Broadband metasurface aberration correctors for hybrid meta/refractive MWIR lenses

Ko-Han Shih; C. Kyle Renshaw; C. Kyle Renshaw; C. Kyle Renshaw

doi:10.1364/OE.460941

1. Introduction

Metasurfaces are two-dimensional, subwavelength, flat optics that enable flexible controls of wavefront at a wavelength scale [1,2]. In recent years, efforts have been made to design metasurfaces to outperform their conventional optical counterparts. For example, achromatic metalenses for on-axis imaging have demonstrated by fine-tuning artificial dispersion to maintain focal length over a broad spectral band [3–5]. However, realizing achromatic metalenses with a large diameter (mm to cm scale) has proven difficult owing to the large group delays required at the edge of a metalens [6]. Meanwhile, monochromatic aberrations caused by nonparaxial effects become increasingly important for wide field-of-view (FOV) designs. Singlet metalenses with wide FOV are achieved by adopting the Chevalier landscape lens design scheme for which a physical aperture is separated from metasurface to control off-axis aberrations [7,8]. The replacement of the aperture by a metasurface further leads to multi-surface aberration-corrected metalenses. Single wavelength operation with diffracted-limited image quality across the whole field of view have been reported [9–11]. Utilization of multiple metasurfaces gives more degrees of freedom and can generally lead to a better aberration control for a large-scale layout. The typical approach to design multi-metasurface (MMS) lenses starts with a phase profile that can be expanded as a superposition of certain smooth-varying basis functions, and raytracing is combined with gradient algorithms to optimize expansion coefficients. However, this approach does not consider the phase dispersion characteristics of meta-atoms making broadband designs difficult to produce. Furthermore, aberration correction tends to be more challenging for longer effective focal length (EFL) designs under a fixed F-number. This is due to aberration scaling with lens dimensions verse a constant diffraction-limited spot size and MMS designs can require many metasurfaces to achieve good image quality for fast, long EFL lenses [12].

Designing broadband, wide-FOV metalenses with large apertures and long EFL is a challenging task. Alternatively, the dynamic phase control of metasurfaces could be employed to correct aberrations from conventional, bulk (i.e., refractive) optics. In a pioneering work, Chen et al. demonstrated spherical and chromatic aberrations of a refractive lens can be simultaneously corrected using a dispersion-engineered metasurface put in front of the lens [6]. In their design, raytracing is used to determine on-axis imaging wavefront errors across a given wavelength range, and the wavefront compensations required for the metasurface can be determined. Centimeter-scale metasurfaces for either spherical or chromatic aberration correction were also demonstrated with a similar design approach [13]. Here, we report a different but versatile approach to design large-scale, broadband, MMS layouts to correct aberrations for a wide FOV refractive lens; we demonstrate design of a metasurface aberration corrector (MAC) for a midwave infrared (MWIR) lens. A scalar field inversed design technique is invoked to efficiently optimize all parameterized physical geometries across a MMS layout with meta-atom phase dispersion taken into consideration.

This paper is arranged as follows: In Section 2 the inversed design scheme for MACs is detailed. In Section 3, we apply the method to optimize a pair of dielectric metasurfaces composed of nanopillars with spatially varying radius to correct optical aberrations from a F/2, 10 mm EFL, 30° FOV refractive lens in MWIR. Image quality and loss analyses are then applied to demonstrate the effectiveness of the proposed method. The working bandwidth of the hybrid lens is limited by the relatively large phase dispersion from pillar meta-atoms. We then demonstrate the benefits of using holey metasurfaces with a reduced meta-atom phase dispersion by virtue of its lower effective index dispersion. Improved focusing efficiency of the hybrid lens over the stand-alone refractive lens can be attained in the wavelength range λ₀ = [3.5–4.5] µm and across the whole FOV. Section 4 gives the conclusions from this work.

2. Design framework

We illustrate the design framework by considering a design example of a hybrid imaging system as shown in Fig. 1(a). A F/2, 10 mm EFL, 30° FOV refractive lens composed of two germanium (Ge) optics is combined with two metasurfaces along its optical axis. The refractive lens is pre-optimized in Zemax OpticStudio for averaged root-mean-square spot sizes over 3 wavelengths λ₀ = {3.75, 4, 4.25} µm and 5 equally spaced field angles between 0° and 30°. Prescriptions of each refractive surface are provided in the Supplementary Information (SI). In a hybrid imaging system combining both refractive and diffractive components, the two groups of components could be co-optimized in a ray optics approach wherein the diffractive surface is described by a generic phase profile and then a meta-atom layout can be performed to achieve the desired phase profile; we refer to this process as “direct design”. However, this direct design process is not informed by real meta-atom transmission, diffraction, and dispersion characteristics; this limits the fidelity of such designs. An effective alternative is to use inversed design techniques enabled by wave optics [14]. However, when diffractive elements are combined with bulk optics, direct full-wave propagation through the hybrid system is too computationally intensive for design work—which requires many iterations for optimization. Efforts to perform co-design of both bulk and metasurface components with reduced computational load could enable full freedom in the design optimization.

