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Abstract
We propose a method to design the exact phase profile of at least one metasurface in a stigmatic singlet that can be made to implement a desired ray mapping. Following the generalized vector law of refraction and Fermat’s principle, we can obtain exact solutions for the required lens shape and phase profile of a phase gradient metasurface to respect particular ray conditions (e.g., Abbe sine) as if it were a freeform refractive element. To do so, the method requires solving an implicit ordinary differential equation. We present comparisons with Zemax simulations of illustrative designed lenses to confirm the anticipated optical behaviour.
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1. Introduction
Metasurfaces are two-dimensional arrangements of subwavelength scatterers that can modify the phase, polarization and amplitude of propagating optical waves. They have recently attracted a lot of attention because they enable unprecedented functionalities within a small form factor compared to traditional optics [1–4]. However, designing these optical surfaces is still very challenging. An emergence of numerical techniques have been proposed in order to design the optimal phase profile and the nanostructures of metasurfaces, such as inverse design using a combination of deep learning and adjoint optimization [5–9]. Deep learning approaches culminate with the realization of end-to-end surrogate solvers for metasurface design, solving for an optimized phase response for a library of meta atoms within a hybrid refractive-diffractive system or in a diffractive-only system [10,11].
For metalens imaging, approaches based on ray tracing and the generalized law of refraction [12] can be insightful when trying to limit the aberrations of a hybrid design [13–15], a singlet [16] or even a doublet metalens optical system [17–19]. Unfortunately, those models are either based on approximate solutions of a required phase profile, are difficult to generalize for utilization inside a complex optical system, or provide non-trivial solutions to work with (e.g., integration of the roots of a fourth-order polynomial [16]). Therefore, adopting an optimization-based approach appears promising as the way forward. To optimize a hybrid design consisting of refractive and metasurface lenses, as for diffractive hybrid design, it is advised to model the metasurface as a planar phase surface to be optimized with the other optical elements in Zemax [20,21]. This procedure will return optimized phase profiles of planar metasurfaces for a given optical system. However, seeking exact analytical solutions is important as they could be used as a baseline performance estimator and benchmark [22]. These exact solutions can perform quick optical design over a few free parameters giving insights into important design parameters [23,24] or can be applied to obtain exact predetermined coefficients in a design [25]. As such, finding exact solutions for the phase gradient of the metasurface is relevant for the field of metasurface design and could help refine the existing optimization processes. This would be especially relevant for applications requiring the optimization of both the freeform of the substrate and the metasurface phase gradient [26–28]. For traditional refractive or reflective optics, the exact freeform lens profile to limit spherical and coma aberrations or to perform a particular ray mapping can be determined analytically according to Fermat’s principle and ray tracing [29,30].
The fundamental but simple idea of the analytical lens design framework is to introduce functional optical properties and relate them to the refraction or reflection of an optical system [29,30]. Applying Fermat’s principle to any ray entering the pupil of a system, it is possible to solve for free parameters, such as lens curvatures, in order to obtain a system without spherical aberration and coma, i.e., stigmatism [30,31]. The framework is not limited to stigmatic systems; it can be made to find the physical properties of a refractive or reflective design to have a given ray mapping in a process that does not require any optimization or numerical approximation.
The scope of this article is to introduce the metasurfaces to the analytical lens design framework. As such, it will be possible to analytically or semi-analytically find the optical element phase profiles and shape based on the generalized vector law of refraction and Fermat’s principle. Firstly, we present the model to obtain a stigmatic lens, a lens without spherical aberration and coma. Then, the method to produce a given ray mapping is described. Some illustrative examples of lens design are presented, and their optical properties are confirmed using Zemax optical design software. Finally, we present more cases where the technique can be applied, such as in non-imaging devices. However, this article is not interested in how to design the meta atoms required to obtain the design metasurface phase profile.
