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Abstract
Metalenses, composed of patterned meta-atoms in various dimensions, offer tailored modulation of phase, amplitude, and polarization for diverse imaging applications across the visible and near-infrared spectra. However, simultaneously achieving achromatic and wide field of view (WFOV) imaging remains a significant challenge. In this paper, we propose a general inverse design framework for metalens-doublets that simultaneously enables broadband achromatic and WFOV imaging. The broadband achromatic and WFOV (BA&WFOV) metalens-doublet comprises a propagation phase metalens and a geometric phase metalens positioned on opposite sides of the substrate. This framework requires only once polarization conversion and mitigates aperture size constraints imposed by the limited group delay (GD) range of meta-atoms. We present a BA&WFOV metalens-doublet with an f-number of 3.9, a full field of view (FOV) of 68°, and a wavelength range from 640nm to 820nm. This metalens-doublet exhibits diffraction-limited focusing with an average absolute focusing efficiency of 16% and an average relative focusing efficiency of 60%. This innovative framework holds significant promise for applications in fields such as phone cameras, VR/AR, and endoscopes.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metalenses have attracted significant attention owing to their flat and lightweight package and flexible modulation of phase, amplitude, and polarization enabled by the patterned meta-atoms in different dimensions. Metalenses exhibit immerse potential across diverse applications, such as biomedical imaging [1–4], 3D sensing [5–7], beam steering [8,9], spectrometer [10,11], and near-eye display systems [12–14].
In the imaging field, achromatic imaging and WFOV imaging are crucial for practical metalenses applications. While studies have demonstrated metalenses’ capabilities in achromatic imaging at discrete wavelengths or broadband within a narrow FOV [15–22] and WFOV imaging at a single wavelength [23–30], simultaneously achieving broadband achromatic and WFOV imaging remains a significant challenge. Various approaches have been explored to simultaneously achieve achromatic imaging at only a few discrete wavelengths and WFOV imaging, such as phase optimization [31], quadratic phase combined with phase optimization [32,33], quadratic phase combined with high order diffraction [34], inverse design based on topology optimization [35,36], and cascading metalenses [37–39]. However, these approaches often suffer from limitations such as limited aperture size, suboptimal performance compared to diffraction-limited imaging, or challenges in manufacturability.
In this work, we propose a general inverse design framework for metalens-doublets, simultaneously enabling broadband achromatic and WFOV imaging. The BA&WFOV metalens-doublet comprises a propagation phase metalens and a geometric phase metalens positioned on opposite sides of the substrate. Prior to joint optimization, an initial structure is established. The radii of the nanopillars on the first metalens are determined based on the required phase profile at the central work wavelength. The coupled nanofins on the second metalens are selected according to the folded GD and their rotation angles are determined based on the required phase profile at the central work wavelength. Subsequently, the joint optimization is performed using the Gcmma algorithm, a globally convergent algorithm. The objective function aims to find a set of radii for the nanopillars on the first metalens and a set of rotation angles for the coupled nanofins on the second metalens, maximizing the total focusing efficiency.
We find that there is a background component without phase modulation mixed in the transmitted light when a circularly polarized light is incident on two cascaded geometric phase meta-atoms. In contrast to two cascaded geometric phase metalenses, our proposed metalens-doublet is conducive to high polarization conversion efficiency. Conventionally, the aperture size is constrained by the limited range of GD provided by the meta-atoms. In this paper, we mitigate this constraint by folding the required GD between $GD_{min}$ and $GD_{max}$ which are bounded by the meta-atoms’ library.
We present a BA&WFOV metalens-doublet with an f-number of 3.9, a full FOV of 68$^{\circ }$ and a wavelength range from $640nm$ to $820nm$. This metalens-doublet exhibits diffraction-limited focusing with an average absolute focusing efficiency of 16% and an average relative focusing efficiency of 60%. Additionally, we present another BA&WFOV metalens-doublet with a moderately large aperture diameter of $200\mu m$ for the first metalens and $619\mu m$ for the second metalens exhibiting diffraction-limited focusing. This metalens-doublet demonstrates that our proposed BA&WFOV metalens-doublet effectively mitigates the aperture size constraint imposed by the limited GD range of meta-atoms. This framework will have great potential in fields such as phone cameras, VA/ARs, and endoscopes.
2. Methods
We propose a general inverse design framework for metalens-doublets shown in Fig. 1(a), which simultaneously enables broadband achromatic and WFOV imaging. The BA&WFOV metalens-doublet comprises a propagation phase metalens and a geometric phase metalens positioned on opposite sides of the substrate.
Fig. 1. (a) Layout of the BA&WFOV metalens-doublet. (b) Schematic of a nanopillar on the first metalens. (c) Schematic of coupled nanofins on the second metalens.