Fig. 1. (a) Schematic of a hybrid lens consisting of two metasurfaces and a conventional lens. (b) A close-up illustration portrays a few unit cells of the metasurface based on the SOI platform; note the wafer substrate is 0.5 mm thick and not draw to scale. The orange box marks the window for full-wave simulations of a meta-atom periodic array to determine transmittance and phase delay maps. (c) The SOI nanorod meta-atom used in inversed design with all the relevant dimensions marked. The Si nanorod radius R is the only parameter used to control local phase delay. (d) Phase delay and (e) transmittance of a SOI nanorod meta-atom periodic array as a function of nanorod radius R and wavelength λ₀ under normal plane wave incidence from the Si substrate. The contour lines on the phase delay plot are equally spaced with a spacing of π/4.

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In the present work, instead of co-optimizing both groups of optical elements simultaneously, we focus on optimizing the diffractive elements only (i.e., metasurfaces). Optical power is mainly carried by the bulk elements and the set of metasurfaces, collectively called a MAC, serve to correct optical aberrations from the former lens and are optimized using an inversed design technique, which will be detailed later. The resultant hybrid imaging system is aimed to maintain the same EFL, image spot positions, and F-number as the stand-alone conventional lens.

Each metasurface imposes a phase discontinuity on incident waves, which is in turn controlled by nanoscale geometric properties of a metasurface [1]. It is important to note an arbitrary number of metasurfaces can be added behind the refractive optics, and each metasurface can be parameterized independently. In this work, all-dielectric metasurfaces are built upon silicon-on-insulator (SOI) platform and are implemented as arrays of silicon (Si) nanorods with circular cross-sections, as shown in Figs. 1(b-c). Phase control is attained by varying nanorod radius locally. The circular cross-section of a nanorod ensures weak dependence on incident wave polarization and can lead to polarization-insensitive metasurfaces for landscape imaging [9]. A small pitch size is selected to attain high sampling rate and to maintain subwavelength characteristic of a metasurface while staying in the off-resonance regime for broadband application [15]. The basic geometry (pillar pitch = 1 µm, pillar height = 4 µm and range of pillar radii $R$ = [200–450] nm) was chosen using trades on these quantities bounded by fabrication constraints while trying to achieve high transmission and full 2$\pi $ phase delay across the MWIR spectrum.

The SOI platform was adopted to facilitate fabrication of large aperture lenses. The SOI wafer has a thin layer of silicon dioxide (SiO₂) on top of a thick Si substrate and buried beneath a thin Si device layer; the device layer will be etched to define high aspect ratio Si nanorods while the buried oxide provides an etch stop to facilitate fabrication of the designed structures. Typical deep reactive-ion etch (DRIE) processes used for high-aspect ratio etching on Si, such as the Bosch process, can be difficult to maintain a consistent etch rate with large variation in feature size, especially over large apertures; this presents a challenge to control pillar height precisely using the etch alone. Fortunately, these etches are highly selective to etch Si over SiO₂ [16]. The buried oxide provides a convenient etch stop to ensure nanorod height uniformity is determined by the uniformity of the device layer thickness rather than the etch process. The downside of this platform is that SOI wafers are more costly than bare Si (>10x at the 100 mm wafer scale) and there is a slight penalty of transmission efficiency due to the additional optical interfaces (c.f. SI, Section 2).

For a largescale metasurface, full-wave electromagnetic simulation for the whole metasurface is generally unfeasible. The standard approach adopts the local uniformity approximation (LUA) for each unit cell (meta-atom) on a metasurface [7–10,17]. In this approximation, each meta-atom is surrounded by an infinite sea of identical meta-atoms to determine its intrinsic transmission efficiency (T) and phase delay (ϕ); the LUA is expected to work well when the phase profile is changing slowly across a metasurface aperture so that adjacent meta-atoms are very close in radii. With this in mind, we simulate the intrinsic SOI meta-atom characteristics in the MWIR spectrum as a function of nanorod radius (Figs. 1(d-e)) using commercial electromagnetic solver CST Studio. Periodic boundary conditions are applied at the lateral sides of a meta-atom and open boundary conditions are applied at the top and bottom surfaces in air and Si, respectively. The top 2 µm thick section of the Si substrate (adjacent to the buried oxide) is included in the simulation volume, the remainder of the thick substrate just contributes a constant (i.e., piston) phase delay that does not affect the metasurface performance. This simulation launches a normally incident plane wave from the Si substrate and the field is sampled at the top of the simulation volume in air. The refractive index data for Si and SiO₂ in MWIR is obtained from [18] and [19], respectively.

Due to fabrication constraints, we limit nanorod radius to be within the range R = [200–450] nm. This gives a minimum gap between adjacent pillars of 100 nm corresponding to a 40:1 aspect ratio which is achievable in our nanofab; meanwhile, we have seen the minimum pillar diameter of 400 nm is sufficiently robust to withstand the etch processing and subsequent clean-up steps. This range provides phase delay coverage larger than 2π across the whole MWIR wavelength range and we focus on the central half of the band in this work with λ₀ = [3.5–4.5] µm (Fig. 1(d)). The averaged transmittance in the given radius and wavelength range is 83.02%. The phase contour manifests phase dispersion dϕ/dλ₀ inherent in the SOI nanorod meta-atom. Moreover, this phase map ϕ(R, λ₀) will serve as a lookup table relating phase distribution to nanorod radius distribution and thusly enables multiwavelength MAC inverse design, as discussed next.

