Leaky wave lenses for spoof plasmon collimation

Anastasios H. Panaretos; Douglas H. Werner

doi:10.1364/OE.24.014654

1. Introduction

Spoof plasmons (SPs) are a class of surface waves (SWs) excited along corrugated metallic surfaces engineered to have specific properties [1–6]. Their name stems from the fact that they are considered the microwave and THz equivalent of their optical analogue, namely the surface plasmon polaritons (SPPs). The distinct difference between these two types of SWs is the realization of the reactance surface (RS) along which they are excited. In the case of SPPs the RS is the surface of a noble metal (which at optical frequencies is characterized by a negative dielectric permittivity) surrounded by a medium with positive dielectric permittivity (for instance free space). In the case of SPs the RS is created using metallic corrugated surfaces, which are characterized by a local surface impedance represented by the input impedance of a shorted transmission line with length less than $λ / 4$ such that the input impedance remains purely inductive. It should be noted that the reason for using metallic corrugated surfaces in order to excite plasmons in the microwave and THz frequency range is simply because the noble metals in these lower frequency ranges behave as good/perfect conductors. The SW guiding properties of corrugated surfaces have been extensively investigated by the electromagnetics community as early as the 1950’s [7–14]. However, the explosive advancements in plasmonics over the past decade have rekindled an interest in grooved surfaces that can support SP [15–28]. This is because SPs allow the techniques and methodologies originally developed for devices operating at optical wavelengths to be extended to lower frequencies.

This paper examines the radiation properties of SPs excited on planar transversely infinite corrugated surfaces. As mentioned previously, SPs are inherently guided waves meaning that they are characterized by purely real wavenumbers with values greater than that in free space. However, any SW can in principle be converted into radiating modes if the RS that supports it is periodically perturbed. This periodic perturbation excites spatial Floquet modes that modify the wavenumber of the SW and under certain conditions its real part may become smaller than that of free space, and thus allowing radiation to be achieved. As a matter of fact, in the case of the SPs discussed in [29], the authors demonstrated that a periodic sinusoidally modulated corrugated RS can efficiently couple SPs into radiating modes. In other words, it has been demonstrated that although SPs are inherently suited for guiding electromagnetic waves, their conversion to leaky wave modes is feasible provided that the supporting corrugated surface is structurally modified in a periodic fashion. This is realized by creating a corrugated structure with a sinusoidally modulated reactance surface (SMRS) profile. The electromagnetic properties of SMRSs were originally studied by Oliner and Hessel where, in their seminal paper [30], they theoretically demonstrated that such structures can support both guided and leaky wave modes. Recently, the interest in SMRSs has been revived because they provide a compact theoretical framework for the design of metasurfaces and leaky wave antennas (LWAs) [31–36].

In this paper we extend the concept of radiating SPs by presenting a leaky wave lens (LWL) operating in the microwave frequency range, which is capable of producing a collimated beam of electromagnetic energy in the Fresnel region. One of the first microwave LWL systems was described by Ohtera in [37]; the structure is based on a slotted rectangular waveguide which functions as a LWA uniformly radiating towards a fixed direction. Moreover, when this structure is bent in a concave fashion, then the radiation from the bent slot exhibits converging characteristics that create a focusing effect. The aforementioned approach, albeit successful, is based on the structural modification of the radiator while the wavenumber along the slot remains constant (provided the bend is not very sharp). A more flexible and scientifically interesting approach is the one where the LWL is realized using a leaky wave structure characterized by a tapered/non-uniform wavenumber. The effectiveness of this approach for the realization of microwave focusing systems has been successfully demonstrated both theoretically and experimentally [38–44].

The methodology described in this paper follows the aforementioned paradigm. In particular, we design a corrugated surface consisting of a non-uniform distribution of tooth lengths. This way we effectively realize a non-uniform RS characterized by a tapered propagation constant, where both the phase constant and the attenuation factor are spatial functions. These are judiciously engineered so that the SPs along the non-uniform corrugated surface are radiated according to a prescribed recipe that creates a focusing effect. For the proposed LWL, the desired non-uniform RS is realized via a non-uniform sinusoidally modulated reactance profile, where both the modulation factor and the mean reactance value are spatial functions. As mentioned previously, a uniform SMRS directs the radiated electromagnetic waves towards a constant radiation angle. Therefore, by introducing a spatial variation across the SMRS, we can independently control the radiating properties (amplitude and phase) of each infinitesimal segment across the RS. Loosely speaking, the proposed methodology is equivalent to the principles of operation of an antenna array with non-uniform amplitude and phase distribution.