2. General stigmatic metasurface system
Consider the general singlet metalens geometry presented in Fig. 1. The singlet is composed of two optical surfaces defined by $(r_a, z_a)$ and $(r_b, z_b)$ that may implement a phase profile $\phi (r_a)$ and $\psi (r_b)$, respectively. $r_a$ is the only independent variable. $l_o, h_o$ and $l_i, h_i$ are the object and image distances and heights to the edge of the lens on the optical axis, respectively. The parameter $t$ is the thickness of the lens when considering symmetric lenses. When considering radially asymmetric lenses, the distance $t$ functions as an effective back focal length, and we need to introduce the length parameter $d$ corresponding to the difference between this back focal length parameter and the projection of the segment $z_b-z_a$ for the chief ray. The stop is considered to be at the first optical surface (i.e. at $z_a=0$), however, this is not required. $r_{max}$ is the maximum height at the pupil.
Fig. 1. General off-axis singlet with two metalens. The first surface is defined by its shape and phase $(r_a, z_a, \phi )$, and the second surface by $(r_b, z_b, \psi )$ and $\Theta$ is the path inside the lens for any ray. The reference ray is denoted by blue, and an arbitrary ray is schemed in light blue. The stop is considered at the first surface.
One of the most important results from [30] was to show that, given the optical properties of the first optical surface, the second surface should be defined as follows:
where $P_r, P_z$ are the direction cosines according to the refraction inside the lens and $\Theta = \sqrt {(r_b-r_a)^2 + (z_b-z_a)^2}$ is the optical path of a given ray inside the singlet. We need to connect $\Theta$ with ray tracing and the target optical properties, such as stigmatism, to determine the appropriate lens designs. However, it is important to mention that Eq. (1) requires either that all rays converge at one point or that they do not intersect within the lens (i.e. producing a smooth bijective mapping). If only a few rays cross each other inside the lens, the method will result in an inhomogeneous and nonphysical optical element [32,33].Considering the geometry, a ray inside the lens is defined by
By ray tracing, it is possible to relate $\vec {k_2}$, the refracted ray, to $\vec {k_1}$, the ray incident at the interface, following [34,35]
Considering the case where the rays are coming from a point situated at $(h_o,l_o)$, the expressions of a given incident ray and of the normal of the first surface are
where, due to space constraints, dependencies (e.g., $z_a$, $\phi (r_a)$ and $z_a'= \frac{\textrm{d}z_a}{\textrm{d}r_a}$) are assumed to be implicit. The last important parameter is the phase gradient $\boldsymbol{\nabla} \phi = \left [\frac {\partial {\phi }}{\partial {r}},0,\frac {\partial {\phi }}{\partial {z}}\right ]$. For the phase gradient to be tangential to the surface defined by $(r_a,z_a)$, the dot product with the normal should be zero, i.e. $\boldsymbol{\nabla} \phi \cdot \vec {n}_a = 0$. The gradient should also satisfy the traditional vector refraction at the interface, following $\vec {n}_a \boldsymbol{\times} (\vec {k_2} - \vec {k_1} - \boldsymbol{\nabla} \phi ) = 0$ [34]. As such, the phase gradient $\boldsymbol{\nabla} \phi$ isUsing Eq. (3), one can compute the ray after the refraction at the surface given the phase gradient $\boldsymbol{\nabla} \phi$ and the surface curvature $(r_a,z_a)$. Thus, we find the relation for the direction cosines after the first interface given that $\frac {\partial {\phi }}{partial{r_a}} = \phi '$, following
The optical path length of every potential path must be equal in order to obtain a stigmatic optical system [37]. This requirement constrains the design possibilities of the optical system, influencing the shape the second interface, whether or not this interface implements a phase term $\psi$. The design is referenced with a ray characterized by a path $\text {OPL}_{ref}$, to which any possible other path $\text {OPL}_{r_a}$ is to be compared. From geometry, the optical path lengths of both the reference and the arbitrary rays are
Thus, the phase profile at the second surface and its curvature (respectively given by $\psi (r_b)$ and $r_b,z_b$) can be computed analytically from Eqs. (1) and (12).
As such, the strategy to obtain a stigmatic design incorporating at least one metasurface is completed. Hence, we are able to find the profiles of $r_b, z_b$ and $\psi$ from the properties of the first surface. As shown, it requires solving for the propagation length inside the lens $\Theta$ and respecting Fermat’s principle for every ray. This is the same procedure as for freeform refractive lenses or reflective mirrors [30]. Analytical solutions are possible for simple geometries. An example of the definition of the path length $\Theta$ of a radially symmetric stigmatic lens in a finite/finite conjugation is presented in Appendix A. If one is interested in finding the phase of the first surface $\phi$ knowing the other parameters, one could invert the interfaces’ order or solve the differential equation defined by the phase gradient when setting the other parameters of the singlet.
