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Abstract
Achromatic bifunctional metasurface is of great significance in optical path miniaturization among advanced integrated optical systems. However, the reported achromatic metalenses mostly utilize a phase compensate scheme, which uses geometric phase to realize the functionality and uses transmission phase to compensate the chromatic aberration. In the phase compensation scheme, all the modulation freedoms of a nanofin are driven at the same time. This makes most of the broadband achromatic metalenses restricted to realizing single function. Also, the phase compensate scheme is always addressed with circularly polarized (CP) incidence, leading to a limitation in efficiency and optical path miniaturization. Moreover, for a bifunctional or multifunctional achromatic metalens, not all the nanofins will work at the same time. Owing to this, achromatic metalenses using a phase compensate scheme are usually of low focusing efficiencies. To this end, based on the pure transmission phase in the x-/y- axis provided by the birefringent nanofins structure, we proposed an all-dielectric polarization-modulated broadband achromatic bifunctional metalens (BABM) in the visible light. Applying two independent phases on one metalens at the same time, the proposed BABM realizes achromatism in a bifunctional metasurface. Releasing the freedom of nanofin’s angular orientation, the proposed BABM breaks the dependence on CP incidence. As an achromatic bifunctional metalens, all the nanofins on the proposed BABM can work at the same time. Simulation results show that the designed BABM is capable of achromatically focusing the incident beam to a single focal spot and an optical vortex (OV) under the illumination of x- and y-polarization, respectively. In the designed waveband 500 nm (green) to 630 nm (red), the focal planes stay unchanged at the sampled wavelengths. Simulation results prove that the proposed metalens not only realized bifunctional achromatically, but also breaks the dependence of CP incidence. The proposed metalens has a numerical aperture of 0.34 and efficiencies of 33.6% and 34.6%. The proposed metalens has advantages of being flexible, single layer, convenient in manufacturing, and optical path miniaturization friendly, and will open a new page in advanced integrated optical systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Metasurface provides a valuable platform as a powerful artificial subwavelength planar array consists of rationally designed nanofins [1–8]. The nanofins of a metasurface can be elaborately designed through transmission phase and geometric phase. By changing the material and structures of the nanofins, the transmission phase of the incidence can be controlled [9]. By altering the angular orientations of the nanofins, the nanofins will carry a geometric phase in the orthogonal direction of the incidence. Taking advantage of the transmission phase and geometric phase, metalens can manipulate over phase, amplitude, and polarization to realize extraordinary functions like phase shift [1,2], polarization test [3–5], hologram [6–8], nonlinear optics [10–12], and super-resolution nanofabrication [13].
The progress in a modern system has raised an urgent demand for a miniaturized and integrated optical device. By utilizing the transmission phase or/and geometric phase, a single metalens can be imparted with different phase profiles for different functions [14–28]. Wang et al. proposed a bifocal metalens using geometric phase [18]. Yan et al. reported a bifunctional adjustable optical vortex (OV) generator using transmission phase in x- and y-axis [23]. Recently, dynamic controllable multiplex optical-devices, which incorporate different tuning mechanisms into static metasurfaces, has become an important field, and has drawn great attention of the researchers [29–31]. Sedeh et al. proposed a time-modulated metasurface, which can control the OV’s topological charge dynamically [29,31]. Compared with a monofunctional metalens, bifunctional/multifunctional metalenses will lead to numerous potential applications where information can be multiplexed without increasing the device volume and will play an important role in miniaturized optical devices.
