Design of unit cells and demonstration of methods for synthesizing Huygens metasurfaces

Available online at www.sciencedirect.com

ScienceDirect Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

Design of unit cells and demonstration of methods for synthesizing Huygens metasurfaces Joseph P.S. Wong, Michael Selvanayagam, George V. Eleftheriades ∗ The Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, 10 King’s College Road, Toronto, Ontario, Canada M5S 3G4 Received 10 March 2014; received in revised form 2 July 2014; accepted 5 July 2014 Available online 11 July 2014

Abstract The systematic design of unit cells for a Huygens metasurface, a particular class of metasurface, is presented here. The design of these unit cells uses transmission-line theory. This is validated through application to 1D refraction and Gaussian-to-Gaussian beam focusing. The 1D refraction is further validated experimentally. These applications demonstrate the practical utility of these Huygens metasurfaces. The Huygens metasurfaces presented here are printed on two bonded boards instead of many stacked, interspaced layers. This simplifies fabrication and enables the scaling down of the metasurfaces to shorter wavelengths. These two bonded boards implement a single, collocated layer of electric and magnetic dipoles. The electric and magnetic dipoles are synthesized using sub-wavelength arrays of printed elements. These printed elements can be manufactured using standard PCB fabrication techniques, and are capable of synthesizing the full range of impedances required. Furthermore, in contrast to frequency-selective surfaces (FSSs) and traditional transmitarrays, which are on the order of a wavelength thick, these designs are only λ/10 thick while incurring minimum reflections losses. © 2014 Elsevier B.V. All rights reserved. Keywords: Metasurface; Equivalence principle; Transmitarrays; Flat lens

1. Introduction Due to the challenges presented by loss, dispersion, and fabrication, metasurfaces, the 2D analog to volumetric metamaterials, have attracted much interest. In many cases, the metasurface can encode the same functionality as its metamaterial counterpart. The interaction of a metasurface with electromagnetic fields can be described in general through an impedance boundary condition collocated with its magnetic dual [1]. The

∗

Corresponding author. Tel.: +1 416 946 3564. E-mail address: [email protected] (G.V. Eleftheriades).

http://dx.doi.org/10.1016/j.photonics.2014.07.001 1569-4410/© 2014 Elsevier B.V. All rights reserved.

same description can be independently arrived at from the perspective of polarizabilities instead of impedances [2]. Examples of the applications of these descriptions to metasurfaces, in terms of impedances or susceptibilities, have included extraction of susceptibilities for both mono and bi-anisotropic scatterers [3,4], electromagnetic shielding [5], circuit modeling of Huygens sources [6], active cloaking [7], interfacial refraction [8,9], Gaussian-to-Bessel beam transformers [9], electromagnetic resonators [10], waveguides [11], exciting surface waves and complex modes [12], leaky-wave antennas [13], tunable metasurfaces [14], and plasmonic metasurfaces [15]. For a detailed review of the literature pertaining to metasurfaces, see [16].

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

The theoretical foundation and experimental verification of a Huygens metasurface has recently been discussed in [6,8,9]. This particular class of metasurface is an application of both the Huygens principle and the equivalence principle. Composed of a single layer of sub-wavelength unit cells containing both a loop and a dipole, it implements an array of Huygens sources. That is, the electric and magnetic responses are collocated in a single layer. The characteristics of a Huygens source provides the flexibility needed to engineer both the electric and magnetic surface currents enforcing the equivalence principle. This in turn provides the flexibility necessary for a wide range of applications. In [9], 1D interfacial refraction and Gaussian-to-Bessel beam transformation are demonstrated. In the case of refraction, suppression of side-lobes and reflections are seen. Furthermore, an accurate Bessel beam is reproduced with near-unity transmission. Until recently, attempts with both metasurfaces and transmitarrays for interfacial refraction have only been partially successful in efficiently coupling to the refracted beam. In the case of metasurfaces, this is because the responses are purely electric or magnetic. Furthermore, in [17] the metasurface operates only on the cross-polarized radiation. For transmitarrays, the implementation of a linear phase profile has been shown to always excite higher-order Floquet modes which manifest as side-lobes [18]. The inclusion of both an electric and magnetic response in a single layer in the Huygens metasurface suppresses these side-lobes and the reflections simultaneously. In [19] optical metasurfaces for 1D refraction and focusing are discussed and are without these issues. In this case, the response is exclusively electric. However, the design requires three interspaced layers of electric dipoles, as opposed to a single, collocated layer of electric and magnetic dipoles. However, to this date not much attention has been given to the discussion of the design of the constituent unit cells in the Huygens metasurface. In [6], a lattice circuit model is proposed for the Huygens source unit cell. Here we aim to incorporate this model to provide a systematic approach to the unit cell design process. This process is then applied to the 1D refraction and focusing of an incident Gaussian beam. In both cases, suppression of side-lobes and reflections is demonstrated. Experimental results for refraction are presented. Partial preliminary results have been disclosed in [20]. In contrast to the Huygens metasurface of [9], this implementation is printed on two bonded boards instead of many stacked, interspaced layers. This considerably simplifies fabrication and allows the scaling of the metasurfaces to millimeter-wave frequencies and beyond.

361

This simple design contains both the electric and magnetic responses, embodied in the printed elements. These printed elements can be manufactured using standard PCB fabrication techniques, and are able to synthesize the full range of impedances required. Furthermore, in contrast to FSSs and traditional transmitarrays, which are on the order of a wavelength thick, these designs are only λ/10 thick while incurring minimum reflections losses. 2. Theory Fig. 1 illustrates the scenario considered for this discussion. The definitions are shown in the figure. From array theory, the electric and magnetic surface currents can be interpreted as arrays of electric and magnetic dipoles [21]. The infinitesimal magnetic dipole is equivalent to an infinitesimal electric loop. Thus, the surface currents located along the same boundary, enforcing the equivalence principle, can be engineered through sub-wavelength arrays of collocated electric dipoles and loops. This sub-wavelength condition is precisely the homogenization required in metamaterials. This allows the homogenization with reactive sheets. Pairs of collocated electric and magnetic dipoles correspond to Huygens sources. Furthermore, these currents can be related to impedances corresponding independently to the electric and magnetic responses [8,9]. Due to both the equivalence principle and the uniqueness theorem, these electric and magnetic impedances constitute a transfer function between a predetermined excitation and the desired response. Practically, we are limited not only by the applications that can be conceived, but also by the impedances that can be synthesized. Ideally, the impedances are passive, lossless, and moderate in value. Often though these are active, lossy, and extreme. Thus, further research would involve both the synthesis of improved unit cells, as well as methods for approximating the ideal transfer functions with more realizable ones.