For validation consideration, we will explicitly define the second surface of the singlet. As the second surface ($r_b,z_b$, $\psi$) is dependent on $r_a$, the normal vector to the second surface $\vec {n}_b$ is defined as
where $z_b' = \frac{\textrm{d}z_b}{\textrm{d}r_a}$ and $r_b' = \frac {\textrm{d}r_b}{\textrm{d}r_a}$. The phase gradient implemented by the second metasurface is defined asThus, the refracted vector after the singlet $\vec {k_3}$ can be computed with Eq. (3) with the definition of $\vec {n_b}$ and $\boldsymbol{\nabla} \vec {\psi }$. Finally, one can validate the designed singlet performance by comparing this $\vec {k_3}$ vector with its geometrical definition,
To ascertain the quality of the stigmatic designs, we compare the refracted vector $\vec {k_3}$ in the image space against its definition $\vec {k_3}_{def}$. To this end, we evaluate the error $E$ by taking the mean of the norm of the normalized difference of the vectors, as defined in Eq. (16).
2.1 Stigmatic examples and validation
In this section, some illustrative stigmatic designs with at least one metasurface will be presented. The designs meet Eq. (1) and have an equal optical path length over the entire pupil. Figure 2 displays various geometries of singlets. The position of the metasurface in the lens, the shape of the metasurface and the refractive surfaces are all changed to obtain stigmatic performances while keeping the same physical properties for every lens (e.g., $\lambda$, $l_o$, $l_i$, $t$, $r_{max}$, $n_o$, $n_s$, and $n_i$ are fixed). We obtain a myriad of designs that are always stigmatic with the same pair of points, even with off-axis deviation in the case of Fig. 2(d). As such, the proposed technique is general and can be applied to a variety of geometries, just as for the analytical lens design framework for refractive lenses. It can reproduce results found in the literature such as correcting the phase profile of a planar metasurface considering the substrate as presented in [16]. In fact, Fig. 2(a) is the same case with different input parameters. As such, we can simply compute the exact phase profile required to reach stigmatism of the singlet without any numerical or paraxial approximation, and even without integration if we are interested in a design consisting of only one metasurface at the exit surface of the singlet as shown in Appendix B. This is done with a general formalism that is easily portable to other geometries as shown in Fig. 2 where it would not be the case with the implementation from [16]. Indeed, using the proposed framework, it is possible to engineer simultaneously the phase gradient and the shape of a freeform metasurface as shown at Fig. 2(c) in order to obtain stigmatic properties. This shows the relevance and the strength of the presented technique.
Fig. 2. Stigmatic singlets composed of a metasurface and substrate with various on-axis design. The orange dashed lines correspond to a metasurface and gray to a refractive surface. For every design, $l_0=-10$ mm, $l_i=9$ mm, $Max[r_a] = r_{max} = 4$ mm, $t=1$ mm and $n_o=n_i=1$, $n_s = 2$.
In the same order of ideas, Fig. 3 presents the phase profile of the planar singlet presented in Fig. 2(a) for various values of distance $l_o$ from the input to the singlet (see Fig. 1). As expected, in the case where the distance tends to be much larger than the focal length $l_i$, the phase profile tends to the well-known hyperbolic phase profile given by $\psi (r_b) = \frac {2 \pi }{\lambda } ( l_i -\sqrt {r_{b}^2+l_i^2} )$ [38,39]. This is precisely the case as shown in Fig. 3.
Fig. 3. Phase profile of a planar singlet with a metasurface on the second surface, as presented in Fig. 2((a). The phase profile should tend to the hyperbolic profile as $l_o \to - \infty$. The lens design parameters are $l_i=9$ mm, $r_{max} = 4$ mm, $t=1$ mm and $n_o=n_i=1$, $n_s = 2$.
By definition, the designs are without spherical aberrations for the stigmatic pair of points between the image and object space, as it is a necessary and sufficient requirement [37]. As no paraxial approximation has been done, this holds true no matter the incidence angle. It only requires that the generalized law of refraction is valid and applicable [40].