We employ nanopillars with a circular cross-section, shown in Fig. 1(b), as the meta-atoms on the first metalens. These nanopillars act as truncated waveguides, introducing varying phase delays based on different radii and heights. The accurate local transmitted fields for a range of the nanopillar radii and heights across the work wavelengths are calculated in advance using FDTD solutions. Subsequently, we construct a surrogate model utilizing polynomial interpolation to rapidly predict the local fields of nanopillars within geometric parameters suitable for fabrication. The surrogate model is time-saving and convenient in the following inverse design framework.
To increase the design freedom, we employ two coupled nanofins as meta-atoms on the second metalens. Their geometric parameters are defined in Fig. 1(c). Generally, the coupled nanofins can be treated as an anisotropic scatter. When a right-handed circularly polarized beam is incident on the coupled nanofins, the transmitted light can be described using the Jones vector [40–45]:
It is more conducive to high polarization conversion efficiency to adapt this combination of a propagation phase metalens and a geometric phase metalens than two cascaded geometric phase metalenses. We derived the transmitted light when a right-handed circularly polarized light is incident on two cascaded geometric phase meta-atoms (see Supplement 1, Section 1 for details). The transmitted light is shown as Eq. (2) if the two cascaded geometric phase meta-atoms have the same geometric parameters. Therefore, their $\tilde {t}_{L}$ and $\tilde {t}_{S}$ are respectively the same.
The initial phase profile for the BA&WFOV metalens-doublet is represented using the even aspheric polynomials ($\varphi (\rho )=\sum _{i} a_{i} \rho ^{2 i}$). The first step in realizing a BA&WFOV metalens-doublet is to utilize the software Zemax, based on ray tracing, to determine the initial parameters. These parameters include the phase coefficients of the two metalenses at the central work wavelength, the thickness of the substrate, and the distance from the second metalens to the image plane.
For the first metalens, the surrogate model helps quickly map the target phase to nanopillars’ radius. Since the phase of nanopillars is considered only at the central work wavelength, some chromatic aberration is introduced here, which will be compensated for in the following joint optimization step.
For the second metalens, the coupled nanofins are selected according to the folded GD and their rotation angles are determined based on the required phase profile at the central work wavelength. We refer to the general Snell Law:
Fig. 2. (a) The required GD (blue line) and folded GD (the red line) for the BA&WFOV metalens-doublet. (b) Phase profiles for different work wavelengths are calculated using the folded GD, exhibiting obviously phase discontinuities at the folded boundaries.
After determining the radii of the nanopillars on the first metalens and selecting the coupled nanofins on the second metalens, the initial point for the joint optimization is established. The second step is the joint optimization based on the inverse design framework starting from the aforementioned initial point. The fundamental principle of inverse design is to commence with the final imaging performance and then optimize it given the constraints of actual applications [20]. In this paper, imaging performance is quantified by the total focusing efficiency across a set of wavelengths and AOIs of interest. The objective function is to find a set of radii for the nanopillars on the first metalens and a set of rotation angles for the coupled nanofins on the second metalens that maximizes the total focusing efficiency. The objective function is described as the following equation:
The light field in the image space, a function of nanopillars’ radii and the coupled nanofins’ rotation angles, is calculated using Rayleigh Sommerfeld diffraction integrals as a forward evaluation of the BA&WFOV metalens-doublet’s performance. The objective function, maximizing the total focusing efficiency, is optimized using the Gcmma algorithm. This algorithm is globally convergent and well-suited for problems involving a substantial number of variables, typically ranging from $10^4$ to $10^5$ [50,51].
3. Results
A parameter sweep of nanopillars and coupled nanofins is simulated using FDTD solutions software to build a library. The material of all the meta-atoms is $TiO_2$. The nanopillars with the same height $h=1.2\mu m$ and the coupled nanofins with the same height $h=800nm$ are equally spaced with a period $p=300nm$. The material of the substrate is $SiO_2$. The GD range provided by the coupled nanofins is approximately $4fs$. The coupled nanofins with a transmission below 40% are excluded. From the remaining coupled nanofins, the folded GD range is determined to be between $3.6fs$ and $4.6fs$. See Supplement 1, Section 2 for details.
