Metasurfaces made from densely packed resonant wavelength-scale particles enable abrupt modulation of impinging electromagnetic radiation within an ultrathin surface. Combining duality symmetry of particles and rotational symmetry of their arrangement led to the development of Huygens’ metasurfaces with perfect transmission. However, so far, when identical particles are considered, only their dipolar multipolar contributions are engineered. There, the achievable phase coverage at a fixed wavelength when modifying the period is smaller than 2π, being a clear limitation for applications. To lift such limitation, we consider dipolar-quadrupolar Huygens’ metasurfaces. They consist of scatterers that require a dipolar and a quadrupolar term to capture their response. We show that such metasurfaces offer access to the desired 2π phase coverage while preserving the perfect efficiency when the conditions of duality and symmetry continue to be met. We also propose core-multishell and disk-multiring particles made from realistic materials to meet the requirements and that can be used to build such metasurfaces. Our results are important as a theoretical basis for large-scale fabrications in imaging and integrated optics.
Huygens’ metasurfaces, i.e. metasurfaces made from an n-fold rotationally symmetric Cn>2 arrangement of resonant scatterers that are electromagnetically dual-symmetric , are characterised by a suppressed back-reflection . Made out of nonabsorbing particles, these metasurfaces provide unity optical transmittance while allowing to tune the phase angle of the transmission , , , , , . Huygens’ metasurfaces are appealing candidates to implement wavefront modulation devices such as holograms in ultrathin surfaces . In Huygens’ metasurfaces, the excited electric and magnetic moments of the scatterers have to be equal. Thus far, emphasis had been put on dipolar metasurfaces, i.e. metasurfaces where the scattering response from its constituents can be explained while considering only the dipolar contribution , , , . This corresponds to the idea of a Huygens’ source that lent the device its name.
For a fixed geometry and by sweeping through the resonance in frequency domain, the phase angle of transmission through Huygens’ metasurfaces can be tuned to cover the entire 2π range . However, the metasurfaces are not operated at various but at a fixed frequency. Therefore, the phase angle of the transmission should be tuned by either locally tuning the geometry of the particles for a fixed arrangement or tailoring the arrangement of otherwise identical particles. At a fixed operational frequency, enforcing an equal electric and magnetic response requires a careful design of the geometry of the particle.
Our study is focused on metasurfaces with identical particles. For large-scale self-assembled metasurfaces, having identical particles is highly demanding. Due to fabrication constraints, the consideration of multiple particles satisfying the required conditions such as unity transmittance and a desired transmission phase angle is daunting. For example, designing Huygens’ metasurfaces with nanocylinders having equal electric and magnetic response under normal incidence requires a specific aspect ratio of radius over height but different absolute sizes for different transmission phase angles. This implies that different heights across the (meta)surface are required for a unity transmittance and full-phase coverage. Such a requirement is clearly not suited for lithography. Therefore, as an additional constraint, we shall require the metasurface to be made from identical particles. And the only tuning parameter that is left over in such configuration is the lattice constant that tunes the arrangement of the particles.
However, as we will demonstrate shortly, by sweeping through the lattice constant and while considering only dipolar particles, it is nearly impossible to cover the entire 2π range of transmission phase angles at a fixed operational wavelength. To entirely lift such limitation, we propose here to exploit quadrupolar resonances in the scatterers along with the dipolar resonances to access the entire phase-shift range of 2π.
It is worth noting here that high-transmittance nonresonant and non-Huygens’ metasurfaces that can provide the full-phase coverage are possible. In these metasurfaces, nonidentical particles can be used , , , , , . In this contribution, our focus is on resonant polarisation-insensitive Huygens’ metasurfaces made out of identical rotational-symmetric particles.
To analyse in this contribution the transmission through dielectric metasurfaces in a generic and systematic approach, we initially consider particles with a scattering response modelled with a Lorentzian dispersion profile in their individual dipolar and quadrupolar multipolar terms. Then, to offer a structural implementation for such particles, we design core-multishell and disk-multiring particles from realistic materials to satisfy the required conditions for the construction of metasurfaces with full-phase-shift coverage. With these implementations at hand, and as an application, we study the effect of a full-phase-shift coverage on the image quality of a metasurface hologram. The discussion and results are covered in the Supplementary information.
