1 Introduction

Consumer-oriented applications, such as next-generation wireless networks (5G), device-to-device communications, wireless personal area networks (WPANs), and chip-to-chip communications [1,2,3], demand cost-affordable terahertz and sub-terahertz power generators [4, 5]. Typically, low-noise [6] room-temperature [7] continuous wave sources are limited on applications because of the insufficient achieved power level for commercial use. A 300-GHz link with data rates up to 48 Gbps is reported in [8]. Due to spectrum availability at frequencies between 100 GHz and 1 THz, simple modulation schemes, such as On-Off Keying, are feasible, which allows high-throughput low-latency point-to-point wireless links. Other applications, such as spectroscopy, and security at millimeter and sub-millimeter wave frequencies became attractive topics in recent years [7, 9, 10].

Photomixer-based terahertz sources use two lasers with different frequencies to generate a photocurrent in the range of millimeter and sub-millimeter wave frequencies [7, 11], as sketched in Fig. 1. The photocurrent is injected into an antenna structured around the semiconductor device. For taking advantage of the potential wideband of optoelectronic terahertz devices [12], UWB antennas, such as log-periodic or log-spiral ones, are chosen. The photomixer and the lithographically structured antenna conform the antenna emitter (AE).

Fig. 1
figure 1

Scheme of a photomixing-based terahertz source (antenna emitter). The two laser beams (orange) are focused onto the small semiconductor device placed between the arms of a log-periodic antenna (pear). The generated THz radiation (red) is radiated into the substrate due to its high permittivity. Typically, a hyper-hemispherical lens (gray) is used for increasing the amount of power extracted from the substrate

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Due to the high permittivity (\(\varepsilon _{_{\text {SC}}}=9.6\)) and electrically large thickness (350 μ m >> λ) of the substrates typically used (for example InP for 1550 nm lasers), most of the generated power is dissipated in the substrate because of total internal reflections. The emission efficiency η for a lossless substrate defined as

$$ \eta \triangleq \frac{P^{^{\text{rad}}}_{_{\text{THz}}}|_{_{Z>0}}}{P^{^{\text{gen}}}_{_{\text{THz}}}} = \frac{P^{^{\text{rad}}}_{_{\text{THz}}}|_{_{Z>0}}}{P^{^{\text{rad}}}_{_{\text{THz}}}|_{_{Z>0}}+P^{^{\text{rad}}}_{_{\text{THz}}}|_{_{Z<0}}+P^{^{\text{tir}}}_{_{\text{THz}}}} $$
(1)

remains low. \(P^{^{\text {gen}}}_{_{\text {THz}}}\) is the total amount of generated THz power; \(P^{^{\text {rad}}}_{_{\text {THz}}}|_{_{Z>0}}\) and \(P^{^{\text {rad}}}_{_{\text {THz}}}|_{_{Z<0}}\) are the amount of power radiated in half-semispace towards Z > 0 and Z < 0, respectively, and \(P^{^{\text {tir}}}_{_{\text {THz}}}\) is the power that does not leave the substrate due to total internal reflection at the substrate-air interface. According to our full-wave simulations, the efficiency can be as low as 29.6% (around 1–2% according to our experimental work) for a plain AE (i.e., without the lens). A silicon lens is usually placed at the bottom of the substrate for increasing the radiated power to Z > 0. It can be seen from [13] that the lens selection has a high impact on the performance.

This paper considers the dielectric rod waveguide (DRW) antennas as a cost-affordable and more compact alternative to dielectric lenses that can be easily integrated in handheld devices. A sketch of an AE assembled to such a dielectric antenna is shown in Fig. 2. When coupled to an AE, DRW antennas are limited in frequency by the existence of higher-order modes in the rod [14].

Fig. 2
figure 2

Dielectric antenna of length LTAPER and thickness T attached to an electrically large substrate layer STHK thick. The inset displays a 3D view of the AE attached to the DRW antenna

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For avoiding such a limitation, the use of a dielectric planar lens embedded into the DRW is proposed [15].

