Tutorial on broadband transmissive metasurfaces for wavefront and polarization control of terahertz waves

The terahertz frequency range that spans from 0.1 to 10 THz is one of the least-utilized portions of the electromagnetic spectrum.¹ This relatively un-tapped region possesses vast under-utilized spectral resources and unique properties for ultrafast-data-rate wireless communications,² material characterization,³ non-invasive imaging,⁴ and security screening.⁵ To harness the capabilities of terahertz waves for those applications, intensive efforts have been made to develop functional devices.^6–10 One prominent category of terahertz devices is metasurfaces.

A metasurface is typically formed by using subwavelength metallic or dielectric resonators arranged as a planar and periodic array. Each individual element on the metasurface interacts locally with incident waves to impose a scattering response shaping the outgoing waves. Due to the strong resonance of these elements, customized radiation and polarization characteristics can be achieved.¹¹ Owing to their exotic capabilities in manipulating terahertz waves, metasurfaces are promising for wavefront engineering and polarization control.

Wavefront engineering aims to create desired beam patterns in the near- or far-field region through tailoring the spatial phase distributions of free-space propagating waves.¹² This concept is of importance for terahertz technology to support a variety of applications. For instance, wavefront engineering techniques can be employed to collimate radiation from a point source¹⁰ so as to provide high-gain radiation to alleviate free-space path loss. Moreover, beam deflection that involves steering an incident beam off a normal path to a predefined angle is of significance for terahertz communications¹³ so that the line-of-sight limitation can be relaxed.¹⁴ Conventional wavefront controlling devices based on geometric optics, for example, lenses, are well-developed and offer a large bandwidth and a low insertion loss. However, practical applications require a high degree of compactness and integrability, along with sophisticated functionality and tunability. Consequently, metasurface-based terahertz components with flat-profile and improved integrability provide a viable route to meet the stringent requirements set by practical scenarios.

Aside from wavefront engineering, polarization control is essential for a wide range of applications in terahertz systems, including studies of chiral structure in biology and chemistry¹⁵ and polarization-division multiplexing for terahertz communications. Polarization control includes polarization conversion^8,16–19 and circular polarization filtering.^20,21 Conventional approaches to achieving polarization conversion involve waveplates that are made of anisotropic crystalline dielectric materials. The operation mechanism of conventional waveplates can be explained in terms of the distinct phase accumulations for electric field components with orthogonal polarizations. By carefully determining the thickness of such a birefringent material, a desired phase difference between the two electric field components can be obtained at a specified frequency. However, waveplates made of such materials typically feature substantial thickness, narrow bandwidth, and high material loss. At oblique incidence, a bulky waveplate exhibits phase deviation and a lateral shift in the beam path, leading to a significant decrease in bandwidth and efficiency. In addition, there is a lack of strongly anisotropic birefringent materials with low loss at terahertz frequencies. As an alternative, planar metasurfaces can be employed for polarization control via tailored phase responses for the two polarization axes. They present improved robustness at oblique incidence resulting from their subwavelength thicknesses. The anisotropic phase response for polarization-control metasurfaces shares the same design principle as the isotropic phase response for beamforming metasurfaces.

Despite the fact that metasurfaces promise attractive properties for wavefront and polarization control, the inherent characteristic of narrow bandwidth associated with resonance impedes their wide adoption. At microwave frequencies, multi-layer structures can be employed for bandwidth enhancement. However, they are practically challenging to scale down for operation at higher frequencies due to fabrication constraints, leading to requirements of a small number of layers for terahertz and optical metasurfaces. In addition to the narrow bandwidth performance, metasurfaces operating at terahertz and optical frequencies are susceptible to non-negligible material losses from metals and dielectrics. Specifically, as compact terahertz sources generate relatively low power and propagating terahertz waves experience moderate atmospheric attenuation, terahertz metasurfaces with high efficiency are thus preferable. At optical frequencies, laser sources provide sufficient level of power to alleviate the material losses. In general, widening the bandwidth and increasing the efficiency of terahertz metasurfaces are great challenges that demand further efforts.