In practice, metalens in broadband wavelength is another challenge for the researchers because of the chromatic aberration. Achromatic bifunctional metasurface is of great significance in miniaturized and integrated optical systems. The chromatic aberration appears with the change of the incident wavelength, which alters the nanofin’s material refractive index continuously. So far, the reported achromatic metalenses mainly use phase compensate scheme [19,20,28,32–36], which uses transmission phase combined with geometric phase. Geometric phase is used to realize the functionality, and transmission phase is used to compensate the chromatic aberration. For example, Wang et al. proposed a broadband achromatic single focal metalens using phase compensate scheme with metal-dielectric-metal nanorod [32]. Yu et al. used single dielectric nanofin employing phase compensate scheme to realize broadband achromatic [35]. However, the phase compensate scheme calls for both geometry phase and transmission phase, which means all the modulation freedoms of a nanofin will be driven at the same time. For this reason, most of the broadband achromatic metalenses are restricted into realizing a single function. And since geometric phase is used to realize the functionality, in the bifunctional/multifunctional achromatic metasurfaces, not all the nanofins will work at the same time. For example, for a bifunctional achromatic metasurface, one part of the nanofins works to realize function 1, and the other part of the nanofins work to realize function 2. So the efficiency will be affected. Moreover, the phase compensate scheme is always addressed with circularly polarized (CP) light because of the adoption of geometric phase. CP light incidence will lead to the need of extra optical devices, like polarizers, in the optical path, and will lead to difficulties in the optical path miniaturization. In addition, the output light of phase compensation scheme contains CP light in both orthogonal directions. For example, a focusing metalens for left-handed circularly polarized (LCP) incident will output focusing right-handed circularly polarized (RCP) light together with LCP light, where the output LCP part acts as background noise. Consequently, the output efficiencies of phase compensate scheme are usually low, no higher than 25% [19,20,28,32–34]. Several pioneering researches on broadband chromatic aberration control breaking the limit of phase compensation scheme have been reported [37–40]. Khorasaninejad et al. realized achromatic single focal metalens in the visible over 60 nm by using optimization of transmission phase [37]. Ou et al. reported polarization-controlled broadband achromatic OV generator using transmission phase in x- and y-axis, realized bifunctional achromatic metadevice in mid-infrared range [38]. Up to now, few polarization-modulated broadband achromatic bifunctional metasurface in the visible with single layer have been reported, which is of great significance in optical path miniaturization and advanced integrated optical systems.
In this work, we designed an all-dielectric polarization-modulated broadband achromatic bifunctional metalens (BABM) that breaks the dependence on CP light incident, thus improved the efficiencies and facilitated the miniaturization of achromatic optical devices. Different from the metalenses using phase compensate scheme, the metalens is designed based on pure transmission phase by simply changing the nanofins’ structures. At the same time, particle swarm optimization (PSO) algorithm is used to optimize the nanofins’ structures, in order to optimize the BABM to work in broadband wavelength. Transmission phases in x- and y-axis are manipulated independently by tailoring the rectangular nanofins’ side lengths in x- and y-direction, respectively. Hence, two independent phases can be applied on one set of the metalens nanofin pattern simultaneously so as to realize single focal and OV function upon orthogonal linear polarization (LP) incidence achromatically. Since the freedom of nanofin’s angular orientation is released, the proposed metalens breaks the dependence on CP incident, all the nanofins can work simultaneously to form independent phase profiles. In this way, the metalens not only realizes independent bifunction achromatically, but also improves the average focusing efficiencies. Simulation results show that by simply changing the direction of the incident x-/y-LP beam, the designed BABM is capable of focusing the incident beam into a single focal spot or an OV at the same focal plane in the wavelength range 500 nm (green) to 630 nm (red). As an achromatic bifunctional metalens, the focusing efficiencies are improved to 33.6% and 34.6%, respectively. The proposed BABM has a numerical aperture (NA) of 0.34 and low cross talk. Compared with other works with the monofunctional or bifunctional achromatic metalenses using phase compensate scheme [19,20,28,32–34], the proposed metalens has the improved efficiencies up to 37%. The single layer and all dielectric properties of the BABM exhibit great advantages in manufacturing and integration [41]. Besides, breaking the dependence on CP incident, the BABM brings a new method for achromatic multifunctional metalens, and will open a new pave in optical path miniaturization of advanced integrated optical systems.
