1. Introduction

Metasurfaces are metamaterials of sub-wavelength thickness, which have been extensively studied for controlling electromagnetic radiation [1–3]. In particular, metasurface lenses have the potential to replace conventional bulk lenses in applications where a high level of integration is required [4]. When operating in transmission, the electric and magnetic responses are often balanced, leading to an impedance matched Huygens’ metasurface [5–7]. In addition to suppressing specular reflection, impedance matching prevents spurious diffraction orders from being transmitted or reflected. Therefore a high efficiency of refraction (a high fraction of incident power refracted in the designed direction), minimizes distortion of the transmitted beam. It has also been shown that magnetoelectric coupling (Omega bianisotropy) must be added into the design to maintain high efficiency at large angles of refraction [8–10].

Although Huygens’ and bianisotropic metasurfaces offer improved efficiency compared to those with a purely electric response, their response typically has a strong frequency dependence. This can be understood by considering a metasurface exhibiting uniform anomalous refraction, which has a spatially periodic structure. In such cases anomalous refraction is equivalent to diffraction in a blazed grating [11], with correspondingly strong frequency dependence of the diffraction angle. In [12], it was shown how careful meta-atom design can maximize the efficiency of diffraction into a particular order over a wide bandwidth. However, a frequency dependent diffraction angle is undesirable for many applications. For example, in a lens, it leads to strong variation of the focal distance with frequency, which is a form of chromatic aberration [13,14].

To overcome these limitations, broadband achromatic metasurfaces were proposed at optical frequencies, initially using plasmonic material dispersion to compensate geometric dispersion [15], then subsequently using a combination of waveguide and grating dispersion in all-dielectric structures [16]. Achromatic optical metasurfaces were subsequently demonstrated experimentally utilizing multiple grating structures of dielectric pillars [17–19] and gap-plasmon resonators operating in reflection [20]. In the microwave frequency range, multi-layer delay line type metasurfaces have demonstrated broadband achromatic operation [21]. For delay-line devices, the operating bandwidth scales with the number of layers utilized, so this approach does not enable broadband operation with a very thin structure. While these works have shown the potential for broadband achromatic metasurface operation, it remains unclear whether the bandwidth performance is limited by design ingenuity and fabrication quality, or if there is some limit imposed by physics.

Here we analyze the physical limits on metasurface bandwidth for achromatic operation, considering a geometry based on three patterned metallic layers, separated by dielectric substrates [8,9,22,23]. This is one of the most practical designs for the microwave frequency range, and it has the advantage that the required response of each layer is specified analytically. This design approach can implement either Huygen’s or Omega-bianisotropic metasurfaces, the distinction being whether or not the structure has mirror symmetry in the direction normal to the metasurface.

In modelling three layer printed circuit metasurfaces, the dielectric layers are treated as transmission lines, and the metallic layers as single port shunt impedances [6, 9]. However, a single port impedance cannot exhibit arbitrary variation with frequency. If it is to be passive, lossless and causal, the impedance must comply with Foster’s reactance theorem [24,25]. This is closely related to the requirement that a dielectric material should exhibit normal dispersion, since anomalous dispersion is intrinsically linked to material dissipation. While lossless, non-Foster compliant meta-atoms have been demonstrated to overcome bandwidth limitations [26,27], the complex active circuits required for their implementation are impractical for most applications.

Here we examine the constraints that Foster’s reactance theorem puts on metasurface operating bandwidth, refraction angle and aperture size, to obtain limits for a passive achromatic metasurface realization. We consider first the case of uniform anomalous refraction, then consider the implications for metasurface lenses. We will show that the maximum size of the considered achromatic metasurfaces is inversely proportional to their operating bandwidth.