Common challenges associated with metasurface-based terahertz devices are limited bandwidth and efficiency. This section reviews factors that limit the bandwidth of reflective and transmissive metasurfaces for wavefront engineering and polarization control. Moreover, the existing techniques for enhancing the bandwidth and efficiency of metasurfaces are also given.

In order to customize near- or far-field radiation, the phase distribution across a radiating aperture needs to be determined accordingly. To this end, terahertz wavefronts can be shaped by a nonuniform metasurface that is constructed by a large number of nonidentical resonators. The phase response of a resonator is predominantly determined by its geometrical parameters. Each resonator introduces a phase discontinuity, and when assembled in an array, the resonators collectively alter the phase front of the outgoing waves. As a result, constructive interference takes place at the desired location or direction, while destructive interference happens elsewhere, leading to a prescribed radiation characteristic in the near- or far-field region. Typically, a complete phase coverage of $360 °$ introduced by a set of resonators at a specified frequency is necessary for full phase front control, and a larger phase range can be obtained by employing phase wrapping. There are two principal factors that limit the bandwidth performance of beamforming metasurfaces.²² First, the employed resonators intrinsically limit the desirable phase range within a narrow band. For instance, reflective metasurfaces composed of microstrip patch resonators typically exhibit a fractional bandwidth of less than $5%$⁠.^22,23 It is noted that a fractional bandwidth is defined as the metasurface absolute bandwidth for given specifications divided by its center operation frequency. The fractional bandwidth is employed as a measure to directly and fairly compare metasurfaces operating at different frequencies. The limited fractional bandwidth of microstrip patch resonators can be attributed to the resonance nature that produces highly non-linear reflection phase curves vs the patch length changes at different frequencies as illustrated in Fig. 1(a). Second, the frequency-dependent spatial phase delay arising from the varying path lengths between the free-space feed and each resonator further degrades the bandwidth. This can be explained in terms of phase compensation achieved by a set of resonators. As shown in Fig. 1(b), the varying paths from the feed to resonators yield frequency-dependent phase delays. However, those resonators are arranged to compensate the phase delays at a specific frequency. Thus, a deviation from the design center frequency leads to phase errors across the metasurface, which can result in beam squint effect.²³ 

It is found that the first limiting factor can be mitigated by developing broadband resonators. As shown in Fig. 2, apart from offering a complete phase coverage of $360 °$ at each operation frequency, the reflection or transmission phase of a broadband resonator topology is anticipated to change smoothly and linearly against one of its geometrical parameters. Importantly, resonator phase profiles at different frequencies are expected to be parallel to each other. Intensive efforts have been made to enhance the resonator bandwidth, leading to the topology development including stacked patches,²⁴ multi-cross loops,²⁵ circular²⁶ or square rings,²⁷ meander-shaped elements,²⁸ parasitic dipoles,^29,30 U-shaped resonators,³¹ and slotted rectangular patches.^32, Figure 3(a) shows a phase-shifter element for constructing broadband reflective metasurfaces at microwave frequencies.³³ Relying on the concept of true-time delay, Fig. 3(b) reveals that the presented resonator provides a linear phase response and more than four complete cycles of phase delay. However, the involvement of multiple layers imposes challenges in micro-fabrication, making it highly difficult to be practically scaled at higher frequencies. To reduce fabrication complexity, a single-layer reflective metasurface consisting of two square loops together with an I-shaped dipole³⁴ was presented at millimeter-wave frequencies as illustrated in Fig. 3(c). The presented structure exploited multiple resonances formed by the dipole and loops to expand the reflection phase coverage, while retaining a reflection loss less than 1.25 dB as shown in Fig. 3(d). At terahertz frequencies, a single-layer stub-loaded resonator was introduced as depicted in Fig. 3(e). It covers a complete phase range of $360 °$ by varying the stub lengths at the design center frequency, while the phase curves are nearly parallel at different frequencies as depicted in Fig. 3(f). The stub-loaded resonator has experimentally demonstrated its broadband performance by configuring a terahertz focusing reflective metasurface that exhibited a relative bandwidth of more than $23%$⁠. These broadband reflective resonators demonstrate that true-time delay techniques^33,35 and multi-resonant behavior³⁴ provide effective routes to enhance the reflection phase coverage in a relatively broad frequency range. The reflection phase of a true-time-delay resonator is directly proportional to its physical line length, while multi-resonant resonators break the theoretical phase coverage limit of a non-resonant or single-resonance element.²² 