Finally, it should be emphasized that there was an important purpose behind the choice to realize the RS through a non-uniform sinusoidally modulated reactance profile. As will be described in the next section, SMRSs offer the unique advantage that their dispersion relation is known analytically [30]. This is a feature that greatly simplifies the proposed LWL design methodology since the dispersion properties of such surfaces can be computed numerically by simply solving the corresponding dispersion relation. This way, any desired wavenumber can be in principle mapped to the profile of a specific SMRS. Consequently, if we know the tapered wavenumber required for the realization of a LWL, then this can be trivially translated into properties of a non-uniform SMRS which can be further mapped onto a corrugated surface with a non-uniform distribution of tooth lengths. In the following sections we provide a complete and thorough theoretical analysis of the various design steps required for the realization of a microwave LWL that can collimate SPs in the Fresnel region. Throughout this paper the $e^{j ω t}$ time convention has been adopted.

2. Preliminary definitions

A. Planar corrugated surfaces

A planar corrugated surface (shown in Fig. 1(a)) consists of a ground-plane with a periodic arrangement of transversely infinite vertical metallic teeth defined along its length. The input impedance of such a corrugated surface is given by [45]

\vec{Z} = \frac{E_{y}}{H_{x}} = j Z_{0} \frac{G}{G + T} \tan (k_{0} l) = j X_{s}

Note here that the preceding expression corresponds to the input impedance of a shorted anisotropic magneto-dielectric slab with length l and relative dielectric permittivity and magnetic permeability defined as

ε_{r}^{y y} = \sqrt{(G + T) / T}

and

\sqrt{ε_{r}^{y y} μ_{r}^{x x}} = 1

. In this study we set the frequency of operation to 9 GHz, T = 0.5 mm, and G + T = 2 mm. Given these design specifications, the normalized input reactance of the corrugated surface is plotted in Fig. 1(b) as a function of the groove depth. It can be seen that the electromagnetic response of the surface remains inductive for teeth lengths up to about 8 mm and then it becomes capacitive. This study is solely concerned with inductive corrugated surfaces; therefore the length of the teeth in the structures to be examined in the following sections never exceeds 8 mm. Such structures naturally support SWs (in this case SPs), which can be trivially proven after computing the wavenumber

k_{y}

along the corrugated surface. This requires that the transverse resonance method (TRM) be applied at the interface defined between free space and the corrugated surface. From Fig. 1(a), it follows that

\overset{\leftarrow}{Z} + \vec{Z} = 0 \Rightarrow j Z_{0} \frac{G}{G + T} \tan (k_{0} l) + \frac{k_{z}}{ω ε_{0}} = 0

where

\overset{\leftarrow}{Z} = \frac{k_{z}}{ω ε_{0}}

denotes the TM input impedance of free space.

Fig. 1 (a) Planar corrugated surface. (b) Normalized input impedance of a corrugated surface with T = 0.5 mm and G + T = 2 mm at 9 GHz as a function of tooth length. (c) Transverse wavenumber along the corrugated surface from (b) as a function of tooth length.

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Note here that since we are examining the possibility of SW guidance along the corrugated surface, k_y should be real and greater than k₀, therefore k_z is purely imaginary, or $k_{z} = - j \sqrt{k_{y}^{2} - k_{0}^{2}}$ . Figure 1(c) displays the wavenumber of the SP as a function of the groove length. Evidently, for all the groove lengths for which the surface remains inductive, k_y is greater than k₀.