Plane wave input fields are sufficient for designing a lens comprised of only a single metasurface—a metasurface singlet. In addition, the ideal phase profile for singlet to perfectly focus axial light is well known [1,2]. On the other hand, for a MAC the “input fields” actually refer to output fields from refractive optics and the required phase profiles to correct wavefront errors in these output fields are typically unknown. A freeform design permits all meta-atom parameters in all metasurfaces to be tailored independently while considering real meta-atom characteristics. However, as thousands of optimization parameters are considered, typical gradient methods become inefficient because cost function gradients are computed serially. On the other hand, adjoint gradient method (AGM) is an iterative local gradient optimization algorithm allowing parallel computation of cost function gradients for all variables and is suitable for largescale metalens designs [14]. Moreover, it allows user-defined input fields and desired output fields, and meta-atom phase dispersion can be considered in the optimization process. All these features are indispensable for multiwavelength MAC design, as illustrated below.

Certain data is required to initialize inverse design for a MAC. First, we need the meta-atom phase map, or library, as a function of meta-atom parameters and wavelength. Here, the SOI meta-atom nanorod radius is the only parameter, so a two-dimensional map as shown in Fig. 1(d) is sufficient. Note this library is generated under the LUA and used for layout; however, post-design analysis goes beyond the LUA to characterize transmission losses and focusing efficiency with improved fidelity as described in Section 3.2. Second, we need a description of the incident field at the start of the MAC for all combinations of wavelength λ_j and field angle θ_i for the system. In this work, we incorporate the scalar wave approximation and require the scalar exiting wavefronts $\textrm{E}_{int}^{i,j}(x,y)$ from the refractive optics which carry information regarding optical aberrations and serve as the inputs for the subsequent optimization step. For a beam with beamwidth much larger than the wavelength and far away from its focal point, diffraction is expected to have only modest impact and field raytracing in Zemax OpticStudio is used to obtain wavefronts behind the refractive lens on a plane transverse to the optical axis (further discussed in the SI Section 3). Third, we define the idealized image intensity $I_{\textrm{ideal}}^{i,j}(x,y)$ for the hybrid lens as a diffraction-limited spot centered at the corresponding spot centroid arises from the stand-alone refractive lens at the image plane. This definition of $I_{\textrm{ideal}}^{i,j}(x,y)$ means we do not intend to correct optical distortions. Last, we define a cost function of the optimization problem in the usual way as

(1)$$C = \sum\limits_k {\sum\limits_{{\lambda _j}} {\sum\limits_{{\theta _i}} {{w_i}} } } {w_j}{[{I_{\textrm{ideal}}^{i,j}({x_k},{y_k}) - I_{\textrm{out}}^{i,j}({x_k},{y_k})} ]^2},$$

with (x_k, y_k) a sampling position on the image plane and $I_{\textrm{out}}^{i,j}(x,y) = {|{E_{\textrm{out}}^{i,j}(x,y)} |^2}$ the image intensity from the hybrid lens; w_i and w_j are weightings for sampling angle and wavelength, respectively. In this work, we set w_i = w_j = 1 for simplicity. This cost function is defined in such way that the image positions under different field angles and wavelengths are unchanged by the MAC, so the F-number and EFL of the system are unchanged, too.

To calculate $I_{\textrm{out}}^{i,j}(x,y),$ we propagate fields from the plane where $E_{int}^{i,j}(x,y)$ are defined to the image plane sequentially through metasurfaces. This process is called forward propagation. For free space propagation, the scalar angular spectrum method is implemented [20]. It should be noted multi-reflections between layers are not considered in this work. For a given nanorod radius distribution R(x, y) on a metasurface, an incoming wave $E_ - ^{i,j}(x,y)$ accumulates a phase discontinuity ${\Phi _j}(x,y) = \exp [i{\phi _j}(x,y)]$ as it passes through the metasurface:

(2)$$E_ + ^{i,j}(x,y) = {\Phi _j}(x,y)E_ - ^{i,j}(x,y).$$

In reality, a reduced amplitude arises due to transmission losses and can be considered within the cost function, in principle. However, we do not consider transmission losses in this work as we expect the phase term dominates optical aberration control. The relation between R(x, y) and ϕ_j(x, y) can be obtained using the meta-atom phase map Fig. 1(d) as a lookup table. Combining free space propagations and phase discontinuities at metasurfaces, $I_{\textrm{out}}^{i,j}(x,y)$ can be calculated. Then a loop through θ_i and λ_j leads to cost function value C of a current MAC design.

The ultimate goal is finding a set of nanorod distributions R(x, y) on the two metasurfaces that minimizes the cost function value (Eq. (1)) for the resultant hybrid lens. As an iterative gradient method, in each iteration AGM finds the cost function local gradient with respect to each nanorod radius dC/dR on both metasurfaces. What sets it aside from typical gradient methods is that in AGM all local gradients can be computed in parallel by combining a forward propagation of incident fields and a back propagation of the adjoint fields. The adjoint field at the image plane is given as [14]

(3)$${a^{i,j}}(x,y) = {w_i}{w_j}E_{\textrm{out}}^{i,j}(x,y){[{I_{\textrm{ideal}}^{i,j}(x,y) - I_{\textrm{out}}^{i,j}(x,y)} ]^2},$$

which is a measure of departure of current design from a target one. Then a backward propagation of adjoint field from the image plane using the same scheme as the forward propagation determines cost function gradients with respect to local phase at a metasurface as