Fig. 1. Huygens metasurface. A Huygens metasurface can be designed such that equivalent surface currents (Js , Ms ) transform an incident field (E1 , H1 ) into a transmitted field (E2 , H2 ). The normal to the boundary is nˆ .

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

Referring to Fig. 1, the interaction of macroscopic electromagnetic fields with a metasurface in a homogeneous medium can be modeled by an impedance boundary condition and its magnetic dual [1]. The impedance boundary condition is expressed as ˆ n × (E2 − E1 ) = 0,

(1)

ˆ n × (H2 − H1 ) = Js ,

(2)

4

1 Impedance (Ω)

362

x 10

0.5 0 −0.5 −1 0

10

20

and ˆ n × E1 = Ze Js .

(3)

Taking the magnetic dual, the admittance boundary condition is ˆ n × (E2 − E1 ) = −Ms ,

(4)

ˆ n × (H2 − H1 ) = 0,

(5)

and



ˆ n × H1 =

1 Zm

Ms .

(6)

E2 + E1 = Ze ˆ n × (H2 − H1 ) 2

and ˆ n×

60

is 10.00 GHz. That is E1 = Einc + Erefl and E2 = Erefr where Einc = E0inc e−jkinc · r ,

(9)

−jkrefl · r

,

(10)

Erefr = E0refl e−jkrefr · r .

(11)

and

Here Ze and Zm are electric and magnetic scalar surface impedances, corresponding to the orthogonal electric and magnetic currents found in a Huygens source, respectively. Note that in general, Ze and Zm are dyadics, corresponding to the anisotropies necessary to relate the polarizations of the input and output fields. Provided that the impedances and the admittances are collocated and do not couple, these boundary conditions can be combined to obtain the more compact description ˆ n×

50

Fig. 2. Electric impedance Ze profile. This in conjunction with Zm transforms a normally incident plane-wave to a 30◦ refracted planewave at 10.00 GHz. Side-lobes and reflections will be suppressed.

Erefl = E0refl e



30 40 Position (mm)

H2 + H1 =− 2



1 Zm

(7)

Here E0 and k are the corresponding amplitudes and wavevectors, r is the radial coordinate vector, and j is the imaginary unit. The time dependence is ejωt and this time-harmonic form will be employed throughout, therefore this notation will be dropped. Following the result in [8], in order to obtain passive and lossless electric and magnetic impedances Ze and Zm , the amplitudes must be uniquely specified. Otherwise, these become active or lossy at different positions along the interface. To aid in this discussion we repeat the result for the amplitudes E0refl =

 ˆ n × (E2 − E1 ).

(8)

1 1 and ˆ n × H2 +H denote the average tanHere ˆ n × E2 +E 2 2 gential fields across the boundary, ˆ n × (E2 − E1 ) and ˆ n × (H2 − H1 ) denote the difference of the tangential fields across the boundary. Since the electromagnetic fields are macroscopic, the distances considered from the metasurface must also be macroscopic for this description to be valid.

3. Design procedure for refraction Here we refer back to Fig. 1. Taking (7) and (8), we stipulate normally incident and reflected planewaves, as well as a 30◦ refracted plane-wave as the field distributions on both sides. The design frequency

E0inc (cos θinc − cos θrefr ) , cos θinc + cos θrefr

(12)

2E0inc cos θrefr . cos θinc + cos θrefr

(13)

and E0refr =

Here θ inc and θ refr are the incident and refracted angles, respectively. Note that the reflected field is in general much smaller than the transmitted field. The application of the design parameters and (9)–(12) into (7) and (8) results in the electric and magnetic impedances shown in Figs. 2 and 3 respectively. Using the lattice cell model in [6], the uncoupled electric and magnetic impedances can be related to the impedance matrix of a standard 2-port network. There the electric and magnetic responses correspond to the shunt and series branches of the lattice network, respectively. Converting this to the scattering matrix representation, the phase of the transmission coefficient

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

363

4

Impedance (Ω)

4

x 10

2 0 −2 −4 0

10

20

30 40 Position (mm)

50

60

Fig. 3. Magnetic impedance Zm profile. This in conjunction with Ze transforms a normally incident plane-wave to a 30◦ refracted planewave at 10.00 GHz. Side-lobes and reflections will be suppressed.

S21 can be compared to the expected result for planewave refraction according to [17,22]. From the expected result a roughly linear phase profile is seen. Furthermore, according to (12) and (13), for the passive and lossless Huygens metasurface, we expect the magnitudes of S11 and S21 to be nearly zero and unity, respectively. The gradient of the linear phase profile determines the spatial periodicity of Ze and Zm . Fig. 4 illustrates the scenario and its definitions to aid the discussion. From either source [22,17], the phase insertion gradient along the infinitesimal section of boundary is given by sin θrefr − sin θinc =

1 dΦ . k dl

(14)

This is from the application of Fermat’s principle. Here k is the wavenumber in the medium, which is the same on both sides. Integrating along the boundary with the chosen θ inc and θ refr , the Huygens metasurface must linearly span 360◦ transmission phase with a spatial periodicity of 60.0 mm. This is also the spatial periodicity of Ze and Zm .

Fig. 4. Determination of spatial periodicity. Shown is an infinitesimal section along the boundary interface. Here dl is an infinitesimal length along the boundary, and Φ is the inserted phase.

Fig. 5. Total electric far-field radiation pattern. This reveals the effects of discretization on the quality of the refracted beam. This also determines the size of the unit cells in the Huygens metasurface. For this design, we settle on a discretization level of λ/10 or 3 mm at 10 GHz.

3.1. Effects of unit cell discretization Furthermore, Ze and Zm are implemented practically by taking the profiles in Figs. 2 and 3 and then discretizing them into unit cells. Taking the discretized versions of Ze and Zm into the lattice model, we utilize the transmission-line full-wave solver developed in [23]. This allows us to determine the quality of the refracted beam in terms of side-lobes and reflections for a given discretization. The merit of this method is that it can be done without first synthesizing an entire profile of unit cells. On the other hand, this does not account for the complex geometries of the unit cells. The solver utilizes the impedance values alone. However, this allows for the investigation of discretization before embarking on the laborious task of synthesis for an entire unit cell profile. More detailed simulations of the entire Huygens metasurface including unit cell geometries are done once an acceptable discretization level is committed to. In terms of S21 , the chosen discretization along the boundary determines the transmission phase that each Huygens source unit cell must provide. Fig. 5 demonstrates the effects of discretization on the quality of the refracted beam. Based upon these investigations, for this design, we settle on a discretization level of λ/10 or 3 mm at 10 GHz. This means that the Ze and Zm profiles are implemented with a period of 20 unit cells. This choice results in the first sidelobe and the maximum reflection being −26.56 dB and −22.81 dB from the main-lobe maximum, respectively. The increased discretization reduces the side-lobes and reflections because of the homogenization condition underlying (7), (8).

