1. Introduction

Dielectric resonator metasurfaces have been shown to be an interesting technology platform for the design of compact optical devices [1–11]. The appealing characteristic of these devices is the possibility to locally control the phase of an incoming wavefront exploiting the resonators’ features, rather than the curvature of surfaces, as it is for conventional refractive lenses. The wavefront shaping has been demonstrated both for transmissive structures [1–6] and for reflective structures [7]. Moreover, it has been demonstrated that the local phase of the two polarization states can be independently controlled, adding new optical functionalities, with respect to conventional devices based on birefringent materials [8, 9]. In the past, most of lenses and optical components based on metasurfaces have been designed considering single fixed incidence angles and single wavelength operation [2,4–8]. There have been some attempts to design one-dimensional achromatic metasurfaces lenses, but the performances have been achieved in a finite set of discrete wavelengths, not on a continuous wavelength range [10,11]. Recently, plasmonic nano-gratings have been used as deflectors and lenses, in reflection mode, and they have shown good performances in terms of BW and FOV [12, 13]. However, for most applications, both broad wavelength range and off-axis illumination have to be taken into account, as well as other performance criteria, such as wavefront aberrations, distortion, Strehl ratio, etc. Regarding these aspects the design techniques for optical components based on metasurface are still not advanced as their refractive counterparts, usually based on commercial tools such as CODE V [14] or Zemax [15]. A systematic approach, considering all the requirements in the design phase, the complete (wavelength and angular) dispersive behavior of the metasurface and the effects of manufacturing errors, is still not available to optical designer. This paper is aiming at contributing in the process of developing these design techniques. In this contribution, we present a general design strategy to optimize the nano resonators distribution in all-dielectric metasurfaces, aiming at satisfying desired optical key performances under given BW and FOV requirements. As an example, the optimization strategy is applied to a challenging design: an array of microlenses to be used in a maskless lithography (ML) setup. This example highlights how additional aspects, such as undesired apodization effects and reduced manufacturing tolerances, must be taken into account in the design phase. In general, it emerges also that due to these effects, it is difficult to use single layer metasurface lenses to meet the requirements of a an advanced optical design.

This clearly emerges when several requirements (e.g. BW and FOV) must be simultaneously taken into account. This brings also to the conclusion that a big effort has to be devoted to the development of efficient modeling and design tools for multilayer metasurfaces. Nevertheless, the robust optimization technique is a useful tool to shape the local distribution of dielectric resonators. The results show that the designed array of microlens is a characterized by a reduced sensitivity to manufacturing errors compared to designs in which the metasurface is designed by simply mapping an analytic phase distribution in a unit cell distribution. Moreover, better performances have been achieved at the edges of the required BW, compared to a theoretical design not taking into account the dispersive behavior of the dielectric resonators. The paper consists of the following parts: in section 2 a description of the maskless imaging application and of the typical requirements and optical solutions involved are presented. The full-wave (FW) and the scalar diffraction theory (SDT) models used to characterize, the local and global behavior of the lens respectively, are described in section 3. In section 4, the different steps of the design strategy are presented together with the results obtained for the specific case under study. Conclusions and remarks on the design strategy are part of the last section.

2. Array of microlenses for maskless lithography

Maskless lithography systems rely on the use of focused beams to selectively expose photoresist areas. The pattern is created by scanning the beam across the photoresist surface, while modulating the intensity of the beam. In order to increase the throughput, multiple focused beams are used in parallel, each illuminating sub-areas of the whole resist surface [16]. Typically these beams are obtained by using arrays of refractive microlenses. In practice, manufacturing tolerances across the microlens array, will result in non-uniformities in the focal properties (un-flat focal plane), that may be detrimental for the global system performance. Since metasurfaces lenses are realized using a reduced number of steps (in principle only one lithographic step to shape the different resonators), they represent a promising technology platform to realize an array of microlenses with reduced focal variations, and therefore be better suited for maskless lithography applications. Apart from the usual lithographic techniques, such as electron beam lithography (EBL) and focused ion beam lithography (FIB), nanoimprinting techniques [17] can play a major role in realizing large area metasurface lenses arrays, such the ones proposed in this paper. In the proposed design, the illumination is assumed to be provided by an array of light-emitting diodes (LEDs), operating at 405 nm. In order to take into account the LEDs intrinsic BW, the non-uniformities in the LEDs array and the possible thermal variations, a BW of 20 nm is assumed. In maskless lithography, the figure of merit for evaluating lens quality is the percentage of energy in the focused spot with respect to total energy reaching the focal plane. Because of that fact, the energy that falls outside the central disk will illuminate the photoresist in areas where no exposure is desired. For this reason, the per cent energy should be at least bigger than 68%, that is the theoretical fractional energy for a lens characterized by a Strehl ratio of 0.8 [18]. For this value, the lens is still considered diffraction limited. The central energy performances should be achieved over a field of view of θ_max = ±60 mrad. The numerical aperture of the lens at the image space is set to NA = 0.19 (focal length f = 1.5 mm) and it should accept an incident beam of radius R_a = 300 μm. Moreover, the optical system must be telecentric in the image space, as it is commonly requested for focusing systems in lithography applications [18]. This is achieved by placing the aperture stop at the front focal plane of the lens. Considering that, under telecentric operation, the impinging beam hits portion of the lens shifted with respect to the optical axis, the total radius of the clear aperture is R_tot = R_a + f|tan θ_max| ≈ 390μm.