2. Principle and design
2.1 Principle of polarization-modulated BABM
The designed BABM consists of a single layer rectangular nanofins with different cross-sections. Each nanofin on the metalens acts as a birefringence waveguide. For a cubic nanofin with no rotation, the nanofin’s side lengths lx and ly will provide two independent phase shifts under the x- and y-polarization incidence. The transmission property of each nanofin can be expressed using spatial Jones matrix [21,38]:
On this basis, we designed a bifunctional metalens. OV beam in our work is introduced to verify the BABM’s two independent functions. As shown in Fig. 1, the bifunctional metalens convergence the incident beam into a single focal spot or OV by phase profile modulation. The total phase profile of the designed bifunctional metalens can be described as a superposition of a spiral phase plate and a converging lens [14,38]:
Fig. 1. Phase profile of bifunctional metalens in (a)x-polarization and (b)y-polarization incidence.
It can be seen from Eq. (4) that the focal length f is wavelength dependent. When the bifunctional metalens is upon the illumination of broadband incidence, the successive wavelength shift will lead to continuous focal plane displacement. At a specific wavelength λm in a broadband wavelength range, the phase function in Eq. (2) should be modified into:
In the design of broadband achromatic bifunctional metalens, f is set to be fixed over continuous wavelength change, so the phase factor Cj(λm) can be different at different wavelength. In order to control the broadband chromatic aberration, Cj(λm) should be optimized by minimizing the summation of phase error between the real and ideal phase across the design waveband and the nanofin coordinate, which can be expressed as:
2.2 Design of BABM
As shown in Fig. 2, the proposed BABM can achromatically focus the broadband incidence at the designed focal planes. The focused OVs carry different topological charge(L|| = 0, L⊥ = 2) depends on the polarization state of the incident LP light. That is, the BABM works as a converging lens when x-axis LP incidence, and it works as OV generator when y-axis LP incidence. The BABM is composed of plentiful TiO2 nanofins with different sizes but fixed angular orientations deposited on a SiO2 substrate and is designed by careful optimization on the nanofins’ transmission phase in x- and y-direction modulated by the nanofin structure parameter through PSO algorithm.
Fig. 2. Schematic diagram of the BABM. The broadband x- or y-polarized incidence is normally incident on the BABM. The transmitted light is achromatically focused into a single focal spot or an OV on the focal planes according to the incidence polarization state.
In order to realize a metalens shown in Fig. 2, we first established a transmission phase library of the birefringent unit cell structures. In subwavelength, for a dielectric nanofin, the presence of both electric and magnetic resonances at the same frequencies allows a 2π phase control. The resonance contains electric dipole, magnetic dipole, higher-order multipoles. As shown in Fig. 3(a) and (b), the unit cell is composed of a TiO2 rectangular nanofin stands on a SiO2 square lattice. The height h of the nanofins is fixed at 600 nm to meet with fabrication capacity limitation and provide sufficient phase accumulation. Rectangular nanofin is chosen over ellipsoid nanorod to maximizes the filling factor range of the nanofins cross section versus the lattice period [37]. Such rectangular act as a truncated waveguide with birefringent effect. The rectangular cross section of the nanofins is changed by the lengths in x- and y-directions lx and ly, result in different effective refractive indices of waveguide modes along x- and y-axis. So, by controlling the same nanofins’ side length lx and ly, the nanofin’s independent phase response in x- and y-directions can be modulated, respectively. The period u must satisfy the Nyquist sampling criterion, meanwhile, the period should be larger than all wavelengths in the nanofin to excite guided mode and should be smaller than all wavelengths in the free space to avoid diffraction of light into high diffraction orders [40], so u is set to be 360 nm. Each nanofin is parallel to x- or y-axis, so when the incident light with polarization along x- or y-axis passes through the nanofin, the nanofin won’t change its polarization state but generates modulation on phase and amplitude only. Figure 3(c) shows the magnetic field distributions in a representative nanofin size lx = 250 nm, ly = 100 nm. The operating wavelength is selected at 600 nm, which is in the range of the operation waveband 500 nm to 630 nm.The white dashed squares show the boundaries of the nanofins. The rectangular nanofin act as a truncated waveguide supporting the Fabry-Perot resonance. The standing wave patterns show that the light field is strongly confined inside the nanofin. Moreover, it is obvious that the magnetic field distribution is polarization-dependent, showing the nanofin’s independent response in x- and y-direction because of its anisotropy. The corresponding transmission phase at λ=600 nm is shown in Fig. 3(d). The yellow dot and the star shape show the phase response of the selected structure upon x- and y-polarization incidence, respectively. Note that because of the symmetry, the phase response of y-polarization incident can be obtained by mirror symmetry with respect to the diagonal line where lx = ly. It can be seen that the nanofin's lx and ly are capable of independent phase response.