To ascertain the quality of the stigmatic designs, we evaluate the error $E$, as defined in Eq. (16). For the designs presented in Fig. 2, we achieved values an average error $E$ of $1.139 \times 10^{-14}$ % with a standard deviation of $0.7135 \times 10^{-15}$. These values could be even smaller, but the numerical solver has convergence issues caused by the highly oscillatory nature of the error $E$.
3. Achieving a particular ray mapping
Until now, the lens designs were limited to stigmatic singlets made with at least one metasurface. The design method can be made to respect more demanding merit functions at the cost of numerical complexity [29,30]. As such, the method is well suited to design optical elements achieving particular and known ray mapping angular relations between the object and image planes, such as the Abbe sine or the ‘Herschel conditions [37]. Typical ray mappings are the Abbe sine condition, the Herschel condition and the condition for constant angular magnification, respectively given by the relations [29,37]
To design a lens respecting these conditions, we need at least one degree of freedom at any given interface (e.g., one phase profile at the first interface and the curvature at the second). One needs to relate the angles before and after the singlet, following the geometry presented in Fig. 4. As the vector $\vec {k}_3$ is dependent upon the properties of every interface of the lens, so is the angle $\theta _3$. As we set a relation between the incidence angle $\theta _1$ and output angle $\theta _3$, it imposes design constraints on the requirements of the properties of the lens in order to respect the optical condition over the whole pupil. Then, one needs to solve, often numerically, the implicit ordinary differential equation that the ray mapping condition imposes over the free parameter. This free parameter can be the phase profile of the metasurface being at the first optical element of the singlet, for example. If a solution exists for the free parameter of the first interface, then the whole optical element is designed following Eqs. (1) and (12), as the singlet needs to be stigmatic while respecting the ray mapping condition [37].
Here is an example of the implicit differential equation that needs to be solved for the Abbe sine condition. From geometry and Eq. (17), we can write
For an arbitrary free parameter at the first surface (e.g., the phase profile $\phi (r_a)$ or the surface shape $z_a$), Eq. (18) becomes an implicit ordinary and non-linear differential equation, as it contains various powers of the derivative of said free parameter (e.g., $\boldsymbol{\nabla} \phi (r_a)$) or $z_a'$) with the independent variable being the height at the pupil $r_a$. As stated earlier, finding a solution for the free parameter enables solving for the entire lens with the proper optical properties. We want to stress that our lens design is conducted without employing any optimization techniques, which would typically be necessary for freeform lenses if the analytical lens design framework weren’t used [32,42,43]. Furthermore, to the best of the authors’ knowledge, there is no existing literature on design methods incorporating general ray mapping conditions for kinoform diffractive lenses which are optical elements that use the same governing ray tracing equation as a metasurface (i.e. Eq. (3)). Previous works were focused on stigmatic diffractive lenses [44] and addressed the Abbe sine condition through optimization [45] or aberration considerations [46,47] ill-suited for other ray mapping conditions.
For cases where a solution of the ordinary differential equations is not continuous, we must conclude that no lens can be designed to fulfill the specific ray mapping function according to the generalized Snell’s law and the Fermat’s principle for the particular geometry defining the calculation.
3.1 Illustrative ray mapping designs
In this section, designs implementing various ray mappings are presented. Analysis performed in the Zemax software are also put forward to corroborate the optical properties of said designs.
Figure 5 presents three designs using two metasurfaces implementing successfully the Abbe sine condition (an aplanatic lens), the Herschel condition and the condition to obtain a constant angular magnification. The two-metasurface geometry was chosen as it is of interest in the community [18,48,49]. As seen in section 2.1, the technique is not limited to this particular geometry. However, considering designs with two metasurfaces corresponding to different conditions is interesting to illustrate the flexibility of the method. As expected, we obtained particular ray mappings via small variations in the phase profile when considering the same physical lens input parameters (i.e. $l_0$, $l_i$, $r_{max}$, $t$, $n_o=n_i$, $n_s$ and $M_{tr}=\sqrt {M_{long}}=1/M_{ang}=0.8$). As one can see in Fig. 5(d) to 5(f), the ray mapping relations are well implemented with slight variation, consisting of a maximum relative error of 1$\times 10^{-7}$ on the magnification over the whole pupil. It is also notable that for the designs presented, the error $E$ is of the order 1.3$\times 10^{-14}$ % following the definition at Eq. (16), suggesting that the observed variation is mainly due to instabilities when integrating the highly nonlinear differential equations implemented in Mathematica. These errors are marginal and can be neglected for practical purposes. The phase profiles $\phi (r_a)$ and $\psi (r_b|_{r_a})$ are computed according to the framework and, as expected for moderate $f_{number}$ designs (here 2.5) [29], are akin but respecting their distinct conditions. Similarly, the point spread functions (PSF) are nearly the same for every design as shown in Fig. 5(j) to 5(l) and they effectively demonstrate diffraction limited performances. These results were obtained by implementing the three designs according to theory in Zemax software using Grid Phase objects. Said Grid Phase objects are used, as the governing ray tracing equation is the same as for the metasurface case. It was also more precise to model the phase profiles compared to the usual Binary 2 power series fit.