B. Sinusoidally modulated reactance surfaces

Sinusoidally modulated reactance surfaces (SMRSs) are characterized by the following periodic impedance profile

Z (y) = j X_{u} [1 + M \sin (\frac{2 π y}{p})]

In the preceding expression p represents the period of the reactance profile, and M is the modulation factor which can assume any value less than one. Finally, X_u is the un-modulated (M = 0) or the mean surface reactance. The electromagnetic properties of such configurations were first studied by Oliner and Hessel in [30]. The most appealing attribute of such surfaces is that their dispersion properties can be described by the following recursive relation:

D (n, 0) - \frac{1}{D (n, + 1) - \frac{1}{D (n, + 2) - \dots}} - \frac{1}{D (n, - 1) - \frac{1}{D (n, - 2) - \dots}} = 0

where

D (n, m) \equiv \frac{2}{M} [1 - \frac{j}{X_{u}} \sqrt{1 - {(\frac{γ}{k_{0}} + \frac{2 π (n - m)}{k_{0} p})}^{2}}]

In Eq. (5),

γ

is the complex propagation constant along the SMRS. Before proceeding with the remainder of our analysis, the following remarks should be made regarding the dispersion properties of a SMRS. An un-modulated SMRS is characterized by a uniform impedance profile

Z (y) = j X_{u}

which supports SWs with a wavenumber

β_{u} > k_{0}

. For any M > 0 the SMRS becomes a periodic structure which according to Floquet theory exhibits a modified wavenumber given by

κ = β_{u} + \frac{2 π n}{p} - j α = β_{n} - j α

where

n

is an integer number indicating the spatial Floquet mode. The imaginary part of Eq. (6) is referred to as the leakage rate. The importance of the periodicity and the corresponding Floquet modes is that for some negative n the condition

β_{u} + \frac{2 π n}{p} < k_{0}

may be satisfied, which indicates the possibility that the SWs guided by the SMRS can be coupled into radiating modes. It should be emphasized that all of the previous analysis is applicable for any periodic RS without them having to necessarily be sinusoidally modulated. Therefore, the question becomes: is there any advantage in the periodic RS being sinusoidally modulated? Stated differently, why should a SMRS be used instead of any other periodic RS configuration? The answer lies in the expression given in Eq. (4): the dispersion properties of a SMRS can be trivially generated by simply solving the aforementioned dispersion relation for any desired combination (X_u, M). Hence, this totally eliminates the need to numerically solve the corresponding electromagnetic eigenmode problem.

The dispersion properties of three SMRS configurations are included in Fig. 2. In particular, the three cases examined correspond to SMRSs characterized by the following (X_u/Z₀, M) combinations: (1, 0.25), (2, 0.25), and (2, 0.5). It should be noted here that Eq. (4) was solved using Davidenko’s method after setting n = 0 and m = 15. Figure 2(a) shows the real part $Re {κ} = β_{- 1}$ of the wavenumber that corresponds to the n = −1 Floquet mode. Also, the imaginary part $Im {κ} = α$ of the wavenumber along a SMRS is plotted in Fig. 3(b) for the three cases examined. The following conclusions can be drawn from the dispersion diagrams: (a) there is a frequency range where $Re {κ} = β_{- 1}$ lies within the triangle defined by the light line clearly indicating that, for certain (X_u, M) combinations, the SMRSs can support radiating modes. (b) By modifying X_u and M we can independently control the leakage rate and the real part of the propagation constant. In other words, by judiciously choosing X_u and M, the dispersion properties of an SMRS can, in principle, be synthesized as desired. It should be noted here that lower modes such the n = −2 may also be radiating, however the most intensely radiating mode is the n = −1. Finally, the angle at which the potential radiating mode n directs its radiation is determined by

Fig. 2 Dispersion properties of three different realizations of a sinusoidally modulated reactance surface. (a) Propagation constant associated with the n = −1 Floquet mode. (b) Attenuation factor.

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Fig. 3 2D aperture antenna exhibiting focusing radiation properties.

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k_{0} \sin (θ_{s}) = β_{u} + \frac{2 n π}{p}

3. Theoretical aspects of a 2D collimating aperture antenna

Let us consider the 2D antenna configuration shown in Fig. 3, which is comprised of an infinite (in the transverse direction, x-axis) magnetic current distribution that lies along the y-axis with its length equal to L. A non-uniform current distribution $K = \hat{x} K (y)$ is assumed to exist along the aperture of the antenna

K (y) = | K (y) | e^{- j ψ (y)}

which creates a TE_x (E_y, E_z, H_x) electromagnetic field distribution. In order for this field configuration to function as a lens, it is required that each elemental magnetic current