Fig. 3. The unit cell design for the BABM. (a) Perspective view of the unit cell, showing the height h of the nanofin. (b) Top view of the nanofin, showing the period u, length lx and ly of the unit cell. (c) Magnetic distribution of the nanofin whose size is lx = 250 nm, ly = 100 nm when x- and y-polarized beam incident at 600 nm. The dashed white squares represent the peripherals of the nanofin. The different magnetic distributions in orthogonal polarized inject directions show the anisotropy of the nanofin. (d) The corresponding transmission phase of the selected structure. The yellow dot represents the x-polarization incidence, and the star shape represents the y-polarization incidence.
The nanofin structure is scanned by finite-difference time-domain (FDTD) method to build library. In the scanning of nanofins structure, periodic conditions are applied in x and y directions, the perfect matched layer (PML) condition is applied in z direction. Broadband x-polarization light from 520 nm to 624 nm is used as incident light. The lx and ly were scanned from 60 nm to 330 nm in steps of 1 nm. Figures 4(a) and (b) show the transmission phase and the amplitude transmission coefficient library for the x-polarized incident light in the designed waveband. It can be observed that the transmission phases not only provide 0-2π coverage, but also possess birefringence property. The library of y-polarization incident can be obtained by mirror symmetry with respect to the diagonal line where lx = ly.
Fig. 4. (a) Transmitted phase and (b) amplitude transmission coefficients library built by nanofin structure scanned under x-polarized incident, wavelength 520 to 624 nm.
As discussed in Section 2.1, in the design of BABM, the real phase profile provided by the selected nanofin structures should match the ideal phase profile required by the BABM as much as possible. Therefore, the problem of finding the most suitable nanofin structure can be transformed into the problem of finding the minimum of $\mathrm{\Delta }\varphi _{\textrm{Total}}^j$ by optimizing Cj(λm). In this work, PSO algorithm is utilized to search the minimum $\mathrm{\Delta }\varphi _{\textrm{Total}}^j$ by iterative optimization.
The flow of PSO algorithm is shown in Fig. 5. In the optimization, PSO updates a group of random particles by constantly iterating and tracking the current location a of the particles. In each iteration, a fitness function will be calculated using the information carried in location a, the best location found by a particle itself is defined as pbest, and the best location found by a particle swarm is defined as gbest, then the iterative search continues. In each iteration, the particle has three attributes: current particle location a, the location of previous pbest, and the particle velocity v. Changes in v and a during one optimization iteration can be expressed as:
Here, we first selected three arbitrary wavelengths (520 nm, 572 nm, and 624 nm) within the design waveband. Next, we put the required phases of each unit cell coordinate at each wavelength into a target list. Then, PSO algorithm is performed in x- and y-polarization scenarios, simultaneously. The computer configuration we used in this work is Windows 10 operating system, Intel Core i5-8300 H CPU@2.3 GHz, GPU NVIDIA Geforce GTX 1050Ti, RAM 12 G. The whole optimization process took 6 hours. Figures 6(a)-(d) show the overlays of the required phase and the optimized phase at three wavelengths. Figures 6(a) and (c) shows the required phase and the optimized phase at three wavelengths when x-polarization incidence while Figs. 6(b) and (d) shows those of y-polarization incidence.