(4)$$\frac{{dC}}{{d{\phi _j}}}(x,y) ={-} 4{w_i}{w_j}\mathrm{{\cal R}}\{ {[E_ - ^{i,j}(x,y)]^\ast }a_ - ^{i,j}(x,y)\} ,$$

where $\mathrm{{\cal R}}\{ \;\} $ denotes the real part of the argument and the asterisk denotes complex conjugation and $a_ - ^{i,j}(x,y)$ is backward propagating adjoint field at the left side of a metasurface. The cost function gradient with respect to radius is

(5)$$\frac{{dC}}{{dR}}(x,y) = \sum\limits_{{\theta _i}} {\sum\limits_{{\lambda _j}} {\frac{{d{\phi _j}}}{{dR}}} } (x,y)\frac{{dC}}{{d{\phi _j}}}(x,y).$$

The term dϕ_j/dR determines how a small change in a nanorod radius will introduce a change in the local phase at wavelength λ_j and can be determined numerically using the phase map Fig. 1(d). However, the cost gradient profiles obtained at this stage are not circularly symmetric. An average process is then implemented to ensure metasurface circular symmetry by dividing a metasurface into concentric rings centered at the optical axis with ring width equal to the meta-atom pitch; an average of cost function gradients weighted by total intensity (summed over all sampling fields and wavelengths) illuminated on each point within a ring is applied and rod radius profiles are updated accordingly. Mathematically the process is expressed as

(6)$$\frac{{dC}}{{dR}}(r) = \frac{{\sum\limits_k {\frac{{dC}}{{dR}}} ({x_k},{y_k})\sum\limits_{{\lambda _j}} {\sum\limits_{{\theta _i}} {{w_i}} } {w_j}I_ - ^{i,j}({x_k},{y_k})}}{{\sum\limits_k {\sum\limits_{{\lambda _j}} {\sum\limits_{{\theta _i}} {{w_i}} } {w_j}I_ - ^{i,j}({x_k},{y_k})} }},$$

where (x_k, y_k) are sampling points within a concentric ring with radius r and $I_ - ^{i,j}(x,y)$ is intensity at the left side of a metasurface under an incident field with field angle θ_i and wavelength λ_j. The nanorod radius is updated as $\Delta R(r) ={-} \alpha dC(r)/dR$ with α a constant to control maximum variation of nanorod radius in a single AGM iteration. Meanwhile, a nanorod radius should be limited within the meta-atom design range R = [200–450] nm. If a radius value falls outside this range after updating, a phase-equivalent radius that stays within the range is chose instead. For single wavelength optimization, this phase-equivalent radius is unique; for multi wavelength optimization, different wavelengths generally have different radius values because meta-atom phase dispersion exists, then a weighted average is applied to determine a shift amount. After this, a new AGM iteration is started using the updated nanorod radius profiles obtained from the prior iteration. This process is repeated until a converged result is reached. Figure 2 is a flowchart to summarize the optimization steps for MAC design.

Fig. 2. Flow chart for metasurface aberration corrector design using the adjoint gradient method as presented in Section 2.

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3. Results and analyses

3.1 Nanorod hybrid lens design

To demonstrate the validity of the proposed method, we present analyses to a multiwavelength MAC designed for the refractive lens shown in Fig. 1(a) using the procedure mentioned in the prior section. The MAC consists of a pair of metasurfaces. The first metasurface is 2 mm behind the last surface of the refractive optics and the second metasurface is 2 mm in front of the image plane. The two metasurfaces are well separated because if they are close together then the cost function gradients will be similar for both metasurfaces. This usually means we do not fully use the degrees of freedom allowed. While we do not discipline the functionality of each metasurface, the second metasurface is put close to image plane as this is a position where different field angles are relatively separated so it could lead to a better control of local wavefront.

For both metasurfaces, a Si substrate thickness of 0.5 mm is also considered. Note refraction at a planar air-Si interface is automatically considered in the scalar angular spectrum method. In the design process, we considered 15 equal-spaced sampling field angles between 0° and 30° and 3 wavelengths λ₀ = {3.75, 4, 4.25} µm in MWIR. To expediate optimization, spatial sampling intervals are set to twice the length of meta-atom pitch. All nanorod radii were initially set to 325 nm and around 600 AGM iterations were used. In each iteration, the maximum absolute variation in nanorod radius is set to α = 2 nm. In the optimization process we did not specify metasurface aperture size. Instead, we prepared a simulation window that is large enough to capture all non-zero cost function gradients.

We found it is crucial to choose sufficient number of sampling field angles, especially when a metasurface is close to the image plane where ray bundles from different field angles are separate. Specifically, there should be sufficient overlap between fields that there is some cross-field correlation in the phase map; otherwise, optimization results in independent domains akin to a lenslet array except that the domains only work at the optimization fields while performance falls off very-quickly between them. To safeguard against this effect, we optimize with a large number of fields but also perform post-design analysis at the optimization fields and at interstitial fields between those points. Figure 3(a) shows the optimized nanorod radius distributions for the two metasurfaces in the MAC. The discontinuities are due to phase wrapping and oscillations are a natural feature of freeform design from a gradient-based algorithm. Importantly, slow-varying nanorod radius distributions in comparison to wavelength are obtained. This justifies the metasurface local periodicity approximation made before. Figure 3(b) shows the corresponding phase profiles in the radial direction for the two metasurfaces at a wavelength of 4 µm. We further compare these profiles to that from an on-axis-only metalens with similar specifications to the hybrid lens design (F/2, 10-mm EFL). It is clear the two metasurfaces in the MAC have smaller phase gradients, as optical power is mainly carried by the refractive optics. As will be discussed later, these small phase gradients also suggest small diffraction loss from the MAC.

