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Abstract
In this paper, we report on the design, simulation and measurement of a flat-top beam shaper (FBS) lens for a millimeter-wave (mm-wave) antenna coupled source. The beam shaper is designed to transform the pseudo-Gaussian beam of the mm-wave source into a flat-top beam of sharp roll-off at a specific distance in the Fresnel region. Firstly, relying on the geometrical optics principles, we propose an analytical formulation for the design of lens profiles. Next, a simple optimization method based on the full-wave simulation of the lens and the mm-wave antenna is suggested to tune the analytically extracted lens profile. The optimized lens profile is then studied to assess its performance sensitivity to various deviations. Finally, a prototype of the proposed FBS lens is fabricated and measured. As an example study, the radiated beam of a 100GHz conical horn antenna is shaped into a flat-top beam of radius 40 cm with a sharp skirt and low ripples at the distance of 3 meters. The radius of the generated flat-top beam is also shown to be easily tunable by displacing the FBS lens with respect to the horn antenna. The proposed procedure for the design of FBS lens results in a high-quality flat-top beam, which is promising for the applications where a uniform irradiance within the field-of-view with negligible spillover is of high importance. Terahertz (THz) and mm-wave imaging systems are potential candidates in this regard.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Quasi-optical techniques have been widely used in terahertz (THz) and millimeter-wave (mm-wave) technologies for the development of various building blocks of sensing, imaging and communication systems [1,2]. The much longer wavelength in the mentioned frequency bands, as compared to the optics regime, necessitates more deliberation in the design of quasi-optical elements. Hence, several recent works have been focused on revisiting the optical techniques and devices for application to THz and mm waves. Beam shaping is one of such techniques that is well developed in the optics regime [3] and is extensible to THz and mm waves. In particular, conversion of a Gaussian laser beam into a flat-top or top-hat beam of uniform distribution and sharp roll-off has extensively been studied in the optics regime [4–12]. However, there are a few reports on the design and realization of flat-top beam shapers (FBS’s) at THz and mm waves, to the best of our knowledge.
In [13], a diffractive low-profile lens in the form of a phase-only hologram has been designed to obtain a uniform beam of 100mm diameter at the distance of 1m at 650GHz for a compact measurement range. The hologram consists of a curved groove pattern on a Teflon plate and is designed to shape the radiated beam of a corrugated horn antenna. In another work, a plano-convex refractive lens made of polypropylene has been proposed to shape the Gaussian beam of a diagonal horn antenna into a top-hat beam at the distance of 3m at 625GHz in an active THz imaging system [14]. An analytical beam shaping method followed by an iterative optimization routine has been used for extracting the profile of the reported Gaussian to top-hat beam shaper lens. In [15], two thin diffractive phase plates has been proposed to transform the non-collimated Gaussian beam of a 300GHz diagonal horn antenna into a flat-top beam of diameter 40mm at the distance of 50mm with respect to the outermost phase plate. The diffractive phase plates have the diameter of 80mm and are spaced 50mm apart.
In the above-mentioned works, the aim of the beam shaping has been to create a flat-top intensity distribution on a plane in the Fresnel (radiative near-field) region which is mostly required in imaging systems. Alternatively, the goal of the beam shaping can be the shaping of radiation pattern in the Fraunhofer (far-field) region which is of interest in communication systems. Examples of such beam shaping scenarios have been reported in [16,17] where an integrated lens antenna has been proposed to generate a flat-top far-field radiation pattern at 28GHz and 60GHz, respectively. In these works, the 3D shaped lens is integrated with a patch antenna and is designed by a hybrid geometrical optics/physical optics (GO/PO) method.
In this paper, we report on the design, fabrication, and measurement of a refractive double curved flat-top beam shaper (FBS) lens for application in a mm-wave stand-off imaging system. THz and mm-wave imaging systems have been developed in various types including active, passive, close range, stand-off, single-pixel scanning, and multi-pixel camera-like systems [18–21]. In active stand-off multi-pixel systems, the illuminated area on the object plane covers several imaging pixels. Hence, uniform illumination of the target region with a low spillover is of prime importance. Recently, an active camera-like stand-off imaging system has been made commercially available by Terasense Inc. [21]. The mentioned imaging system is based on 100GHz IMPATT diode sources and a 2D array of detectors referred to as the camera. The IMPATT diode source is provided with a conical horn antenna for which a FBS lens is proposed in this work. The proposed FBS lens belongs to the category of Fresnel region beam shapers and is designed to transform the divergent quasi-Gaussian beam of the conical horn antenna into a flat-top beam of sharp roll-off and small ripples.