Fig. 6. Phase matching diagram of x and y dimensions at 520 nm, 572 nm, and 624 nm when (a) (b) x-polarization incidence corresponding to single focal and (c) (d) when y-polarization incidence corresponding to OV.
3. Numerical simulation and discussions
Following the above method, a polarization-modulated BABM is designed by matching the required BABM phase with the simulated nanofins’ transmission phase in orthogonal axis using PSO algorithm. Figure 7(a) shows the layout of the designed BABM, which is of diameter of 29.16µm, the scale bar is 2µm. The designed metalens is arranged in circular-shape. Both diameters in x and y directions consist of 80 nanofins. The focal length f is designed as 43µm. The parameters of the BABM comply with the relation equation of NA, maximum achievable diameter, and the achromatic performance in [42]. The detail of the designed BABM [marked with the white frame shown in (a)] is shown in Fig. 7(b), the scale bar is 2µm.
There have been several experimental realizations of all dielectric metasurfaces [43,44]. The metasurface with TiO2 nanofins posted on SiO2 substrate can be realized by using conformal filling process based on electron beam lithography (EBL). In this way, the proposed unit cell can be implemented by steps of EBL, atomic layer deposition (ALD), etching and exposure. First, a double-side glass substrate should be coated with one layer of the electron beam resist. The thickness of the resist should be the same as the designed nanofin height, h. Then, the sample should be coated with a layer of 10 nm thick aluminum (Al) with the aid of thermal evaporation to define the pattern during the following EBL step. Afterward, the sample should be patterned using EBL system and then be developed in hexyl-acetate. Next, the patterned sample can be coated with TiO2 using ALD. After the ALD, the overcoated TiO2 layer should be etched by the inductively coupled plasma reactive ion etching (ICP-RIE) with gas mixture of Cl2 and BCl3. Finally, the samples should be exposed to UV irradiation, followed by the removing of the resist. Thus, the TiO2 nanofins with predesigned structures can be fabricated. Due to the conformal property of ALD, the structures of nanofins can be directly determined by the morphology of the electron beam resist, and the dielectric structures with high aspect ratio can be obtained.
Figures 8 and 9 show the simulation results of the designed BABM performed by the FDTD method. In the simulation, the perfect matched layer (PML) condition is applied in x, y, and z directions. The meshing step is set as 0.25 nm. Broadband LP lights from 500 nm to 630 nm in x- and y-direction are used as incident light. The light field of x-z plane and x-y plane are recorded. Focal plane is defined as the plane where the maximum power locates on the z-axis. Figures 8(a) and (b) show the simulated normalized intensity distribution along the x-z plane for the x-and y-polarized incidence, respectively. The white dashed line marked the distance of 40µm in z axis. It can be seen that the focal planes of generated single focal spot and the OV almost stay constant within the wavelength range. Figures 9(a) and (b) show the x-y plane of the focal spots. The focal spots show the polarization independent property of the designed BABM. That is, different topological charge within different incident polarization.
Fig. 8. Normalized intensity profiles along the x-z plane within the wavelength band from 500 nm to 620 nm for (a) x-polarized incidence and (b) y-polarized incidence.
Fig. 9. Simulated focal normalized intensity distributions within the wavelength band from 500 nm to 620 nm for (a) x-polarized incidence and (b) y-polarized incidence.
