Quasioptical Fresnel-based lens antenna with frequency-steerable focal length for millimeter wave radars

Abstract

This article presents the design of a dielectric lens antenna that utilizes the concept of a stepped Fresnel lens for focusing electromagnetic millimeter waves. Based on the quasi-optical properties of these waves, a Cartesian Oval is optimized and employed as a focusing lens. Multiple such lenses are combined to two different Fresnel-based lens antennas. We survey these newly designed lens antennas and compare them with a focusing lens antenna based on a Cartesian oval and a far-field lens antenna. Simulations and measurements with a frequency-modulated continuous-wave (FMCW) radar validate the effectiveness of the new design, demonstrating an even improved focus size while significantly reducing the size and weight of the lens antenna by up to 53% and by nearly 48 %, respectively. Additionally, the Fresnel-based lens antennas reveal a frequency dependency, enabling frequency-based steering of the focal length over a wide relative tuning range of 177%, which we thoroughly investigate for various bandwidths and center frequencies.

Introduction

A previous version of this paper was presented at the 52^nd European Microwave Conference and published in its Proceedings [1]. Radar systems offer many possible applications and are often customized to meet specific requirements. One of the most important aspects is to guide the electromagnetic beam in a desired direction, as demonstrated in [2], and to shape it to achieve a focused beam or a collimated beam with a plane wavefront and high gain. This is, for example, utilized in reflector antennas with the transmitter or the receiver in the focal spot [3, 4]. Focusing is also applied in material characterization to selectively illuminate specific parts of a sample, as illustrated in [5] and [6]. Additionally, shaping electromagnetic beams finds applications in medical therapy for hyperthermia cancer treatment [7] and to enhance the range resolution of a radar system [8].

Redirecting or shaping of electromagnetic beams relies on passive components such as lenses and mirrors. Shaping through mirrors is either done by a curved mirror, as shown in [5] and [6] or by utilizing reflect arrays, which consist of arrays of microstrip patches [9], making reflect arrays thinner than curved mirrors.

Lenses, on the other hand, can be external systems, as used in [7, 8, 10–12] or lens antennas, as shown in [13–17]. Lens antennas are dielectric lenses designed to be directly mounted to devices like a radar system, offering the advantage of compactness and easy alignment compared to external components. In contrast, external lenses are positioned after an antenna, e.g., a horn antenna. This results in a more extended setup. To reduce the thickness of external lenses, similar to the use of reflect arrays instead of curved mirrors, Fresnel lenses are employed, as shown in [10, 11].

Requirements for focusing systems differ in their focal length and focal spot size. Some applications require relatively large focus distances compared to the employed electromagnetic wavelength and the desired focal spot size. These requirements arise from specific needs, such as maintaining a safety distance from moving parts in industrial environments [2], or because of distant targets within the human body, as discussed in [7]. Because of diffraction limits, this leads to large dimensions, no matter which focusing system is selected. Particularly lens antennas, where length and diameter depend on each other, become very large and therefore heavy. This increases material costs and makes manufacturing and subsequent handling more difficult. An alternative to conventional shaped lenses offer grooved and multi-dielectric Fresnel lenses as the authors of [10] and [11] use.

In this paper, we introduce a focusing lens antenna for millimeter waves that utilizes the idea of a stepped Fresnel lens, which is a new design approach to the best of the authors’ knowledge. The presented Fresnel-based lenses are smaller and therefore lighter than comparable focusing lens antennas and even achieve a smaller focal point radius. Furthermore, they have got a frequency dependence, allowing for steering the focal length.

Following in second section, we introduce essential quasioptical considerations for focusing lens antennas. Afterward in third section, we present the new Fresnel-based lens design and the manufactured lenses. In section “Simulation and Measurement Results,” we show simulation and measurement results of these lenses and compare their different sizes and designs. After that, we investigate the frequency-dependence of the Fresnel-based lenses and examine how it can be utilized to steer the focal length.

