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Abstract
Multifocal and/or extended depth-of-focus designs are widely used in many optical applications. In most of them, the optical configuration has axial symmetry. A usual design strategy consists of exploring the optimal wavefronts that emerging out of the optical system would provide the desired multifocal properties. Those properties are closely related to light concentration on caustic surfaces. We present a systematic analysis of how to obtain those multifocal wavefronts given some prescriptions on the locations of caustics. In particular, we derive several multifocal wavefronts under archetypical prescriptions in the sagittal caustic alone, or combined with the tangential one at certain points, with some emphasis on visual optics applications.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Multifocal and/or extended depth-of-focus optical beams and designs are experiencing a boom in optics with the advent of techniques allowing spatial control over the phase of light beams (for instance, thanks to spatial light modulators), in what is called wavefront or phase-front shaping. The diversity of applications is high: data storage [1], microscopy [2], computational photography [3], ophthalmic corrections for presbyopia [4], optical micromanipulation [5], or nondiffracting beams for non-linear and atom optics [6].
Wavefront shaping consists of exploring the optimal wavefronts that emerging out of the optical system would eventually provide the desired multifocal and/or extended depth-of-focus properties. This can be carried out either by theoretical and computer simulations, experiments, or both [4,6]. Once the optimal wavefront has been found, the next step is to design an optical system configuration coupling an incoming wavefront with that optimal wavefront. This can be carried out in several ways: a holographic technique, a diffractive optical element, such as a phase plate, or through a set of refractive elements. From a mathematical point of view, it has been proved that such configurations exist and can be built as long as the optical surfaces do not contain points of caustic surfaces belonging to the incoming or outgoing wavefronts [7,8].
Multifocality aims at providing light intensity distributions not concentrated around a single point (monofocality) but instead around certain locations or along an extended region. Using an anatomical metaphor [9], the intensity spatial structure of a light beam depends on a wave-field flesh over a skeleton determined by ray-optics caustics. Within ray optics theory, caustics are associated with spatial locations where two or more rays intersect, hence providing those locations with a higher concentration of light. Although in most of the through-focus applications wave optics theory is mandatory, for accurate simulation of beam propagation or evaluation of optical performance, the ray-optics skeleton offers a reasonable approach for, at least, characterizing some basic properties or exploring possible design solutions [10–12]. Furthermore, an asymptotic (ray-optics) analysis of wave optics reveals finite and non-homogeneous intensity at caustics depending on the density of rays crossing them [13]. Particularly, in an axially symmetric system, ray density and hence light intensity at the sagittal caustic (caustic sheet along the optical axis) is greater than that at the tangential one [14]. From all this, caustic analysis emerges as a natural tool to explore the multifocal properties of wavefronts.
Although in recent times the major interest in caustics has come from the hand of non-diffractive beams [6,11,12] caustic analysis also offers a high potential for multifocal optical design problems where light is intended to be confined in a segment along the optical axis. Some previous works are proposing caustic surfaces for optical design [15–19]. More recently, it has been proved the potentials of diapoints (closely related to caustics) as an initial metric in axially symmetric optical design [20]. However, these studies have not led to a systematic line of research, and what is more relevant for this work, caustic analysis, to our knowledge, has not been applied for multifocal applications. All this, even though caustics have a long history in optics, firstly studied in Renaissance times by Maurolycus [21].
The general goal of this work is to offer a theoretical framework to predict the existence or not of wavefronts with prescribed multifocal properties. Additionally, we describe the fundamentals of a methodology to design multifocal solutions given some caustics restrictions. We shall limit our analysis to axially symmetric wavefronts, considering that is a very common restriction due to manufacturing and assembly limitations. First, in section 2 we review the formulas that provide the caustics associated with the wavefront located at the exit pupil. Although they are not difficult to be deduced, we do it for completeness. In section 3 we explore some options when restrictions are only imposed on the so-called sagittal caustic (more intense than the tangential one). The theory of ordinary differential equations allows us to obtain some valuable results on the existence and uniqueness of solutions. Additionally, we provide an example in the field of visual optics. In section 4 we expand our analysis to joint restrictions on the sagittal and tangential caustics. In this case, general statements are more difficult to prove so we will limit ourselves to giving some numerical solutions. Finally, in 5 we will summarize major results and discuss some technical issues not handled in this work.
