5. FPC-LWAs based on nematic liquid crystals

#### 5.1. NLC properties

Nematic liquid crystals (NLCs) are among the most widely studied and used liquid crystals (LCs). As is known [17], in addition to the solid crystalline and liquid phases, LCs exhibit intermediate phases (mesophases) where they flow like liquids (thus requiring hydrodynamical theories for their complete description), yet possess some physical properties characteristic of solids (thus requiring the elastic continuum theory for their complete description). As a function of temperature, or depending on the constituents, concentration, substituents, and so on, LCs exist in many so-called mesophases: nematic, cholesteric, smectic, and ferroelectric [17].

From an engineering point of view, LCs are attractive since, near the isotropic nematic phase transition temperature, the LC molecules become highly susceptible to external fields, and their responses tend to slow down considerably [17]. In particular, let us consider a layer of uniaxial NLCs sandwiched between two substrates covered by electrodes. It is known that [17] if an alignment film is deposited on the substrates and no bias is applied (Vb ¼ 0 V, unbiased state), the optical axis of the LC molecules is parallel to the substrate plane, in the direction given by the alignment film (horizontal axis), whereas, when a sufficiently large driving voltage (Vb ¼ V∞, biased state) is applied across the LC, the optical axis is fully tilted and parallel to the applied electric field direction (vertical axis) [55]. While the maximum achievable tuning range can be inferred from the knowledge of these two limiting states, the voltage-dependent tunable properties of LCs require a rigorous study of the LC dynamics. This task can be performed by a numerical implementation of the Q-tensor theory of liquid crystals, a convenient theoretical formulation for the study of the LC orientation in confined geometries [56].

Among the different kinds of NLCs, we consider here the nematic mixture 1825 because of its high birefringence at THz frequencies [56]. The dielectric properties of such a material are described by a complex permittivity tensor ε<sup>r</sup> . As a uniaxial crystal, the permittivity tensor of an NLC in its unbiased and fully biased states reduces to a diagonal matrix, whose diagonal elements are the ordinary ε<sup>o</sup> and extraordinary ε<sup>e</sup> relative permittivities of the NLC. For intermediate states, the NLC permittivity tensor is no longer a diagonal matrix. However, a thorough analysis of the Q-tensor has revealed that for an NLC aligned along the horizontal z-axis (see the reference frame in Figure 7), when a low-driving voltage is applied [56] across a fishnet unit-cell, the optical axis of the NLC tilts in the xz-plane exhibiting a negligible rotation over both the yzand the xy-plane. Thus, at first approximation8 , the NLC can still be locally modeled as a uniaxial crystal whose complex permittivity tensor ε<sup>r</sup> ð Þ Vb is a diagonal matrix diag εxxð Þ Vb ; εyy; εzzð Þ Vb where εyy ¼ εxxð Þ¼ 0 εzzð Þ¼ V<sup>∞</sup> ε<sup>o</sup> ≃ 2:42 and εxxð Þ¼ V<sup>∞</sup> εzzð Þ¼ 0 ε<sup>e</sup> ≃ 3:76 for a z-oriented NLC in the THz range [56], respectively. It is worth noting here that the assumption of uniaxial crystal for the NLC layer allows for easily describing its behavior with a relatively simple equivalent circuit model [57]. On the contrary, when the off-diagonal components are nonnegligible, more complicated equivalent networks are needed [57, 58], also accounting for the dielectric tensor spatial distribution [55].

#### 5.2. NLC-based designs

broadside (x<sup>0</sup> ¼ 0:82h1) does not lead to the best configuration in terms of radiation efficiency (x<sup>0</sup> ¼ h<sup>1</sup> or x<sup>0</sup> ¼ 0). However, the radiation performance improvement in terms of directivity of a GSS with respect to a GPW is paid at the expense of a moderate reduction of radiation efficiency. To give some numbers, a GPW shows D<sup>0</sup> ≃14 dB and er ≃ 69%, whereas an optimized

15, 30, 45<sup>∘</sup> . The chemical potentials for the GPW and the GSS are reported in the legend [19].