Fig. 3. Nanorod radius distribution for (a) metasurface 1 and (b) metasurface 2 in the MAC. (c) The corresponding phase at a wavelength λ₀ = 4 µm for the two metasurfaces. Also shown is the ideal phase profile $\phi (r) ={-} 2\pi \sqrt {{r^2} + {f^2}} /{\lambda _0}$ for a F/2, 10 mm EFL metalens for on-axis imaging.

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Figure 4 shows sliced images along the y-z plane of focusing beam propagating through the MAC at two representative field angles θ_i = {0°, 30°}. The locations of the metasurfaces are marked. Meanwhile, the incoherent point spread function (iPSF) on the image plane resulted from the hybrid lens are compared to that of the stand-alone refractive lens to illustrate the effectiveness of the MAC. For the hybrid lens, the on-axis spot has a shape close to an Airy pattern, and at the maximum field angle slight astigmatism is observed. Still, improvements in terms of spot size and shape over the stand-alone refractive lens are manifest. It is worth mentioning the centroid position of the off-axis spot is not moved even with the introduction of the MAC (Fig. 4(e) and (f)). As noted, the MAC is designed in such a way that spot locations are not significantly shifted. Hence the F-number and EFL are largely unaffected. Next, we compare the modulation transfer functions (MTF) of the two systems along the sagittal and tangential directions for the two representative field angles, the result is given in Fig. 5. Improvements at high spatial frequency are manifest. It is noted the lens is anamorphic, so different cut-off frequencies alone the x- and y-directions should be applied for off-axis MTF analysis. These cut-off frequencies can be determined from working F-number of a beam. From the corresponding iPSF (Fig. 4(e)) we readily see the spot width in the y-direction is relatively narrow. We further corroborate this result by loading the phase profiles at λ₀ = 4 µm into Zemax to simulate the hybrid lens. As shown in SI (Fig. S2), the MTF analysis by Zemax shows good agreement for both field angles.

Fig. 4. (a) yz-slice of beam propagation for the hybrid lens at a field angle θ_i = 0°. Only the parts after the refractive lens are shown. The white and black dash lines indicate the positions of metasurfaces and Si substrate boundaries, respectively. (b) and (c) is incoherent point spread function for the hybrid lens and the stand-alone refractive lens, respectively, on the image plane. (d-f) are the corresponding results for θ_i = 30°. In all simulations, the wavelength is set to λ₀ = 4 µm and the optical axis is at (x, y) (0, 0).

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Fig. 5. Modulation transfer function for the hybrid lens and the stand-alone bulk lens at a field angle θ_i of (a) 0° and (b) 30°. Both the sagittal (x) and tangential (y) directions are displayed. Also shown are the diffraction-limited results with off-axis anamorphic effect considered. The wavelength is set to λ₀ = 4 µm.

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During AGM iterations, we usually need a way to monitor performance of the latest optimized result and to check convergence. The cost function value (Eq. (1)) is a possible candidate. Nonetheless, a more intuitive option is focusing efficiency, which in this work is defined as

(7)$$\eta \equiv \frac{{\int_{{R_{\textrm{Airy}}}} {{I_{\textrm{out}}}(x,y)dS} }}{{\int_{{R_{\textrm{Airy}}}} {{I_{\textrm{ideal}}}(x,y)dS} }},$$

where R_Airy denotes an Airy disk centered at the spot centroid for the given wavelength and F-number. We stress the output intensity ${I_{\textrm{out}}}(x,y)$ as calculated from the steps in Section 2 is under the LUA and does not consider transmission loss and diffraction loss from Fresnel zone boundaries. This definition is chosen to allow a direct comparison of image spot size with and without the MAC. This directly parallels conventional optimization routines using sequential ray-tracing packages to minimize spot size without considering reflection losses, ghost images, light scattering, etc. – these require deeper analysis typically performed after initial optimization using non-sequential ray tracing and/or physical optics packages. In Section 3.2 we develop a simple method to analyze these losses beyond the LUA; the method is simple because it does not require computationally intensive full-wave simulations of large lens structures in order to predict their performance. Figure 6(a) shows evolution of focusing efficiency of the 15 field angles as AGM runs. Because of the trade-off between different field angles and the implementation of weighted average to maintain metasurface circular symmetry at the end of each iteration, focusing efficiency does not rise monotonously in the optimization process. Generally speaking, the inherent symmetry of the on-axis spot leads its focusing efficiency to rise rapidly at the first few AGM iterations, but will later compromise in order for the performance of other field angles to improve. Still, a converged result can be obtained after sufficient number of iterations. Lastly, Fig. 6(b-c) compares focusing efficiency of the hybrid lens and the bulk lens using a finer field angle and wavelength sampling. The averaged focusing efficiency over the entire wavelength range and FOV before and after adding the MAC is 37.05% and 55.85%, respectively. Strong oscillations are observed at the edges of the wavelength range in between optimization fields. We will show these oscillations are due to strong phase dispersion inherent in the SOI nanohole meta-atom (Fig. 1(d)). Nevertheless, for single wavelength operation at λ₀ = 4 µm, the averaged focusing efficiency over the entire FOV before and after adding the MAC is 35.35% and 64.74%, respectively. In principle, narrowband performance may be increased further by reducing the spread of optimization wavelengths which currently span λ₀ = 3.75 µm to 4.25 µm; however, our goal is increasing the operational bandwidth while maintaining significant improvement over the stand-alone refractive lens.