The main purpose of this work is to demonstrate that additional degrees of freedom in the design of Huygens’ metasurfaces can be unlocked when considering particles with a more complicated scattering response as their constituents. The particles we propose are exemplary, and many more particles can be considered, driven from the analytical insights. Only recently, the opportunities in exploiting the entire multipolar response of individual particles have been appreciated to tailor their scattering response , , , , , , , . And with this work, we extend such considerations to Huygens’ metasurfaces thereof.
2 Lorentzian particles
The polarisation of particles by an electromagnetic excitation can be quantified by the multipolar polarisability in a Cartesian basis , . The particles considered in this article are initially isotropic, and hence, the polarisabilities are scalar. Later, we also present a design for a nonisotropic particle, but this does not change the argumentation. We approximate the response of the particles by their electric and magnetic dipolar and/or quadrupolar polarisabilities. For the initial numerical analysis, we have used a Lorentzian dispersion model for the resonances of the polarisabilities that are corrected for the radiation losses to satisfy the optical theorem , , , , . In such model, we can easily assume that the electric and magnetic response is equal. The dispersion models used for the dipolar αd and quadrupolar αq polarisabilities are defined as follows:
where k=2π/λ is the wavenumber in free space, α0d and α0q are related to the oscillator strength, ω0d and ω0q are the resonance frequencies, and γΩd and γΩd are the Ohmic losses of the dipole and quadrupole resonances, respectively. Considering only nonabsorbing particles, the Ohmic losses throughout the article are always set to zero, i.e. γΩd=γΩq=0. The electromagnetic duality symmetry of these particles is easily enforced by choosing the free parameters in the electric and magnetic polarisabilities to be the same. Properties between dipolar and quadrupolar terms, e.g. resonance frequency or oscillator strength, can be distinct, though.
To describe the light scattering from particles with such polarisabilities and to study reflection and transmission through arrays thereof, we rely on a scattering theory that explicitly considers these polarisabilities when defining the T-matrix linked to the scatterer. In the multipole expansion method , , the T-matrix relates the multipolar scattering coefficients to the incident ones. The T-matrix describes the electromagnetic properties of particles , . For an isotropic particle, the corresponding T-matrix in electric/magnetic basis is diagonal, and the elements are the Mie coefficients an and bn , . Here, an and bn are multipolar electric and magnetic Mie coefficients, with n denoting the multipole order. A particle having equal electric and magnetic response is said to have electromagnetic duality symmetry, and the particle is called a dual particle , . For the isotropic particles considered here, a particle is dual if and only if an=bn for n=1, 2, …. A dual particle preserves the helicity of the impinging light in scattering, i.e. it does not mix the right and left polarisation of the incident light.
The Mie coefficients can be related to the polarisabilities by the following equations :
where we call and the electric and magnetic Mie angles, respectively. Such a model has the least number of degrees of freedom, i.e. only the Mie angles, while it still offers all the characteristic features. This model is particularly useful when systematically driving the properties of the particles through their resonance because only the angle needs to be tuned. For nonabsorbing particles as considered here, are real angles covering a range from −π/2 to π/2. An angle equal to zero corresponds to the resonance of the Mie coefficient. Negative/positive angles correspond to the red/blue side of the spectrum of the resonance, respectively.
3 Dipolar and quadrupolar Huygens’ metasurfaces
Relying on a T-matrix method formalism for multipolar analysis , , we have employed a tool for periodically arranged scatterers to study the transmission through two-dimensional lattices of particles. It is an exact approach to calculate reflection and transmission from arrays of identical particles that are characterised by their T-matrix. The latter is being renormalised due to the interaction in the array that we correctly consider in the Ewald summation. We use for our purpose an in-house code, but a description of a program with a relatively similar approach can be found in .
To explore the transmission through a periodic arrangement of identical meta-atoms, we use the Lorentzian models in (1) and (2) for the polarisabilities. The corresponding Mie coefficients are shown in amplitude and phase in Figure 1A. While tuning through the resonance, the phase in each coefficient changes up to π. When we arrange such identical particles, characterised by a resonant dipolar or quadrupolar polarisability in either the electric or magnetic multipole moment, in a periodic array and sweeping the illumination wavelength through the resonance, an overall π phase-shift coverage in the spectrum of the transmission is seen (not shown here). Directly in resonance, the transmittance goes to zero. In the complex plane (i.e. real and imaginary transmission), this response corresponds to a circle with a radius of 0.5 and a centre of 0.5+0i, which touches the origin 0+0i at resonance.