Details of the FBS lens design are presented in Section 2 where we begin from the basic principles of geometrical optics (GO) and derive expressions for the beam shaper profile. In Section 3, we show that the proposed design methodology is applicable to difference scenarios of flat-top beam shaping and study the sensitivity of FBS lens design to various parameters. We also show that the flat-top beam radius is easily tunable by displacing the lens with respect to the horn antenna. In Section 4, the fabrication and measurement of a FBS lens prototype are presented. Finally, some concluding remarks are given in Section 5.
2. Design of the flat-top beam shaper (FBS) lens
Various numerical and analytical methods can be used to design the FBS lens [4,10]. In this work, we found that a simple design procedure based on the principles of the GO is effective and provides satisfactory results. The beam shaper should transform the divergent beam of a conical horn antenna into a flat-top beam at a specified distance. To realize such a beam shaper, we design a rotationally symmetric double curvature dielectric lens. The profile of the one side of the lens is designed to convert the divergent spherical beam of the horn antenna into a collimated pseudo-Gaussian beam. And, the profile of the other side is designed to transform the resultant collimated beam into the required flat-top beam. In this regard, we follow a four-step procedure to accomplish the lens design; (i) calculating the emitted beam profile and the phase center of the horn antenna by using the full-wave simulation, (ii) designing the collimator profile using the basics of GO, (iii) calculating an initial estimate of the beam shaper profile analytically, and (iv) optimizing the beam shaper profile to obtain a flat-top beam of sharp roll-off and low ripple by numerical simulation of the lens and the horn antenna.
2.1 Design of the collimator lens profile
The mm-wave IMPATT diode source is coupled to free-space via a conical horn antenna [21]. The radiated beam of this horn antenna is transformed to a collimated beam of plane wavefront by a plano-convex lens. As shown in Fig. 1(a), the focal point of the lens coincides with the phase center of the horn antenna denoted by $O$. If the incident wavefront is approximated by a spherical wave emanating from the phase center of the horn antenna, the profile of the collimator lens can be extracted using the Fermat’s principle as:
in which, $R$ is the distance of an arbitrary point on the collimator surface from the focal point, $L$ represents the focal length, $n$ denotes the refractive index and $\theta$ is the angle with respect to the lens axis, i.e. the $z$-axis. The lens surface is created by the rotation of the derived profile around the $z$-axis corresponding to $\theta =0$.
Fig. 1. (a) Geometry of the horn antenna and the collimator lens profile. (b) Simulated profile of the collimated beam at 100GHz versus the radial distance from the lens axis in the E- (solid line) and H-planes (dashed line) together with the fitted Gaussian profile (dotted line).
The horn antenna is fed by a cylindrical waveguide of diameter $D_{wg}$=2.3mm and its length and output diameter are $L_h$=30mm and $D_h$=12mm, respectively. Full-wave simulation of this horn antenna using the CST Microwave Studio reveals that the directivity and beamwidth of the antenna are approximately 20dBi and $16^\circ$ at 100GHz. Moreover, the phase center is estimated to be $d_{pc}$=9mm behind the horn aperture. Based on this information, the collimator profile is designed.
We choose the PTFE ($n\approx 1.45$) as the dielectric material of the lens due to its low loss at mm-wave frequencies. To keep the overall structure as compact as possible without deteriorating its performance, we assume the diameter and the focal length of the collimator lens as $D_l$=10cm and $L$=5cm, respectively. These values guarantee that the collimator lens is sufficiently far from the horn antenna ($L\approx D_{h}^2/\lambda$) and it captures most of the radiated power of the horn antenna. Specifically, the collimator profile subtends the angular range of $-31.5^\circ <\theta <+31.5^\circ$ which is nearly four times the beamwidth of the horn antenna. Based on the mentioned assumptions, the collimator profile is readily obtained from Eq. (1). To assess the performance of the designed collimator, the horn antenna and the collimator lens shown in Fig. 1(a), are simulated by the integral equation (IE) solver of the CST Microwave Studio. The structure is simulated at the central frequency of 100GHz. Due to the large dimensions of the structure relative to the wavelength, the multi-level fast multipole method (MLFMM) has been utilized to make the computational cost affordable. The simulated magnitude of the electric field at the output aperture of the collimator lens, called the collimated beam profile hereafter, is then recorded in both the E- and H-planes of the horn antenna. The simulated collimated beam profile in the mentioned planes is illustrated in Fig. 1(b). In this figure, the intensity of the electric field is plotted versus the radial distance ($r$) from the axis of the lens. We take the E-plane beam profile into consideration and find the best fitted Gaussian curve as $E_0exp[-(r/r_g)^2]$ in which the $1/e$ beam waist radius is approximately $r_g$=11.9mm. The fitted Gaussian curve is depicted in Fig. 1(b) by a dotted line and is used in the next section to extract the beam shaper profile.