Fig. 6. (a) Hybrid lens focusing efficiency of the 15 field angles as a function of AGM iterations. The values are averaged over the wavelengths λ₀ = {3.75, 4, 4.25} µm. Focusing efficiency at different wavelengths λ₀ and field angles θ_i for (b) the stand-alone refractive lens and (c) the optimized hybrid lens. The same color axis limit is used for both plots for easy comparison.

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3.2 Post-design loss analysis

The performance of a metasurface can be degraded by imperfect transmission and diffraction. This is particularly true when the desired phase profile has fast variations and/or a meta-atom unit with large pitch is used [21]. To shed light on this aspect and its potential impact on the real performance of our hybrid lens, we perform numerical simulations to model both transmission and diffraction loss of a metasurface. Here, we model a circularly symmetric metasurface by discretizing it along the radial direction into concentric rings. For each ring, we apply an adiabatic approximation and estimate losses from a one-dimensional blazed grating (Fig. 7(a)) with a diffraction angle and grating period that locally match the zone [22]. The ideal phase profile of a blazed grating implemented by meta-atoms and within the adiabatic approximation is [23]

(8)$$\phi (x) = \frac{{2n\pi }}{{mL}}x + {\phi _0},$$

where m is the number of meta-atoms that comprise the grating period, L is the meta-atom pitch, n is the desired grating order for diffraction, and ϕ₀ is a reference phase. Note that the grating period is equal to mL. This idealized phase profile provides a grating wavenumber ${k_d} = 2n\pi /mL$ to an incident plane wave. For normal incidence, this leads to a deflected wave with an angle ${\theta _d} = {\sin ^{ - 1}}(n{\lambda _0}/{n_t}mL)$ with respect to the surface normal. Here n_t is refractive index of the output region. As n and m are integers, diffraction angles that can be realized by meta-atom blazed grating with a fixed pitch size are discrete. Through these simulations we first determine the fraction of incident power that is transmitted past metasurface and this is the transmittance, T. Next, we consider that the transmitted light is not directed entirely into the desired direction θ_d. To quantify this diffraction loss, we define diffraction efficiency η_d of a blazed grating as the ratio of the transmitted power at the designed diffraction order with respect to the total power of the transmitted light.

Fig. 7. (a) Illustration of a unit cell of nanorod meta-atom blazed grating which corresponds to n/m = 1/6. Transmission coefficients for transmitted mode orders are recorded in each simulation. (b) A database of nanorod meta-atom blazed grating diffraction efficiency. In each set of simulation, averaged (over ϕ₀, see main text for details) transmittance for the designed grating order ${\bar{T}_n}$ and averaged total transmittance $\bar{T}$ are recorded under y-polarized plane wave normal incidence at a wavelength λ₀ = 4 µm. The ratio gives averaged diffraction efficiency ${\bar{\eta }_d} = {\bar{T}_n}/\bar{T}$ of a blazed grating. Also shown for reference is the ideal diffraction efficiency for a m-staged staircase grating [24].

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To investigate diffraction loss of the MAC design, we first construct a database of 60 blazed grating simulations with distinct fractional numbers n/m (so distinct k_d). All simulations are performed using CST Studio with a y-polarized (normal to grating vector direction) plane wave with a wavelength of 4 µm normally incident from the Si substrate. For a given fractional number n/m and phase reference ϕ₀, a blazed grating period is composed of m SOI meta-atoms, and the meta-atom phase map Fig. 1(d) is used to map the ideal phase profile (Eq. (8)) to a set of m nanorod radii. The same boundary condition as the meta-atom simulation stated in Section 2 is applied. The output field is sampled at the air region (i.e., top boundary) where transmittance into the designated grating order T_n and total transmittance T are recorded. Since T_n and T are also functions of ϕ₀, we sampled the range ϕ₀ = [0, 2π) with 35 equidistant values for each blazed grating configuration (i.e., each k_d). We average these results to estimate ${\bar{T}_n}$ and $\bar{T}$ for a given n/m and report the results in Fig. 7(b). From this data, we also calculate and plot the averaged diffraction efficiency ${\bar{\eta }_d}$. Data for very small grating wavenumber is missing due to computing constraints because this corresponds to increasingly large number of meta-atoms in the simulation; for example, the smallest non-zero k_d = 0.1429k₀ (k₀: free space wavenumber) corresponds to n/m = 1/28. However, this should not severely affect our subsequent analysis because we expect a smooth transition from our lowest k_d to the local uniformity approximation (shown at k_d = 0) and we interpolate losses through this region. Importantly, total transmittance T is only weakly dependent on grating wavenumber, while transmittance for the desired grating order T_n decreases more rapidly as grating wavenumber increases. Hence, we can infer the dominant loss mechanism for a blazed grating with a small grating wavenumber k_d is transmittance of the nanostructure, while for a large grating wavenumber unwanted diffraction plays a larger role.