Aside from the resonators developed for reflective metasurfaces, various topologies have been investigated to realize broadband transmissive metasurfaces, such as double square loops,^37,38 spiral dipole,³⁹ cross-slot element,⁴⁰ slotted metallic resonator,⁴¹ I-shaped resonator,⁴² and concentric loop scatters.^43, Figure 4(a) illustrates an annular-slot resonator for constructing a metasurface-based transmissive beam deflector.³⁶ Measured results reveal that the realized metasurface was capable of deflecting an incident beam to a prescribed angle off normal path with a relative bandwidth of $23%$ and an efficiency above $25%$⁠. In theory, the maximum transmission efficiency for metasurfaces formed by a single non-magnetic layer is limited to $50%$ without accounting for material losses.⁴⁴ To enhance efficiency, an I-shaped resonator⁴² was presented as shown in Fig. 4(b). The resonator design was employed to form a metasurface for beam deflection at $∼$0.9 THz, and the metasurface exhibited a measured peak efficiency of $44%$⁠. Figure 4(c) presents a three-layer resonator based on square-loop slots.⁴⁵ A planar lens constructed using this resonator as a building block maintained the desired focusing capability from 0.75 to 1.00 THz, equivalent to a relative bandwidth of $29%$⁠. However, the realized lens exhibited a limited measured efficiency of $45%$ at 0.95 THz, which requires further enhancement. The reported broadband transmissive resonators confirm that multi-layer Huygens' structures with identical outer metallic layers⁴² offer a large transmission phase range together with a moderate transmittance.

Different from wavefront engineering that is typically realized by nonuniform metasurfaces, polarization control basically relies on uniform metasurfaces. Such a polarization-control metasurface is formed by identical resonators that collectively impose distinct phase responses for two orthogonal linear polarizations. A broadband reflective or transmissive metasurface is capable of maintaining a desired phase difference between the two polarizations over a wide bandwidth.

At terahertz frequencies, a variety of resonators have been developed for constructing broadband reflective metasurfaces to realize polarization control. Figure 5(a) shows a broadband linear polarization converter consisting of slanted metallic T-shaped resonators on a grounded dielectric substrate.¹⁹ The operation mechanism of the structure can be explained by decomposing an incident $x$-polarized electric field into two orthogonal components along $u$- and $v$- axes as shown in Fig. 5(a). The main arm of the T-shaped resonator couples with the $E v$ component and is on resonance at 0.38 THz as indicated in Fig. 5(b). On the other hand, the reflection phase profile for the $E u$ component indicates that the resonator is not on resonance along the $u$-axis over the frequency range of interest. Thus, a single resonance along the $v$-axis leads to a large phase difference of $∼ 180 °$ between the two orthogonal electric fields over a wide bandwidth. As a result, the constructed polarization converter was shown to rotate an $x$-polarized incident wave into a $y$-polarized output wave with a measured relative bandwidth of $100%$⁠. Figure 5(c) illustrates a reflective quarter-wave metasurface relying on coupled mode theory.¹⁷ The reflective metasurface was made of rectangular pillars coated by gold on their top surfaces and on the rest of the substrate. Through properly tailoring the off-resonance phase dispersion of the birefringent resonator, a nearly constant phase difference of $− 90 °$ between two orthogonal linear polarizations can be achieved over a broad bandwidth as illustrated in Fig. 5(d). Consequently, the presented metasurface maintained an experimentally confirmed 3-dB axial ratio over a relative bandwidth of $80%$⁠. As an alternative to metallic resonators, Fig. 5(e) depicts a terahertz half-wave mirror employing silicon resonators with sinusoidal edges supported by a gold ground plane.¹⁶  Figure 5(f) suggests that the presented structure induced three resonances for the two orthogonal polarizations, and an approximately constant phase difference of $180 °$ between them was achieved. Thus, the half-wave mirror was demonstrated to rotate $45 °$ linearly polarized incident waves by $90 °$ upon reflection over a measurable fractional bandwidth of $53%$⁠, while it maintained the handedness of incident circularly polarized waves. These reported broadband reflective anisotropic resonators reveal that single or multiple resonances for two orthogonal polarizations are beneficial to achieving a large reflection phase difference between them.