364

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

Fig. 6. Unit cell and Huygens metasurface. (a) Example of a unit cell implementing a Huygens source. The collocated infinitesimal electric and magnetic dipoles are approximated by sub-wavelength electric dipoles and loops. The printed capacitors synthesizing the impedances are visible. (b) One period of the fabricated Huygens metasurface.

3.2. Unit cell design The Huygens source can be envisioned as collocated infinitesimal electric and magnetic dipoles. Provided that the unit cell is sub-wavelength, the magnetic dipole can be approximated by an electric loop. In addition to this, the orientations of the dipoles must correspond with the polarizations of the input and output fields in (7), (8). This reflects the anisotropies of the impedances required to enforce the surface currents. Keeping the above in mind, unit cells were synthesized in the HFSS commercial full-wave electromagnetics solver to obtain the required electric and magnetic impedances. Here infinite array analysis was used. This result makes sense in view of the transmission-line description of infinite array analysis [24]. Furthermore, it is consistent with the earlier conversion of the lattice model for the Huygens source into transmission-line scattering parameters. Though the Huygens metasurface is in fact inhomogeneous, this analysis still provides an accurate result as the geometrical variations are slight along the interface. This is demonstrated in later sections in the simulations and experimental results. Fig. 6(a) and (b) show an example unit cell and one period of the fabricated metasurface. The printed capacitors synthesizing the impedances are visible. The Huygens source unit cell implements a

loop with the outer conductor layers, while the dipole is contained in the conductor within. Note that the 1D refraction is in the horizontal plane, and the unit cell is compatible with the field polarizations stipulated in this problem. The input and output electric field polarizations are vertically oriented. The input and output magnetic field polarizations are in the horizontal plane. The unit cells are designed as an entirely printable metasurface, and are implemented on a Rogers RO3003 substrate [εr = 3.00, tan(δ) = 0.0013]. In order to implement the entire range of impedances and admittances in Figs. 2 and 3, sufficient unit cell area must be retained. Otherwise, not all the required values can be patterned in the conductor. This limits the discretization that can be achieved. The unit cell dimensions are λ/10 wide by λ/5 tall by λ/10 thick. Note that since refraction is in the horizontal plane, the relevant dimensions for discretization are the horizontal and normal directions. This provides greater surface area to synthesize a wider range of impedance values. Since the Huygens source unit cell collocates the dipole and loop, coupling between these structures is minimal. This can be demonstrated by application of the lattice circuit model of [6] and parametric sweeps of the geometries. The lattice model is reproduced here in Fig. 7 to aid this discussion.

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

Fig. 7. Lattice circuit model. The two-port lattice network description of [6]. The electric impedances Ze correspond to the shunt branches, while the magnetic impedances Zm correspond to the series branches.

The two-port impedance matrix of the symmetric and reciprocal lattice circuit model is related to the shunt Ze branches and the series Zm branches through Ze =

Z11 + Z21 , 2

(15)

and Zm = 2(Z11 − Z21 ).

(16)

Here Z11 and Z21 belong to the impedance matrix of a standard two-port network. For the parametric sweeps, the geometries of the printed dipole and loop are swept in the unit cell including the substrate but excluding the other component. Fig. 8(a) and (b) show the geometries of a typical printed dipole and loop, as well as the parameters being varied. Fig. 9(a) and (b) illustrate the parametric sweeps of Ze and Zm from the lattice model as a function of the capacitor gap spacing in the printed dipole. The different curves represent various capacitor widths. Fig. 10(a) and (b) illustrate the parametric sweeps of Ze and Zm as a function of the capacitor width in the printed loop. Here it is seen that for the printed dipole, there is variation in Ze but not in Zm . This demonstrates that the printed

365

dipole responds electrically but not magnetically. This is ideal, since this causes Ze and Zm in the combined Huygens source unit cell to be uncoupled and therefore independent. However it is seen that while the printed loop responds primarily magnetically, there is still a slight electric response. Note that the substrate alone was simulated as well, and in this case Ze = −176.47  and Zm = 268.42 . Since the printed dipole has Zm almost identical to that of the substrate, this reinforces the notion that it responds purely electrically. Zm of the printed dipole can be viewed as belonging to the dielectric-filled transmission line it is immersed in. The conductor comprising the printed dipole has itself no contribution. The printed dipole has no closed-loop area for the changing magnetic flux to generate a response. In contrast to this, the printed loop, which does have a closed-loop area for the changing magnetic flux, responds magnetically in a dramatic fashion. In fact, there is a strong magnetic resonance in Zm . However, it can be seen in Fig. 10(a) that there is a weak electric response in Ze . This is because the printed loop forms a shunt element capacitively coupled to the transmission-line, albeit weakly. These electric and magnetic responses for the printed loop are similar to those found for a split-ring resonator (SRR), which this loop essentially is. This shunt element shows up in the lattice model as the electric component of the response, Ze . Thus the dynamic range of Zm is much larger than that of Ze in the printed loop. Furthermore, this agreement of the results with intuition demonstrates that the best model for an infinite array of dipole, loop, or Huygens source unit cells is the lattice network. To a good approximation then, the combined Huygens source unit cell can be treated as having the magnetic response exclusively due to the printed loop. Taking the electric response to be purely due to the printed dipole provides a good starting point, but because the printed loop has an electric response similar in dynamic range, the Ze of the resulting unit cell can deviate noticeably

Fig. 8. Printed element geometries. (a) The geometry of the printed dipole. The capacitor width (w) and gap spacing (g) are varied to achieve Ze . (b) The geometry of the printed loop. The capacitor width (w) is varied to achieve Zm .

366

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

(a)

(b)

150 0.7620 mm 100

1.7780 mm

Impedance (Ω)

2.7940 mm

Impedance (Ω)

0.7620 mm

272

1.7780 mm

50 0 −50

2.7940 mm

271 270 269

−100 268 −150 0.5

1

1.5

0.5

2

1

1.5

Capacitor Gap Size (mm)

Capacitor Gap Size (mm)

Fig. 9. Printed dipole responses. (a) The electric response of the dipole component of the Huygens source unit cell alone. Ze is plotted against the capacitor gap size for the various capacitor widths. (b) The magnetic response of the dipole component of the Huygens source unit cell alone. Zm is plotted against the capacitor gap size for the various capacitor widths.