Quasioptical considerations

To obtain a focusing lens antenna, the geometric shape of the lens must guide the electromagnetic waves in such a way that they converge. As in [15], we utilize a Cartesian oval to meet these requirements. A Cartesian oval possesses two focal points, one inside the oval and the other one outside. Figure 1 illustrates the cross-section of a focusing lens antenna that is based on a Cartesian oval. By rotating the 2-dimensional shape becomes a 3-dimensional object. The focal point inside the lens is used for feeding from a rectangular waveguide into the lens antenna. To achieve optimal matching, we utilize a stepped impedance transformer at the antenna feed, as the authors do in [13]. After propagating through the lens, the waves refract at the interface between the lens and air and converge to the focal point outside the lens, which is the desired geometric focus for the application.

Figure 1. Cross-section of a focusing lens antenna, which is based on a Cartesian oval.

The lens in Figure 1 is a Cartesian oval with the length l and the diameter D. The focal length f, which is the distance between the geometric focus and the largest diameter of the lens, is set by the shape of the Cartesian oval (see [15]). The resulting shape determines the front focal distance $s_\mathrm{F}$, which is the distance between the beam waist and the vertex of the lens.

The former design considerations refer to the model of geometric optics, which describes the propagation of electromagnetic waves as rays but ignores the wave properties and effects by diffraction. This leads to deviations between the properties given by geometric optics and the actual behavior of the beam. A more precise description for focused electromagnetic millimeter waves is the paraxial solution of the wave equation as a Gaussian beam, which is valid for a beam radius $w_0 \gt 0.9\cdot \lambda$ [18]. Figure 1 also depicts the resulting Gaussian beam of a Cartesian oval, illustrating that the minimum radius

(1)\begin{equation} w_0 = \frac{2\lambda}{\pi}\cdot F_{\#}, \ \: \mathrm{with}~F_{\#}=\frac{f}{D}, \end{equation}

occurs at the waist of the Gaussian beam and is not infinitely small. The beam waist w ₀ is defined as the width from the axis of propagation to $\frac{1}{\mathrm{e}^2}$ of the maximum intensity or $\frac{1}{\mathrm{e}}$ of the maximum amplitude of the beam (see Figure 2). Equation (1) indicates that in order to achieve a narrow beam width, the F-number $F\#$ must be small. Conversely, a larger F-number results in a wider beam profile and a lower amplitude [18]. Figure 2 illustrates this aspect, displaying the Gaussian amplitude profiles at the beam waist for lenses with the same diameter D but increasing focal lengths in the z-direction, which leads to a bigger beam waist. To compensate for this effect, the diameter of the lens must increase, as equation (1) indicates.

Figure 2. Amplitude profiles of Gaussian beam waists of focusing lenses with identical diameters ${D}$ but increasing focal lengths in ${z}$-direction.

Furthermore, the Gaussian beam differs from the geometric optics approach in that the waist of the former is positioned closer to the lens antenna than the geometric focus, as Figure 1 illustrates. According to [19], this difference can be quantified as follows:

(2)\begin{equation} \Delta f = f-z_0= \frac{z_\mathrm{R}^2}{z_0}. \end{equation}

In equation (2), z ₀ represents the distance from the largest diameter of the lens to the beam waist (as Figure 1 shows), and $z_\mathrm{R}$ denotes the Rayleigh range. The Rayleigh range corresponds to the distance from the minimum area of the beam cross-section at the beam waist until it increases by a factor of two. It also characterizes the distance over which a Gaussian beam remains primarily collimated before it starts to diverge, and it indicates the point where the radius of curvature of the electromagnetic waves reaches its minimum. In [18] the author provides the following expression for the Rayleigh range:

(3)\begin{equation} z_\mathrm{R} = \frac{\pi\cdot w_0^2}{\lambda}. \end{equation}

Inserting equation (3) with (1) into equation (2) gives

(4)\begin{equation} \Delta f = \frac{16\cdot\lambda^2\cdot f^4}{\pi^2\cdot D^4\cdot z_0}, \end{equation}

which implies that a small deviation $\Delta f$ requires a short focal length or a large lens diameter, too.