Resonators for transmissive metasurfaces have also been developed for polarization control. Figure 6(a) shows a transmissive terahertz quarter-wave metasurface based on a single metallic layer.⁴⁶ The presented metasurface was developed based on asymmetric cross slots that support two orthogonal resonant modes. A phase difference of $90 °$ between the two orthogonal electric fields was achieved in a narrow band, leading to a measured 3-dB axial ratio relative bandwidth of $5%$ with a total efficiency above $30%$⁠. Since a single-layer metallic pattern cannot provide a complete phase control together with a high transmission,⁴⁷ transmissive metasurfaces cascading multiple metallic layers have been developed to enhance bandwidth and efficiency. Those multi-layer metasurfaces could be made of resonators, such as Jerusalem cross grids,^48,49 strip lines,⁵⁰ split-ring resonators,⁵¹ metal gratings,⁵² meanderline and patch,⁵³ and cut-wire pairs.^54, Figure 6(b) depicts a three-layer metasurface-based linear polarization converter operating in transmission.⁸ The structure employed orthogonal metal gratings on the top and bottom layers, while split-ring and H-shaped resonators were implemented on the middle layer. Due to multiple resonances introduced by the resonators, the structure rotated a linearly polarized incident wave by $90 °$ in a measured ultra-wide fractional bandwidth of $133%$ with a conversion efficiency higher than $80%$⁠. It is found that transmissive resonators supporting multiple resonant modes contribute to the bandwidth enhancement by providing a large phase difference between two orthogonal electric field components. Moreover, complementary structures⁴⁶ typically feature limited efficiency resulting from their narrow transmission window.¹¹ In addition to the afore-mentioned metasurfaces that are based on metallic resonators, various all-dielectric structures have been developed for polarization control in transmission. For instance, achromatic quarter-wave plates employing multiple pieces of birefringent materials were presented at terahertz frequencies.^55–57 The presented all-dielectric waveplates involved optimization algorithms to determine the thickness and rotation angle of each layer so as to achieve a partial cancellation of spectral variations of phase retardation from different layers, thus allowing to maintain the desired phase shift over a broad bandwidth.

Despite the fact that various resonator topologies have been developed, there exists a room for further bandwidth enhancement. To improve the bandwidth of reflective metasurfaces, a systematic method was presented in Ref. 58. The presented method derived the required admittances of the metallic layers at different frequencies for a specified wavefront functionality. The required frequency-dependent admittances were then fitted by equivalent LC circuits, and those frequency-independent reactance values were eventually translated into physical structures.