(a)

(b)

−80

−90

Impedance (kΩ)

Impedance (Ω)

30

−100

−110

20 10 0 −10

−120 1

2

0.5

Capacitor Width (mm)

1

1.5

2

2.5

Capacitor Width (mm)

from the profiles in Figs. 2 and 3. In addition, another source of coupling between the printed dipole and loop comes from the interaction of currents with the patterns of the conductors themselves. Here an HFSS optimization analysis was run to fine tune each unit cell to its required point on the impedance profiles in Figs. 2 and 3. Fig. 11 shows one period of the realized electric and magnetic impedances corresponding to each unit cell superimposed on their ideal profiles. The impedance profiles agree well except near the resonances, where the more extreme impedances are difficult to synthesize exactly.

Impedance (Ω)

Fig. 10. Printed loop responses. (a) The electric response of the loop component of the Huygens source unit cell alone. Ze is plotted against the capacitor width. (b) The magnetic response of the loop component of the Huygens source unit cell alone. Zm is plotted against the capacitor width.

6000

Zshunt

4000

Zshunt Ideal Zseries Zseries Ideal

2000 0 −2000 −4000 −6000 −30

−20

−10

0

10

20

30

Position (mm)

4. Simulated and experimental results for refraction To experimentally characterize this design, we use a near-field scanner in conjunction with an ATM 90-441-6

Fig. 11. Realized and ideal impedances. One period of the realized electric and magnetic impedances corresponding to each unit cell superimposed on their ideal profiles. This implements the Huygens metasurface which refracts a normally incident plane-wave in the 30◦ direction.

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

Fig. 12. Experimental setup. Includes the near-field system, horn, lens, and Huygens metasurface.

horn antenna and a 203.2 mm double-convex hyperbolic Rexolite (εr = 2.53, tan(δ) = 6.6E − 4 at 10.00 GHz) lens. The output after the lens launches an approximate Gaussian beam with a measured waist size of 27.5 mm which is small enough to impinge 99.9% of the power upon the metasurface at its waist. Fig. 12 shows the experimental setup used. Using this system, we captured the frequency response, the response to oblique incidence, as well as the near-fields with a near-field scanner. Here we will present both the simulated and experimental results for comparison. Fig. 13 presents the total electric far-field radiation patterns of the HFSS simulated results at the design frequency of 10 GHz, and the measured results at the optimal frequency which has shifted to 10.35 GHz. The simulated results include dielectric and conductor losses, surface roughness effects, and the patterning of the conductors. This simulation utilizes a maximum of 25.5 GB of random access memory (RAM), and on an Intel Core i7-3820 running on 2 cores, takes about 45.3 h

Amplitude (dB)

0 −10 −20 −30

Simulated Raw Measured Averaged Measured

−40 −50

−60

−40

−20

0

20

40

60

Azimuth (°) Fig. 13. Simulated and measured total electric far-field. Simulated results at the design frequency of 10 GHz, and measured results at the optimal frequency of 10.35 GHz. The simulated maximum sidelobe level is −17.13 dB, while the measured maximum side-lobe level is −15.17 dB.

367

to converge to within 2% relative energy error. These characteristics are typical of the frequency response and oblique incidence response simulations as well. For the measurements, due to limitations in the setup, only the response for the range of angles shown in the forward direction could be captured. The measured optimal frequency is taken to be the frequency with the minimum side-lobe level, after smoothing of the ripples in the measurement. For the simulated result, the maximum side-lobe level and reflection are seen to be −17.13 dB and −9.66 dB from the main-lobe maximum, respectively. For the measured result the maximum side-lobe level is seen to be −15.17 dB from the main-lobe maximum, while the reflections remain to be characterized experimentally. The locations of the main-lobe and sidelobes are seen to be approximately identical between the simulation and the measurement after smoothing of the measured data. It must be stressed that the measured first side-lobe level should be taken qualitatively. Meaningful quantitative results require the presence of the ripples to be addressed. These ripples in Fig. 13 could be attributed to higher-order Gaussian modes exciting surface currents upon the metasurface. This is due to the poor coupling of the ATM 90-441-6 horn used to the fundamental Gaussian mode. In addition to this, the waist of the fundamental Gaussian mode in the ATM 90-441-6 horn is outside the limit of the paraxial approximation. Truncation effects at the lens cause a change in the waist position and size, which need near-field diffraction calculations to account for. In the far-field, truncation causes non-Gaussian components, which could be another source of ripples. See [25] for a detailed description of these factors. In addition to this, absorbers which perform optimally in the far-field may in fact become resonant at near-field distances. Given the strong agreement between the results, and comparison with [9], it seems plausible that the refraction performance in terms of reflections should likewise be promising. This is left for a future experimental investigation. Figs. 14 and 15 show the simulated and measured frequency responses for several representative frequencies, including the design and optimal frequencies. The simulated results reveal that the first side-lobe level and the reflections are best at the design frequency of 10.00 GHz, being −17.13 dB and −9.66 dB, respectively. The side-lobe levels are −14.37 dB and −12.00 dB at 10.35 GHz and 10.50 GHz, respectively. The reflections degrade to −4.02 dB and −1.50 dB at 10.35 GHz and 10.50 GHz, respectively. The response is narrowband in nature, which can be accounted for by the highly dispersive nature of the SRR-like loops. The

368

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

−30 −40

−40 −60 −80 10.00 GHz 10.35 GHz 10.50 GHz

−100 −120

−50

0

50

100

150

200

Amplitude (dB)

Amplitude (dB)

−20

−50 −60 −70 −80

−20° −10° 0° 10° 20°

−90

250

Azimuth (°)

−100

Fig. 14. Simulated total electric far-field frequency response. Given for several representative frequencies. The first side-lobe level and reflections deteriorate away from the optimal frequency of 10.35 GHz. The main-lobe location shifts left with increasing frequency.

main-lobe shifts slightly left as the frequency increases, as per (14). The measured results have been averaged here to remove ripples. These results are thus only useful for observing qualitative trends. More refined quantitative results, and their comparisons with simulations will be left to future work. The measured results reveal that the first side-lobe level is best at the optimal frequency of 10.35 GHz. The main-lobe location shifts left as expected. However, the main-lobe is seen to also narrow as the frequency increases. Most likely this is because the gain of the ATM 90-441-6 increases appreciably over the 0.5 GHz range. In the simulations, the characteristics of the excitation are frequency invariant, so the effect is not noticeable there. Figs. 16 and 17 show the simulated and measured responses to oblique incidence for several representative angles of incidence. The simulations reveal that, as the angles of incidence increases, the location of