K (y + d y) d y

across the aperture emits at a different angle

θ = θ (y)

in such a way that the emanating cylindrical electromagnetic waves will collimate at some focal point F(y_F, z_F), as graphically illustrated in Fig. 3. Our approach is based on realizing the non-uniformly radiating aperture through a tapered leaky wave antenna which is characterized by a spatially-dependent complex wavenumber, or

γ (y) = β (y) - j α (y)

In particular, a non-uniformly modulated corrugated RS will be devised which is capable of radiating SPs at different angles so that the desired focusing effects can be achieved. The phase variation

ψ (y)

along the antenna is related to the local phase constant

β (y)

as [14]

ψ (y) = - \int_{0}^{y} β (y^{'}) d y^{'} \Rightarrow β (y) = - \frac{d ψ (y)}{d y}

where

y \in [0, L]

. Since

β (y)

represents the phase constant of a fast travelling wave, it is related to the free space wavenumber as

β (y) = \sin θ (y) k_{0}

. Moreover, based on the geometric interpretation of the term

\sin θ (y)

(shown in Fig. 3), the phase constant can be written as [38,46]

\sin θ (y) = \frac{β (y)}{k_{0}} = \frac{y_{F} - y}{\sqrt{{(y_{F} - y)}^{2} + z_{F}^{2}}}

In order to complete the description of the wavenumber in Eq. (9), the non-uniform leakage rate

α (y)

must be determined. For a LWA, this is defined as [47]

α (y) = \frac{1}{2} \frac{{| K (y) |}^{2}}{\frac{1}{η} \int_{0}^{L} {| K (y^{'}) |}^{2} d y^{'} - \int_{0}^{y} {| K (y^{'}) |}^{2} d y^{'}}

where

η

is the desired radiation efficiency of the radiating structure (

η

= power radiated/power supplied).

Although the amplitude of the current distribution $| K (y) |$ can be defined arbitrarily (uniform, cosine on a pedestal, etc.), it is usually specified according to the desired far-field radiation pattern. In this study, and without loss of generality, a uniform far-field pattern (uniformly radiated power per unit angle) is assumed for $θ \in [θ_{2}, θ_{1}]$ , where the two angles are defined in Fig. 3. Based on this assumption, and using reciprocity arguments, the focal point creates a cylindrical wave that propagates electromagnetic power according to the $1 / ρ$ law (with $ρ$ being the distance between the focal point and a spatial point $(y, z)$ ). Therefore, the current amplitude along the antenna aperture is described by

| K (y) | = \frac{1}{{[{(y_{F} - y)}^{2} + z_{F}^{2}]}^{\frac{1}{4}}} = \frac{1}{\sqrt{ρ}}

with

y \in [0, L]

. If Eq. (13) is inserted into Eq. (12), it becomes trivial to show that the corresponding leakage rate is given by

α (y) = \frac{1}{2 ρ} {[(\frac{1}{η} - 1) \sin h^{- 1} (\frac{y_{F}}{z_{F}}) + \frac{1}{η} \sin h^{- 1} (\frac{L - y_{F}}{z_{F}}) - \sin h^{- 1} (\frac{y - y_{F}}{z_{F}})]}^{- 1}

At this point the design equations for determining the non-uniform wavenumber associated with the radiating aperture have been completely defined.

4. SP leaky wave lens design methodology

A. Devising a non-uniform corrugated reactance surface

The first step in the design of a LWL is the extraction of the corresponding dispersion map. This requires that the values of X_u and M be simultaneously calculated within properly chosen ranges. Next, the dispersion relation in Eq. (4) is solved, and finally the corresponding propagation constant pairs $(β_{- 1}, α)$ are extracted. In this study we let $10^{- 3} \leq X_{s} / Z_{0} \leq 2.5$ and $10^{- 3} \leq M \leq 1$ , where the resulting dispersion maps are shown in Figs. 4(a) and 4(b). It should be noted here that since we are interested only in the radiating modes of the SMRS, the useful range of the (X_u/Z₀, M) values in Fig. 4 are those for which the condition $β_{- 1} < k_{0}$ is satisfied.