2.2 Design of the beam shaper profile
After extracting the collimator lens profile, we design a beam shaper lens that can convert the pseudo-Gaussian profile of the input beam into a flat-top profile at a specific distance from the lens. The geometry of the beam shaping scenario is depicted in Fig. 2. A collimated symmetrical Gaussian beam of waist radius $r_g$ is incident on the input plane where the plano-concave beam shaper lens is situated. The intensity of the incident beam is given by $I_i(r)$ in which the radial distance from the axis of the lens is denoted by $r$ in the input plane. The beam shaper deflects the parallel rays of the incident beam in such a way that a flat-top intensity distribution is obtained at the output plane. The output plane is located at the distance of $d$ and the intensity of the shaped beam in this plane is represented by $I_o(r^\prime )$ in which $r^\prime$ denotes the radial distance from the beam shaper axis, i.e. the $z$-axis.
Fig. 2. Geometry of the beam shaping scenario. A collimated pseudo-Gaussian beam is incident on the input plane where the beam shaper lens is located. The beam shaper lens deflects the incident rays in such a way that a flat-top intensity distribution on the output plane is obtained. The overall structure is rotationally symmetric around the $z$-axis.
Applying the conservation of power in the introduced rotationally-symmetric geometry and assuming an ideal flat-top output beam, one can write:
The phase profile of the lens can be expressed as $\phi (r)=k_0(n-1)h(r)$ in which $n$ is the refractive index and $h(r)$ is the height profile of the lens. In addition, the collimated beam profile can be approximated by a Gaussian profile of waist radius $r_g$, as noted in the previous section. Hence, by solving Eq. (3), the height profile of the beam shaper lens can be obtained as:
Equation (4) provides an estimate for the beam shaper profile based on several simplifying assumptions. In this work, an acceptable flat-top beam should meet the following criteria: (i) the roll-off should be sharp so that the intensity reduces 10dB in less than one fourth of the beamwidth, and (ii) the intensity ripples inside the beamwidth should be less than 1.0dB. In order to optimize the lens profile for an acceptable performance based on the mentioned criteria, we adopt full-wave simulation of the designed lens together with the horn antenna by the IE (MLFMM) solver of CST Microwave Studio. It is found that by just scaling the beam shaper profile and without changing the collimator profile, the lens performance can be tuned to the required specifications.
As shown in Fig. 3(a), by concatenating the collimator and beam shaper profiles a double curved lens, referred to as the FBS lens, is obtained. The lens is located at a specific distance from the horn antenna which is determined by the focal length of the collimator lens. As mentioned before, PTFE is chosen for the dielectric of the lens and the refractive index is assumed $n=1.45$. The collimator profile is determined by Eq. (1). The distance of the output plane ($d$) is a design parameter for the beam shaper profile. The beam shaper profile is determined by scaling the initial estimate of Eq. (4) as $\alpha _{opt}h(r)$. The optimum scaling factor ($\alpha _{opt}$) is found by full-wave simulation of the structure shown in Fig. 3(a) and assessing the quality of the generated flat-top beam. As an example, a FBS lens is designed for the generation of a flat-top beam of radius $r_{ft}$=40cm at the distance of $d$=3m. The diameter of the lens and the focal length of the collimator are assumed as $D_l$=10cm and $L$=5cm, respectively. The dimensions of the horn antenna are similar to those mentioned in the previous section. In this case, the optimum scaling factor is found as $\alpha _{opt}=0.88$. The cross-section of the designed lens is shown in Fig. 3(b) and the simulated flat-top beam intensity at 100GHz is illustrated in Fig. 3(c). In these figures, both the initial estimate of the beam shaper profile and the optimized one along with the resulting output beams are shown. It is clear that even the initial estimate provides a flat-top beam of good shape. After optimization, maximum ripple of the obtained beam is $\pm$0.6dB and the roll-off sharpness is satisfactory as Fig. 3(c) shows. In this figure, the resultant beam for the lens diameter of $D_l$=20cm is also included which shows a lower ripple and sharper roll-off. The proposed method can readily be applied to other scenarios with a different lens diameter, flat-top beam radius, output plane distance, etc. More design examples are presented in the next section.