−110

−50

0

50

100

150

200

250

Azimuth (°) Fig. 16. Simulated total electric far-field response to oblique incidence. Given for several representative angles of incidence. The location of the main-lobe shifts right as expected. The side-lobe level is best at angle of incidence −10◦ , but the reflections are best at 0◦ .

the main-lobe shifts right as expected. This can be predicted from (14). Unexpectedly, the first side-lobe level decreases monotonically as the angle of incidence varies from −10◦ , instead of 0◦ . The back-lobe level relative to the main-lobe is still best at normal incidence, which we designed for. The measured results will again be taken for their qualitative trends. The location of the main-lobe shifts right with increasing angle of incidence, as expected. Due to ripples and the averaging process, the side-lobe performance here is difficult to interpret, even qualitatively. Since the results for an angle of incidence of −20◦ and 0◦ are close, it is difficult to say which is superior. It does look like the result for an angle of incidence 0

0

−20 −30 10.00 GHz 10.35 GHz 10.50 GHz

−40 −50

−60

−40

−20

0

20

40

Amplitude (dB)

Amplitude (dB)

−10 −10

−20

−30

−20° −10° 0° 10° 20°

−40

60

Azimuth (°) Fig. 15. Measured total electric far-field frequency response. Given for several representative frequencies. This should be taken qualitatively for its trends, which can be verified to be the same as the simulations. The only major difference is an extra narrowing of the main-lobe observed here, which can be accounted for.

−50

−60

−40

−20

0 20 Azimuth (°)

40

60

Fig. 17. Measured total electric far-field response to oblique incidence. Given for several representative angles of incidence. The trends here agree generally with the simulations.

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

of 0◦ is better, as expected, but there is room to question this because the margin is small. Generally speaking however, the measurements still agree well with the simulations. Finally, Fig. 18(a) and (b) show the simulated transmitted near-field at the design frequency of 10 GHz, and the measured transmitted near-field at the optimal frequency of 10.35 GHz. That the refraction of the incident radiation is primarily in the desired direction is clearly seen in both results.

Fixing the waist size at the metasurface, from the paraxial approximation of a Gaussian beam, the corresponding minimum waist size and position are not unique. That is, the field profile with its 50.2 mm waist impinging on the Huygens metasurface can correspond to another Gaussian beam, with a different minimum waist size and position. The paraxial approximation for the electric field component of the Gaussian beam as used here is given as x2 w0 − wx22(z) −jkz−jk 2R(z) +jξ(z) . e w(z)

E(x, z) = E0

5. Gaussian-beam to Gaussian-beam focusing Here we design a Huygens metasurface for 1D Gaussian-to-Gaussian beam focusing with highly suppressed reflections. Instead of utilizing (7), (8), the paraxial approximation and the definition of the transmission coefficient S21 are relied upon. It will be demonstrated that for waist sizes, the procedure is reliable beyond the limits of where the paraxial approximation does not typically require corrections. For 0.45λ ≤ w0 ≤ 0.9λ, first-order corrections are required [25], and we demonstrate a design in the middle of this range where w0 = 0.723λ or 21.7 mm. Here w0 is the minimum waist size. Furthermore for this design, Ze and Zm are purely imaginary, and the Huygens metasurface exhibits highly suppressed reflections. However, the minimum waist position cannot be stipulated in addition to its size. For the characterization of this 1D lens, the relevant parameters for the X-band conical horn used at 10.00 GHz are an aperture size of 132.08 mm, and a 16 . 5◦ half-power beamwidth. This corresponds to a calculated waist of 50.2 mm at the aperture, where the 1D lens will be placed. This is to minimize truncation effects. The minimum waist is 114.6 mm into the horn where its size is 42.1 mm. Similar to the case of plane-wave refraction, the electric field polarization is taken to be perpendicular to the plane of focusing. In contrast to the ATM90-441-6 horn used earlier, w0 is well within the paraxial approximation. Furthermore, this conical horn provides 91% coupling efficiency to the fundamental Gaussian mode [25]. Simulation results for this 1D lens are presented. Detailed experimental results will be the subject of future work. 5.1. Design procedure for focusing The field pattern at the input of the metasurface is that of the conical horn antenna. This input Gaussian beam has its minimum waist size and position, and its waist size at the metasurface, as described in Section 5.

369

(17)

Here E0 is the amplitude of the Gaussian beam, x is the transverse coordinate, z is the normal coordinate, and w(z), R(z), and ξ(z) are defined as   2 z , (18) w(z) = w0 1 + zR   2  zR R(z) = z 1 + , (19) z and ξ(z) = arctan

z . zR

(20)

Finally, zR is the Rayleigh range, given as π w20 . (21) λ The definition of the transmission coefficient S21 is  E2+  S21 = −  . (22) E1 E− = 0 zR =

2

Here E1− is the incident electric field to the input port, and E2+ is the reflected electric field to the output port. The output port is matched, therefore E2− which is the incident electric field to the output port, is zero. This reveals that to obtain an S21 profile with unity magnitude across the Huygens metasurface, the amplitudes of the input and output field profiles must be the same on both sides. The action of the Huygens metasurface is then to provide the phase difference between the field profiles. Furthermore, because S21 has unity magnitude, the reflections are expected to be highly suppressed. Thus for focusing, given the input Gaussian beam, an output Gaussian beam can be found such that the field profiles across the entire Huygens metasurface are the same. The minimum waist sizes and positions in the regions across metasurface need not be the same. Then S21 has a magnitude profile of unity as just described. Furthermore, the S21 phase profile can be found from the phases of the field profiles at the Huygens metasurface.

370

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

Fig. 18. Transmitted near-fields. (a) Simulated and (b) measured near-field results after normalization, showing clearly that refraction of the incident radiation is primarily in the desired direction. In both figures the metasurface is at the right boundary, and the incident beam is from the right. The dimensions of the boundaries are 14.13 λ × 21.33 λ, and the scale applies to both plots.