Considering equations (1) and (4), it is evident that a large lens diameter D is necessary to achieve a narrow beam waist and minimize the deviation $\Delta f$, especially when the front focal distance, and thus the focal length, is large due to application-specific requirements. However, a large lens diameter also leads to an undesirably increased length l and, consequently, to a higher overall weight.

Fresnel-based lens design

To overcome the disadvantage of large and heavy focus lens antennas, we propose a new lens design based on a stepped Fresnel lens as it is commonly used in optics to reduce the mass and volume of a conventional lens. The idea of a stepped Fresnel lens is to remove material that does not contribute to beam forming since only the surface of the lens refracts the beam. Therefore, the lens is divided into multiple circular areas and unnecessary material is removed, which forms the steps and makes the lens thinner.

Figure 3 presents our design concept for a Fresnel-based lens antenna. In contrast to a conventional Fresnel lens in optics, the introduced focus lens is based on a Cartesian oval and has two focal points. For all circular steps, all inner and all outer focal points must overlap constructively. Consequently, we designed each step as a circular area from a new lens with a different length l but the same distance between the antenna feed and the focal point outside the lens. Therefore, the length difference $\Delta l$ between two consecutive lenses must be designed to achieve a constructive overlap, even though the electromagnetic waves have various propagation speeds and wavelengths through the air and the lens material. Since the Cartesian oval inherently ensures constructive interference for all beam paths between both focal points for one lens, it is sufficient to consider the most straightforward beam path between the inner and outer focal points to match multiple lenses to each other. This beam path is the direct one along the propagation axis, and it leads to a length difference between the individual lenses of

(5)\begin{equation} \Delta l = \lambda_\mathrm{air}(n_1+\frac{\Phi}{2\pi})=\lambda_\mathrm{lens}(n_2+\frac{\Phi}{2\pi}), \end{equation}

Figure 3. Design concept of the Fresnel-based lens antenna: Multiple focus lenses, with differing lengths ${l}$ separated by $\Delta l$, are merged and subsequently trimmed to the length of the shortest lens.

where $\lambda_\mathrm{air}$ and $\lambda_\mathrm{lens}$ are the wavelengths outside and inside the lens, n ₁ and n ₂ are the respective numbers of integer wavelengths within $\Delta l$. Φ is the phase shift, which has to be equal for the propagation inside and outside the lens.

With $\Delta l$, we designed all lenses, starting from the original-sized focus lens, which is the largest and outermost, down to the smallest one. Whereas the smallest lens must be larger than the distance from the antenna feed to the largest diameter of the original-sized focus lens to retain the largest diameter D and the focusing properties. As equation (4) indicates, the difference $\Delta f$ increases with decreasing lens diameters and increasing focal lengths, both of which applying to the shorter focus lenses. To compensate for this, we utilized CST Microwave Studio to optimize the focus length of each focus lens, ensuring that their waists are positioned at the same location. Subsequently, we added all focus lenses together and cut the lenses to the size of the smallest lens. To prevent an inner smaller lens from shadowing an outer bigger lens, the transitions between the lenses are cut at an angle towards the feed of the antenna.

Figure 4 presents the in this work further investigated lenses. We designed all lenses for a center frequency of $f_{\mathrm{c}}=80\,\mathrm{GHz}$ and used polytetrafluoroethylene (PTFE) as a dielectric material, because of its very low dissipation factor of $\mathrm{tan}(\delta)\approx 0.0002$, a relative permittivity of $\epsilon_r=2.1$ and good machine processing properties. All lenses are manufactured through a turning process, and their properties are summarized in Table 1. Lens (1) is a small far-field lens, as the authors of [14] present. Lens (2) is a focus lens based on a Cartesian oval as Figure 1 depicts and presented in [15]. Lenses (3) and (4) are Fresnel lenses we made out of ten and five different-sized focus lenses, respectively, according to the design concept Figure 3 illustrates. Their outermost lenses have the same length as lens (2), before being cut to the size of the smallest lens. Lenses (2) to (4) have the same front focal distance of $s_\mathrm{F} = 300\mathrm{\,mm}$, and therefore, lenses (3) and (4) have a shorter length z ₀ than lens (2) and a slightly smaller diameter D (see Table 1). Together, this results in an up to 18 % smaller Gaussian beam radius $w_\mathrm{0,calc}$, which we calculated with equation (1). Based on $w_\mathrm{0,calc}$, we calculated the Rayleigh range $z_\mathrm{R}$, which exhibits an even larger disparity between lens (2) and the Fresnel lenses. Lenses (2) to (4) differ not only in their focusing properties but also in their weight and length. Specifically, lenses (3) and (4) are 48% and 42% lighter and 54% and 48% shorter than lens (2).