The manuscript is organized as follows. In Section 2, a brief description of DRW antenna is given. Both simulation and measurement results are provided. Section 3 introduces the planar dielectric lenses embedded in a DRW. Both hyper-hemispherical approximation and elliptic lens integration in a DRW are theoretically discussed. A DRW antenna with an embedded lens is presented in Section 4. Full-wave simulation results are provided for estimating the frequency range improvement. Both hyper-hemispherical and elliptic lens are considered. Finally, a proof-of-concept prototype is presented in Section 5. A polypropylene dielectric rod waveguide antenna is manufactured and characterized. The original band is extended from 6–12 GHz to 6–30 GHz.

2 Dielectric Rod Waveguide Antennas

Figure 3 shows a packaged and pigtailed AE. It is composed of a three-period n-i-pn-i-p photomixer [16] placed in the gap of a logarithmic-periodic broadband antenna. Then, it is assembled with a high resistivity (HR) silicon (Si) DRW antenna designed for 150 GHz. The photomixer substrate is STHK = 350 μ m thick. The taper length LTAPER is 8 mm and WROD = 0.5 mm. The rod is cut out of a silicon wafer of 0.5 mm thickness (T).

Fig. 3
figure 3

Photomixer-based terahertz source. The inset shows a closer view of the AE assembled to the DRW antenna before packaging

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The normalized radiation pattern is measured with a Golay cell, as sketched in Fig. 4a. A laptop controls (black lines) both the motion controller (MC) and the tunable lasers. It is further connected to a Standford Research Systems SR830 DSP lock-in amplifier. Both lasers (blue lines) are combined in a polarization maintaining (PM) 3 dB fiber splitter. The optical pump is chopped and fed to the integrated photomixer plus DRW antenna package (device under test, DUT). The chopper reference signal is sent to the lock-in amplifier as well as the Golay cell output (green lines).

Fig. 4
figure 4

Sketch (a) and photograph (b) of the measurement setup

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Both E-plane (XZ plane, black) and H-plane (YZ plane, red) total E-field (in dB) are displayed in Fig. 5. The measured radiation through 𝜃 = ± 90∘ can be explained as the THz signal not coupled into the rod that is scattered into the substrate (\(P^{^{\text {tir}}}_{_{\text {THz}}}\)).

Fig. 5
figure 5

Measured radiation pattern at 150 GHz. Both E (XZ plane, black) and H (YZ plane, red) planes are shown. 𝜃 is swept between ± 90∘. Radiation pattern is normalized, in dB

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According to our simulations, 77% of the generated power is collected into one single main lobe featuring a maximum at 𝜃 = 0 at 150 GHz. This is a significant improvement in the performance comparing with the plain AE without any lens or DRW antenna. For comparison purposes, the AE is simulated with two different hyper-hemispheric silicon lenses: a small one of 0.75 mm radius and hyper-hemispherity H = 217 μ m; and a larger one of 3 mm radius and H = 870 μ m. Table 1 summarizes the simulated efficiencies at 150 GHz.

Table 1 Comparative of efficiencies
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Despite of their lower efficiency, DRW antennas are an appealing alternative when considering other factors. Silicon lenses can be in the order of hundreds of euros while rods can be made for less than one euro. Moreover, DRW antennas are electrically compact, which makes them appealing for array configurations [17], since electrically small lenses are less efficient (see Table 1).

The bandwidth is limited by the presence of higher-order modes inside the DRW, since they degrade the radiation pattern [14]. This happens despite of the radiation taper, which is able to cover a higher bandwidth. An AE emits spherical waves inside an electrically thick substrate, which creates higher-order modes inside the rod because of the internal reflections, as illustrated in Fig. 6, where the E-field amplitude in the XZ plane is simulated at 150 GHz (Fig. 6a), 200 GHz (Fig. 6b), and 300 GHz (Fig. 6c). According to our simulations, a single-mode regime is achieved as long as \(W_{\text {ROD}}<\lambda _{\text {DRW}}\), being λDRW the wavelength of the fundamental mode inside the DRW (Fig. 6c). As reported in [18], the DRW fundamental mode \(E^{^{x}}_{_{11}}\) does not have a cutoff frequency, but for lower frequencies, most of the power goes outside the rod. In such case, the DRW antenna is unable to extract the power from the AE substrate, which defines an effective cutoff frequency for this dielectric structure.