Meanwhile, the local diffraction angle from a metasurface with continuous one-dimensional phase profile ϕ(x) under normal incidence is [25]

(9)$$\theta (x) = {\sin ^{ - 1}}\left( {\frac{1}{{{n_t}{k_0}}}\frac{{d\phi }}{{dx}}(x)} \right),$$

For an arbitrary θ_, we can find a blazed grating with a close deflection angle θ_d. Applying Eq. (9) to the radial phase profiles of the two metasurfaces in the MAC, one can obtain local diffraction angle profiles under normal incidence. We then estimate transmittance and diffraction efficiency as a function of radial position for the two metasurfaces using the blazed grating database, as shown in Fig. 8. Piecewise interpolation was implemented to estimate missing points in the database. We see that because of their small phase gradients, diffraction efficiency close to unity is estimated, and loss in the MAC is largely attributed to finite metasurface transmittance. Here we only present the results for y-polarized plane wave incidence. An analysis using x-polarized plane wave gives similar results (c.f. SI Section 6). For comparison, we also plot the losses of a metasurface singlet lens (i.e., metalens) matching our hybrid lens (10 mm EFL, F/2). We observe increased losses arising from the larger phase gradients required when the metasurface is tasked to provide the primary power rather than correct aberrations. Please note that the focusing efficiency of the metalens (not shown) is significantly degraded away from the axis.

Fig. 8. (a) Diffraction efficiency across MAC surfaces estimated from the blazed grating model and, for comparison, a metasurface singlet (i.e. simple metalens) which has much faster phase variation resulting in larger diffraction loss. (b) Transmittance profiles across each surface in the local uniformity approximation (solid lines) and the average value estimated from the blazed grating model (dashed lines). Results are for on-axis operation at a wavelength of 4 µm.

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3.3 Nanohole hybrid lens design

We conclude the major loss in the MAC is metasurface transmission. Moreover, a major roadblock to broadband MAC design is the strong phase dispersion inherent in the SOI nanorod meta-atom. In view of these, we further propose a meta-atom design with a lower phase dispersion and a higher transmittance in MWIR compared to the archetypical design (Fig. 1(c)). Through some quick optimization studies, we arrived at the new meta-atom geometry provided in Fig. 9(a). We switch from nanorod to nanohole design with an anti-reflection (AR) layer [26]. In this new design, a 0.5 µm thick zinc sulfide (ZnS) AR layer is added, and the buried oxide thickness is increased to 1.4 µm. The refractive index of ZnS is obtained from [27]. Similar to its counterpart, the nanohole radius R is considered as a parameter to locally control phase delay, and a bound R = [200–450] nm is applied. LUA maps for phase delay and transmittance are shown in Fig. 9(b) and (c), respectively. An averaged transmittance of 91.77% can be attained in the wavelength range λ₀ = [3.5–4.5] µm. It is noted an even higher averaged transmittance up to 96.62% can be attained if the buried SiO₂ layer is removed. However, we choose to retain this layer for fabrication consideration as discussed in Section 2. Importantly, phase coverage larger than 2π can still be attained for the whole wavelength range and a smaller phase dispersion dϕ/dλ₀ is observed. This can readily be confirmed by comparing contours in Fig. 9(b) and in Fig. 1(d). To better understand the benefits of having a small meta-atom phase dispersion, we redo the MAC inversed design using the method mentioned in the previous section with the same initial condition and parameter setting; the only difference is this time the SOI nanohole meta-atom phase delay map is incorporated. Performing focusing efficiency analysis to the new MAC design, the result is shown in Fig. 9(d). This result should be contrasted with Fig. 6(c). It is clear a higher and smoother result is obtained. The averaged focusing efficiency on the given range is 62.99%. In Fig. S4 we show the nanohole radius distribution for the two metasurfaces in the MAC. The distributions are highly similar to our previous design using the nanorod meta-atom shown in Figs. 3(a, b). This suggests the phase profiles are similar for these two designs, which in turn means difference in broadband performance is largely attributed to difference in meta-atom phase dispersion. Also, this means the new MAC design is expected to have a small diffraction loss, as we analyzed before.

Fig. 9. (a) The SOI nanohole meta-atom with all the relevant dimensions marked, and its (b) phase delay and (c) transmittance as a function of nanohole radius R and wavelength λ₀ under normal plane wave incidence from the Si substrate. The contour lines on the phase delay plot are equally spaced with a spacing of π/4. (d) Focusing efficiency at different wavelengths λ₀ and field angles θ_i for the optimized hybrid lens using SOI nanohole meta-atom. The same color bar limit as that in Fig. 5(b) is used for direct comparison.