2. Achromatic refracting metasurfaces

2.1. Equivalent circuit model

As shown in Fig. 1(a), we consider a two-dimensional problem with refraction in the x − z plane. The incident wave arrives at angle θ_in and is refracted to angle θ_out by a metasurface located at z = 0, and we consider cases where θ_in ≠ θ_out. This anomalous refraction is the fundamental mechanism which underlies more complex metasurface devices such as lenses, which we will consider in Section 3. Assuming the metasurface is surrounded by a homogeneous medium of permittivity ∊ and permeability μ, the transmission phase Φ_t is a linear function of both position x on the metasurface and frequency ω. This can be formulated as

To maintain a fixed refraction angle over a broad bandwidth, the transmission phase Φ_t(x, ω) must be engineered to obey Eq. (1) over the frequency range of interest, whilst maintaining full transmission. In Fig. 1(a) we show schematically three plane waves of different frequencies, which should all be refracted in the same direction. The corresponding phase response as a function of position is plotted in Fig. 1(b), for each of the three frequencies. It can be seen that the period of Φ_t is different for each frequency components, meaning that an achromatic metasurface cannot be created with a periodic structure. At each position on the metasurface, Eq. (1) shows that Φ_t is a linear function of frequency, with a slope that depends on position x. Thus, each element of the resulting metasurface should implement a specified constant delay over the operating bandwidth.

 

Fig. 1 (a) Metasurface with several different frequencies of incoming plane wave in refracting operation. (b) The Required phase response as a function of position, is shown for three different frequencies, where λ₀ is the wavelength of the center frequency. (c) Equivalent circuit model of the three-layered structure.

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We utilize a two-port impedance matrix model to obtain relations between the tangential fields at the two sides of the metasurface. In the bianisotropic case, we must account for the different incoming and outgoing wave impedances due to the differences in the incident and transmitted angles [9]. In a Huygens’ metasurface, the difference between incoming and outgoing wave impedances is not taken into account, with a consequence of lower efficiency and the presence of reflections, but with simpler symmetric unit-cell structure. For transverse electric (TE) waves, the local wave impedances are $Z_{\text{in}}=\sqrt{\left(\mu /∊\right)}\text{sec}{\theta }_{\text{in}}$ and $Z_{\text{out}}=\sqrt{\left(\mu /∊\right)}\text{sec}{\theta }_{\text{out}}$ . For transverse magnetic (TM) waves, the local wave impedances are $Z_{\text{in}}=\sqrt{\left(\mu /∊\right)}\text{cos}{\theta }_{\text{in}}$ and $Z_{\text{out}}=\sqrt{\left(\mu /∊\right)}\text{cos}{\theta }_{\text{out}}$. The bianisotropic metasurface is designed to match both of these impedances. In the Huygens’ case, perfect matching is impossible, but matching both the input and output ports to Z_out (i.e. setting Z_in = Z_out in all design formulas) has been shown to be the best compromise [7].

For both Huygens’ and bianisotropic metasurfaces, the corresponding spatially-dependent two-port impedance parameters can be written as [9]

The blue curves in Fig. 2 show the impedance for both Huygens’ and bianisotropic cases. The horizontal axis variable Φ_t can be considered as a normalization of the frequency, since by Eq. (1) they are linearly proportional. We consider a normally incident TE wave refracted at an angle of 50°, a lossless substrate with thickness 0.017λ₀ and permittivity ∊_s = 6.2, where λ₀ is the wavelength at the center frequency. These figures are calculated at a position x = 11.26λ₀ while varying Φ_t, corresponding to a frequency range of 20% of the center frequency. Since we consider a passive, lossless metasurface, the imaginary part of the impedances are zero, so only the reactance X_n = Im{Z_n} is shown. It can be seen that the reactance has sharp resonant peaks at certain frequencies, showing that strongly resonant meta-atoms are required to achieve achromatic operation.

2.2. Realizability of impedances

In order to create a broadband achromatic metasurface, it is necessary to find an arrangement of metallic layers which closely approximates the impedance functions shown in Eq. (3) and Fig. 2. Creating a design to implement these impedance functions is clearly a challenging problem, which we do not address here. However, before any attempt is made to implement these impedance functions, we must first establish whether they are physically realizable. In particular, if the functions are not causal and passive, then they can only be implemented with complex active elements, which are impractical for many applications.