Fig. 6
figure 6

Simulated E-field distribution on XZ plane at 150 GHz (a), 200 GHz (b), and 300 GHz (c). Radiation zones are detailed

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The dispersion chart for TEM modes can be estimated following the procedure explained in [18]. Figure 7 shows the modal chart for a DRW with a cross-sectional area \(T \times W_{\text {ROD}} = 0.5 \times 0.5 \text {mm}^2\). Due to its square cross section, both \(E^{^{y}}_{_{pq}}\) and \(E^{^{x}}_{_{pq}}\) family modes have exact axial propagation constant kz. Figure 7 is generated by giving values to p and q (p,q= 1,2, ...). The ordinate corresponds to

$$ \frac{k^{2}_{\mathrm{z}} - {k^{2}_{0}}}{k^{2}_{_{\text{SC}}}-{k^{2}_{0}}} $$
(2)

which takes values from 0 to 1, where k0 = 2π/λ0 and \(k_{_{\text {SC}}}\) the propagation constant inside the semiconductor medium. The abscissa is a function of the wavelength λ0 that takes into account the DRW thickness and permittivity.

Fig. 7
figure 7

Normalized axial propagation constant kz for a DRW with a cross-sectional area T × WROD = 0.5 × 0.5mm2

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$$ \frac{2\cdot T}{\lambda_{0}}\sqrt{\varepsilon_{_{\text{SC}}}-1} $$
(3)

At f = 150 GHz (abscissa value of 1.65), a negligible amount of power is expected in higher-order modes, as confirmed from the simulated E-field distributions (Fig. 6a).

The taper of the rod ensures a smooth transition in order to radiate into air. The width of the rod, WROD(z), where the DRW antenna starts radiating strongly depends on the excited mode. When multiple modes with different critical width \(W_{\text {ROD}}(z)|_{min}\) are present in the DRW antenna, many radiation zones exist (Fig. 6c). In this scenario, a radiation pattern with a poor side-lobe level (SLL) is achieved. For some frequencies, there is destructive interference, which leads to a null in the Z-axis, as illustrated in Fig. 8 for a polypropylene DRW antenna (Fig. 9) designed to operate in the single-mode in the 6–12-GHz band [15]. Measurement was performed at 25 GHz. For exciting higher-order modes, the rectangular waveguide (RWG) can be directly attached to the rod without matching taper [19].

Fig. 8
figure 8

Measured radiation pattern of the polypropylene DRW antenna at 25 GHz. Higher-order modes are present in the rod. Interference of the modes results in a null in the Z-axis, and an SLL of 4 dB. Both E-plane (red) and H-plane (black) co-polar (solid) and cross-polar (dashed) components are shown. 𝜃 is swept between ± 90∘. Radiation pattern is normalized, in dB

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Fig. 9
figure 9

Manufactured polypropylene DRW antenna (lower inset) as a proof-of-concept in the 6–12-GHz band. It was characterized at 25 GHz. LTAPER is 190 mm, WROD 24 mm, and thickness 5 mm. An HP R281A waveguide-to-coax adapter is used as a DRW antenna feeder (upper inset)

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3 Ultra-Wideband DRW Antennas

An AE creates spherical waves inside the DRW for an electrically large width WROD. Total internal refection along the radiation taper lead to higher-order modes, as shown in Figs. 6c and 8. The integration of a planar lens inside the rod is proposed for rectifying the generated wavefront of the spherical waves (Fig. 10) and, therefore, avoiding the presence of non-TEM modes. It is worth to mention that such modes with non-zero Ez field component cannot be described by the Marcatilli’s approximation [18].

Fig. 10
figure 10

Sketch of dielectric rod waveguide antenna with an embedded planar lens

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Reflection losses at the lens-DRW interface are calculated according to the angle of incidence to the interface between media. Planar lenses must be E-plane oriented instead of H-plane, since it minimizes the reflections in the refractive interface. This is because the module of the power reflection coefficient of a p-oriented E-field \(\left | {\Gamma }_p(\beta _{_{\text {SC}}})\right |\) is less than or equal to an s-oriented E-field reflection coefficient \(\left | {\Gamma }_s(\beta _{_{\text {SC}}})\right |\) for \(\beta _{_{\text {SC}}} \in \left [0,\beta _{\mathrm {c}}\right ]\), being \(\beta _{_{\text {SC}}}\) the angle of incidence of the medium change (see Figs. 11 and 12) and βc the critical angle. Both \(\left | {\Gamma }_p(\beta _{_{\text {SC}}})\right |\) and \(\left | {\Gamma }_s(\beta _{_{\text {SC}}})\right |\) can be calculated with the Fresnel coefficients