We present a BA&WFOV metalens-doublet with an f-number of 3.9, a full FOV of 68$^{\circ }$, and a wavelength range from $640nm$ to $820nm$. The aperture diameters for the first and second metalens are $40\mu m$ and $103\mu m$, respectively. These structural parameters, as well as the phase coefficients, are obtained using the softwave Zemax, based on ray tracing. These two metalenses are patterned on opposite sides of a substrate. The thickness of the substrate is $79\mu m$. To save simulation time and computing resources, the BA&WFOV metalens-doublet is designed as a 2D cylindrical lens (invariant along the $y$-axis). Then the second step is the joint optimization based on the inverse design framework, starting from the aforementioned initial parameters. The objective function is to find a set of radii for the nanopillars on the first metalens and a set of rotation angles for the coupled nanofins on the second metalens that maximizes the total focusing efficiency. The code was running on MATLAB(2022b) on a server with Intel Xeon Platinum 8160 CPU @2.10GHz. It takes about 14 hours before we achieve satisfactory results. The optimization results can be seen in Supplement 1, Section 2. We simulate the proposed BA&WFOV metalens-doublet using the FDTD solutions software. It takes about 2 hours to simulate this structure at a single wavelength and AOI. The BA&WFOV metalens-doublet exhibits diffraction-limited focusing with an average absolute focusing efficiency of 16% and an average relative focusing efficiency of 60%. The absolute focusing efficiency is defined as the ratio of the power within a circle with a radius three times that of the Airy disk around the position of theoretical imaging point to the total incident power. The relative focusing efficiency is defined as the ratio of the power within a circle with a radius three times that of the Airy disk around the position of theoretical imaging point to the total transmitted power just behind the second metalens. Notably, both chromatic and monochromatic aberrations are effectively corrected.
An ideal BA&WFOV metalens-doublet and a singlet chromatic metalens are simulated for comparison. For the ideal BA&WFOV metalens-doublet, the required phase profile at the central work wavelength is designed based on ray tracing over the target AOIs, the same as our proposed BA&WFOV metalens-doublet. Subsequently, the phase profiles for both the first and second metalens at other work wavelengths are obtained according to Eq. (4), ensuring fixed angles of refraction for a specific AOI across the entire work wavelength band. All other parameters for both the ideal and our proposed metalens-doublet are kept the same. The performance of the ideal BA&WFOV metalens-doublet at other work wavelengths is expected to resemble that at the central work wavelengths. For the singlet chromatic metalens, the required phase profile is only satisfied at central work wavelength. The focal length is inversely proportional to the wavelength.
When a plane wave is incident, the transmitted light fields on the $x$-$z$ plane of our proposed BA&WFOV metalens-doublet are all focusing tightly, as shown in Fig. 3. The simulation results before joint optimization are presented in Supplement 1, Section 3. The additional results for the ideal BA&WFOV metalens-doublet, the singlet chromatic metalens, and our proposed BA&WFOV metalens-doublet, not presented here, can be found in Supplement 1, Section 4.
Fig. 3. The light intensity around focusing point on the $x$-$z$ plane of the BA&WFOV metalens-doublet. The scale bar indicates $20\mu m$.
In Fig. 4, the relative intensity sections or point spread function (PSF) profiles are acquired at the best focusing plane of the corresponding metalens (with the same $z$-axis position) and normalized to their maximum intensity. The PSFs of our proposed BA&WFOV metalens-doublet exhibit superior performance compared to the singlet chromatic ones and closely resemble the ideal ones.
Fig. 4. (a)-(c) PSFs of the ideal BA&WFOV metalens-doublet. (d)-(f) PSFs of the singlet chromatic metalens. (g)-(i) PSFs of our proposed BA&WFOV metalens-doublet.
The theoretical full-width at half-maximum (FWHM) is calculated as $FWHM= 0.514*\lambda _{central}/NA$, where $NA$ is the numerical aperture, and $\lambda _{central}=730nm$ [52]. In Fig. 5, the FWHMs are consistently below or around the theoretical value (dashed line), indicating nearly diffraction-limited focusing. Meanwhile, the FWHMs of the ideal BA&WFOV for the AOIs larger than 27$^{\circ }$ at the wavelength of $820nm$ are slightly larger than the theoretical value (see Fig. S8 of Supplement 1 for details).
In Fig. 6, the solid lines indicate the absolute focusing efficiencies and the dashed lines indicate relative focusing efficiencies. The average absolute focusing efficiency is 16% and the average relative focusing efficiency is 60%.
Fig. 6. Focusing efficiencies of our proposed BA&WFOV metalens-doublet. The solid lines indicate the absolute focusing efficiencies. The dashed lines indicate the relative focusing efficiencies.
Although the focusing positions along the $z$-axis for all the on-axis FOVs (Fig. 7(a). solid line) are not highly concentrated on the horizontal black dashed line (obtained by the ideal BA&WFOV metalens-doublet), all the aforementioned focusing properties (PSFs, FWHMs, and focusing efficiencies) are all obtained on the same imaging plane and perform well. This is attributed to the inverse design, as its objective function emphasizes the focusing performance on the target image plane rather than the focusing positions. Moreover, the focusing positions along the $z$-axis of our proposed BA&WFOV metalens-doublet deviate less from the ideal line than those of the singlet chromatic metalens. The focusing positions along the $x$-axis (Fig. 7(b)) are concentrated around the ideal line (black dashed line, obtained by the ideal BA&WFOV metalens-doublet at the central work wavelength). The maximum deviation of the focusing position on the $x$-axis from that of the central work wavelength occurs at the wavelength of $640nm$ and AOI of 34$^{\circ }$. The maximum deviation is $3.6\mu m$, which is close to the distance separation of two stars that can just be resolved according to the Rayleigh criterion ($1.22*\lambda _{central} F_{number}=3.5\mu m$) [53].