3. Local and global electromagnetic modelling of metasurfaces

Dielectric resonators metasurfaces allows shaping any incoming wavefront by means of usually high contrast dielectric resonators that exhibit strong resonant responses, not influenced by the mutual coupling between the resonators [1]. In order to characterize the local response of the array of resonators, full wave simulations are used. For this work the finite element method commercial software COMSOL Multiphysics has been used [19]. Under the assumption of an infinite array of identical cylindrical resonators, the transmission characteristics under plane wave excitation can be computed. Propagation of only the zeroth order of diffraction is ensured if the periodicity a of the resonators’ lattice is chosen to be smaller than [20]:

 

Fig. 1 (a) Artistic impression of a dielectric resonators metasurface lens applied in maskless lithography, (b) unit cell of the designed metasurface lens with dimensions. For the sake of clarity the 500 μm thick back layer of fused Silica is not shown in both pictures.

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Fig. 2 Transmission characteristics of the array of resonators vs. resonators radius, for different wavelengths and angles of incidence: (a,b,c) transmittance, (d,e,f) transmission coefficient phase.

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Once the transmission characteristics of the resonators are found, the lens can be designed by positioning the resonators in the array forming the lens, according to the phase distribution, using the mapping that relates the phase of the transmission coefficient with the resonator radius. The global modelling of the optical performances for the designed lens can be done by applying common formulas, based on scalar diffraction theory [25]. Depending on the size of the lens, its focal length and the lattice period of the resonator, the Angular Spectrum Method, Fresnel propagation or Fraunhofer propagation can be used [26]. For the design of the ML lens, considering the reduced NA, a Fresnel propagation integral, numerically implemented by the fast Fourier transform (FFT), algorithm has been used. Since a fine sampling is needed at the metasurface plane, coarse sampling is expected at the image plane. Interpolation with spline interpolants is used to get smoother profiles at the image plane.

4. Design for wavelength band and extended field of view through robust optimization

As anticipated before, if the design of the lens is based only on the resonators’ characteristics for the central wavelength under normal incidence, the lens may suffer from different aberrations when it operates under different conditions. This is because the phase shift, given by each resonator, varies with incidence angle and wavelength. Moreover, by choosing the resonator distribution on the basis of the desired phase distribution mapping only, a non uniform transmittance distribution (ring-shaped) across the metasurface is obtained. This distribution can be seen as an unwanted apodization that will modify the focusing performance of the lens. Effectively, the diffraction effects due to the presence of this additional transmittance mask will add to the effect of phase modulation of the dielectric resonators, giving as result and image intensity distribution that can be substantially different from the case when the transmittance modulation is not taken into account. Therefore any amplitude distribution must be accurately described in the design process. To take into account all these effects in the design, a particle swarm optimization (PSO) routine is used. The PSO algorithm defines a swarm of particles in the search parameter space. The cost function is evaluated for each one of these particles. Each particle moves in the search space influenced by its best historical position and the best global position of the whole swarm. After some iterations, all the particles converge to the global optimum [27]. The PSO algorithm has been preferred over local optimization techniques, e.g. gradient-descent methods, because it is a global optimization technique and, owing to its stochastic nature, it does not suffer of the problem of local optima. The lens is studied under different operating conditions, e.g. wavelengths λ = 395 nm, 405 nm, 415 nm, angles of incidence 0 mrad, 30 mrad and 60 mrad. For off-axis incidence, a linear phase distribution, accounting for the tilting, is added to the phase distribution of the lens, before numerically propagating the wavefront. The transverse intensity profiles (point spread functions, PSFs) along one of the main axes at the image plane are compared with the ideal profiles, e.g. Airy’s profiles, centered at the Gaussian image point (GIP). Different cost functions can be defined, depending on the target of the optimization. For our purpose, we use a cost function defined as:

4.1. Influence of wavelength and angular dispersion on performances

First, the procedure described above has been used to asses the use of a single theoretical diffractive surface (not affected by the dispersive behavior associated to a particular technological implementation) to meet the required performances. The theoretical phase surface implements only a phase distribution, without apodization. Moreover, no mapping between a given phase distribution and the resonators has been implemented, therefore the theoretical phase surface must be assumed free from errors due to quantization and without additional dispersive effects except from those typical of a diffractive element. To take into account these effects, the phase distribution is recomputed rescaling the phase for each wavelength [28]:

 

Fig. 3 Focal plane intensity distribution of the theoretical phase surface for different wavelengths and angles of FOV: (a,b,c) wavelength λ = 395 nm, FOV = 0, 30, 60 mrad; (d,e,f) wavelength λ = 405 nm, FOV = 0, 30, 60 mrad; (g,h,i) wavelength λ = 415 nm, FOV = 0, 30, 60 mrad; (j) per cent energy in the central disk for different wavelengths with the respect to the FOV angles.

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Fig. 4 a) Per cent energy vs. field angles of the theoretical phase profile mapped into dispersive (wavelength and angular) resonators; b) zoom of the plot in a) in the range of energies related to the extreme wavelengths in the BW.

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4.2. Influence of manufacturing errors on performances

In addition to the variations due to wavelength shifts, for a fully functional metasurface lens, it is worthwile to take into account manufacturing inaccuracies during the design process. The most harmful inaccuriacies are errors related to the realization of the resonators with the right cross section. These errors can affect both the local phase and transmittance distribution. Moreover, this wavefront error is difficult to predict using the classical optical definition of aberrations. In fact, the highly nonlinear behavior of the relation between the resonator radius and the phase of the wavefront makes hard giving a prediction of aberration with radial functions. For such a reason, in order to design a lens, whose performances are robust with respect to variations in the resonators shape, we have extended our design strategy following a Monte Carlo approach. For each set of optimization variables, C, K, C_i, in each operational condition, a number of different resonator distributions, N_MC, are defined. These distributions have been obtained from the nominal distributions, adding an error on each resonator radius. The error is taken from a Gaussian distribution with zero mean value and variance σ². The cost term for each wavelength and field angle is then defined as an average between all the cost values for the samples. With this procedure, the resonator distribution is optimal in a robust sense, meaning that the lens optical performances are maintained even if there is an error of maximum ±σ in the manufacturing of the resonators. The extended robust optimization procedure has been used to design a lens according to the requirements previously described. The lens characteristic parameters are reported in Table 1.

Table 1. Characteristic parameters of designed telecentric metasurface lens

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The PSO routine has run for 150 iterations, taking into account 20 different particles in each iteration. For the Monte Carlo procedure, N_MC = 10 samples for each operational condition have been considered. A cut of the optimal phase profile found with the PSO routine is plotted in Fig. 5. It is worth mentioning that an “a posteriori” check for aliasing issues on the phase profile has been made, ensuring that the following condition is satisfied [29]:

 

Fig. 5 Optimized phase profile of the dielectric resonator metasurface lens.