5.2. Demonstration of design procedure for focusing Given the aperture waist of 50.2 mm, this can be focused to an w0 in the range of 10.9 mm to 50.2 mm. The lower bound is obtained from the full-width halfmaximum of a diffraction-limited sinc, which is 0.603 λ. The diffraction limit is below the w0 = 0.45 λ limit for first-order corrections to the paraxial approximation. In fact, at this point, the Gaussian beam model fails completely [25]. We demonstrate a design where w0 = 0.723λ or 21.7 mm. This is in the middle of the range of where the paraxial approximation requires first-order corrections. Despite this, with respect to synthesizing a desired w0 , the paraxial approximation is still an accurate tool. Following the design procedure outlined in Section 5.1, we specify an aperture waist of 50.2 mm and an

w0 of 21.7 mm. Then the paraxial approximation yields a minimum waist location of 101.1 mm from the Huygens metasurface. The amplitude of the Gaussian at the output is predicted to be 1.94 times that of the Gaussian at the input. From the definition of the S21 , a magnitude profile of unity is obtained. Thus S11 and the reflections are expected to be highly suppressed. Fig. 19 shows the corresponding phase profile. The profile is quadratic, and the focal spot predicted from ray-optics is 96.2 mm from the metasurface. Taking the lattice model, the scattering parameters can be converted to their corresponding Ze and Zm profiles across the Huygens metasurface. Figs. 20 and 21 illustrate the profiles. Note that the impedances are 200

Phase (°)

This is parabolic as expected from ray-optics theory. Stipulating a desired output w0 , the paraxial approximation dictates the minimum waist position and maximum field amplitude. Taking the lattice model in [6], the scattering parameters are then converted to Ze and Zm , which are found to be purely imaginary. At this stage, the unit cells can be synthesized according to the procedure outlined in Section 3.2.

100 0 −100 −200

−100

−50

0

50

100

Position (mm) Fig. 19. S21 phase profile across the Huygens metasurface. The profile is quadratic.

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375 0

Real Imag

3 2 1 0 −1 −100

−50

0

50

100

Fig. 20. Ze profile across the Huygens metasurface. The impedances are purely imaginary.

10

Impedance (kΩ)

−20 −40

−60

Position (mm)

Real Imag

5 0 −5

−10

Amplitude (dB)

Impedance (kΩ)

4

−2

371

−100

−50

0

50

100

Position (mm) Fig. 21. Zm profile across the Huygens metasurface. The impedances are purely imaginary.

purely imaginary. The Huygens metasurface is passive and lossless as desired. 5.3. Simulation using transmission-line model Using the transmission-line full-wave solver provides a quick verification prior to unit cell synthesis. Fig. 22 shows the normalized complex magnitude of the electric field. There minor reflections are seen. However, as demonstrated in Fig. 23, these decay at far-field

Fig. 22. Normalized complex magnitude of the electric field. Obtained with the transmission-line model. The output minimum waist size obtained is 21.3 mm, 0.013 λ away from the design value. Its position is 80.8 mm from the Huygens metasurface. The input minimum waist size is 42.1 mm, and its position is 114.6 mm from the Huygens metasurface. The near-field reflections decay into the far-field.

−150 −100 −50

0

50

100

150

Azimuth (°)

Fig. 23. Far-field radiation pattern of the electric field. Obtained with the transmission-line model. The pattern is typical of a Gaussian beam in the far-field. The reflections are seen to be small in the far-field.

distances. There it is also seen that the first side-lobe and reflections are −26.72 dB and −28.62 dB from the main-lobe, respectively. Note also from Fig. 23 that the far-field pattern is that typical of a Gaussian beam. The w0 obtained is 21.3 mm, 0.013 λ away from the desired w0 of 21.7 mm. Considering just the w0 , this demonstrates the validity of this approach even when the paraxial approximation requires first-order corrections. However, the obtained minimum waist position is 80.8 mm, 0.67 λ away from the desired position of 101.1 mm. This is due to ignoring the first-order corrections to the paraxial approximation. We demonstrate this assertion in the following section. In addition to this, V the output amplitude relative to the input is 1.4059 m V compared to the expected 1.94 m . This deviation is also due to ignoring the first-order corrections in the paraxial approximation. This also demonstrates that specifying w0 , the position and maximum amplitude cannot be controlled. 5.4. Further investigation of design procedure for focusing This method is further investigated in the manner of Sections 5.1–5.3 for the designed output w0 in the range of 21.7 mm to 49.0 mm. That is, for the middle of the range where the paraxial approximation requires firstorder corrections to about the limit of the aperture waist. Table 1 demonstrates the accuracy of w0 obtained for this range of design values. The obtained w0 values are from the transmission-line solver. It is seen that the obtained w0 is always within 0.027 λ of the design value. The second row of Table 1 refers to the limit of the paraxial approximation where typically no corrections apply, and the last two rows to the aperture waist limit. Table 2 demonstrates the accuracy of fout obtained for the range of w0 design values. Here fout refers to the position of the minimum waist. The fout predictions are from (17) to (21). The obtained values are from the TLM

372

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

Table 1 Desired and obtained w0 . The obtained w0 is always within 0.027 λ of the design value. This is from the middle of the range where the paraxial approximation requires first-order corrections to about the aperture waist. w0 Desired (mm)

w0 Obtained (mm)

w0 Deviation (λ)

21.7 27.0 30.0 33.0 36.0 39.0 42.1 45.0 48.0 49.0

21.3 26.2 29.3 32.8 35.8 38.8 41.8 44.7 47.8 48.8

0.013 0.027 0.023 0.0067 0.0067 0.0067 0.01 0.01 0.0067 0.0067

Table 2 Desired and obtained fout . The deviation in fout improves as the paraxial approximation becomes increasingly valid. Here fout refers to the position of the minimum waist. w0 Desired (mm)

fout Predicted (mm)

fout Obtained (mm)

fout Deviation (λ)

21.7 27.0 30.0 33.0 36.0 39.0 42.1 45.0 48.0 49.0

101.1 117.2 123.6 127.4 128.0 124.5 114.6 97.0 61.0 36.0

81.0 101.0 109.3 116.3 117.3 116.0 108.8 93.8 60.5 45.7

0.67 0.54 0.47 0.37 0.36 0.28 0.19 0.107 0.0167 0.323

5.5. Simulation using HFSS solver. It is observed that the deviation in fout improves as the paraxial approximation becomes increasingly valid. The last row is a discrepancy from this trend. In Section 2, the final comment made is that the distances from the metasurface must be macroscopic in order for the description to be valid. This is because the fields themselves are macroscopic, as is the Gaussian beam considered here, which is also a far-field distribution. Close to the metasurface, the near-fields begin to dominate the description, and the last predicted fout in Table 1 is approaching a wavelength from the metasurface. This illustrates the constraint given in Section 2. The same trends as for fout exist for the output amplitude relative to the input. In addition to this, from the transmission-line solver, the reflections are small in the far-field for all entries in the tables. The far-field patterns are also typical of Gaussian beams, as in Fig. 23.

