2.1. Correction of Off-Axis Monochromatic Aberration
The phase profiles of the aperture metalens and focusing metalens are as follows:
where
represents the phase profile of the aperture metalens,
represents the phase profile of the focusing metalens, a
i and b
i are coefficients of the above polynomials, which are optimized by ray-tracing optimization,
is the radial coordinate of each nanopost at each metalens,
x and
y are position coordinates of each nanopost with respect to the center of each metalens, and
R is the radius of the doublet metalens [
5,
6]. The number of coefficients of the above polynomials can be chosen from 3 to 8. Considering the tradeoff between focusing performance and optimization time, the number is set to 5.
We designed a MIR doublet metalens with a diameter of 100 μm and a focal length of 120 μm, giving NA of 0.38 or f-number of 1.2, with an operating band from 3.1 to 3.5 μm. Here, the substrate is fused silica and the thickness of the substrate is 70 μm. The flowchart of optimization procedures is shown in
Figure 2, including ray-tracing optimization and particle-swarm optimization. In the ray-tracing optimization procedure, the initial values of
ai and
bi were all set to zero, for
i = 1, 2, 3, 4, 5. The target of the ray-tracing optimization is to minimize the focal spots of incident rays with incident angles of 0°, 2.5°, 5°, 7.5°, and 10° at the fixed plane of Z = 120 μm (i.e., the focal plane). The optimizations were carried out at wavelengths of 3.1 μm, 3.2 μm, 3.3 μm, 3.4 μm, and 3.5 μm, respectively. After optimization, the optimized coefficients a
i and b
i were obtained, and are listed in
Table 1 and
Table 2, respectively. Phase profiles of the aperture metalens and focusing metalens according to Equations (1) and (2) at wavelengths 3.1 μm (lower limit of operating band), 3.3 μm (center wavelength), and 3.5 μm (upper limit of operating band) are shown in
Figure 3a–f, respectively. The phase profiles of the aperture metalens indicate that the aperture metalens works as a concave lens, which diverges incident rays. On the contrary, the phase profiles of focusing metalens indicate that the focusing metalens works as a convex lens, converging rays to the focal plane. The aperture metalens and focusing metalens work together to correct the monochromatic aberrations induced by oblique incidences, thus realizing diffraction-limited focusing on the same focal plane with different incident angles from 0° to 10°, corresponding to 20° FOV. In general, FOV is double the largest incident angle considering the symmetry of metalens [
5,
6].
2.2. Correction of Chromatic Aberration
From
Table 1 and
Table 2, we know that the phase profiles vary with different wavelengths and cannot be simultaneously satisfied by the same doublet metalens, thus resulting in chromatic aberrations when broadband incident light propagates through the doublet metalens. To be more specific, if we design spatial distribution of silicon nanopillars with various sizes according to the phase profiles of the doublet metalens at one particular wavelength, it cannot satisfy the phase profiles at other wavelengths. Therefore, broadband incident light propagating through the doublet metalens has different foci along the longitudinal direction at different wavelengths, which means chromatic dispersion. In order to minimize chromatic aberrations and thus realize diffraction-limited achromatic focusing, the phase profiles at each wavelength should be engineered to tailor chromatic dispersion.
The required phase profiles of the aperture metalens and focusing metalens to achieve achromatic diffraction-limited focusing for incident light are as follows:
where
λ is wavelength,
represents the required phase profile of the aperture metalens at each wavelength,
represents the required phase profile of the focusing metalens at each wavelength,
represents the obtained phase profile of the aperture metalens at each wavelength based on Equation (1) after ray-tracing optimization,
represents the obtained phase profile of the focusing metalens at each wavelength based on Equation (2) after ray-tracing optimization, and
and
are phase corrections of the aperture metalens and focusing metalens, respectively, which are radial coordinate-independent but wavelength-dependent. The role of
and
is to significantly reduce the difficulty of finding an adequate spatial distribution of silicon nanopillars with various sizes that simultaneously satisfies the required phase profiles at all wavelengths in the target operating band [
7]. The phase corrections
and
are optimized by multiobjective optimization based on a particle-swarm optimization algorithm, which we explain later.