2. Caustics of an axially symmetric wavefront
There are two, formally equivalent, ways of defining caustics. First, setting a ray mapping between two-dimensional space coordinates, caustics are the set of points in the image space where the Jacobian of the ray mapping is zero [9,22]. Second, as the geometric locus of centers of principal curvatures of the wavefront [23]; since there are two principal curvatures, each wavefront generates two caustic sheets. In this work, we follow the second definition that apart from geometrically being more appealing leads to more straightforward algebraic expressions.
Let’s consider the space of smooth axially symmetric wavefront functions $W$ (z: axis of symmetry) parametrized as:
The unit normal vector to the wavefront surface is:
and the so-called tangential ($k_t$) and sagittal ($k_s$) principal curvatures are given by:Setting for convenience the meridional plane with $\theta = 0$, we reduce the problem to two-coordinates vector value functions, from now on $(x, z)$, being z the coordinate along the optical axis, but with a single independent variable $x$: $\textbf {w} (x) = \left (x, y(x)\right )$ and $\textbf {n}(x) = \frac {(- y'(x), 1)}{\sqrt {1+y'(x)^2}}$. The caustic surfaces, as the locus of centers of principal curvatures of the wavefront, can be obtained from the wavefront surface simply computing the segments along the wavefront normals which modulus are the inverse of the principal curvatures [23]. Therefore, the equations providing caustic curves, are:
As shown by Eq. (4), axially symmetry imposes that the sagittal caustic ($\textbf {c}_s(x)$) degenerates into the segment of a straight line along the z-axis that is called a spike.
Equation (4) shows two coupled non-linear differential equations for function $y(x)$: the wavefront. In other words, Eq. (4) can be seen as integrability conditions for the construction of a multifocal wavefront under prescribed caustics. However, in general, setting simultaneously $\textbf {c}_t(x)$ and $\textbf {c}_s(x)$ does not lead to an admissible solution. In the following sections, we will study some solutions for some archetypical caustic prescriptions.
3. Prescribed sagittal caustic
As already mentioned in section 1, sagittal are considerably brighter than tangential caustics. Therefore, in many cases of practical interest, one could construct the wavefront with only prescribed sagittal caustics, while ignoring the tangential. Although, after wavefront construction, it should be checked that the tangential caustic location is within some quality metric criteria. Then, the wavefront construction is simplified to a single first-order ordinary differential equation. As the sagittal curve is a rectilinear line along the optical axis, the simplest and most obvious prescription is to set the sagittal line segment’s boundaries.
When those limits correspond to rays coming from the pupil center and edge, that is equivalent to a wavefront design where pupil center and edge focuses are selected. Such a wavefront, where the most intense light concentration is within a closed segment along the optical axis, is called an axicon [24]. Axicons are used in a great diversity of optical systems such as laser inspection, optical coherence tomography, or optical trapping (see, e.g. references in [25]).
Let’s define the ray mapping between points on the wavefront $\textbf {w}(x)$ and the sagittal caustic: $z \equiv g(x): [0, \xi ] \rightarrow [z_c, z_e]$, $z_c$ and $z_e$ setting the limits of the z-coordinate along the sagittal caustic. Then, the second of Eq. (4) can be rewritten as:
Equation (5) becomes an exact, or total, differential equation if and only if $g(x)=c$ (being $c$ an arbitrary constant). In other words, if all the rays converge to a single point; thereby, multifocality degenerates into monofocality. Excluding this degenerate case, Eq. (5) becomes a Cauchy initial value problem given that we can arbitrarily set a zero value of the wavefront at the pupil center ($y(0)=0$).
By Peano’s existence theorem [26], Eq. (5) has at least one solution in $x \in [0, \xi ]$ if $g(x)$ is continuous and bounded and also $g(x) \neq y(x) \quad \forall x \in [0, \xi ]$. If additionally $g(x)$ is Lipschitz continuous then the solution is unique (Lipschitz’s uniqueness theorem, p. 68 [26]). But both mathematical conditions must be met because for the function $g(x)$ to be an admissible geometrical optics mapping, it must be a symplectic transformation [27], that is, a smooth function and thereby Lipschitz continuous.
Deriving $y'(x)$ in Eq. (5) and then substituting $y''(x)$ and $y'(x)$ in the tangential caustic expression of Eq. (4) we obtain:
Therefore, if a numerical solution for $y(x)$ is obtained, the tangential caustic of such wavefront could be numerically obtained by recursively computing first Eq. (5) and subsequently Eq. (6).