Figure 4. Investigated lenses: (1) far-field lens, (2) focus lens, (3) Fresnel lens with ten steps, (4) Fresnel lens with five steps.

Table 1. Properties of the investigated lenses from Figure 4

Simulation and measurement results

We simulated and measured all lenses as shown in Figure 4. For the simulations we used CST Microwave Studio and for the measurements, we used a radar, which is an enhanced version of the in [20] and [21] proposed ultra-wideband FMCW radar with a center frequency of 80 GHz, and a corner with a side length of 20 mm as a target. The radar and the corner are each mounted to a different linear track. The linear tracks are mounted so they move perpendicular to each other, as Figure 5 presents. In this manner, the corner can be moved through the yz-plane of the investigated lenses.

Figure 5. Measurement setup: FMCW radar with investigated lens antenna and a corner reflector with 20 mm side length on two perpendicular mounted linear track.

Figures 6 and 7 present the simulated and measured normalized amplitudes in the yz-plane, with the reference point being at the apex of the respective lens in the rotational center, as illustrated in Figure 1. As one expects from the radar range equation, lens (1) has an amplitude drop with increasing distance. Conversely, lens (2) has a different behavior with a high amplitude at the focal length. The focal spot of lens (2) is limited in y- and z-direction, having a small width but an expanded length. In comparison, the focal spots of the Fresnel lenses are significantly shorter in the z-direction, as expected due to their shorter Rayleigh range, and have a smaller width (refer to Table 1). Comparing the measured results with the simulated ones demonstrates a strong agreement. The main lobes of the focusing lenses (2) to (4) are clearly visible, and the focal spots are comparable in size and amplitude. Only lens (2) exhibits a longer front focal distance in the measurement than expected based on the design and the simulated results. This discrepancy may be attributed to manufacturing deviations. The large size and the weight of lens (2) cause greater manufacturing challenges than the intricate contours of the Fresnel lenses (3) and (4), and even slight variations in the shape can affect the behavior of a lens of this size. Another difference between the simulated and measured results is the visibility of smaller amplitudes, which can be seen in the simulations but are not present in the measurements, resulting in missing side lobes, particularly noticeable in the focusing lenses. As the amplitude of the far-field lens (1) is lower than the one of the focusing lenses, the measurement results of lens (1) are also attenuated.

Figure 6. Simulated amplitudes of lens (1) to (4) in the ${yz}$-plane with a center frequency of $f_{\mathrm{c}}=80\,\mathrm{GHz}$, normalized to the maximum of lens (2) [1].

Figure 7. Measured amplitudes of lens (1)–(4) in the ${yz}$-plane with a center frequency of $f_{\mathrm{c}}=80\,\mathrm{GHz}$ and a bandwidth of $B= 5\,\mathrm{GHz}$, normalized to the maximum of lens (2) [1].