Figure 6 (a) Normalized phase constants and attenuation constants of the fundamental TM (in blue) and TE (in red) leaky modes of a GPW (dashed lines) at f ¼ 0:922 THz and of a GSS (solid lines) with graphene placed at the optimum position x<sup>0</sup> ¼ 0:82h<sup>1</sup> at a fixed frequency f ¼ 1:132 THz, as a function of the chemical potential in the range 1 > μc > 0 eV. (b) H-plane and (c) E-plane radiation patterns, normalized to the overall maximum (achieved at broadside), vs. elevation angle θ for the GSS antenna (solid lines) and for the GPW (dashed lines). The scanning behavior at a fixed frequency (f ¼ 1:132 THz for the GSS and f ¼ 0:92 THz for the GPW) is shown for four theoretical pointing angles θ<sup>0</sup> ¼ 0,

The dispersion analysis of a GSS and its radiating performance are reported in Figure 6(a) (see blue and red solid lines for the dispersion curves of the TM and the TE leaky modes of the GPW, respectively) and Figures 6(b) and (c) (see colored solid lines). As manifested by comparing the dashed and solid colored lines in Figures 6(b) and (c), the GSS shows considerably narrow beamwidths (thus higher directivities) over the entire beam scanning range. However, it should be noted that the GSS achieves the same angular range of a GPW over a wider range of variation of the chemical potential. This aspect reveals the slightly weaker degree

Nematic liquid crystals (NLCs) are among the most widely studied and used liquid crystals (LCs). As is known [17], in addition to the solid crystalline and liquid phases, LCs exhibit intermediate phases (mesophases) where they flow like liquids (thus requiring hydrodynamical theories for their complete description), yet possess some physical properties characteristic of solids (thus requiring the elastic continuum theory for their complete description). As a function of temperature, or depending on the constituents, concentration, substituents, and so on, LCs

From an engineering point of view, LCs are attractive since, near the isotropic nematic phase transition temperature, the LC molecules become highly susceptible to external fields, and their responses tend to slow down considerably [17]. In particular, let us consider a layer of uniaxial

exist in many so-called mesophases: nematic, cholesteric, smectic, and ferroelectric [17].

GSS shows D<sup>0</sup> ≃18:5 dB and er ≃ 56%.

104 Metamaterials and Metasurfaces

5.1. NLC properties

of reconfigurability of the GSS with respect to the GPW.

5. FPC-LWAs based on nematic liquid crystals

The proposed device (see Figure 7(a)) consists of a multistack of alternating layers of thin alumina layers and of NLCs placed above a GDS. The choice of Zeonor for the substrate layer has been motivated by the index matching between its relative permittivity εr<sup>1</sup> ¼ 2:3 and the ordinary relative permittivity ε<sup>o</sup> ¼ 2:42 of the NLC layer in the THz range [56], as it is required to properly enhance the resonance condition in an FPC [29].

As we have seen in Section 2.4, the alternation of high- and low-permittivity layers, with thicknesses fixed at odd multiples of a quarter wavelength in their respective media, allows for obtaining a narrow beam radiating at broadside [29]. In the proposed device, the innovating feature is represented by the possibility of exploiting the tunable properties of the NLCs [56], here representing the low-permittivity layer. In particular, the application of a common control signal to the NLC layers allows for changing their dielectric properties, thus achieving beam-steering capability at a fixed frequency, similar to the one obtained with the graphenebased FPC-LWAs shown in Section 4.

<sup>8</sup> A more accurate model should take into account the non-zero values of the εxz and εzx components.

A circuit model (see Figure 7(a)) has been developed for the dispersion analysis of such a structure, taking into account the voltage dependence of the NLC layers. Hence, when no bias is applied (Vb ¼ 0 V), the NLC molecules are aligned along the horizontal z-axis (Figure 7(a)), i.e., εzzð Þ¼ 0 εe, promoted by a few tens nm-thin alignment layer, which does not affect the electromagnetic properties of the device. When a sufficiently large driving voltage (V∞) is applied across the LC layers, the NLC molecules reorient along the vertical x-axis, i.e., εzzð Þ¼ V<sup>∞</sup> εe, thus providing the maximum reconfigurability [55].

As a consequence, with reference to the transverse transmission line of Figure 7(a), the characteristic admittances and the normal wavenumbers of the NLC layers for both the TE and the TM polarizations (with respect to the xz-plane) are functions of Vb. Their expressions are given by,

$$\hat{\mathbf{Y}}\_{0}^{\text{TE}} = \hat{\mathbf{k}}\_{\text{x}}^{\text{TE}} / \eta\_{0\prime} \ \hat{\mathbf{k}}\_{\text{x}}^{\text{TE}} = \sqrt{\varepsilon\_{yy} - \hat{\mathbf{k}}\_{\text{z}}^{2}} \tag{11}$$

As expected, when Vb <sup>¼</sup> <sup>V</sup><sup>∞</sup> (blue curve), the splitting condition <sup>β</sup>^

that β^

6. Conclusion

fixed frequency through bias voltage.

at broadside and at the maximum pointing angle [22].