3.1 Uniform light intensity along the sagittal caustic
In some cases, the desired target might be an axicon having a uniform light intensity along with the sagittal caustic. Within the asymptotic ray-optics solution of wave optics, the intensity along a geometrical caustic is proportional to the density of rays falling on per unit area [13]. Although the uniform light intensity axicon can be solved using the geometric law of energy conservation [28], here we follow a different approach.
Considering that the ray density is proportional to the Jacobian of the ray mapping, the uniform ray density condition leads to a linear ray mapping equation:
which after substituting in Eq. (5) gives:As mentioned before, in the degenerate case of monofocality $z_c=z_e$, Eq. (8), with initial condition $y(x)=0$, sets a Cauchy problem with an explicit solution:
which, as expected, is the circumference equation centered in the wavefront vertex. For other values of $z_e$ and $z_c$, apparently Eq. (8) only has numerical solutions.As an example of application, we obtained a uniform intensity axicon for a visual optics application. In the human eye, multifocal lenses (either contact or intraocular lenses) provide a common solution to compensate for the presence of presbyopia in combination with other refractive errors such as myopia. A standard eye model has a mean optical power of 60 diopters. A typical pupil size might be $\xi = 3$ mm [29], and an addition for presbyopia correction might be 3 diopters. For these numbers we get $z_c = 16.7$ mm and $z_e = 15.9~mm$. For this example, we implemented the numerical algorithm to solve Eq. (8) with the help of ode45 MATLAB function.
Figure 1 shows the solution obtained: the wavefront elevation $y(x)$ and the difference in elevation with respect to a spherical wavefront of radius of curvature: $z_c = 16.7$ mm. Figure 2 shows the shape of the tangential caustic curves when imposing a uniform density of rays along the sagittal caustic. A segment 0.8 mm of length in the sagittal caustic induces a tangential caustic 0.97 mm of length and 0.05 mm of height.
Fig. 1. Wavefront elevation (mm) and wavefront difference with respect to reference sphere for a uniform light intensity axicon.
Fig. 2. Sagittal (blue line) and tangential (red line) caustics curves associated with a uniform intensity along the sagittal caustic axicon.
3.2 Non-uniform density of rays along the sagittal caustic
In other cases, the target might not be a uniform intensity along the z-axis. For instance, in visual optics, major importance is given to far and near vision foci (pseudo-bifocal multifocality) or to a single focus with some extended depth-of-focus. Those design targets could be met with the following $g$ mapping:
with $n$ being a positive integer number. If $n=0$ we get the uniform-intensity axicon; for $n>1$ we get a wavefront that gives more weight to center and edge pupil foci over the rest; and for $0<n<1$, only a focus is generated but with a extended depth-of-focus. Using the same visual optics example from before we computed both caustics for the following values of n: 0.5, 1, 1.5 and 2.Figure 3 shows that increasing the concentration of rays in the two ($n>1$) extreme focuses makes larger the length of the tangential caustic. On the other hand, for $n<1$ the length of the tangential caustic is substantially decreased, although, at the expense of reducing the number of prioritized focuses to one. We note that the relevant focus is located at the intersecting point between the tangential and sagittal caustic, due to the presence of an umbilical point (equal principal curvatures) at the wavefront.
Fig. 3. Sagittal (blue line) and tangential caustic curves for different concentration of rays along the sagittal caustic.
4. Joint restrictions on sagittal and tangential caustics
Sometimes, when the effect of the sagittal caustic can not be ignored, simultaneous conditions on both the sagittal and tangential caustics may be required. Then, the only reasonable strategy is either to impose a bound to the maximum separation between both caustics or to impose perfect focusing at certain points of the wavefront (umbilical points). In the following, we follow the latter strategy. In this section we analyze several cases of this scenario. We first note the impossibility of a perfect geometrical axicon generated by a smooth (continuity up to the second derivative) wavefront, i.e. one where both caustic curves are coincident along a z-axis segment. Such wavefront would be a surface with a varying principal curvature and zero cylinder (difference between principal curvatures), which is not admissible by the compatibility integration conditions of surface theory [30]. Relaxing the smooth condition and within the paraxial approximation there are special wavefronts (the so-called Durnin’s beam) that only have one caustic sheet [31].