However, things change drastically when the condition of duality is met. For a dipolar particle, i.e. a particle characterised by only dipolar polarisabilities, and if the electric and magnetic polarisabilities are equal (i.e. dual), an overall 2π phase-shift coverage in the transmission spectrum td is observed (Figure 1B, C), while, as discussed earlier, the transmittance stays unity. In the complex plane, this corresponds to a circle of radius 1 with a centre in origin 0+0i (Figure 1B, the red curve) . This response corresponds to a full control over the phase shift.
One step further, and to include higher-order multipoles, if a periodic arrangement of dual dipolar-quadrupole particles is assumed, i.e. particles with a1=b1=ad, a2=b2=aq, the complex transmission td+q as a function of the wavelength will be a double winding circle around the origin 0+0i, covering an overall 4π phase shift through the four resonances (Figure 1B, C).
However, any device that is geared toward the modulation of the wavefront is operated at a fixed frequency. There, the phase angle of the transmission is tuned by sweeping the lattice constant of the arrangement. Tuning the lattice constant renormalises the polarisabilities, making the lattice constant a parameter to tune the phase shift of the transmission , . For dual dipolar particles, fixed to operate at the resonance wavelength of the polarisabilities, it turns out that a full 2π phase coverage is not possible while tuning the lattice constant. This is shown in Figure 2 when plotting td. Eventually, only a range of about is covered (i.e. 75%). This argument is generally valid and independent of the structural implementation of the particle, as the analysis at this stage is done using a generic model for the polarisability of the particles.
To mitigate such limitations, we suggest that exploiting quadrupolar resonances in the scatterers along with the dipolar resonances helps to cover the whole phase-shift range of 2π by tuning the periodicity. The larger phase space that could be covered when tuning through the wavelength was a strong indication that this is feasible. Indeed, for a selected example of a square array of particles, a phase coverage slightly larger than 2π can be seen in Figure 2, where td+q is shown. Again, the particles are operated at the resonance wavelength, which is equal for the dipolar and the quadrupolar terms, and we only tune the lattice constant.
To obtain a more systematic insight into the accessible phase coverage when combining particles with different dipolar and quadrupolar dual resonances and not only those where the operational wavelength is fixed to the resonance wavelength, we resort to the generic model for the Mie coefficients presented in (5) and (6). In particular, we can change there the electric and magnetic Mie angles, and and can study the accessible range for the phase angle in transmission when changing the lattice constant. Selected examples for the phase are shown in Figure 3. Note that as all combinations considered are dual, the transmittance is always unity and for simplicity not shown. In Figure 3A, only dual dipolar particles are considered, while in Figure 3B, dual dipolar-quadrupolar particles are examined. The larger phase space that is accessible for the dipolar-quadrupolar particles is evident. In Figure 3C, the phase-shift coverage for all dual combinations is shown. The phase-shift coverage corresponds here to the difference in maximum and minimum phase value of the transmission when changing the lattice constant between 1/3 of the operational wavelength and the operational wavelength. In this figure, the slope of the phase dispersion is not shown, and it should be noted that the slope can be very steep at certain points. It remains to mention that this is not always a disadvantage, but this can be beneficially used in optical sensing devices. However, from this figure, we can safely extract a parameter range that covers a range up to around 3π for the phase angle in transmission when tuning through the lattice constant. We can pick any of the related points in this space that offer this functionality and can build a metasurface from such particles. We only need to find a particle that is characterised by the respective Mie coefficient.
4 Multilayered metasurfaces
In this section, we design and propose actual particles that offer the required Mie coefficients necessary to verify the results derived for Lorentzian particles. We show that by sweeping the lattice constant within the physical constraints, i.e. the lattice constant has to be smaller than the wavelength to preserve the nondiffracting nature of the metasurface, a metasurface made out of identical dipolar particles cannot give us a full-phase-shift coverage in the transmission and unity transmittance. However, we show that using identical designed dipolar-quadrupolar particles, the aforementioned requirement can be satisfied.
First, we design a dipolar core-multishell particle, abbreviated in the following as c1, to meet dual dipole resonance at λ=1500 nm, i.e. a1=b1=1 and an>1≈bn>1 ≈0. For the design of the core-multishell particles, we have used a particle swarm optimisation method , , . We consider a core-multishell particle with up to two shells. The core and the shells were allowed to be made from isotropic nonmagnetic materials with a refractive index in the range from 1 to 7. For the optimised particle, the Mie coefficients as a function of the wavelength are shown in Figure 4A. Clearly, at the desired operational wavelength of λ=1500 nm, the dipolar coefficients are at unity, while higher-order Mie coefficients are negligible. The transmission through the metasurface made from such particles at the resonance wavelength λ=1500 nm as a function of the normalised lattice constant is shown in Figure 4C and D (). As expected, the results are very similar to td in Figure 2; the phase coverage is about 75% of the full 2π, and as can be seen, it is not possible to have the full-phase coverage within the possible lattice constants. Note that the lattice constant is bound by the physical size of the actual particle (a=2×346.65=693.30 nm) at the lower end and by the diffraction limit (a=λ=1500 nm) on the upper end.