Unit cells were synthesized in HFSS using the method described in Sections 3.1 and 3.2. These are shown in Fig. 24, and the printed inductors and capacitors synthesizing the impedances are visible. However, instead of working with the impedances Ze and Zm , it is more straightforward to work with the transmission coefficient S21 . This is because the S21 magnitude profile is unity, while the variation is in its phase. For passive and lossless plane-wave refraction, as described in Section 3 and [8], small reflections must exist. In that case, the impedances Ze and Zm are more straightforward to work with. The S21 magnitude and phase profiles, though respectively close to unity and roughly linear, are not as apparent. Figs. 25 and 26 show the desired and synthesized S21 magnitude and phase profiles. Note that in the center of the screen, where most of the incident Gaussian beam energy is concentrated, the match between the profiles

Fig. 24. Unit cells for focusing. Unit cells synthesized in HFSS using the method described in Sections 3.1 and 3.2 for this lens design. The printed inductors and capacitors synthesizing the impedances are visible.

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

373

1.02 1 0.98

|S11| Ideal |S11| Realized

0.3

Magnitude

Magnitude

|S21| Ideal |S21| Realized

0.25 0.2 0.15 0.1 0.05

0.96

0 −100 0.94

−100

−50

0

Position (mm)

50

100

Fig. 25. Desired and synthesized S21 magnitude profiles. In the center of the screen, where most of the incident Gaussian beam energy is concentrated, the match between the profiles is most important.

is most important. This ensures the highly suppressed reflections, while providing the desired output beam properties. The magnitude matches closely across most of the screen, but most importantly in the center. The phase is seen to match closely across the entire screen. Fig. 27 shows the desired and synthesized S11 magnitude profiles. Note that in the center of the screen, where most of the incident Gaussian beam energy is concentrated, S11 should be as close to zero as possible to minimize reflections. This is demonstrated here in our design. There are stronger reflections at several positions along the metasurface, namely ± 66, 70, 74, 106 mm. From Fig. 20, the Ze profile at these points is extreme in value. These impedances are more difficult to synthesize, thus the sub-optimal S11 and S21 at these points. Examining Figs. 9(a) and 10(a), the typical Ze response is seen to be non-resonant in nature. The combination of Ze from the dipole and loop would need to be resonant to achieve such extreme shunt impedances. 200

∠S21 Ideal ∠S21 Realized

150

−50

0

50

100

Position (mm) Fig. 27. Desired and synthesized S11 magnitude profiles. In the center of the screen, where most of the incident Gaussian beam energy is concentrated, S11 should be as close to zero as possible to minimize reflections.

Note however that in contrast, Fig. 10(b) shows the Zm response of the loop to be resonant. Thus, there is no difficulty in achieving the required Zm profile in Fig. 21. However, among these positions with stronger reflections, at the positions closest to the center (± 66 mm), the amplitude of the Gaussian profile is 18% that at the center. Fig. 28 presents the normalized complex magnitude of the total electric field profile obtained in HFSS. This simulation utilized a maximum of 25.4 GB of RAM, and took about 51.0 h to complete for the same convergence criteria as in the refraction design. The Huygens metasurface lies along the interface of the discontinuous electric field profiles. Note that across this interface, the field profiles are about identical, as would be expected when S21 has unity magnitude along the boundary. The minimum waist size w0 obtained is 24.5 mm, which is . 093 λ and . 107 λ off from the analytical and transmission-line model results. These were 21.7 mm and 21.3 mm, respectively. The minimum waist position is 81.5 mm or 0.017 λ off from the transmissionline model result. The deviation in the minimum waist

Phase (°)

100 50 0 −50 −100 −150 −200

−100

−50

0

50

100

Position (mm) Fig. 26. Desired and synthesized S21 phase profiles. In the center of the screen, where most of the incident Gaussian beam energy is concentrated, the match between the profiles is most important.

Fig. 28. HFSS normalized complex magnitude of total electric field. The output minimum waist size obtained is 24.5 mm, and its position is 81.5 mm from the Huygens metasurface. The input minimum waist size is 42.1 mm, and its position is 114.6 mm from the Huygens metasurface.

374

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375

transmission-line solver compared to HFSS. For instance, the effect of interactions between the electromagnetic fields with the complex geometries of the unit cells is not accounted for. 6. Conclusion

Fig. 29. HFSS normalized complex magnitude of scattered electric field. This demonstrates clearly that most of the incident power is redirected to the output.

position from the analytical result is much greater as discussed in Section 5.3. Fig. 29 presents the normalized complex magnitude of the scattered electric field profile in HFSS. This demonstrates clearly that most of the incident power is redirected to the output. Fig. 30 shows the far-field radiation pattern of the synthesized lens. In the forward direction, the shape of the far-field pattern is similar but not quite that of a typical Gaussian beam. In the backward direction, the reflections are seen to have decayed in the far-field. The first sidelobe level and reflections are −23.31 dB and −12.02 dB from the main-lobe, respectively. Comparing Figs. 22 and 23 with Figs. 28–30, there are some discrepancies between the result of the transmission-line solver and the HFSS results. Clearly, the realized scattering parameters obtained in Figs. 25–27 for the unit cells are not quite ideal. However, using the synthesized S11 and S21 in Figs. 25–27 in the transmission-line solver, w0 and fout are identical to before, and the far-field pattern is typical of a Gaussian as in Fig. 23. Thus, the discrepancies can be attributed to the simplified analysis of the

Fig. 30. HFSS far-field radiation pattern of synthesized lens. The pattern in the forward direction is similar but not quite that of a typical Gaussian beam. In the backward direction, the reflections are seen to have decayed in the far-field.

The design of unit cells to synthesize Huygens metasurfaces, utilizing a lattice network and transmission-line theory, has been described. The concept is validated in our applications to 1D refraction and focusing. Simulations and experiments for 1D refraction show suppressed side-lobes and reflections, as well as a predictable angle of refraction. Simulations for 1D focusing show Gaussian-to-Gaussian beam focusing with suppressed side-lobes and reflections, and a reliable minimum waist size w0 at the output. The minimum waist position becomes reliable as the paraxial approximation becomes increasingly valid. Factors causing discrepancies could include discretization levels, sub-optimal unit cells designs, corrections to the paraxial approximation, and limitations in the experimental setup. However, the results are sufficient to demonstrate the validity of the concepts presented in this work. Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of Canada, grant number CGSM-442030-2013. References [1] S. Tretyakov, Analytical Modelling in Applied Electromagnetics, Artech House, 2003. [2] E.F. Kuester, M.A. Mohamed, M. Piket-May, C.L. Holloway, Averaged transition conditions for electromagnetic fields at a metafilm, IEEE Trans. Antennas Propag. 51 (10) (2003) 2641–2651, http://dx.doi.org/10.1109/TAP.2003. 817560. [3] A. Dimitriadis, N. Kantartzis, I. Rekanos, T. Tsiboukis, Efficient metafilm/metasurface characterization for obliquely incident TE waves via surface susceptibility models, IEEE Trans. Magn. 48 (2) (2012) 367–370, http://dx.doi.org/10.1109/ TMAG.2011.2175374. [4] A. Dimitriadis, N. Kantartzis, T. Tsiboukis, Consistent modeling of periodic metasurfaces with bianisotropic scatterers for oblique TE-polarized plane wave excitation, IEEE Trans. Magn. 49 (5) (2013) 1769–1772, http://dx.doi.org/10.1109/ TMAG.2013.2238517. [5] U. Paoletti, T. Suga, H. Osaka, Average transmission cross section of aperture arrays in electrically large complex enclosures, Asia-Pac. Symp. Electromagn. Compat. 677 (680) (2012) 21–24, http://dx.doi.org/10.1109/APEMC.2012. 6237888.