THz, which is the design frequency. Even more interestingly, once the frequency is fixed, e.g.,

This is confirmed by the radiating patterns on the E-plane (the NLCs tunability affects only TM modes), calculated by means of leaky-wave theory (black lines) and CST full-wave simulations (blue circles), as shown in Figure 7(c). Results have been reported in Figure 7(c) considering radiation at broadside (biased status) and at the maximum pointing angle (unbiased status). It should be noted that the radiation efficiency of this device is limited to values around 40% due to the nonnegligible losses introduced by the three NLC cells. However, if one reduces the number of NLC cells, the beam-steering capability of the device is reduced as well [22]. The interested reader can find a comparison of different FPC-LWAs based on NLCs in [22].

Fabry-Perot cavity leaky-wave antennas (FPC-LWAs) represent a valid solution for designing fully-planar, low-cost, high-gain antennas in the THz range. In this Chapter, starting from the conventional design of FPC-LWAs, the typical technological constraints of the submillimeter radiation have been thoroughly addressed to characterize the design criteria of THz FPC-LWAs. On this ground, three different THz-FPC LWA designs have been extensively discussed. First, an FPC-LWA based on a homogenized metasurface has been shown to produce high gain without requiring neither high complexity nor high fabrication cost. This has been made possible due to the employment of a fishnet unit-cell, a very interesting element for the impedance synthesis of metasurfaces in the THz range. This structure exhibits an excellent radiation performance, but it

<sup>z</sup> > α^<sup>z</sup> by simply lowering the bias voltage, whereas the value of the normalized attenuation constant α^<sup>z</sup> remains almost the same. As a consequence, the dispersion diagram of Figure 7(b) reveals the possibility to steer the beam with a quasi-constant beamwidth at a

at <sup>f</sup> <sup>¼</sup> <sup>0</sup>:59 THz, it is possible to change the value of the normalized phase constant <sup>β</sup>^

Figure 7. (a) 2-D cross section and transmission line model of the NLC-based FPC-LWA. (b) Dispersion curves (β^

vs. f) of the fundamental TM leaky mode, when the NLC layer is biased at V<sup>∞</sup> (blue lines) and when it is unbiased at 0 V (yellow lines). Colors gradually shade from blue to yellow as Vb decreases from V<sup>∞</sup> to 0 V. (c) Radiation patterns considering the broadside operation (solid) and a scanned beam at the maximum pointing angle (dashed). The radiation patterns have been calculated by means of leaky-wave theory (black lines) and CST simulations (blue circles) for radiation

Terahertz Leaky-Wave Antennas Based on Metasurfaces and Tunable Materials

http://dx.doi.org/10.5772/intechopen.78939

<sup>z</sup> ≃ α^<sup>z</sup> is achieved for f ¼ 0:59

<sup>z</sup>, such

<sup>z</sup> and α^<sup>z</sup>

107

$$\mathbf{Y}\_0^{\rm TM} = \left(\hat{\boldsymbol{k}}\_{\rm x}^{\rm TM} \boldsymbol{\eta}\_0\right)^{-1} \boldsymbol{\varepsilon}\_{\rm zz}(V\_b)\_{\prime} \hat{\boldsymbol{k}}\_{\rm x}^{\rm TM} = \sqrt{\frac{\boldsymbol{\varepsilon}\_{\rm zz}(V\_b)}{\boldsymbol{\varepsilon}\_{\rm xx}(V\_b)} \left(\boldsymbol{\varepsilon}\_{\rm xx}(V\_b) - \hat{\boldsymbol{k}}\_{\rm z}^2\right)}\tag{12}$$

Since εyy is the only component of the NLC which does not depend on Vb, only the TM leaky modes will be affected by the application of the bias; thus, the following discussion will be limited to the study of the fundamental TM leaky mode<sup>9</sup> . The dispersion equation of the TM modes is therefore computed as in Eq. (6), i.e., by replacing the YPRS with the input admittance seen looking upward the interface, and by modeling the characteristic admittances and the normal wavenumbers of each NLC transmission line segments with those given in Eq. (12).

In order to give a proof of concept, we have considered a layout with N ¼ 8 layers (see [22] for this and other possible layouts), comprising three NLC cells and four alumina layers with thicknesses given by Eq. (9), assuming that ε<sup>r</sup> ¼ ε<sup>o</sup> as the thickness of the NLC cell. In this specific design, the choice of f <sup>0</sup> ¼ 0:59 THz is dictated by the thickness of commercially available alumina thin layers (127 μm), which exhibit a relative permittivity εr<sup>2</sup> ¼ 9 and a loss tangent of about tan δ≃ 0:01 at 0:59 THz [22]10.