$$ \begin{array}{@{}rcl@{}} \left|{\Gamma}_{s}(\beta)\right| &=& \left|\frac{n_{_{\text{SC}}}\cos \beta_{_{\text{SC}}}-n_{_{\text{ANT}}}\cos \beta_{_{\text{ANT}}}}{n_{_{\text{SC}}}\cos \beta_{_{\text{SC}}}+n_{_{\text{ANT}}}\cos \beta_{_{\text{ANT}}}}\right|^{2} \end{array} $$
(4a)
$$ \begin{array}{@{}rcl@{}} \left|{\Gamma}_{p}(\beta)\right| &=& \left|\frac{n_{_{\text{SC}}}\cos \beta_{_{\text{ANT}}}-n_{_{\text{ANT}}}\cos \beta_{_{\text{SC}}}}{n_{_{\text{SC}}}\cos \beta_{_{\text{ANT}}}+n_{_{\text{ANT}}}\cos \beta_{_{\text{SC}}}}\right|^{2} \end{array} $$
(4b)

where \(n_{_{\text {SC}}} = \sqrt {\varepsilon _{_{\text {SC}}}}\) and \(n_{_{\text {ANT}}} = \sqrt {\varepsilon _{_{\text {ANT}}}}\) are the refractive indices of the two media for non-magnetic materials with \(\mu _{_{\text {SC}}} = \mu _{_{\text {ANT}}} = 1\). The radiation taper avoids reflections between the intermediate medium and the air when LTAPER is large enough by converging the effective permittivity εeff(z) progressively to ε0.

The lens dimensions must be chosen by taking into account the beamwidth of the AE. In order to prevent total internal reflection at the lens-DRW interface, the beam waist of the antenna must be smaller than the critical angle, \(\beta _{\mathrm {c}}\) (− βc<βSC<βc). Further, the lens must be large enough to collect most of the THz power.

It can be shown [20] that an elliptic pattern is the exact solution to the problem, according to geometric optics (GO). Hyper-hemispheric lenses, however, are easier to produce and represent a good approximation to an elliptic interface.

3.1 Elliptical lenses

The embedded lens is a single refracting surface elliptical lens [20]. Its shape can be analytically determined as follows

$$ \begin{array}{@{}rcl@{}} X(\alpha) &=& \frac{(q-1) f \sin(\alpha)}{q \cos(\alpha)-1} \end{array} $$
(5a)
$$ \begin{array}{@{}rcl@{}} Z(\alpha) &=& \frac{(q-1) f \cos(\alpha)}{q \cos(\alpha)-1} \end{array} $$
(5b)

where \(q=\sqrt {\frac {\varepsilon _{_{\text {ANT}}}}{\varepsilon _{_{\text {SC}}}}}\) takes into account the ration between relatives permittivities, X(α) and Z(α) describe the shape of the elliptic lens when giving values to the angle α, and f = STHK + Z(0) is determined for fitting the lens inside the DRW antenna and maximizing the collected THz power. The geometry is given in Fig. 11. AE is placed in one of the ellipse focus.

Fig. 11
figure 11

Geometry of an embedded elliptical planar lens

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3.2 Hyper-Hemispherical Approximation

Figure 12 sketches the geometry of a hyper-hemispherical lens embedded in a DRW antenna. The planar lens is defined by a circumference of center C and radius R.

Fig. 12
figure 12

Geometry of an embedded hyper-hemispherical planar lens

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The lens hyper-hemisphericity is H = STHK + ΔZ. Several series of simulations were defined in order to approximate the wavefront phase error inside the DRW radiation taper to the one obtained with an elliptic lens. We appreciated that this phase error was reduced in at least a 1 : 10 frequency range when forcing TA ray (red) to be parallel to the Z-axis for \(\beta _{_{\text {SC}}} = 45^{\circ }\). Assuming the same refractive index for the lens and the photomixer substrate, hyper-hemisphericity is determined by the following expression:

$$ H = \left[\tan\left( \sin^{-1}(\frac{q}{\sqrt{2}})+\frac{\pi}{4}\right)-1\right]\frac{R}{\sqrt{2}} $$
(6)

Figure 13 shows the design chart according to (4a, b). Designs above the blue line are not feasible, since ΔZ > R. Points below the red line are lenses with hyperhemispherities smaller than the AE substrate thickness STHK, which are unachievable too. The insets sketch aspects of lens-based DRW antenna designs for different chart regions.