Fig. 4 Dispersion properties of a SMRS at 9 GHz with p = 30 mm as a function of the modulation factor M and the normalized mean reactance X_u/Z₀. (a) Propagation constant of the n = −1 Floquet mode. (b) Attenuation factor.

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In this example we defined the length of the lens as $L = 10 p$ and target the focal point to be at $F (11 p, 10 p)$ . Note here that in this study p = 30 mm, which is approximately equal to one wavelength at 9 GHz. If we set the radiation efficiency $η = 0.8$ , then the required non-uniform phase constant and leakage rate can be determined by using Eqs. (11) and (14). The variation of these two quantities along the aperture is shown plotted in Fig. 5(a). Note that the value of the radiation efficiency was chosen so that the resulting values of the leakage rate are with the range shown in Fig. 4(b). Notice that the propagation constant monotonically decreases or equivalently the leaky mode accelerates, which is a direct consequence of the fact that for the problem under consideration we have $θ_{2} > θ_{1}$ .

Fig. 5 (a) Propagation constant and attenuation factor variation along the LWL. (b) Radiating angle variation of the leaky wave mode along the LWL. (c) Ray-optics representation of the LWL radiation and focusing effect.

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Figure 5(b) shows the range of the corresponding radiation angles, while Fig. 5(c) presents a ray optics illustration of the lens functionality. In the next step, the values of $β (y)$ and $α (y)$ are mapped to a non-uniform SMRS with an impedance profile given by

Z (y) = j X_{u} (y) [1 + M (y) \sin (\frac{2 π y}{p})]

The distinct difference between Eq. (15) and Eq. (3) is that the latter assumes spatial dependency for both the modulation factor

M (y)

and the mean surface reactance

X_{u} (y)

. Therefore, the objective becomes to map the pairs

(β (y), α (y))

shown in Fig. 5(a) onto the properties of a non-uniform SMRS represented as

(X_{u} (y), M (y))

, for all

y \in [0, L]

. Towards this goal we employ a 2D interpolation scheme based on the dispersion map shown in Fig. 4. In particular, within one period of the SMRS there exist 15 corrugations, therefore across the length of the lens we define the sequence of points

y_{n} = n (G + T)

with

n = 0, ..., 149

. Next, the parameter pairs

(β (y_{n}), α (y_{n}))

are evaluated at each of the 150 points as described in the previous section (Fig. 5(a)), and following the application of the 2D interpolation scheme, the corresponding set of values

(X_{s} (y_{n}), M (y_{n}))

are extracted. After inserting these values into Eq. (15), the non-uniform normalized reactance

X_{s} (y) / Z_{0}

is obtained, which is depicted in Fig. 6(a). In the same graph, the dashed line represents the spatially dependent mean normalized reactance

X_{u} (y) / Z_{0}

. Finally, the dotted line indicates the envelope of the non-uniform sinusoidally varying reactance, where again it is evident that the amplitude of the sinusoid

M (y)

follows a non-uniform profile.

Fig. 6 (a) Non-uniform reactance profile along the lens. WG denotes a non-radiating waveguide section. LWA denotes a uniform leaky wave antenna section. LWL denotes the actual leaky wave lens. (b) Teeth length variation along the lens system.

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According to the reactance plot shown in Fig. 6(a), the lens system is completed with a waveguide (WG) and a uniform leaky wave antenna (LWA) section placed on the left and right of the LWL, respectively. The length of the WG section is equal to one sinusoidal period $p$ , and it is realized by an unmodulated corrugated surface (M = 0). This length is set equal to that of the first corrugation of the LWL section. In other words, the WG section corresponds to a SMRS with properties $(X_{s}, M) \equiv (X_{s} (y_{0}), 0)$ for all $y \in [- p, 0]$ . The purpose of the WG section is to ensure the smooth transition of the SPs, from the source to the lens sections of the device.

We note that this WG section could also have been used to terminate the LWL. However, numerical experimentation revealed that this type of termination resulted in strong reflections, with detrimental effects in the overall performance of the system. For this reason, a uniform leaky wave antenna (LWA) section was used instead, with a length equal to $3 p$ .