We can determine whether these impedances are to be passive, lossless and causal by checking whether they conform to Foster’s reactance theorem [24]. This theorem is the circuit analog of the normal dispersion regime for refractive index, and it requires the reactance to increase monotonically with frequency. This leads to the requirement $\frac{\partial X_n}{\partial \omega }>0$ for all values of x and ω, with the appropriate limits taken in the vicinity of the poles. These derivatives are plotted in red in Fig. 2, and it is clearly seen that in some range they become negative (the yellow shaded regions). This indicates that achromatic refracting metasurfaces cannot have arbitrarily large operating bandwidth.

A metasurface is physically realizable over some frequency range where all three of its layers implement physically realizable impedances, i.e. where X_1,2,3 all conform to Foster’s reactance theorem. Noting that Fig. 2 is calculated for one specific position on the metasurface, it would be tedious to evaluate these derivatives at every single location on the metasurface to determine the frequency range over which the impedance is Foster-conforming. However, we can simplify the problem considerably if we assume that the dielectric substrates are electrically thin, i.e. β_st ≪ 1. This approximation is consistent with experimentally realized structures, and its accuracy will be verified by comparison with exact calculations. To derive an expression valid for arbitrary position x and arbitrary values of incident and refracted angle, we note that the transmission phase Φ_t is linearly proportional to ω. Without loss of generality, if we assume that x > 0 and (sin θ_out − sin θ_in) > 0, we arrive at the normalized condition for Foster-conforming impedances of each surface

In the bianisotropic case, Z₁ and Z₃ differ since Z_out ≠ Z_in. One of the two impedance sheets will have Z̄ > 1 so that Eq. (6) will have no solutions, hence this layer could be implemented with passive meta-atoms. For the other layer, −1 < Z̄ < 1 is satisfied, equation Eq. (6) has real solutions, and there will always be a region of non-Foster impedance, since the derivative changes periodically from positive to negative. As in the Huygens’ case, the middle layer always has regions of non-Foster behavior.

 

Fig. 2 The imaginary component of impedance (reactance) for each layer (blue curve) as a function of Φ_t (which is proportional to frequency ω) for (a) Huygens and (b) bianisotropic metasurfaces. According to Foster’s theorem, the frequency derivative of the reactance (red curve) must be positive to enable a passive realization. The shading indicates the regions of non-Foster reactance and the crosses mark the transition between Foster and non-Foster regions calculated from Eqs. (5) and (6).

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In Fig. 2, we use crosses to plot the boundary between the Foster and non-Foster regions, calculated according to Eqs. (5) and (6). The impedance curves shown in blue are based on the exact expression, including the influence of the finite substrate thickness. We see that these points match the calculations with the exact expressions for reactance, validating the assumption that the influence of the electrically thin dielectric substrate can be neglected. We have further validated this agreement for other realistic values of substrate permittivity.

For both Huygen’s and bianisotropic cases, the impedance of the center layer Z₂ is not Foster-compliant in some frequency range. For the Huygen’s case both outer layers Z₁ and Z₃ are Foster-compliant, but for the bianisotropic case only one of them is, which one depends on whether the polarization is TE or TM and whether the refraction angle is larger or smaller than the incident angle. As can be seen in Fig. 2, for the bianisotropic case, the non-Foster regions of the inner and outer layers occur in different frequency ranges. Since the physically realizable frequency range must have Foster-compliant impedances for all layers, it is clear that the achromatic operating bandwidth for a bianisotropic metasurface is lower than that for a Huygens’ metasurface having the same parameters.

2.3. Bandwidth and size limits of refracting metasurfaces

Given that the impedance of each layer can have regions of both Foster and non-Foster behavior, we now consider the limits that this imposes on the total size and operating bandwidth of passive achromatic metasurfaces. A passive metasurface must be designed to operate within the unshaded regions for all of the relevant curves in Fig. 2.