Next, we want to design a core-multishell particle that can provide the full-phase coverage of the transmission phase angle in the metasurface with lattice constant tuning. As we showed in the previous section, a suitable particle would require both dipolar and quadrupolar responses. An extra requirement, here for a hologram as a direct application, is that the phase-shift change as a function of the lattice constant is smooth and not steep. This is experimentally important to be able to use the full potential of the lattice sweep.
Based on our simulations, to achieve full-phase-shift coverage and a smooth phase-shift variation, a particle with the designed Mie coefficient angle of is deemed suitable. The dispersion of the phase shift is already shown in Figure 3B (the red dash-dotted line).
Then, based on the required parameters, using the particle swarm optimisation, we have designed a core-multishell particle to fulfil at a wavelength of 1500 nm. The refractive index of the shells is bound between 1 and 6. At this stage, the large refractive index has been chosen to clearly demonstrate the feasibility. Further below, the design of particles is presented that are made from materials with properties offered by naturally available materials in near infrared (NIR). The designed particle in Figure 4B is abbreviated in the following as c2. The Mie coefficients and the lattice constant sweep of the transmission are shown in Figure 4B–D (). The result verifies our initial expectation that with dipolar-quadrupolar particles a full 2π phase-shift coverage is achievable. The simulations are done in free space. In a homogeneous environment with refractive index ne, the results remain the same if the particle radii and the corresponding refractive indexes are down-scaled and up-scaled by a factor of ne, respectively.
Although nonabsorbing natural materials with a refractive index of up to 7 can be found at frequencies up to the mid-infrared spectrum , at NIR, we are limited to a maximum refractive index of 4 offered by natural materials. In this refined analysis, we would like to propose structures that can be used to realise Huygens’ dipole-quadrupole metasurfaces at NIR frequencies. Therefore, we have limited the refractive index to a range of 1 to 4 in the optimisation procedure in the following. We continue to assume a design wavelength of 1500 nm, and we have used again the particle swarm optimisation. We designed two different structures. First was a core-multishell particle considering these more realistic material properties. Second, we designed a disk-multiring particle that can indeed be realised with top-down nanostructure technology. In both cases, we considered a homogeneous dielectric environment as the embedding medium (Figure 5A, B). The electromagnetic properties of the individual optimised structures are shown in Figure 5C and D. The calculations of the optical response from the disk-multiring structure are done with JCMwave, a finite element solver for Maxwell’s equations. Although the particles are not perfectly dual, they have a strong quadrupole component. The results for the transmission through an arrangement of such particles are shown in Figure 5E. The results show promising results. The core-multishell and disk-multiring metasurface covers 97% and 90% of the total phase coverage, while their transmittance keeps near unity.
We have shown that with identical dipolar particles achieving the full-phase shift with lattice constant tuning is practically not possible in Huygens’ metasurfaces. However, using optimised dipolar-quadrupolar particles, metasurfaces can be designed to cover the full 2π phase shift. We have designed core-multishell and disk-multiring particles to support our results. Holograms made from such dipolar-quadrupolar particles show an improved performance, as documented in the Supplementary information.
This research has been funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy via the Excellence Cluster 3D Matter Made to Order (EXC-2082/1–390761711) and under the priority program SPP 1839 Tailored Disorder (RO 3640/7-2 and STA 1426/1-2 under project no. 278747906). A.R. also acknowledges support from the Karlsruhe School of Optics and Photonics (KSOP). R.A. acknowledges financial support provided by the Alexander von Humboldt Foundation. The authors are grateful to the company JCMwave for their free provision of the FEM Maxwell solver JCMsuite with which the simulations on the disk-multiring structures in this work have been performed. They acknowledge support by the DFG and Open Access Publishing Fund of Karlsruhe Institute of Technology.
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The online version of this article offers supplementary material (https://doi.org/10.1515/nanoph-2019-0239).
©2019 Aso Rahimzadegan et al., published by De Gruyter, Berlin/Boston
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