We have then evaluated the dispersion curves of the relevant TM leaky mode in the range 0:5 � 0:75 THz for different bias states. As shown in Figure 7(b), the color of the curves gradually shades from yellow to blue when the bias Vb is changed from 0 (unbiased state) to a threshold voltage V<sup>∞</sup> (biased state), which can be accurately calculated through the method described in [22, 55]. For the proposed NLC cell, values below 20 V are sufficient to practically cover almost the whole switching range. In this simplified analysis, the relative permittivities are assumed to linearly vary with Vb. Thus, while the unbiased and biased states are always correctly predicted, the dynamic variation of β^ <sup>z</sup> and α^<sup>z</sup> for intermediate values of Vb could significantly change once the voltage dependence of ¯ ε is computed.

¯

<sup>9</sup> Note that the assumption of uniaxial crystal allows for decoupling the TM fields from the TE fields. This is not generally true when an anisotropic layer is at the interface with another medium, since in the most general case, more complicated networks are needed to describe its behavior [57, 58].

<sup>10</sup>We note here that, at <sup>f</sup> <sup>¼</sup> <sup>0</sup>:59 THz, the dielectric constant of Zeonor is the same, but the loss tangent is slightly higher, i.e., tan δ≃0:006 [22].

Figure 7. (a) 2-D cross section and transmission line model of the NLC-based FPC-LWA. (b) Dispersion curves (β^ <sup>z</sup> and α^<sup>z</sup> vs. f) of the fundamental TM leaky mode, when the NLC layer is biased at V<sup>∞</sup> (blue lines) and when it is unbiased at 0 V (yellow lines). Colors gradually shade from blue to yellow as Vb decreases from V<sup>∞</sup> to 0 V. (c) Radiation patterns considering the broadside operation (solid) and a scanned beam at the maximum pointing angle (dashed). The radiation patterns have been calculated by means of leaky-wave theory (black lines) and CST simulations (blue circles) for radiation at broadside and at the maximum pointing angle [22].

As expected, when Vb <sup>¼</sup> <sup>V</sup><sup>∞</sup> (blue curve), the splitting condition <sup>β</sup>^ <sup>z</sup> ≃ α^<sup>z</sup> is achieved for f ¼ 0:59 THz, which is the design frequency. Even more interestingly, once the frequency is fixed, e.g., at <sup>f</sup> <sup>¼</sup> <sup>0</sup>:59 THz, it is possible to change the value of the normalized phase constant <sup>β</sup>^ <sup>z</sup>, such that β^ <sup>z</sup> > α^<sup>z</sup> by simply lowering the bias voltage, whereas the value of the normalized attenuation constant α^<sup>z</sup> remains almost the same. As a consequence, the dispersion diagram of Figure 7(b) reveals the possibility to steer the beam with a quasi-constant beamwidth at a fixed frequency through bias voltage.

This is confirmed by the radiating patterns on the E-plane (the NLCs tunability affects only TM modes), calculated by means of leaky-wave theory (black lines) and CST full-wave simulations (blue circles), as shown in Figure 7(c). Results have been reported in Figure 7(c) considering radiation at broadside (biased status) and at the maximum pointing angle (unbiased status). It should be noted that the radiation efficiency of this device is limited to values around 40% due to the nonnegligible losses introduced by the three NLC cells. However, if one reduces the number of NLC cells, the beam-steering capability of the device is reduced as well [22]. The interested reader can find a comparison of different FPC-LWAs based on NLCs in [22].

#### 6. Conclusion

A circuit model (see Figure 7(a)) has been developed for the dispersion analysis of such a structure, taking into account the voltage dependence of the NLC layers. Hence, when no bias is applied (Vb ¼ 0 V), the NLC molecules are aligned along the horizontal z-axis (Figure 7(a)), i.e., εzzð Þ¼ 0 εe, promoted by a few tens nm-thin alignment layer, which does not affect the electromagnetic properties of the device. When a sufficiently large driving voltage (V∞) is applied across the LC layers, the NLC molecules reorient along the vertical x-axis, i.e.,

As a consequence, with reference to the transverse transmission line of Figure 7(a), the characteristic admittances and the normal wavenumbers of the NLC layers for both the TE and the TM polarizations (with respect to the xz-plane) are functions of Vb. Their expressions are given by,

<sup>x</sup> ¼

<sup>x</sup> ¼

Since εyy is the only component of the NLC which does not depend on Vb, only the TM leaky modes will be affected by the application of the bias; thus, the following discussion will be

modes is therefore computed as in Eq. (6), i.e., by replacing the YPRS with the input admittance seen looking upward the interface, and by modeling the characteristic admittances and the normal wavenumbers of each NLC transmission line segments with those given in Eq. (12).