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The maximum phase slope for the optimal phase distribution has been found to be max [∇ϕ(r)] = 2.095 · 10²deg/μm, while the sampling step δ is the lattice period. The transmittance and phase distributions for 405 nm are shown in Fig. 6. The focal plane intensity distributions and the per cent energy in the central disk are reported in Fig. 7. The results show smaller values of the per cent energy in the central disk, with respect to the required value of 68% and also to the result obtained with the theoretical phase surface, within the BW and FOV considered. Looking separately at the different cases, it can be seen that the spot shapes are close to the expected ones, for λ = 405 nm and 415 nm, whereas at the lower wavelength λ = 395 nm, the spot is larger. However, ripples around the central spot considerably compromise the final performances. The effects causing this deviation are: the apodization, due to the non-uniform transmittance distribution; the phase quantization, due to the finite accuracy in mapping the required phase shift with the resonator shapes; the manufacturing errors. The apodization effect can be avoided, if the resonators are characterized by less pronounced resonance dips. In fact, although not interesting for the ML application, another equivalent design at λ = 625 nm, 635 nm, 645 nm has been designed, and it is shown in the appendix. This presents performances closer to the required values in terms of per cent energy in the central spot (in the range of 47% − 57% at the central wavelength). For that band, the resonators used, made of silicon and embedded in fused silica, show less strong dips, confirming that the additional not uniform apodization is detrimental for the final performances of the lens. For this reason, it must be taken into account in the design of an optical device. While the effect of the apodization can be reduced (if at the desired band, high contrast material with reduced losses are available), the effects of manufacturing variations must be always taken into account. Unfortunately, their effect is quite severe, especially if the relation transmission phase vs. resonator radius is characterized by a steep slope. The value of the variance σ² has been estimated on the basis of experimental data for a similar design manufactured in the past [5]. By improving the manufacturing accuracy, the interval of variation, in which the optical design must be robust, decreases, determining an improvement of the optical performances. Furthermore, in order to evaluate the validity of the robust procedure, a Monte Carlo analysis has been performed on two metasurface lenses, based on the phase profiles optimized with an without the robust procedure. In Fig. 8, the per cent energy of the theoretical surface mapped onto a resonator distribution and of the design derived from the robust optimization procedure, are compared, at the different wavelengths and field angles, with the average of a Monte Carlo analysis. This analysis considered 50 different error distributions on top of the nominal resonators’ shapes. The normal distribution used to generate the manufacturing tolerances is characterized by zero mean and variance σ² = 5 nm. The comparison of these results leads to two main conclusions. First: the use of the wavelength dispersive behavior in the optimization process produces a design whose performances are more uniform over the considered BW. This has been achieved by improving the per cent energy in the central disk, at the two extremes of the BW (∼4% to ∼8% maximum at 395 nm, ∼1.5% to ∼4% maximum at 415 nm), although at the cost of a reduction of performances (∼60% to ∼24% maximum) at the central wavelength. Second: the robust procedure succeeds in finding a design more tolerant to manufacturing inaccuracies. The per cent variations in performance are definitely reduced from a maximum of 8% (central wavelength, on axis) for the not robust design to 3.7%. Moreover, as highlighted in Fig. 8, the performance deviations are more uniform within the given BW for the robust optimal design, with respect to the not robust theoretical one.

 

Fig. 6 Transmittance and phase distributions on the optimized metasurface for different angles of FOV at 405 nm. (a,b) transmittance and phase distribution for on axis excitation; (c,d) 30 mrad,; (e,f) 60 mrad. The distributions at the other wavelengths show similar behaviors. The shift of the beam for off axis angles is explained by the telecentric aperture stop placed at the front focal plane of the lens.

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Fig. 7 Focal plane intensity distributions for different wavelengths and angles of FOV for the dielectric resonator metasurface lens optimized with the robust optimization procedure considering the manufacturing tolerances: (a,b,c) wavelength λ = 395 nm, FOV = 0, 30, 60 mrad; (d,e,f) wavelength λ = 405 nm, FOV = 0, 30, 60 mrad; (g,h,i) wavelength λ = 415 nm, FOV = 0, 30, 60 mrad; (j) per cent energy in the central disk for different wavelengths with the respect to the FOV angle.

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Fig. 8 Per cent energy in the central disk for nominal designs vs. the average behaviour out from a Monte Carlo analysis with manufacturing tolerances with σ² = 5 nm, for different wavelengths: a), b), c) nominal design derived from the phase distribution of the theoretical phase surface (same design as Fig. 4): a) 395 nm, b) 405 nm, c) 415 nm; d), e), f) nominal design derived from the robust optimization procedure (same design as Fig. 7 d) 395 nm, e) 405 nm, f) 415 nm. The arrows in each plot show the maximum per cent variation between the nominal design and the Monte Carlo average behaviours for each wavelength.