J.P.S. Wong et al. / Photonics and Nanostructures – Fundamentals and Applications 12 (2014) 360–375 [6] M. Selvanayagam, G.V. Eleftheriades, Circuit modelling of Huygens surfaces, IEEE Antennas Wirel. Propag. Lett. 12 (2013) 1642–1645, http://dx.doi.org/10.1109/LAWP.2013.2293631. [7] M. Selvanayagam, G.V. Eleftheriades, An active electromagnetic cloak using the equivalence principle, IEEE Antennas Wirel. Propag. Lett. 11 (2012) 1226–1229, http://dx.doi.org/10.1109/LAWP.2012.2224840. [8] M. Selvanayagam, G.V. Eleftheriades, Discontinuous electromagnetic fields using Huygens sources for wavefront manipulation, Opt. Expr. 1493 (12) (2013) 5679–5689, http://dx.doi.org/10.1364/OE.21.014409. [9] C. Pfeiffer, A. Grbic, Metamaterial Huygens surfaces: tailoring wave fronts with reflectionless sheets, Phys. Rev. Lett. 110 (19) (2013) 197401, http://dx.doi.org/10.1103/PhysRevLett. 110.197401. [10] C. Holloway, D.C. Love, E. Kuester, A. Salandrino, N. Engheta, Sub-wavelength resonators: on the use of metafilms to overcome the λ size limit, IET Microw. Antennas Propag. 2 (2) (2008) 120–129, http://dx.doi.org/10.1049/iet-map:20060309. [11] C. Holloway, E.F. Kuester, D. Novotny, Waveguides composed of metafilms/metasurfaces: the two-dimensional equivalent of metamaterials, IEEE Antennas Wirel. Propag. Lett. 8 (2009) 525–529, http://dx.doi.org/10.1109/LAWP.2009.2018123. [12] C.L. Holloway, D.C. Love, E.F. Kuester, J.A. Gordon, D.A. Hill, Use of generalized sheet transition conditions to model guided waves on metasurfaces/metafilms, IEEE Trans. Antennas Propag. 60 (11) (2012) 5173–5186, http://dx.doi.org/10.1109/TAP. 2012.2207668. [13] C.L. Holloway, E. Kuester, Surface waves complex modes and leaky waves on a metasurface, USNC-URSI 21 (21) (2013) 7–13, http://dx.doi.org/10.1109/USNC-URSI.2013.6715327. [14] J.A. Gordon, C.L. Holloway, J. Booth, Y.W. Sung Kim, J. Baker-Jarvis, D.R. Novotny, Fluid interactions with metafilms/metasurfaces for tuning, sensing, and microwave-assisted chemical processes, Phys. Rev. B 83 (2011) 205130, http://dx.doi.org/10.1103/PhysRevB.83. 205130. [15] P.-C. Li, Y. Zhao, A. Alu, E.T. Yu, Experimental realization and modeling of a subwavelength frequency-selective plasmonic metasurface, Appl. Phys. Lett. 99 (22) (2011) 221106, http://dx.doi.org/10.1063/1.3664634.

375

[16] C. Holloway, E.F. Kuester, J. Gordon, J. O’Hara, J. Booth, D. Smith, An overview of the theory and applications of metasurfaces: the two-dimensional equivalents of metamaterials, IEEE Antennas Wirel. Propag. Mag. 54 (2) (2012) 10–35, http://dx.doi.org/10.1109/MAP.2012.6230714. [17] N. Yu, P. Genevet, M.A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, Z. Gaburro, Light propagation with phase discontinuities: generalized laws of reflection and refraction, Science 334 (6054) (2011) 333–337, http://dx.doi.org/10.1126/science.1210713. [18] J.Y. Lau, S.V. Hum, Reconfigurable transmitarray design approaches for beamforming applications, IEEE Microw. Mag. 60 (12) (2012) 5679–5689, http://dx.doi.org/10.1109/ TAP.2012.2213054. [19] N.M.E. Francesco Monticone, A. Alù, Full control of nanoscale optical transmission with a composite metascreen, Phys. Rev. Lett. 110 (2013) 203903, http://dx.doi.org/10.1103/PhysRevLett. 110.203903. [20] J. Wong, M. Selvanayagam, G.V. Eleftheriades, A thin printed metasurface for microwave refraction, IEEE Microw. Symp. Dig. (2014), http://dx.doi.org/10.1109/MWSYM.2014.6848308. [21] H.A. Wheeler, Simple relations derived from a phased-array antenna made of an infinite current surface, IEEE Trans. Antennas Propag. 2 (1965) 506–514, http://dx.doi.org/10.1109/APS. 1964.1150148. [22] H. Steyskal, A. Hessel, J. Shmoys, On the gain-versus-scan tradeoffs and the phase gradient synthesis for a cylindrical dome antenna, IEEE Trans. Antennas Propag. 27 (6) (1979) 825–831, http://dx.doi.org/10.1109/TAP.1979.1142192. [23] M. Zedler, G.V. Eleftheriades, Anisotropic transmission-line metamaterials for 2-D transformation optics applications, Proc. IEEE 99 (10) (2011) 1634–1645, http://dx.doi.org/10.1109/ JPROC.2011.2114310. [24] Q. Wu, F.-Y. Meng, M. feng Wu, J. Wu, L.-W. Li, Research on the negative permittivity effect of the thin wires array in left-handed material by transmission line theory, Prog. Electromagn. Res. Symp. 1 (2005) 196–200, http://dx.doi.org/10.2529/ PIERS110821103100. [25] P. Goldsmith, Quasi-optical techniques, Proc. IEEE 80 (11) (1992) 1729–1747, http://dx.doi.org/10.1109/5.175252.