In order to give a proof of concept, we have considered a layout with N ¼ 8 layers (see [22] for this and other possible layouts), comprising three NLC cells and four alumina layers with thicknesses given by Eq. (9), assuming that ε<sup>r</sup> ¼ ε<sup>o</sup> as the thickness of the NLC cell. In this specific design, the choice of f <sup>0</sup> ¼ 0:59 THz is dictated by the thickness of commercially available alumina thin layers (127 μm), which exhibit a relative permittivity εr<sup>2</sup> ¼ 9 and a loss

We have then evaluated the dispersion curves of the relevant TM leaky mode in the range 0:5 � 0:75 THz for different bias states. As shown in Figure 7(b), the color of the curves gradually shades from yellow to blue when the bias Vb is changed from 0 (unbiased state) to a threshold voltage V<sup>∞</sup> (biased state), which can be accurately calculated through the method described in [22, 55]. For the proposed NLC cell, values below 20 V are sufficient to practically cover almost the whole switching range. In this simplified analysis, the relative permittivities are assumed to linearly vary with Vb. Thus, while the unbiased and biased states are always

Note that the assumption of uniaxial crystal allows for decoupling the TM fields from the TE fields. This is not generally true when an anisotropic layer is at the interface with another medium, since in the most general case, more complicated

We note here that, at f ¼ 0:59 THz, the dielectric constant of Zeonor is the same, but the loss tangent is slightly higher,

ε ¯ is computed.

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>ε</sup>yy � ^k<sup>2</sup> z

εzzð Þ Vb εxxð Þ Vb

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� � <sup>s</sup>

<sup>ε</sup>xxð Þ� Vb ^<sup>k</sup>

, (11)

, (12)

2 z

. The dispersion equation of the TM

<sup>z</sup> and α^<sup>z</sup> for intermediate values of Vb could

<sup>x</sup> <sup>=</sup>η0, ^kTE

<sup>ε</sup>zzð Þ Vb , ^kTM

εzzð Þ¼ V<sup>∞</sup> εe, thus providing the maximum reconfigurability [55].

YTE <sup>0</sup> <sup>¼</sup> ^kTE

TM <sup>x</sup> η<sup>0</sup> � ��<sup>1</sup>

limited to the study of the fundamental TM leaky mode<sup>9</sup>

tangent of about tan δ≃ 0:01 at 0:59 THz [22]10.

correctly predicted, the dynamic variation of β^

networks are needed to describe its behavior [57, 58].

9

10

i.e., tan δ≃0:006 [22].

significantly change once the voltage dependence of ¯

YTM <sup>0</sup> <sup>¼</sup> ^<sup>k</sup>

106 Metamaterials and Metasurfaces

Fabry-Perot cavity leaky-wave antennas (FPC-LWAs) represent a valid solution for designing fully-planar, low-cost, high-gain antennas in the THz range. In this Chapter, starting from the conventional design of FPC-LWAs, the typical technological constraints of the submillimeter radiation have been thoroughly addressed to characterize the design criteria of THz FPC-LWAs. On this ground, three different THz-FPC LWA designs have been extensively discussed. First, an FPC-LWA based on a homogenized metasurface has been shown to produce high gain without requiring neither high complexity nor high fabrication cost. This has been made possible due to the employment of a fishnet unit-cell, a very interesting element for the impedance synthesis of metasurfaces in the THz range. This structure exhibits an excellent radiation performance, but it does not offer pattern reconfigurability. To this purpose, two extremely interesting materials, namely graphene and nematic liquid crystals, have been employed for the design of two different reconfigurable THz FPC-LWAs. A closer look to the design and the radiation performance of these two novel reconfigurable THz FPC-LWAs has revealed that the reconfigurable properties of these classes of antennas are paid at the expense of a reduced radiation efficiency and an increase of the fabrication cost and complexity. However, THz technology is a continuously growing field, and the current technological constraints are expected to relax in the upcoming years, thus paving the way for the realization of new interesting reconfigurable and efficient THz FPC-LWAs as well.

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Terahertz Leaky-Wave Antennas Based on Metasurfaces and Tunable Materials

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