 

Fig. 3 Relationship between maximum metasurface size and fractional operating bandwidth for a purely passive realization in (a) Huygens and (b) bianisotropic refracting metasurfaces.

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Consider first the case of a Huygen’s metasurface, which is limited only by the Foster-compliant region of the inner layer Z₂. From Eq. (5) we observe that the boundaries between Foster and non-Foster impedance occur periodically. Hence, the distance between these points, which gives the width of the Foster-compliant impedance range, is the same for all unit cells, and is given by

Consider that the metasurface operates between frequencies f_min and f_max, where Δf = f_max − f_min, and it has a total width Δx = x_max −x_min. In normalized coordinates the corresponding range is given as ΔΦ_t = Φ_t,max − Φ_t,min. We also allow for the case where (sin θ_out − sin θ_in) < 0, which results in a simple sign change. From the linearity of Eq. (1) in x and ω = 2πf, the limit on the bandwidth and total size of a Huygens’ metasurface is

Here Z̄_min is obtained by substituting the TE or TM input and output impedances into Eq. (7). For both polarizations the limiting value is identical, and is determined by whichever of the incoming or outgoing angles is lower

Fig. 3 shows limits on achromatic metasurfaces obtained from Eqs. (9) and (11) for several values of θ_out, with θ_in = 0°. We see that the allowable metasurface bandwidth and size are inversely proportional, requiring designers to make a trade-off between the two, which becomes more severe for large angles of refraction. We also see that the bianisotropic metasurfaces have a much worse trade-off compared to Huygens’ metasurfaces, due to the strongly varying impedance required to match both the incident and transmitted waves. The trade-off between metasurface size and bandwidth can be understood from the fact that a larger metasurface requires a larger delay at its edge, as discussed in [17,18]. This must be realized by sub-wavelength passive structures, which are limited in the maximum achievable group delay. On the other hand, if a metasurface is allowed to have arbitrary thickness, then there is no limit on the group delay which can be achieved. However, sub-wavelength thickness underlies the key advantages of metasurfaces, namely their compactness and ease of fabrication with planar fabrication techniques.

 

Fig. 4 (a) Metasurface with several different frequencies of incoming plane wave in focusing operation. (b) The Required phase response as a function of position, is shown for three different frequencies, where λ₀ is the wavelength of the center frequency.

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It can also be observed that the limits given by Eqs. (9) and (11) contain no quantities which are specific to the considered structure made from three metallic layers. Given that the physics of Huygens’ metasurfaces is essentially the same regardless of the structure used, it is quite likely that similar limits would apply for any type of refracting Huygens’ or bianisotropic metasurface with sub-wavelength thickness.

3. Limits on achromatic metasurface lenses

The anomalous refraction which we analyzed in Section 2 is the building block for more complex metasurface devices such as lenses. Given the physical limits on the size and bandwidth for uniform refraction, it is clear that there should be a corresponding limit on the performance of a lens. The key difference is that the angle of refraction θ(x) is dependent on position, thus the transmission phase profile differs from that in Eq. (1). As shown in Fig. 4(a), we consider normally incident plane waves, with the transmitted wave focused at a distance F from the metasurface. The transmission phase has a hyperbolic dependence [13] and can be written as,

 

Fig. 5 Relationship between the radius and fractional bandwidth for a metasurface lens with (a) Huygens and (b) bianisotropic realizations. Results are shown for different values of normalized focal length.

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As with uniform refraction, maintaining a constant focal length F over the operating bandwidth requires implementing a specified constant phase delay at each location. This hyperbolic phase profile can be analyzed by the spatially-dependent two-port impedance parameters in Eq. (2), enabling the constraints on bandwidth to be obtained. From Eq. (14) it is clear that a larger delay is required for larger apertures (i.e. large values of x) and for shorter focal lengths [13].