Before optimizing the above mentioned phase corrections, we build the database of unit cells of the metalens. Here, the unit cell of the metalens is a silicon nanopillar on a fused silica substrate, shown in
Figure 4a, with the size of unit cell (i.e., the lattice period) set as 1.8 μm and the height of the silicon nanopillar set as 8 μm. The target operating band is set from 3.1 to 3.5 μm and discretized into five equally spaced wavelengths. The phase shifts were calculated by the finite-difference time-domain (FDTD) method using the commercial simulation software EastWave, with the nanopillar diameters swept from 0.2 to 1.8 μm in steps of 10 nm. The incident source in the simulation is an x-polarized plane wave. The simulation boundary is periodic in x-direction and y-direction, and perfectly matched layer (PML) in z-direction. The calculated phase shifts (folded between 0 and 2π) as functions of the nanopillar diameters at wavelengths of 3.1 μm (red line), 3.2 μm (yellow line), 3.3 μm (green line), 3.4 μm (blue line), and 3.5 μm (black line) are shown in
Figure 4b, clearly showing that both multiple 2π phase coverage and anomalous dispersions are achieved at these wavelengths, which are crucial to the realization of broadband achromatic metalens [
7]. Here, multiple 2π phase coverage provides more choices to find adequate silicon nanopillars in the database to meet the phase requirements at different wavelengths. That is to say, silicon nanopillars with several different diameters provide the same phase shift at one wavelength (e.g., 3.1 μm), while silicon nanopillars with these diameters provide different phase shifts at another wavelength (e.g., 3.5 μm). Therefore, we can choose one of these diameters to meet both the required phases at wavelengths of 3.1 μm and 3.5 μm. Furthermore, we can choose one diameter to meet all the required phases at wavelengths from 3.1 to 3.5 μm (in steps of 0.1 μm), as long as the database is rich enough. Besides, the abovementioned anomalous dispersions provide another degree of freedom to design the metalens. Here, anomalous dispersions mean that unfolded phase shifts do not always decrease or increase monotonically with nanopillar diameters, while phase shifts at some diameters experience a contrary variation tendency to most others.
After building the above database of unit cells of the metalens, the phase corrections
and
were optimized by multiobjective optimization based on particle-swarm optimization. In the optimization procedure, the initial values of
and
were all set to zero, for
λ = 3.1 μm, 3.2 μm, 3.3 μm, 3.4 μm, and 3.5 μm. Then,
and
were optimized separately. At each cycle of the optimization, the sizes and spatial distributions of silicon nanopillars at the aperture metalens or focusing metalens were designed according to both the database and the required phase profiles at wavelengths from 3.1 to 3.5 μm (in steps of 0.1 μm) based on Equation (3) or Equation (4). To be specific, at each cycle of the optimization, the design of the sizes and spatial distributions of silicon nanopillars and the calculation of phase difference is as follows. At each position
of the aperture metalens or focusing metalens, at each wavelength
, the absolute difference between the phase in the database and the required phase based on Equation (3) or Equation (4) is calculated as:
where subscript
m represents different positions of the designed aperture metalens or focusing metalens, subscript
n represents different wavelengths in the operating band, and subscript
k represents different nanopillars in the database. The phase differences at all wavelengths in the operating band are summed as:
At each position, the minimum of
in the whole database is found, which represents the best optimization between all wavelengths. The corresponding nanopillar in the database is designed at this position. The phase induced by the designed nanopillar at each position and at each wavelength is
. Then, the total phase difference (referred to as
in Equation (7)) between the required phase profiles based on Equation (3) or Equation (4) (referred to as
in Equation (7)) and the phase profiles induced by the designed aperture metalens or focusing metalens (referred to as
in Equation (7)) at all positions and all wavelengths were calculated at each cycle of the optimization. The total phase difference is calculated as follows:
where subscript
m represents different positions of the designed aperture metalens or focusing metalens and subscript
n represents different wavelengths in the operating band. The target of the optimization is to minimize the total phase difference. By comparing the total phase difference at each cycle with the minimum value of the total phase difference in previous cycles, we updated the values of
or
based on the rules of the particle-swarm optimization algorithm and repeated the above procedures. After multiple optimization cycles, the optimal values for phase corrections
or
were obtained while the total phase difference was minimized. The phase corrections of the aperture metalens and focusing metalens are listed in
Table 3 and
Table 4, respectively. The comparisons of the required phase profiles (blue lines) and the phase profiles induced by the designed aperture metalens and focusing metalens (red dots) as functions of radial coordinates at wavelengths 3.1 μm, 3.3 μm, and 3.5 μm are shown in
Figure 5a–f, respectively. As is clear from
Figure 5, both the designed aperture metalens and focusing metalens satisfy the required phase profiles well. The average differences (abbreviated to ave. dif. in
Table 3 and
Table 4) between the required phase profiles and the phase profiles induced by the designed aperture metalens and focusing metalens at each wavelength are calculated and listed in
Table 3 and
Table 4, respectively. Setting 2πrad as the unit, the average differences between the required phase profiles and the phase profiles induced by the designed aperture metalens and focusing metalens are merely 0.0527 and 0.0588 at 3.1 μm, 0.0352 and 0.0472 at 3.2 μm, 0.0519 and 0.0721 at 3.3 μm, 0.0589 and 0.0684 at 3.4 μm, and 0.0643 and 0.0733 at 3.5 μm. According to the well-known Rayleigh criterion, the optical system is perfect when the maximum wave-front aberration is less than a quarter of the wavelength. That is to say, the imaging quality of the optical system is not significantly different from that of the ideal optical system. Here, the average difference is equivalent to the wave-front aberration in the Rayleigh criterion. Such small average differences (all much less than a quarter of wavelength) verify that the designed doublet metalens satisfy the required phase profiles well, and thus can realize simultaneous reduction of both chromatic and off-axis monochromatic aberrations.