Transmissive metasurfaces employing a single metallic layer typically suffer from limited bandwidth and efficiency. Multi-layer structures provide more degrees of freedom to manipulate the resonant modes for bandwidth and efficiency enhancement but at the cost of increased design and fabrication complexity. To simplify the design procedure, an analytical approach employing a generalized Bloch analysis and transmission-line theory was presented to model and optimize multi-layer transmissive metasurfaces.⁵⁹ However, the presented approach relied on the simulated frequency-dependent admittance of each metallic layer to begin with, and thus, the optimized metasurface performance was dependent on the selected starting metallic patterns. A design strategy relying on the concept of “Huygens' surface” was introduced by Pfeiffer and Grbic to simplify the multi-layer metasurface design and simultaneously improve the bandwidth and efficiency.^18,60,61 Each unit cell of the Huygens’ surface consists of crossed electric and magnetic dipoles of equal strength. In general, a three-layer structure employing identical top and bottom metallic layers with the same dielectric spacer thickness can function as Huygens' surface. By properly tuning the resonances of the crossed electric and magnetic dipoles, a large transmission phase coverage together with a high transmittance can be achieved.⁶² The concept of Huygens' surface yielded various metasurfaces for wavefront and polarization control.^63–66 In order to facilitate the synthesis of transmissive Huygens' metasurfaces, a design method aiming at solving the optimal frequency-dependent admittances at the design center frequency was presented in Ref. 21. The optimal admittances at a specified frequency can be physically realized by diverse metallic patterns. However, only a small portion of those patterns maintain the desired performance over a wide bandwidth, leading to uncertainties for wideband metasurface designs. In Ref. 48, a circuit-driven methodology was presented to optimize Huygens' metasurface designs. However, the proposed synthesis procedure provided accurate evaluation on the equivalent admittances of the metallic patterns only for unit cell sizes of less than $λ 0/5$⁠. As a result, the range of achievable admittance was constrained by such a relatively small unit cell, and the performance of the physically realized metasurface was not optimal. In Ref. 44, a nanocircuit concept was applied to designing Huygens' metasurfaces at optical frequencies. To maintain a low transmission loss and a large transmission phase coverage of $360 °$⁠, the desired reactances of each layer were derived at the design frequency. Subsequently, the desired reactances were approached by tuning the filling ratio of a building block, which comprised plasmonic and dielectric materials.

This section summarizes a semi-analytical approach to designing broadband transmissive metasurfaces,^62,67 which is developed on the basis of the existing narrowband method presented in Ref. 21. The broadband approach determines frequency-independent optimal circuit parameters so that the transmissive metasurface maintains an optimal performance over a wide bandwidth.⁶² Section III A introduces a circuit-level optimization incorporating network analysis and a genetic algorithm. Specifically, the network analysis investigates the scattering characteristics of anisotropic transmissive metasurfaces, while the genetic algorithm determines the optimal circuit parameters. Section III B presents physical realizations of each metallic layer to reproduce these optimal circuit parameters. Section III C presents broadband transmissive quarter- and half-wave metasurfaces^62,68 as examples of structures rigorously designed following the procedure in Secs. III A and III B.

Considering now the inverse problem, in order to obtain a stipulated $S$-parameter specification over the frequency range of interest, frequency-independent optimal circuit parameters need to be determined accordingly. However, the achievable pattern-equivalent inductances and capacitances within a certain size of unit cell cover a relatively large range, and a three-layer metasurface encompasses a large number of adjustable parameters. This large number of combinations makes it actually impractical to seek optimal circuit parameters by investigating the scattering characteristics through an exhaustive search.

In order to facilitate the design procedure and optimize the metasurface performance, a genetic algorithm⁷⁰ is employed in the search process to rapidly determine the optimal circuit parameters that best fit the objectives. More specifically, the optimization process begins with an initial population consisting of a large pool of individuals. Each individual is a possible solution to the problem and carries genes formed by circuit parameters. As the circuit parameters are frequency-independent, the scattering characteristics of each individual can be numerically investigated over the frequency band of interest by employing Eqs. (1)–(5). Subsequently, an application-defined cost function is designed to evaluate the fitness of each individual based on its scattering characteristics. Individuals with high fitnesses have more opportunities to be selected for passing their genes to the next generation. In the process of producing offspring, crossover and mutation may occur to form new genes to maintain diversity of the population. After an iterative process, the algorithm terminates with a converged population that contains the fittest individual, which provides transmission performance that closely approaches the objectives within the frequency range of interest. An overall circuit-based design selection procedure⁷¹ of the optimal frequency-independent circuit parameters is illustrated in Fig. 9.

As an illustrative example, Fig. 10 shows parallel transmission phase curves separated by a constant phase difference of $90 °$⁠. Such progressive staggering of phase profiles together with unity transmittance function provides a set of objectives from 220 to 330 GHz. Following the procedure of circuit-level optimization presented in this section, the optimal circuit parameters are found as listed in Table I. These parameters are bounded by realizable physical values and are scaled with operation frequency. From the optimal parameters, we can observe that a capacitive response is required for phase lagging, while an inductive response is demanded for phase advance. It should be noted that the outer metallic layers, along with the spacers, are not necessarily identical. This is to provide more degrees of freedom to achieve specified transmission phase values.