Fig. 5. Metasurface doublet achieving various ray mapping relations. (a) to (c) present the layout of the Abbe sine, the Herschel and constant angular magnification conditions respectively with $M_{tr}=\sqrt {M_{long}}=1/M_{ang} = 0.8$. The dashed lines correspond to the metasurfaces. (d) to (f) present the phase profiles of the metasurfaces in order to obtain particular ray mapping relations. The purple dot and dashed line correspond to the first metasurface and orange to the second. (g) to (i) present the design relations for every ray. Note that the magnification constants $M_{tr}=\sqrt {M_{long}}=1/M_{ang} = 0.8$ have been extracted of the plot in order to highlight the small computational discrepancies. (j) to (l) are the calculated point spread function cross-sections from the designed lenses imported into Zemax. For every design, $l_0=-10$ mm, $l_i=10$ mm, $r_{max} = 4$ mm, $t=1$ mm and $n_o=n_i=1$, $n_s=1.4$.
Figure 6 presents spot diagrams for transverse and longitudinal shifts also using Zemax software similar to the PSF shifts analysis in [29]. From this figure, one can notice that the aplanatic design is less sensitive to a transverse shift than to a longitudinal shift. In fact, the lens is diffraction limited for a small volume around the object’s stigmatic point as shown from the spot diagrams, being smaller than the Airy disk for transverse shifts smaller than 100 microns and smaller than 5 microns for longitudinal shifts. The RMS radius for on-axis illumination starts at 0.007 $\mu$m, well below the Airy radius of 0.7226 $\mu$m for the presented design in Fig. 6. For a 100 micron transverse shift, the RMS radius increases to 0.599 $\mu$m while a 5 micron longitudinal shift takes the RMS radius to 1.223 $\mu$m. This clearly shows one of the expected behaviours of an optical system respecting the Abbe sine condition: it is more sensitive to a longitudinal shift than to a transverse one [37,41]. It is also notable that the characteristic comet shape of the coma aberration is missing in Fig. 6. Indeed, the coma aberration is negligible in the design dominated with astigmatism, further confirming that the meta-doublet is not only stigmatic, but also aplanatic.
Fig. 6. Spot diagrams of the aplanatic design implemented in Zemax with various transverse and longitudinal shifts. The back circle represents the Airy disk (radius of 0.7226 $\mu$m) and shows diffraction limited performances for small distances about the object stigmatic point of the lens. The design parameters are the same as in Fig. 5.
Thus, the presented framework is an efficient way to exactly design singlets with at least one metasurface inside an optical system including or not freeform without approximation, and it can be done to achieve particular ray mappings.
4. Non-imaging designs
The attention of this article has been put on imaging optical systems. However, the framework of analytical lens design is not limited to these [30]. In this section, we provide a brief overview of how to apply the analytical lens design framework in a non-imaging scenario.
One of the most common non-imaging optical device is the beam shaper, or in its less general form, the collimator. As such, it comes without surprise that metasurfaces have been extensively demonstrated for beam shaping and collimation applications [50–53].