4.1 Multiple umbilical points
Here, we have studied solutions where both caustic curves only touch at certain points along the z-axis. Therefore, at those points the wavefront is umbilical. Let’s denote $\Delta _n := {(x_1, y(x_1)) \cdots, (x_n, y(x_n))}$ a sequence of the $y(x)$ curve generating the wavefront surface. For each point of the set $\Delta _n$, we prescribe that both caustics curves intersect at points along the z-axis ${z_1,\ldots, z_n}$. Then, we can write:
Therefore, finding $y(x)$ becomes a interpolation or approximation problem with conditions given by Eq. (12).
4.1.1 Two perfect focus (umbilical points)
Let’s choose a wavefront with two umbilical points: $\Delta _n := [(0, z_c), (\xi, z_e)]$. Then, conditions in Eq. (12) become:
The problem can be solved by setting a perturbation solution. Particularly, $y(x)$ is modeled as a reference sphere plus a polynomial with two coefficients (perturbation term):
Equation (14) fulfils the conditions in the center of the pupil: $y(0) = y'(0) = 0$ and $z_c = \frac {1}{y''(0)}$. The values of $c_1$ and $c_2$ are obtained imposing conditions on $x = \xi$ to Eq. (14) and subsequent numerically solving with the help of fsolve MATLAB function.
As an example, we solved the two-umbilical wavefront problem using data from section 2: $z_e = 15.9$ mm and $z_c = 16.7$ mm. Figure 4 shows the obtained wavefront expressed as a difference function with respect to the reference sphere of radius $16.7$ mm. Figure 4 also shows both principal wavefront curvatures. Notably, the tangential curvature presents what is called a ridge point, i.e a stationary point of the principal curvature [32].
Fig. 4. (a) Tangential and sagittal curvatures (D) as function of radial coordinate (mm). (b) Wavefront difference ($\mu m$) with respect to reference sphere. Two-umbilical points wavefront.
Figure 5 shows the caustic properties of the obtained wavefront. Green solid lines represent the rays near the caustics. For reference, special rays corresponding to pupil center, middle pupil and pupil edge are depicted with black solid lines. Red and blue solid lines represent the tangential and sagittal caustics, respectively. It is shown that the solution is obtained thanks to the presence of a rib point (a singular point in the caustic curve where its first derivative changes its sign) in the sagittal caustic associated to $x = 2.35$ mm. Such caustic rib points are known to be associated with ridge points of the wavefront [32]; in our case the ridge point of the tangential curvature shown in Fig. 4.
Fig. 5. Rays (green line), and tangential (red lines) and sagittal (blue lines) caustics of a two-umbilical-focus wavefront.
4.1.2 Three perfect focus (umbilical points)
We have also computed an example of a wavefront providing three perfect foci: $(0, z_c)$, $(\xi /2, z_n)$, $(\xi, z_c)$. Again, we used parameters from the example of section 3. As before, we set a polynomial solution and searched for the optimal (least square sense) polynomial coefficients with the help of fsolve from MATLAB. Now, the number of terms of the polynomial is four:
Figure 6 shows the obtained wavefront expressed as a difference function with respect to reference sphere and additionally its principal curvatures. For this wavefront the tangential curvature presents two ridges points, located at: $x = 1.1$ mm and $x = 2.4$ mm.
Fig. 6. (a) Tangential and sagittal curvatures (D) as function of radial coordinate (mm). (b) Wavefront difference ($\mu m$) with respect to reference sphere. Three-umbilical points wavefront.
Figure 7 shows the caustic properties of the obtained wavefront. As before, green solid lines represent the rays near the caustics. For reference, special rays corresponding to pupil center, middle pupil and pupil edge are depicted with black solid lines. Red and blue solid lines represent the tangential and sagittal caustics, respectively. There are two rib points associated with the ridge points of the wavefront shown in Fig. 5.
Fig. 7. Rays (green line), and tangential (red lines) and sagittal (blue lines) caustics of of a three-umbilical-focus wavefront.