Fig. 13
figure 13

Design chart of embedded planar lenses. Limits of achievable hyper-hemispherity values due to the wafer thickness (red) and the ΔZ > R constraint (blue) are shown. Insets show aspects of lens-based DRW antennas for different regions of the chart. When ΔZ = R (blue), the lens is tangent to the junction between wafer and DRW. Curves for R = 3,2,1.5,1,0.5,0.25,0.20,0.15,0.10, and 0.05 mm are given (black). The design simulated in Section 4 is highlighted (D)

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The relative permittivity \(\varepsilon _{_{\text {ANT}}}\) is usually restricted to a few sets of values due to manufacturing issues. When \(\varepsilon _{_{\text {ANT}}}\) is close to \(\varepsilon _{_{\text {SC}}}\) (that is, q → 1), the design of the lens becomes difficult, specially when wafer thickness STHK is non negligible.

In Section 4, two examples, an elliptic and a hyper-hemispheric lens, are evaluated via full-wave simulations. Section 5 shows the measurement results of a prototype with an elliptic lens design.

4 Simulation Results

Both embedded lenses, the hyper-hemispherical and the elliptic ones, are compared in this section. For this experiment, silicon (\(\varepsilon _{_{\text {SC}}} = 12\)) is assumed for both AE wafer and lens. Both lens designs are integrated in a DRW antenna of dielectric constant \(\varepsilon _{_{\text {ANT}}} = \sqrt {\varepsilon _{_{\text {SC}}} \varepsilon _{_{0}}} \approx 3.5\). That is, q is approximately 0.54.

We have chosen a rod length LTAPER = 17 mm and width WROD = 2.2 mm, which leads to an electrically large DRW antenna for frequencies beyond 150 GHz. Thickness T is assumed to be 300 μ m for both the wafer and the DRW antenna.

4.1 Elliptical Lenses

The profile of the elliptic lens is calculated with equations (5a) and (5b). The distance between the AE and the refracting surface along the Z-axis is f = 1.5 mm, which allows for focusing 46.8% of the power by the lens into the DRW.

Simulated E-field distributions in the XZ plane are shown in Fig. 14 at 150 GHz (Fig. 14a), 300 GHz (Fig. 14b), and 800 GHz (Fig. 14c). Elliptic lens transforms the generated spheric wave reducing excitation of higher-order modes in the DRW, even for frequencies on which the rod is electrically large. The number and the importance of secondary radiation zones along the antenna is reduced; however, side lobes are still visible at higher frequencies. Secondary radiation zones are marked in Fig. 14c.

Fig. 14
figure 14

Simulated E-field distribution on XZ plane at 150 GHz (a), 300 GHz (b), and 800 GHz (c) for an elliptic lens as described in the text

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Fig. 15
figure 15

Simulated radiation pattern for the DRW antenna with an elliptic lens at 150 GHz (a), 300 GHz (b), and 800 GHz (c). Both E-plane (black) and H-plane (red) co-polar (solid) and cross-polar (dashed) components are shown

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Calculated radiation patterns are given in Fig. 15.

4.2 Hyper-Hemispherical Approximation

A lens of radius R = 2 mm is designed according to the procedure detailed in Eq. 4 for q = 0.54. It is marked with a D in the design chart (Fig. 13), with ΔZ = 0.7 mm, which leads to a hyper-hemispherity H = 1 mm.

Figure 16 shows the simulated E-field distribution inside the DRW antenna in the XZ plane. The lens generates a phase front similar to the ones achieved with the elliptic lens at frequencies where \(W_{\text {ROD}}>>\lambda _{\text {DRW}}\). Results are shown for 150 GHz (Fig. 16a), 300 GHz (Fig. 16b), and 800 GHz (Fig. 16c). The majority of the power is indeed radiated along the main lobe at 𝜃 = 0∘. However, there are still side lobes caused by higher-order modes at 300 and 800 GHz (Fig. 16c).