The main result of the analytical lens design framework [30] is to be able to relate the properties of the incident rays at a given surface to the ones after it via Eq. (1). For beam shaping purposes in an infinite to infinite conjugation, these equations take the form
Fundamentally, beam shaping is performed by an element to remap the intensity distribution of a beam at the entrance of the device to another profile at its output. This is similar to how we implemented a particular ray mapping condition in section 3. Indeed, the only variation is how we would construct the implicit nonlinear differential equation in order to obtain the target optical properties. Before, we applied a relation between incident and output angles. Now, the relation between the incident and output rays is obtained with the conservation of energy argument as used for refractive beam shaping lens design following [54,55]
This is very similar to how we implemented a particular ray mapping condition in section 3. Indeed, the only difference is how we would construct the implicit ordinary differential equation in order to obtain the target optical properties. Before, we applied a relation between incident and output angles. Now, the symmetric relation between the incident and output rays is obtained with the conservation of energy argument as used for refractive beam shaping lens design following [54–56]
Fig. 7. Illustrative design of a beam shaper, transforming a Gaussian beam to a uniform beam distribution. (a) outlines of the general geometry of the beam shaper, the insets are Zemax geometric image analysis of the irradiance before and after the lens. (b) and (c) illustrate the target normalized irradiance and its comparison to Zemax geometric image analysis. For this design, $-l_o = l_i \to \infty$, $r_{a,max} = 1$ mm, $r_{b,max} = 2$ mm, $t=2$ mm, $n_o=n_i=1$ and $n_s=2$.
The collimator case is even simpler, as it only requires one freeform surface or metasurface [51]. Indeed, to design a collimator, we only need to put the light source (e.g., LED, laser) at one of the stigmatic point while the other is at infinity. This is either taking the limit as $l_o\to \infty$ or $l_i\to \infty$ in the design relations given by Eq. (19). Similarly, more complex non-imaging systems could be designed. Indeed, it would be possible to apply the analytical lens design framework to design freeform metasurface singlets generating aberration-free patterns over diverse and general surfaces, akin to freeform refractive and reflective designs [57].
5. Conclusion
To conclude, we presented an avenue to design stigmatic singlets with at least a metasurface that can be tailored to implement a particular ray mapping condition. Analytical solutions can be obtained for the exact phase gradient of the metasurface of simple designs of stigmatic singlets following a vector generalized law of refraction formalism without any optimization. The ray mapping is a relationship between the image ray angle to the ray angle coming from the object. As examples, we designed various stigmatic lenses and planar metasurface doublets satisfying the Abbe sine, Herschel, and constant angular conditions. The Abbe sine lens design was further investigated and compared with Zemax simulations of the designed lenses to confirm the aberration behaviour. Finally, we presented a beam shaping design and validated its properties with Zemax. We did not investigate the design of the nanostructures required to realize the various optical designs.
This work expands on the analytical optical design framework enabling the design of metasurfaces and kinoform diffractive lenses. The method allows to determine efficiently the required shapes and phase profiles to design robust optical systems.
Appendix A: Solution for path length $\Theta$ for a finite to finite conjugation
Analytical solution of the optical path of a given ray inside a singlet $\Theta$ is tedious in the most general case, but feasible. Here is an example of the analytical relation of $\Theta$ following the geometry of Fig. 4 considering that the system is stigmatic (i.e. using Eqs. (10) and (11) and that the chief ray is confined to the optical axis
In this case, we need to take the plus sign to obtain physical solutions.
Appendix B: Example of stigmatic planar singlet design with one metasurface
We present the simplified equations used to design the stigmatic meta-singlet presented at Fig. 2(a). Both interfaces are planar and the metasurface is at the second surface. Thus, we are interested to solve for the phase function of the metasurface $\psi (r_b)$ at the second surface. The geometry imposes the physical parameters $z_a = z_a' = 0, z_b = t, \phi (r_a)= 0$. Thus, Eqs. (7) to (9) simplify to
where $l_o$ is the distance from the source to the pupil. $n_o, \ n_s$ are the indices of refraction of the object space and the substrate. At this point, we either use Eq. (21) directly or try to simplify it, as it should be possible from its geometrical definition. The planar refractive interface will implement a light path $\Theta = t/P_z$. It is the case and it further validates the proposed design method. Thus, the height $r_b$ of the ray at the second surface followsThen, Eq. (12) gives us directly the solution of the required phase in dependence of $r_a$, the height of the ray in the pupil following
Funding
Natural Sciences and Engineering Research Council of Canada (RGPIN-2016-05962).
Acknowledgment
The authors thank Guillaume Allain for the many useful discussions, time, and support of the project.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.
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