The characteristics of the designed BABM are analyzed in Fig. 10. Figure 10(a) shows the focal length variation with the change of incident wavelength. As can be observed in Fig. 10(a), the simulated focal lengths showed minor discrepancies, 3.16% and 6.3%, for x- and y-polarized incidence, respectively. The focal length discrepancy is defined by the ratio of the focal length difference between the maximum and minimum focal length relative to the mean focal length. The results proved that the BABM can effectively control the chromatic aberration of the focal beams under different polarization direction of the incident light. Figure 10(b) show the vertical cut (y direction) of the focal plane normalized intensity distribution when x- and y-polarized incidence, along with the normalized intensity distribution of ideal Airy disk which implying the ideal diffraction limit at wavelength of 560 nm. It can be seen that the intensity distribution of the single focal when x-polarization incidence is of high similarity with that of ideal Airy disk, proving the high focusing performance of the proposed BABM. It can also be seen that there is no interference between the normalized intensity of the single focal and the OV, implying low cross talk between the light field of x- and y-polarized incidence. The extracted full widths at half maximum (FWHM) of the single focus spots when x- polarized incidence are shown in Fig. 10(c) left axis. The FWHMs of the focal spots are calculated to be 1.61λ-1.88λ, showing the focal diameters are effectively controlled in the working wavelengths. The Strehl Ratio (SR) of the single focal spots at the sampled wavelengths when x-polarization incidence is calculated. The SR is defined as the ratio of central intensity of the recorded focal spot to the ideal Airy spot [34]. The calculated SRs are plotted in Fig. 10(c) right axis. The SR from 500 nm to 560 nm is stable with an average of 0.76, while the SRs from 600 nm to 620 nm relatively decreased. This is mainly due to the imperfect realization of the phase function and non-uniform nanofins’ scattering amplitudes across the metasurface. The focusing efficiency is shown in Fig. 10(d). Here, since the intensity distribution includes OV beam distribution, the focusing efficiency is defined as the ratio of the optical power of the focal plane versus the power incident on the BABM. It can be seen that the designed BABM achieves achromatic focusing while having stable focusing efficiency with averages of 33.6% and 34.6% in the working waveband. The fluctuation of the focusing efficiency is caused by the change of TiO2’s transmission in the visible wavelength and the matching error of the ideal phase and simulated phase. The efficiencies can be further improved by taking amplitude response into optimizing progress, using optimized nanofin shapes with higher anisotropy or optimized square lattice period for lower modal interference between the overlapped phase profiles in the nanofin, using more wavelength selection in the optimizing algorithm, and more efficient sampling mechanisms, such as hexagonal lattice. The mode purities of the generated L = 2 vortex beams on the focal planes at each wavelength can be calculated by the expansion with spiral harmonics exp(iLφ) [45]. As shown in Fig. 10(e), the mode purity is stable in the designed waveband.
Fig. 10. Characteristics of the designed BABM. (a)Focal lengths of the designed BABM when x- and y-polarized incidence at sampled wavelengths. (b)The vertical cut (y direction) of the normalized intensity distributions when x- and y-polarized incidence at 560 nm. The dashed line is the ideal diffraction limit at 560 nm. (c) The extracted FWHMs (left axis) and SRs (right axis) of focal spots at the sampled wavelengths when x-polarization incidence. (d)The focusing efficiencies at the sampled wavelengths for x- and y-polarization incidence. (e) The calculated mode purities of generated vortex fields (L = 2) when y-polarized incidence.
4. Conclusion
In summary, an all-dielectric polarization-controlled BABM is designed in the visible. The BABM is designed based on pure transmission phase in x- and y-axis provided by the side lengths lx and ly of the rectangular nanofin structure. PSO algorithm is used to optimize the nanofin structures to make the BABM work in broadband wavelength. The BABM not only achromatically realizes independent bifunction, but also breaks the dependence of CP incidence. Simulation results show that the BABM can achromatically focus the light into single focal and OV when x- and y-polarization incident in waveband from 500 nm (green) to 630 nm (red). The designed BABM has NA of 0.34. Compared with other achromatic multifunctional metalenses, the BABM has improved focusing quality (33.6% and 34.6%). Moreover, it is of single layer and simple nanofin structure, so it is convenient in manufacture. In addition, the BABM’s designing method provides new idea for miniaturizing modern optical system.
Funding
Shanghai Leading Academic Discipline Project (S30502); National Natural Science Foundation of China (61975125, 92050202).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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