Once the optimal circuit parameters have been determined, a series of simulations should be conducted to physically realize them. Full-wave electromagnetic simulations can be performed separately for each metallic layer with commercial software, such as ANSYS HFSS. Figure 11 shows the simulation settings for a unit cell of each metallic layer in periodic settings. Floquet ports are employed as an excitation to impose incident plane waves and also collect scattered waves. Phase de-embedding is applied to the pattern surface, so as to acquire the intrinsic phase response of the pattern, without involving propagation effects in the media. A periodic and infinite planar array is assumed by implementing master–slave boundary conditions to bound the unit cell transverse planes. To minimize material loss, gold can be employed for the metallic layers. The surface impedance of gold at terahertz frequencies can be described by a Drude model.⁷² 

As illustrated in Fig. 13, metallic patterns can be developed to realize the optimal circuit parameters listed in Table I for the four phase curves, and their simulated responses are compared with the objectives in Fig. 14. A reasonable agreement between realized curves and objectives is achieved, accompanied by relatively high transmittances, thus demonstrating the functionality of the circuit-level optimization. In Fig. 13(a), metallic patches are arranged with a separation to introduce a capacitive response along the $x$-polarization. It is found that a smaller separation between larger metallic patches produces higher equivalent capacitances. In Figs. 13(c) and 13(d), a metallic strip is oriented along the $x$-axis to introduce an inductive response, noting that thinner strips lead to larger equivalent inductances. Generally, the minimum metallic patch separation and the strip width that can be physically realized limit the maximum achievable capacitance and inductance, respectively. The metallic patterns developed in Fig. 13 provide the desired transmission phases for the $x$-polarization, and they can be potentially applied for wavefront engineering. However, metasurfaces employed in polarization control are expected to yield customized phase responses for both the $x$- and $y$-polarizations. Thus, the metallic patterns need to be carefully designed to provide the desired responses for each polarization but a minimized effect on their cross-polarizations. As a result, the achievable equivalent inductance and capacitance for each metallic layer and each polarization are constrained.

As illustrative examples, this section presents quarter- and half-wave metasurfaces that are rigorously designed following the optimal circuit parameters from the broadband approach.^62,68 Both metasurface-based waveplates broadly rely on the concept of “Huygens' metasurface” that separately tune the electric and magnetic dipole resonances, thus leading to a broadband and high-efficiency performance.

A quarter-wave metasurface realizes linear-to-circular polarization conversion. In the microwave regime, circularly polarized waves are preferable in line-of-sight indoor communications⁷³ and satellite communications⁷⁴ to alleviate multipath fading and reduce polarization mismatch. In the terahertz domain, circularly polarized terahertz waves can be employed for chiral structure characterization in chemistry,⁷⁵ wireless communications,⁷⁶ and biological sensing.^77,78 At optical frequencies, circularly polarized light has found applications in circular dichroism spectroscopy,⁷⁹ magnetic recording,⁸⁰ and target detection.⁸¹ To obtain circular polarization with high purity and high efficiency, the quarter-wave metasurface is expected to provide near-unity transmittances for two orthogonal linear polarizations and a constant phase difference of $90 °$ between them. Following the design procedure introduced in Secs. III A and III B, a broadband three-layer quarter-wave metasurface is designed and fabricated as shown in Fig. 15. The electromagnetic responses of the three metallic layers along the $x$- and $y$-polarizations can be described by parallel $LC$ circuits. To experimentally investigate the performance of the fabricated quarter-wave metasurface, a Keysight Precision Network Analyzer (PNA) combined with VDI WR-3.4 extension modules is employed to probe the metasurface from 205 to 340 GHz. As indicated in Fig. 16, four lenses are implemented for collimating and focusing the terahertz beam.⁶² As the sample is positioned inside the Rayleigh range of the incident beam over the entire measurement frequency range, a normally incident plane wave can thus be assumed.