For further consideration, Figure 8 shows the amplitude over the y-axis at the front focal distance $s_\mathrm{F} = 300$ mm for different bandwidths of the radar. Lens (2) at B = 20 GHz and lens (3) at B = 5 GHz have an equal maximum, but lens (3) has an over 24% smaller focus radius than lens (2), as it has been expected due to the shorter z ₀ and the calculated beam radius $w_\mathrm{0,calc}$. Lens (4) at B = 5 GHz also has a 20 % smaller focus radius than lens (2) but a smaller amplitude. The measured beam radius $w_\mathrm{0,meas}$ of 9 mm, 6.8 mm and 7.2 mm for lens (2) at B = 20 GHz and lenses (3) and (4) at B = 5 GHz match well with the theoretical values in Table 1. Another visible aspect is the higher side lobe level (SLL) of lens (4) compared to lens (3). A noticeable effect for lens (4) appears at approximately $\pm 10\,\mathrm{mm}$ with an SLL of −26.6 dB, whereas the SLL for lens (3) is $ \gt -20\,\mathrm{dB}$. The SLL of lens (1) and (2) cannot be determined, as their side lobes are not clearly identifiable.

Figure 8. Measured amplitudes of lens (1)–(4) over ${y}$ with different bandwidths ${B}$, using a center frequency $f_{\mathrm{c}}=80\,\mathrm{GHz}$ and a front focal distance $s_\mathrm{F}=300$ mm.

As we presented in third section, the design rule in equation (5) is based on the wavelength to determine the length difference $\Delta l$ between focus lenses the Fresnel lenses are made of. Consequently, the Fresnel lenses have a frequency dependence with varying properties for frequencies other than the design frequency of 80 GHz, as Figure 8 demonstrates. For higher bandwidths than $B=5\,\mathrm{GHz}$ and greater deviations from the center frequency, the focus radius increases while the amplitude decreases, implying a worse focus capability.

For further investigations on the frequency dependence of the Fresnel lenses, we examined lens (3) and lens (4) at various center frequencies $f_{\mathrm{c}}$ ranging from 70 GHz to 88 GHz with a small bandwidth of $B=4\,\mathrm{GHz}$. Figures 9 and 10 show the amplitude along the z- and y-axis, respectively. All amplitudes are normalized to the maximum of the measurement with a center frequency $f_{\mathrm{c}} = 80\,\mathrm{GHz}$.

Figure 9. Measured amplitudes of lenses (3) and (4) over ${z}$ with different center frequencies $f_{\mathrm{c}}$ from 70 to 88 GHz and a constant bandwidth of ${B}$ = 4 GHz, normalized to the maximum amplitude at $f_{\mathrm{c}}=80$ GHz.

Figure 10. Measured amplitudes of lenses (3) and (4) over ${y}$ with different center frequencies $f_{\mathrm{c}}$ from 70 to 88 GHz and a constant bandwidth of ${B}$ = 4 GHz, normalized to the maximum amplitude at $f_{\mathrm{c}}=80$ GHz.

In Figure 9 the measurements along the z-axis at y = 0 mm demonstrate the significant frequency dependence. For a center frequency of 80 GHz, which is the chosen frequency for the lens design, the peak is at the intended front focal distance of $s_\mathrm{F}=300\mathrm{\,mm}$. However, for increasing and decreasing center frequencies, the front focal distance also increases and decreases over a total range of 530 mm for lens (3) and 460 mm for lens (4). This represents a remarkably relative tuning range of 177% and 153%, respectively. In this regard, the amplitude decreases for larger front focal distances, and the focal spot becomes longer. The measurements in the y-direction at the corresponding focal points on the z-axis confirm this observation and demonstrate a widening of the beam waist for larger distances, as Figure 10 indicates. These results largely coincide with the theoretical considerations from Figure 2. Nevertheless, in both Figures 9 and 10, the highest amplitudes and narrowest beam waists are not achieved at the shortest focal length, but rather at a slightly higher center frequency.