For this section, as evident in Fig. 6(a), both $M$ and $X_{s}$ are constant; their values are set equal to those of the last corrugation in the LWL section, or $(X_{s}, M) \equiv (X_{s} (y_{149}), M (y_{149}))$ for all $y \in [L, L + 3 p]$ . With the addition of the two sections, the total length of the device is equal to $14 p$ , which consists of 210 total corrugations. The length of each tooth can be trivially extracted by interpolating the response shown in Fig. 1(b), while the resulting length variation in the teeth across the LWL is depicted in Fig. 6(b). Note that the length of all teeth in the device is less than 8 mm, signifying the inductive electromagnetic properties of the corrugated surface [see Fig. 1(c)].

B. Lens performance

The LWL system described in the previous subsection was modeled using the ANSYS-HFSS finite element based full wave solver. The corresponding 3D CAD model is shown in Fig. 3(a). The size of the computational domain in the y-z plane is 23λ × 18λ (at 9 GHz), while in the transverse direction (x-axis) the length of the domain is 0.03λ. In order to create the required 2D electromagnetic field configuration the two faces of the computational domain perpendicular to the transverse direction are defined as perfect magnetic conductors (PMCs). The LWL is assumed to reside on an infinite ground plane, therefore the face of the computational domain associated with the base of the structure is defined as a perfect electric conductor (PEC). Radiation boundary conditions are defined on the remaining three faces of the computational domain. In Fig. 7(a), Port #1 denotes the source of the lens system while Port #2 denotes the termination load. The geometrical details of these ports are shown in Figs. 7(b) and 7(c), respectively. For Port #1, a tapered parallel plate TEM waveguide was utilized in order to minimize reflections at the source, while for Port #2 a uniform parallel plate TEM waveguide was employed.

Fig. 7 (a) CAD model of the computational domain defined to full wave simulate the lens performance. (b) Detailed view of Port #1. (c) Detailed view of Port #2.

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Figure 8 summarizes the performance of the lens system. In particular, Fig. 8(a) shows the variation in the magnitude of the real part of the electric field along the y-z plane. It can be clearly seen that the wave-fronts emanating from the lens exhibit a concave profile, gradually converging at the focal point. In this figure, as well as in Fig. 8(b), the first two vertical white lines (moving from left to right) indicate the LWL section. The intersection of the third white line with the vertical line indicates the focal point. Figure 8(b) shows the complex amplitude distribution of the electric field, where a high intensity region is observed that clearly indicates the collimation of the electromagnetic field to a focal point. The focusing accuracy is demonstrated in Figs. 8(c) and 8(d), where the electric field intensity is plotted along two cuts parallel to the major axes that cross the focal point. The location of the latter is indicated in the figures by the red vertical line. It is evident that the proposed LWL system can very accurately focus SPs at the desired point. The observed minor discrepancies are due to the focal drift that typically characterizes such systems [48]. Figure 8(e) shows the reflection $S_{11}$ and transmission $S_{21}$ performance for this LWL (the parameters ${| S_{11} |}^{2}$ and ${| S_{21} |}^{2}$ indicate respectively the ratio of the power reflected back to Port #1 and the ratio of the power delivered to Port #2 divided by the power supplied to Port #1). The $S_{11}$ response indicates that the LWL is very well matched to the source; from the $S_{21}$ response it can be concluded that a minute portion of the supplied power is delivered to Port #2 indicating that most of it has been radiated (before it reaches Port #2). Finally, Fig. 8(f) reports the normalized far-field radiation intensity of the LWL. The vertical green lines indicate the angle interval $[θ_{1}, θ_{2}] \equiv [6 °, 48 °]$ defined in Figs. 3 and 5(b). Within this interval the full-wave predictions are in excellent agreement with the analytical results, while some discrepancies are observed outside the aforementioned interval. The origins of these discrepancies are the following: (a) the minor lobes exhibited in the full-wave predictions for $θ < 0$ are due to radiation from the $n = - 2$ leaky mode. (b) The two TEM waveguides partly radiate, therefore they interfere with the far field pattern of the LWL. (c) The RS of the LWL (top level of the teeth) is elevated with respect to the PEC ground, therefore there is an ensuing imaging effect that has an impact on the far-field pattern. An important clarification for the analytical calculation of the radiation intensity is provided in the Appendix. Finally, Fig. 9 summarizes the performance of a lens system where, in this case, the focal point is defined closer to the lens at $F (11 p, 5 p)$ . All the geometrical characteristics of the lens are the same as those that were examined in the previous case. The only difference lies in the tooth length distribution where, following the methodology described in the previous sections, it has been modified so as to achieve the desired focusing response. Evidently, the performance of the system is quite good.