The largest delay, and hence the tightest constraint on the achromatic bandwidth, comes from the largest difference in normalized optical path length $\mathrm{\Delta }{l}_{\text{max}}^{\left(\text{norm}\right)}$ at position x = R, corresponding to Δl_max indicated in Fig. 4(a). Substituting this value into Eq. (8) gives the constraints imposed by the inner layer Z₂, which is the overall limit for a Huygens’ metasurface lens

For bianisotropic metasurfaces, there is a dependence of the outgoing local wave impedance Z_out on position, in which the difference is maximum at the edge of the aperture, x = R. At this location, the outgoing wave impedance for TE waves is $Z_{\text{out}}=\sqrt{\left(\mu /∊\right)}\text{cos}{\theta }_{\text{max}}$ and for TM waves is $Z_{\text{out}}=\sqrt{\left(\mu /∊\right)}\text{sec}{\theta }_{\text{max}}$. The maximum refraction angle ${\theta }_{\mathit{max}}=\text{arctan}\left(\frac{R}{F}\right)$ is shown in Fig. 4(a). Since we assume a normally incident beam, for both TE and TM waves the incoming wave impedance is $Z_{\text{in}}=\sqrt{\left(\mu /∊\right)}$.

As with anomalous refraction, the overall limits on a bianisotropic lens come from the overlap of Foster-compliant regions of all impedances, given by Eq. (10), leading to

The bandwidth and size limitations for metasurface lenses are plotted in Fig. 5. Similar to uniform anomalous refraction, there is a clear trade-off between lens radius and operating bandwidth, which is more severe for bianisotropic metasurfaces. This analytical result is in line with the results shown in [17,18] where in an achromatic metasurface lens the maximum delay increases with the lens aperture. We show curves for different values of normalized focal length $\frac{F}{R}$, and from Eq. (14) it is clear that a shorter focal length also results in a tighter limit, since it increases the maximum optical path length difference $\mathrm{\Delta }{l}_{\text{max}}^{\left(\text{norm}\right)}$. The focal length is related to the maximum refraction angle by $\frac{R}{F}=\text{tan}{\theta }_{\text{max}}$, therefore this result is consistent with the limits on uniform refraction which are more stringent for larger refraction angles. We can express the normalized focal length in terms of the numerical aperture, which has the values 0.5, 0.766 and 0.939 respectively for the three curves shown in Fig. 5.

We note that achromatic meta-lenses reported in the literature use relatively low values of numerical aperture, 0.106 to 0.15 in [19], and 0.02 and 0.2 in [18]. The mechanism of these metasurfaces is somewhat different, being based on the Pancharatnam-Berry phase imparted on the wave during circular polarization conversion. In [19] the achromatic operating bandwidth did not significantly change with numerical aperture, however increased numerical aperture was accompanied by reduced efficiency.

4. Conclusion

We analyzed the bandwidth constraints on achromatic metasurfaces with fixed refraction angle and high transmission efficiency, realized with a three layer printed circuit design of sub-wavelength thickness. We showed that there is a physical limit on the combination of size, bandwidth and refraction angle of the metasurfaces, that cannot be overcome regardless of the meta-atom design or fabrication quality. This limit arises because the meta-atoms are represented by one-port equivalent impedances, which must comply with Foster’s reactance theorem to be passive and causal.

It was shown that increasing operating bandwidth comes at the cost of decreased size, and that bianisotropic metasurfaces have much worse trade-off compared to Huygens’ metasurfaces, despite their better refraction efficiency. The same analysis was applied to an achromatic metasurface lens, showing a similar trade-off between aperture size and operating bandwidth, which becomes more severe for shorter focal lengths (i.e. high numerical aperture).

Although our results were derived for three layer printed circuit designs, the physical principle of all sub-wavelength Huygens’ and bianisotropic metasurfaces is quite similar. Therefore we predict that other metasurface technologies should be subject to similar performance constraints. We also note that the limits were derived based on the assumption of full efficiency operation. Thus it may be possible to design a metasurface which overcomes these limits, but it would significantly distort the transmitted beam.