Figure 11 depicts the ratio between the measured beam waist radius $w_\mathrm{0,meas}$ and the theoretical value $w_\mathrm{0,calc}$. Lens (3) and lens (4) exhibit only a minor deviation from the theoretical value for center frequencies $f_{\mathrm{c}} \gt 80\,\mathrm{GHz}$. For $f_{\mathrm{c}}=82\,\mathrm{GHz}$, the measured beam radius of lens (3) is even smaller than the theoretical one, possibly attributable to the measurement setup’s influence. In contrast, at center frequencies below 80 GHz, resulting in smaller front focal distances, the deviation increases noticeably. Nonetheless, despite the increasing deviation, the measured beam radius still exhibits a slight decrease, as Figure 12 demonstrates. It shows the measured beam radius as a function of the center frequency for both Fresnel lenses. Only for the lowest center frequencies the beam radius does increase again. Lens (3) has its minimum beam waist radius at a center frequency of 74 GHz and lens (4) at 72 GHz. These values correspond to the center frequencies with the highest amplitudes in Figures 9 and 10. For most center frequencies, lens (3) has a smaller beam radius compared to lens (4) and is closer to the theoretical value (see Figure 11). The smaller beam waist is due to shorter length z ₀ of lens (3) compared to lens (4). However, for center frequencies under 73 GHz lens (4) achieves a smaller beam radius, although it has longer front focal distances for center frequencies $f_{\mathrm{c}} \lt 80\,\mathrm{GHz}$ than lens (3) as seen in Figure 9. One reason for this unexpected behavior might be the no longer correct shape of the individual Cartesian ovals of each Fresnel lens for front focal distances, which are a lot shorter than $300\,\mathrm{mm}$.

Figure 11. Measured waist radius $w_\mathrm{0,meas}$ of lenses (3) and (4) over different center frequencies at a bandwidth of 4 GHz, referring the corresponding theoretical waist radius $w_\mathrm{0,calc}$.

Figure 12. Measured beam waist radius $w_\mathrm{0,meas}$ of lens (3) and lens (4) over different center frequencies at a bandwidth of 4 GHz.

The measurement results in Figures 9 and 10 illustrate the different focal lengths and beam radii for different center frequencies explaining the behavior in Figure 8 for higher bandwidths. As a higher bandwidth is a sweep through a larger spectrum of frequencies, more different focal lengths and beam radii occur in this measurement and add up to a wider beam with a lower amplitude. Figure 13 illustrates the result in ${z}$-direction for different bandwidths at a center frequency of 80 GHz for the Fresnel lenses. The superposition can be seen here as well. For a small bandwidth of 5 GHz, the front focal distance remains at 300 mm. However, as the bandwidth increases, the focus size becomes longer while the amplitude and the front focal distance decrease, although Figure 2 indicates higher amplitudes and a smaller beam waists for shorter front focal distances. An explanation for the decreasing front focal distances with increasing bandwidths are the higher amplitudes for shorter front focal distances at smaller bandwidths as Figure 9 shows. Hence, the shorter front focal distances contribute more to the superimposed outcome.

Figure 13. Measured amplitudes of lenses (3) and (4) over ${z}$ with different bandwidths ${B}$ and a center frequency of $f_{\mathrm{c}}=80\,\mathrm{GHz}$, normalized to the maximum amplitude at ${B}$ = 5 GHz.

Overall, these results highlight the excellent focusing properties of both Fresnel lenses and their ability to adjust their front focal distance over a vast range, although this frequency dependence admittedly leads to a decreased performance for higher bandwidths. Because of its smaller weight and size, narrower beam waist for most center frequencies, and a larger range for frequency steering, we suggest lens (3) for most applications.

Conclusion

We presented a novel concept for focusing lens antennas, based on a stepped Fresnel lens. Our approach uses the quasioptical characteristics of millimeter waves to design multiple focusing lenses based on geometric optics, which we optimized by simulation and joined together to a focusing Fresnel-based lens antenna. Our new design reduces the weight by up to 48% and the length by up to 54% compared to a Cartesian oval-based focusing lens with the same diameter and front focal distance. Subsequent measurements verify the focusing capabilities of the Fresnel lenses, even surpassing the comparable focus lens in terms of the minimum beam waist radius.

Furthermore, the special frequency-dependent design enables a steerable front focal distance by means of the center frequency over a considerable relative tuning range of up to 177%. The focus remains narrow for shorter front focal distances and stays close to the theoretical limit for higher front focal distances. The frequency steering allows for a rapid and simple adjustment of the focal length without any hardware change, which can be useful for various applications in research, industry and medicine.