Fig. 8 (a) Electric field distribution. (b) Distribution of the complex magnitude of the electric field. (c) Complex amplitude of the electric field along a horizontal cut that crosses through the focal point. (d) Complex amplitude of the electric field along a vertical cut that crosses through the focal point. (e) Reflection and Transmission performance. (f) Normalized far-field intensity.

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Fig. 9 (a) Electric field distribution. (b) Distribution of the complex magnitude of the electric field. (c) Complex amplitude of the electric field along a horizontal cut that crosses through the focal point. (d) Complex amplitude of the electric field along a vertical cut that crosses through the focal point. (e) Reflection and Transmission performance. (f) Normalized far-field intensity.

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5. Conclusions

We have theoretically demonstrated the feasibility of collimating SPs using a non-uniform sinusoidally modulated corrugated RS. The proposed methodology relies on the fact that SPs are SWs. Therefore if the RS that supports them is periodically perturbed, then they can be effectively converted to leaky modes. Moreover, if this perturbation is designed so that a tapered wavenumber is created along the corrugated surface, then both the radiation angle and the leakage rate of the SPs can be locally controlled. This enables a radiation beam to be formed that exhibits converging properties. The realization of the desired tapered wavenumber is achieved by mapping the reactance profile of a non-uniform SMRS onto the corrugated surface. The performance of two different lenses was examined and the theoretically expected response of the proposed leaky wave lens system was numerically verified.

Appendix

The radiation intensity of the lens is calculated by computing the far-field integral of the current along the radiating aperture, or

U (θ) = G (θ) \int_{0}^{L} | K (y^{'}) | e^{j ψ (y^{'})} e^{- j k y^{'} \sin (θ)} d y^{'}

In the preceding expression

G (θ)

can be set equal to 1 since it represents the radiation pattern of an infinite magnetic line source above an infinite PEC ground plane. With respect to Fig. 6(a), this integration would normally involve only the LWL section of the lens system. However, for the cases examined here the corrugated surface, apart from the LWL section, consists of a WG and a LWA section. Although the WG section does not radiate, the LWA does, therefore the expression in Eq. (16) requires modification so that the integration involves both the LWL and the LWA sections. A ray optics representation of the radiation from both the LWL and LWA sections is shown in Fig. 10(a). This requires that the amplitude

| K (y) |

and the phase

ψ (y)

of the current along the LWA section be determined.

Fig. 10 (a) Ray-optics representation of the radiation produced by the LWL and LWA sections. (b) Propagation constant and attenuation factor variation along the LWL and LWA sections. (c) Current amplitude along the LWL and LWA sections. (d) Current phase along the LWL and LWA sections.

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As mentioned in our analysis, the characteristics of the SMRS for the LWA section are the same as that of the last corrugation of the LWL portion, or $(X_{s}, M) \equiv (X_{s} (y_{149}), M (y_{149}))$ for all $y \in [L, L + 3 p]$ . Therefore, the propagation constant and attenuation factor shown in Fig. 5(a) must be modified as shown in Fig. 10(b). Given this modification, the amplitude of the current along the LWA section is computed using the following expression:

{| K (y) |}^{2} = α (y) e^{- 2 \int_{0}^{y} α (y^{'}) d y^{'}}

while the phase is computed from Eq. (10). The resulting distributions are shown in Fig. 10(c) and 10(d). Consequently, for the calculation of the radiation intensity, the integration in Eq. (16) should be from 0 to

L + 3 p

, with the associated integrand as shown in Fig. 10(b).

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Leaky wave lenses for spoof plasmon collimation

Abstract

1. Introduction

2. Preliminary definitions

A. Planar corrugated surfaces

B. Sinusoidally modulated reactance surfaces

3. Theoretical aspects of a 2D collimating aperture antenna

4. SP leaky wave lens design methodology

A. Devising a non-uniform corrugated reactance surface

B. Lens performance

5. Conclusions

Appendix

References and links

Cited By

Figures (10)

Equations (17)

Optics Express