An excellent agreement between the calculated, simulated, and measured transmission coefficients is achieved as shown in Fig. 17. A close agreement between the calculated and simulated results indicates negligible coupling effects in the realized quarter-wave metasurface, resulting from an electrically large spacing (⁠ $∼$0.18 $λ 0$ at 275 GHz) between the metallic layers. The deviation between the calculated and simulated results can be attributed to the material losses accounted for in the simulation, frequency-dependent reactances resulting from the fringing effects, and the difference between the optimal and realized pattern-equivalent reactances. The discrepancy between the simulated and measured results can be explained in terms of unavoidable fabrication and measurement tolerances. The former case includes resonator dimension variations, while the latter one involves a non-ideal incident plane wave due to experimental misalignments. In Fig. 17(b), the presented metasurface experiences two resonances that occur at a lower or higher frequency for the $x$- or $y$-polarization, leading to a $90 °$ phase difference together with high transmittances in the frequency range between the two resonances. As a result, a measured 3-dB axial ratio can be maintained from 205 to at least 340 GHz, with a total efficiency higher than $70.2%$⁠. The presented quarter-wave metasurface covers an entire WR-3.4 waveguide frequency range from 220 to 330 GHz,⁸² which can be potentially used for point-to-point terahertz communications.⁸³  Table II indicates that our fabricated quarter-wave metasurface exhibits the highest bandwidth and efficiency compared to notable published terahertz designs. The superior performance can be attributed to the implementation of the systematic design methodology, which yields optimal circuit parameters that collectively maintain the desired transmission coefficients over a wide bandwidth.

A half-wave metasurface converts linearly or circularly polarized incident waves into their orthogonal counterparts, and it is useful for imaging, spectroscopy, and wireless communications. A broadband and highly efficient half-wave metasurface delivers near-unity transmittances for the $x$- and $y$-polarizations, while the phase difference between them should be $180 °$⁠. Figure 18 shows the fabricated half-wave metasurface, where predominantly inductive and capacitive responses are obtained for the $x$- and $y$-polarizations, respectively. The characterization of the manufactured half-wave metasurface shares the same experimental setup with that of the quarter-wave metasurface as depicted in Fig. 16. Figure 19 implies that a reasonable agreement between the calculated, simulated, and measured transmission coefficients of the fabricated half-wave metasurface is achieved. The deviations between them can also be explained by the factors that contributed to the result discrepancy of the quarter-wave metasurface in Sec. III C. The electric and magnetic dipole resonances are tuned to different frequencies so as to achieve the desired complex transmission coefficients in between the two resonances as shown in Fig. 19(b). Consequently, the fabricated half-wave metasurface enables a measured 15-dB extinction ratio from 220 to 303 GHz with a cross-polarization transmission efficiency above $76.7%$⁠. Table III suggests that the manufactured half-wave metasurface presents a superior bandwidth and efficiency performance compared to other terahertz realizations. It is noted that a metal-mesh-based metasurface exhibits similar performance with our half-wave metasurface. However, our three-layer metasurface has an advantage of reduced fabrication complexity compared to the six-layer metal-mesh-based structure.

We have presented a tutorial on a semi-analytical approach to yielding broadband transmissive metasurfaces for wavefront and polarization control. The broadband approach combines network analysis with a genetic algorithm to collectively determine frequency-independent optimal circuit parameters for obtaining the desired complex transmission coefficients over a wide bandwidth. These parameters can then be physically realized by developing specific metallic patterns for each layer. To illustrate the detailed design process, two metasurface-based transmissive waveplates have been presented. Due to the involvement of the systematic design procedure and the concept of “Huygens’ surfaces,” both quarter- and half-wave metasurfaces present optimized performance in terms of bandwidth and efficiency. It is envisioned that the broadband approach is capable of forming mechanically tunable transmissive metasurfaces by implementing air gaps in between the dielectric substrates. Moreover, the broadband approach can be readily adapted to design reflective metasurfaces by modeling a grounded dielectric substrate as a shorted transmission-line section.^58,90 Importantly, the broadband approach can be further developed to account for interlayer near-field coupling⁹¹ so as to improve its accuracy.

This work was supported by Australian Research Council Discovery Projects (Grant No. ARC DP170101922).

The authors have no conflicts to disclose.

The data that support the findings of this study are available from the corresponding author upon reasonable request.