4.2 Designing transition zones in concentric designs
In this section, we address the so-called multifocal concentric designs, which are widely used in the field of visual optics. In these designs, the pupil of the optical system is divided into several concentric rings. Then, the optical system is designed so that the associated wavefront, within each section, converges to a single focus [33]. A major drawback of these designs is how to tackle the unavoidable transition between different sections. This transition can be established to be abrupt, thereby the wavefront present discontinuities between concentric rings, or smooth, so that the wavefront is a smooth function also at those joints. Here, following the second requirement (specifically, we ensure infinitely differentiable functions), we present a caustic-prescriptions procedure to construct such transition zones, which is formally equivalent to the two-umbilical foci construction we presented before.
Say, we want to design the transition between a pupil section with a focus at $z_{1} = 1.4$ mm and another at $z_{2} = 1.6$ mm. We arbitrarily set a transition zone in the interval $(x_{1}, x_{2})$. Therefore, we have four caustic conditions given by:
As in the case of the two-umbilical focus design, we solved the problem setting a polynomial with two coefficients, which is added to the spherical wavefront associated with the first focus ($z_1$), and subsequently numerically solving the set of equations. Figure 8 shows the obtained wavefront expressed as a difference function with respect to the reference sphere (radius $z_1$) and its principal curvatures. For this wavefront, the sagittal curvature changes smoothly (in a sub-linear manner) but the tangential curvature presents a ridge point at $x = 1.52$. As mentioned above the undesired difference between caustics is unavoidable and mainly depends on how long is the segment of the transtion zone: short transitions induce higher differences than long transitions. The reason is that the difference in principal curvatures depends on the ratio of change of the varying principal curvature [34]. It is worth noting that the smooth character of these transition zones, as revealed by Fig. 8, ensures the feasibility in the manufacturing.
Fig. 8. (a) Tangential and sagittal curvatures (D) as function of radial coordinate (mm) in the transition zone $x\in [1.4,1.6],$ of a concentric ring design. (b) Wavefront difference ($\mu m$) with respect to reference sphere.
5. Discussion
Caustics determine the locations of ray intersections and therefore regions of maximum light flux concentration. In consequence, prescribing caustic locations is a blueprint to, at least, pre-design multifocal wavefronts. Of course, as we already mentioned in section 1, obtaining the optimal wavefront is just the first step in a real design. A subsequent step is to derive an optical system configuration providing that wavefront which can be achieved using a single refractive/reflective surface [7,8,35].
In this work, we have not explicitly considered intensity distributions along the caustic surfaces, something that may be required in some design methods. Within wave optics theory, the conventional procedure is to compute the optical field through diffraction integrals. However, in some cases it is possible to obtain simpler relations using the phase stationary principle [9], additionally establishing a bridge between wave and ray optics. One alternative, strictly confined within geometrical optics theory, is to define intensity over a caustic surface as proportional to the density of tangent rays over that surface [14]. Therefore, our multifocal wavefront design approach might be extended to consider intensity at caustics.
We note that our analysis could be substantially simplified if paraxial approximations were applied to caustic formulas. The paraxial approximation, within this context, implies that $1+y'(x)^2 \approx 1$ in Eq. (2) and Eq. (3) and in the subsequent equations. Indeed, such approximation is justified in visual optics applications [4] and has led to some successful explanations of visual phenomena such as starbursts [36,37]. However, the paraxial approximation might introduce considerable errors in other multifocal designs that typically use high numerical apertures, for instance in microscopy or data storage.
With regard to the problem of designing transition zones in concentric multifocal designs for visual optics, it is known that it is not possible to vary a principal curvature without introducing cylinder [34], except at some isolated lines in special surfaces. Therefore, an open and challenging mathematical problem is to design such transition zone minimizing a metric quantifying the distance between tangential and sagittal caustics. A problem that has been formalized, in the context of progressive lens design, by Jian et al. [30].
6. Conclusions
We have presented a systematic analysis of the procedures required to design wavefronts for multifocal applications based on caustics, although under the restriction of axially symmetric optical systems. We have derived several multifocal wavefronts under archetypical prescriptions in the sagittal caustic alone, or combined with the tangential one at certain points. The latter one is specially suited for visual optics applications. In particular, we have presented a method to design transition zones in concentric multifocal designs.
Funding
Spanish Ministerio de Ciencia e Innovacion (Grant PID2020-113596GB-I00).
Acknowledgments
This work was supported by grant PID2020-113596GB-I00 from the Spanish Ministerio de Ciencia e Innovacion. We thank Javier Portilla for his valuable discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data availability. No data were generated or analyzed in the presented research.
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