Fig. 7. (a) Focusing positions for all the on-axis FOVs along the $z$-axis of our proposed BA&WFOV metalens-doublet (blue solid line), the ideal BA&WFOV metalens-doublet (black dashed line) and the singlet chromatic metalens (green dashed line). (b) Focusing positions along the $x$-axis. Colorful dots indicate the results of our proposed BA&WFOV metalens-doublet. Black dashed line indicates the results of the ideal BA&WFOV metalens-doublet at the central work wavelength ($\lambda =730nm$). Green dashed line indicates the results of the singlet chromatic metalens at the central work wavelength ($\lambda =730nm$).
In summary, the performance of our proposed BA&WFOV metalens-doublet far exceeds that of the singlet chromatic metalens and is almost as good as the ideal BA&WFOV metalens-doublet.
To demonstrate the generality of our framework, we design another BA&WFOV metalens-doublet with an f-number of 2.5, a wavelength range from $720nm$ to $780nm$, a full FOV of 52$^{\circ }$, and aperture diameters of $80\mu m$ and $144\mu m$ for the first and second metalens, respectively. The thickness of the substrate is $110\mu m$. This design also achieves diffraction-limited focusing with an average absolute focusing efficiency of 46% and an average relative focusing efficiency of 78% (see Supplement 1, Section 5 for details).
We intend to demonstrate that our framework can mitigates the aperture size limitations imposed by the limited GD range provided by meta-atoms. According to the Ref. [16], the achievable aperture radius of a singlet achromatic metalens is determined by Eq. (6):
where $\partial \varphi / \partial \omega$ is the absolute value of GD, $R$ is the aperture radius of the metalens, NA is the numerical aperture, and $c$ is light velocity. Our meta-atoms library can provide a maximum GD range of approximately $4fs$, akin to the range in Ref. [16]. For example, the aperture diameter can extend to $48\mu m$ for a moderate NA of 0.1 (equivalent to an f-number of 5). In this paper, we design another BA&WFOV metalens-doublet with an f-number of 5, a wavelength range from $720nm$ to $780nm$, a full FOV of 48$^{\circ }$, and aperture diameters of $200\mu m$ and $619\mu m$ for the first and second metalens, respectively. The thickness of the substrate is $887\mu m$.This design also achieves nearly diffraction-limited focusing (see Supplement 1, Section 6 for details). The required continuous GD exceeds the GD range that meta-atoms can provide (see Fig. S25 of Supplement 1 for details). The aperture size in our simulation is constrained primarily by our computing resources rather than the GD range. There are 1365 variables in total to be optimized. The aperture size in our example significantly surpasses the result calculated using Eq. (6).4. Discusion
In this paper, we propose a general inverse design framework for metalens-doublets, simultaneously enabling broadband achromatic and WFOV imaging. Three BA&WFOV metalens-doublets with different f-numbers, all exhibiting nearly diffraction-limited focusing, demonstrate the effectiveness of our inverse design framework. The proposed metalens-doublets exhibit broadband achromaticity, displaying no wavelength selectivity within the spectral range. However, the investigation of the achievable spectral and FOV range of our proposed BA&WFOV metalens-doublet remains a subject for future exploration. There typically exists a trade-off between the spectral and FOV range.
We derive the transmitted light when a circularly polarized light is incident on two cascaded geometric phase meta-atoms and find that there is a background component without phase modulation inevitably mixed in the transmitted light. Therefore, a propagation phase metalens combined with a geometric phase metalens is a good structural choice. This structure achieves higher focusing efficiency than two cascaded geometric phase metalenses. Moreover, this structure has more design freedom than two cascaded propagation phase metalenses.
Our proposed BA&WFOV metalens-doublet effectively mitigates the aperture size constraint imposed by the limited GD range. Regardless of the aperture size, our metalens-doublet selects meta-atoms from the available library within a limited GD range.
5. Conclusion
In conclusion, we propose a general inverse design framework for metalens-doublets, simultaneously enabling broadband achromatic and WFOV imaging. We present a BA&WFOV metalens-doublet with an f-number of 3.9, exhibiting diffraction-limited focusing across a wavelength range from $640nm$ to $820nm$ and a full FOV of 68$^{\circ }$. This represents a significant advance in state of the art for metalens, offering benefits such as broadband achromatic, wide FOV, only once polarization conversion, and the mitigation of the aperture size constraint. The BA&WFOV metalens-doublet can promise applications in diverse areas, including phone cameras, VR/ARs and endoscopes.
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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