_y)$. For $99$ input rays, a lens with index of refraction of $1.5$ and a vertex placed at a distance sufficient to catch the reflected rays before intersecting each other, the resulting aspherics are shown in Fig. 6.

 

Fig. 6. Aspherical lens that corrects the spherical aberration of the spherical mirror. (a) Source, input and output rays, and the aspheric positioned at $V=0$. (b) Graph of the solution as $V$ varies between $-1$ and $0.25$. The algorithm allows solutions closer to the mirror to extrapolate its surface, but they were cut out.

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It is worth noticing that in this example, and in any system needing correction, the resulting beam acquires an inhomogeneous energy density in the vertical direction and, by the optical reversibility, an incoming beam would need such ray distribution to illuminate the focal point uniformly. This can only be circumvented with optical elements designed for apodization.

A parabolic mirror with the same parameters for vertex and curvature (the focal point at $(-1/2,0)$) was also tested, resulting in a plane aspheric (Fig. 7(a)) (as correcting it is redundant). Since the spherical aberration is guaranteed to be corrected and the optical path begins at the desired focus, we gain the ability to change the focal point and even make it off-axis by changing the point source, resulting in various aspherics, as it is shown in Figs. 7(b) and 7(c).

 

Fig. 7. Solutions for the parabolic mirror. (a) $F=-R/2$, a plane lens (here, the starting vertex position is irrelevant). (b) $F=-3R/4$ and $V=-R/2$, a convergent lens. (c) $F=-R/4$ and $V=0$, a divergent lens.

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As shown in Fig. 8, a simulation where the user can change the starting vertex, focal point, start and end angles of the source, relative index of refraction, output function, and the conic constant (from spherical to parabolic) of the mirror was made available in GeoGebra, where the mathematical constructs for testing this algorithm were also created. It can be accessed in https://www.geogebra.org/m/qt7aegza. The number of vertices was set to 25 to optimize browser performance since GeoGebra’s dynamic calculations often slow down the application.

 

Fig. 8. GeoGebra simulation of the iterative assembly employed to correct the spherical aberration of mirrors. The points and sliders allow the user to change parameters by dragging them.

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5. Exploring other applications

In the previous section, only plano-aspheric lenses were generated. However, the algorithm should be applicable to design aspheric-aspheric lenses as well. On the former, the $\theta _{\text {out}}(\theta _{\text {in}})=0$ imposition can collimate the beam inside the lens if the aspheric is the anterior surface. As the posterior surface, it can multiply the increase/decrease in angle caused by the anterior surface by a factor of $k$ by defining $\theta _{\text {out}}=k\theta _{\text {in}}$ or also force the rays to become collimated outside the lens.

Similarly, in the aspheric-aspheric case the deflection can be divided between the surfaces: the first aspheric is defined by $\theta _{\text {out}}=k\theta _{\text {in}}$, where $1 > k > 0$ while the second one by $\theta _{\text {out}}=0$. If $k = 1/2$, each surface reduces the original input angles by a half. The lens can also magnify a point image by a desired factor if each surface has $k > 1$.

Theoretically, an arbitrary amount of aspherics could be created in succession to the point they could simulate a discretized GRIN medium.

The major challenge is, perhaps, to make an aspheric capable of creating a perfect focus for a given input that is not collimated by finding the correct $\theta _{\text {out}}$ function. This is far from being simple as the cases shown so far for two reasons: all ray structures depend on their angle and position, and in this case we do not have control over either aside from the starting vertex; there are two unknowns, and only $\theta _{\text {in}}$ is given. The choice of the output function changes the vertex positions and normals of the aspheric itself, so it is impossible to make one of the unknowns constant. For example, if the input is a point source, the real solution must necessarily be a convex, converging lens for $n>1$. The function $\theta _{\text {out}} = -\theta _{\text {in}}$ seems a good starting guess, as it should make all rays (which are diverging from the source) to converge. This would work if the aspheric could be a plane lens, but we know each input angle requires a different normal angle to be properly refracted. Perhaps one way to start searching for this solution would be to define $\theta _{\text {out}}$ as a Taylor series around $\theta _{\text {in}}=0$ (with $a_0 =0$) and use an optimization algorithm to minimize the longitudinal spherical aberration. Non-polinomial functions can be tested, and the ones that give closer results can have their Taylor coefficients extracted and used as initial guesses. Figure 9 shows two attempts at finding the solution, the second one being manually adjusted until a good convergence was found, although some negative spherical aberration can be noticed on higher rays, indicating $\theta _{\text {out}}$ must grow slightly faster with respect to $\theta _{\text {in}}$ .

 

Fig. 9. Two attempts at finding an aspheric surface capable of refracting light from a point source into another point. The ratio between the $x$ and $y$ axes is 2.2:1.

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It should be impossible to contemplate all angles from $0$ to $90^{\circ }$ since the higher the angle, the lower the slope of its corresponding solution - at some point, the next ray will not hit the lens. From that, we can conclude this curve should have an oblique asymptote like the hyperbolas or grow even faster towards the $x$ axis. In the first case, the last ray will hit the lens infinitely far away and tangent to the surface, where the maximum refraction will occur. For those rays to be refracted toward the focal point, the output angles must be inclined backward, which would violate the law of refraction. The only way to conciliate that issue with the central rays would be by pushing the focal point to infinity, causing all output angles to be equal to zero, which is exactly the same case we used to test this algorithm and the opposite of what we wanted. In any case, this example shows that the solution cannot exist if the last ray does not hit the lens at a finite distance. Since all functions involved in this framework are smooth, it is unlikely a solution would work only for an interval of input rays and not all of them.

6. Conclusion

The iterative assembly proved to be a simple and intuitive method to obtain aspherical curves for optical systems with collimated paths. The angular relationship that underlies its operation, despite giving up control over spatial properties aside from the starting vertex, makes it possible to directly manipulate the angular variables of an input of rays. Even in situations where solutions require more advanced methods and analytical solutions are impossible, the algorithm allows functions to be tested and properties of the real solution to be inferred. This algorithm can be compared or used in conjunction with other 2D aspheric generation methods, and furthermore generalized to three dimensions, allowing for freeform lenses to be generated.