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To better understand the origin of the difference in phase dispersion of the two meta-atom designs, we implement effective index modeling. To have a fair comparison, we make the two designs to have a consistent geometry by adding a 0.5 µm ZnS antireflection layer on the nanorod meta-atom and increasing the SiO₂ layer thickness to 1.4 µm. Applying effective medium theory, the phase delay introduced by a meta-atom periodic array is

(10)$$\phi (R,{\lambda _0}) = {k_0}[{n_{eff,\textrm{ZnS}}}(R,{\lambda _0}){t_{\textrm{ZnS}}}\textrm{ + }{\textrm{n}_{eff,\textrm{Si}}}\textrm{(R,}{\lambda _0}\textrm{)}{t_{\textrm{Si}}}] + {\phi _0}({\lambda _0}),$$

where ϕ₀ is a reference phase to maintain $\phi (R = 200\;\textrm{nm},{\lambda _0}) = 0$; n_eff_,ZnS (t_ZnS) and n_eff_,Si (t_Si) is the effective index (thickness) for the ZnS-air layer and Si-air layer, respectively. It should be noted phase delay is mainly originated from the Si-air nanostructured layer because of its larger thickness and higher index contrast. The effective indices are obtained from Bloch eigenmode analysis [28]. In short, for each periodic layer a square matrix is constructed based on the coupled-coefficient equations that is also used in two-dimensional rigorous coupled-wave analysis (RCWA); in the subwavelength regime, the zero-order eigenvalue of the matrix determines effective index of a layer. Figure 10(a) shows a comparison of phase delay for both meta-atom designs determined by the effective index model to full-wave simulation. The close correspondence between the two methods suggests the validity of using Bloch eigenmode effective index to describe the meta-atoms considered in this work. Small discrepancy is mainly due to multi-reflections are not considered in the effective index model. Comparing effective index dispersion dn_eff_,Si/dλ₀ in the Si-air nanostructured region for the two meta-atoms reveals that a stronger dispersion exists in the nanorod meta-atom, especially for the short wavelength end in the MWIR, as shown in Fig. 10(b). This difference largely leads to the difference in phase dispersion we observed before. We also notice reducing index contrast can reduce this difference in effective index dispersion between holey and pillar meta-atoms. Taking the ZnS-air AR layer for example, we find |dn_eff_,ZnS/dλ₀| < 0.06 µm⁻¹ for the given range (Fig. S5). However, for a low index contrast meta-atom the nanostructure height has to be extended in order to maintain a 2π phase coverage. As a final remark, we noticed a smaller meta-atom phase dispersion leads to a broader working bandwidth of the hybrid lens, and we may further estimate the limit of focusing efficiency when metasurface phase dispersion is absent. Here, we assumed the two metasurfaces of the MAC have their respectively optimized phase profile at λ₀ = 4 µm as obtained previously (Fig. S4(c)). These phase profiles are then used to calculate focusing efficiency of the hybrid lens for all wavelengths in the interested range. The averaged focusing efficiency thusly obtained is 65.02% (Fig. S6), slightly higher than the one considered nanohole meta-atom phase dispersion.

Fig. 10. (a) Comparing phase delay from SOI nanorod and nanohole meta-atom periodic arrays obtained from full-wave simulation to Bloch eigenmode analysis. (b) Comparing effective index dispersion in the Si-air nanostructured layer for the SOI nanorod and nanohole meta-atom periodic arrays.

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4. Conclusions

We introduce a design framework for metasurface aberration corrector aimed to improve image quality of a refractive lens while specifications such as F-number and effective focal length are maintained. The approach is versatile in that an arbitrary number of metasurfaces can be used and for each metasurface the geometry of constituent meta-atom can be independently parameterized. The validity of this approach is corroborated by designing a F/2, 10 mm EFL, 30°-FOV hybrid lens composed of a MWIR conventional lens followed by two SOI nanorod metasurfaces. The two metasurfaces have small phase gradients because optical power of the hybrid lens is mainly carried by the refractive optics. Diffraction losses from the two metasurfaces are expected to be minuscule based on the loss analysis using a simple meta-atom blazed grating model. However, the relatively large phase dispersion from the nanorod meta-atom limits the bandwidth of the hybrid lens. The averaged focusing efficiency over the wavelength range λ₀ = [3.5–4.5] µm and full field of view before and after adding the metasurface aberration corrector is 37.05% and 55.85%, respectively. To reduce this bandwidth problem, a holey meta-atom design with a smaller phase dispersion in MWIR is chose instead. Bloch eigenmode analysis shows this meta-atom design has a smaller effective index dispersion in comparison to its pillar counterpart. A hybrid lens designed using this nanohole meta-atom further improves averaged focusing efficiency on the given range to 62.99%. It should be clarified these values are calculated under the LUA and do not consider meta-atom transmission or diffraction loss.

In this work, MAC surfaces are constrained to lie behind the refractive elements (i.e. in image space) wherein the angular spectrum method is used for both forward and backward propagations during inverse design. A generalization of this method is possible so that interleaved diffractive and refractive elements can be optimized. However, this requires efficient propagation of the adjoint information through refractive surfaces, ideally using geometric optics. We are currently working on this extension so a metasurface can be inserted at any location in a lens and then optimized via inverse design. Furthermore, in this work we limit to metasurface optimization alone, further improvement through co-optimization of refractive and diffractive elements is possible by combining the inverse design scheme with standard raytracing optimizers.

Funding

Air Force Office of Scientific Research (FA8650-16-D-5408).

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.

Supplemental document

See Supplement 1 for supporting content.

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Broadband metasurface aberration correctors for hybrid meta/refractive MWIR lenses

Abstract

Corrections

1. Introduction

2. Design framework

3. Results and analyses

3.1 Nanorod hybrid lens design

3.2 Post-design loss analysis

3.3 Nanohole hybrid lens design

4. Conclusions

Funding

Disclosures

Data availability

Supplemental document

References

Supplementary Material (1)

Data availability

Cited By

Figures (10)

Equations (10)

Optics Express