Fig. 16
figure 16

Simulated E-field distribution on XZ plane at 150 GHz (a), 300 GHz (b), and 800 GHz (c)

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The radiation pattern of the antenna is deteriorated. This can be solved by modifying the lens shape by scaling it and, if its possible, by reducing q in order to have a lens of bigger diameter and lesser hyper-hemispherity H.

Radiation patterns are given in Fig. 17. Since not all the power is radiated at the DRW antenna tip, there are many secondary lobes of comparable levels with the main beam. Furthermore, there is a degradation in the cross-polar discrimination (XPD) for the higher frequencies. The presence of nulls along the Z-axis is avoided, although there can be destructive interference in the radiation pattern for the endfire direction.

Fig. 17
figure 17

Simulated radiation pattern for the DRW antenna with a hyper-hemispheric lens at 150 GHz (a), 300 GHz (b), and 800 GHz (c). Both E-plane (black) and H-plane (red) co-polar (solid) and cross-polar (dashed) components are shown

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It can be appreciated that elliptic lens achieves a better radiation pattern when comparing Figs. 15 and 17 due to the reduction of power radiated in secondary radiation zones. The exact GO solution leads to a lower SLL and less destructive interference at 𝜃 = 0∘, which ensures a radiation level between 0 and − 3 dB along the Z-axis in the 0.1–1-THz frequency band.

When manufacturing capabilities allow embedding planar lenses of arbitrary shape patterns, an elliptic one should be chosen. In other cases, a hyper-hemispheric design is preferable, in particular for inexpensive designs where the lens is accommodated in a drilled hole in a low-permittivity DRW antenna.

5 Measurement results

The concept is validated by manufacturing and characterizing a polypropylene (εr = 2.2) prototype of equal dimensions as in Section 2 (Fig. 9). A high-permittivity (AD-1000, εr = 10) embedded elliptic lens is placed inside the rod. Dimensions are the same of the polypropylene prototype of Section 2. The DRW antenna is shown in Fig. 18. A waveguide-to-coax transition is used for feeding the prototype.

Fig. 18
figure 18

Manufactured polypropylene DRW antenna with an elliptic lens made of AD-1000

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According to our measurements, the working band, defined by the presence of a radiation maximum in the 𝜃 = 0∘ direction, is extended to beyond 30 GHz. Figure 19 shows the measured radiation patterns from 25 to 32 GHz in steps of 1 GHz. A pronounced main lobe along 𝜃 = 0∘ is recorded (Fig. 19a), in contrast to the design without a lens shown in Fig. 8 and in Fig. 19b.

Fig. 19
figure 19

Measured radiation pattern from 25 GHz to 32 GHz in steps of 1 GHz. Both E-plane (red) and H-plane (black) co-polar components are shown for the lens-based (a) and the homogeneous (b) DRW antenna. 𝜃 is swept between ± 90∘. Radiation pattern is normalized, in dB

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Figure 20 shows the simulated efficiency η for the polypropylene DRW antenna with (solid) and without (dashed) the lens. The dielectric planar lens avoids the radiation null shown at 25 GHz in Fig. 8 and increases the overall efficiency along the simulated frequency band.

Fig. 20
figure 20

Calculated efficiency η for the homogeneous DRW antenna (dashed) and for the same design with an embedded elliptic planar lens (solid) via full-wave simulations

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6 Conclusion

Table 1 summarizes the achieved efficiencies for the same AE design. The use of DRW antennas can increase the efficiency of a plain AE, despite it does not reach the silicon lens performance. When considering costs, they become appealing since rods can be between one and two orders of magnitude less expensive than silicon lenses due to its ease of manufacture by simple cutting from planar dielectrics of any kind.

Furthermore, DRW antennas are especially suited for arrays of photomixer sources. They can combine coherent sources efficiently, so its lower performance is compensated by the increase of power. Its cross-dimensions are electrically smaller than lenses, which can be due to the low mutual coupling between them, they can be used for beam steering by controlling the phase of each source or integrating DRW-based phase shifters.

A DRW antenna prototype has been assembled to a photomixing-based n-i-pn-i-p terahertz source. Radiation pattern measurements at 150 GHz have been carried out.

A solution for increasing the bandwidth of DRW antennas is presented by integrating a lens within the DRW.

Two different design strategies have been proposed and compared via full-wave simulations. The concept has been validated through measurements in a low-frequency proof-of-concept with measurements in the 25–40-GHz band.