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Despite the final performances do not meet the initial requirements, it must be highlighted that they have been achieved with a device based on a single surface. Refractive or hybrid refractive/diffractive devices, with the required performances over the desired BW and FOV, are composed of at least two surfaces. Therefore they exploit one more degrees of freedom with respect to the designed lens. The scope of this study was to assess the design of single surface lenses that can be realized with a reduced number of lithographic steps. It must be noted also that the requirements for the lens are quite tight, especially for the BW. Moving towards the use of purer spectral sources, such as lasers, it will ask for smaller BWs, over which better performances can be achieved more easily.

The procedure described is general and can be applied to the design of any optical device. The main point is that it treats rigorously the interaction of light with the metasurface lens even under off axis excitation and for different wavelengths, and it exploits this information to end up with an optimized design based on particular requirements. In addition to the design of a lens, the following procedure can be used for the design of devices for beam shaping or holograms. Furthermore, by considering the two independent polarizations, also polarization sensitive devices can be designed, taking into account the not equal dispersive effect through the metasurfaces of the two independent polarizations.

5. Conclusion

The design of a dielectric resonator metasurface microlens for an array to be employed in ML applications has been described in this paper. The dielectric resonator metasurfaces have been analyzed over a finite BW of 20 nm centered at 405 nm and for different angles of incidence, within a ±60 mrad of FOV. The design process exploits a PSO to find the optimal phase distribution at the metasurface, considering the different operational conditions, and the effect of the transmittance distribution in the final focal properties of the array. The results presented show clearly that all the information available about wavelength and angular dependent behavior of the dielectric resonators must be included in the design process of optical devices based on this technology. Furthermore, the design has been optimized to be robust against manufacturing errors in the shape of the dielectric resonators, within a given range, applying a Monte Carlo approach in the cost function evaluation. The performance of the designed lens does not meet the given requirements (maximum of 27% of energy on axis for the central wavelength, when more than 68% is required over the whole BW and FOV). Despite this, the design procedure proposed is able to reduce significantly the variation due to manufacturing tolerance, generating a more robust design within the given BW and FOV. This study represents a first effort to bring the metasurface concept inside a typical optical design loop. The optimization procedure, in fact, takes into account typical optical requirements, e.g. FOV, BW, telecentricity, Strehl ratio, etc, that are familiar for optical designers. This helps to finally fill the gap between optical design and metasurface concepts. Although the design procedure, presented in this paper, is shown for a single device, it is general and can be applied directly, or easily extended, to different devices, such as lenses, diffractive optical elements, holographic surfaces and polarization sensitive structures.

Appendix: Design of metasurface lenses centered at 635 nm

An array of microlenses, with the same optical requirements as the lens described above, has been designed to work at wavelengths λ = 625 nm, 635 nm, 645 nm. Although not useful for ML lithography, where sources in the ultraviolet (UV) range are used, this design shows that results closer to the specification may be achieved when high-contrast material are available within the desired bandwidth with the procedure explained in this contribution. The typical features of the designed metasurface lenses are reported in Table 2:

Table 2. Characteristic parameters of designed telecentric metasurface lens

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The refractive index values have been found respectively in [22,30]. The transmission characteristics (transmittance and phase) are reported in Fig. 9. The transmittance and transmission phase distributions are reported in Fig. 10 and the PSFs with the per cent energy in the central disk in Fig. 11.

 

Fig. 9 Transmission characteristics of the array of resonators with respect to resonators radius for different wavelengths and angles of incidence: (a,b,c) transmittance, (d,e,f) transmission coefficient phase.

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Fig. 10 Transmittance and phase distributions on the metasurface for different angles of FOV at 635 nm. (a,b) transmittance and phase distribution for on axis excitation; (c,d) 30 mrad,; (e,f) 60 mrad. The distributions at the other wavelengths show similar behaviors. The shift of the beam for off axis angles is explained by the telecentric aperture stop placed at the front focal plane of the lens.

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Fig. 11 Focal plane intensity distribution of the optimized dielectric resonator metasurface lens for different wavelengths and angles of FOV: (a,b,c) wavelength λ = 625 nm, FOV = 0, 30, 60 mrad; (d,e,f) wavelength λ = 635 nm, FOV = 0, 30, 60 mrad; (g,h,i) wavelength λ = 645 nm, FOV = 0, 30, 60 mrad; (j) per cent energy in the central disk for different wavelengths with the respect to the FOV angles.

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Funding

This work has been supported by the TNO Early Research Program “3D Nanomanufacturing Instruments”.

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