Fig. 3. (a) Geometry of the horn antenna and the FBS lens. The FBS lens is created by concatenating the collimator profile and a scaled version of the beam shaper profile. (b) The cross section of the designed FBS lens and (c) the simulated output flat-top beam at 100GHz for $d$=3m, $L$=5cm, $D_l$=10cm, $r_{ft}$=40cm, before and after optimization ($\alpha _{opt}$=0.88) and similar result for $D_l$=20cm ($\alpha _{opt}$=1.15).
3. More studies and sensitivity analysis of the FBS lens design
In the previous section, a straightforward procedure for the design of FBS lens is proposed. Then, the designed lens together with the horn antenna is simulated and the efficacy of the proposed design procedure is proved. The design shown in Fig. 3(b) is considered for the prototype fabrication the measurement results of which are presented in the next section. However, the proposed design procedure is not limited to that scenario and can also be used to obtain different flat-top beam radiuses at various distances. To prove this claim, we tried different designs based on the conical horn dimensions given before. In Figs. 4(a) and 4(b), the designed beam shaper profile for various flat-top radiuses at the distance of $d$=3m and the simulated beam profiles are illustrated. In Figs. 4(c) and 4(d), we assumed a fixed flat-top beam radius and designed the FBS lens to generate such a beam at different distances. In all of these figures, only the initial estimate given by Eq. (4) is demonstrated. It reveals that the FBS lens can be designed for different distances and output beamwidths. In addition, Fig. 4(d) shows that at a fixed beam radius, the roll-off is sharper at the shorter distances. It should be added that to generate a flat-top beam of high quality in some scenarios one may need also to assume different dimensions for the horn antenna and the lens diameter.
Fig. 4. Designed beam shaper profile for generating various flat-top radiuses of 30, 40, and 60cm at the distance of 3m and the simulated results for the designed FBS lenses at d=3m (a) and (b). Design of the beam shaper for a fixed beam radius of 40cm at different distances of 2, 3 and 4m and the simulated results at the mentioned distances (c) and (d).
In addition, we study the sensitivity of the performance of the designed FBS lens to the change of frequency, refractive index and distance to the horn antenna. For this purpose, we reconsider the example presented in the previous section and study the quality of the output beam upon change of frequency, refractive index, and distance to horn antenna ($L$). The FBS lens design is based on Eqs. (1) and (4) which are independent of frequency at first sight. However, the waist radius of the collimated beam, i.e. $r_g$, in Eq. (4) reflects the radiation characteristics of the horn antenna which is frequency dependent. As a result, the input beam of the FBS lens and accordingly the output flat-top beam change with frequency. To evaluate the extent of this dependency, we simulated the designed FBS lens at various frequencies around 100GHz and found that the performance remains acceptable in the frequency range of 90GHz to 110GHz. The simulated flat-top beam at 90, 100, and 110GHz is illustrated in Fig. 5(a). It is evident that the beam radius and the roll-off sharpness are almost unchanged. However, some degradation in the ripples at 90GHz is observable.
Fig. 5. Sensitivity of the generated flat-top beam to the changes of frequency (a), refractive index (b), and distance to the horn antenna (c) in a FBS lens designed at 100GHz for $n$=1.45, $L$=5cm, $D_l$=10cm, $r_{ft}$=40cm, and $\alpha _{opt}$=0.88.
Figure 5(b) illustrates the impact of refractive index change on the generated beam. The lens profile is designed for the refractive index of $n$=1.45. If the designed profile is fabricated using a dielectric of slightly different refractive index, the shape of the output beam will change. As this figure shows, the quality of the flat-top beam remains acceptable when the refractive index changes to 1.5 or 1.4, equivalent to a deviation of $\pm 3.5\%$. However, a noticeable alteration in the radius of the beam occurs. The lens is designed for a flat-top beam radius of $r_{ft}$=40cm. For the mentioned $\pm 3.5\%$ deviation in the refractive index, it is observed that the flat-top beam radius is nearly altered to 40cm$\pm$10cm which is equivalent to a change of $\pm 25\%$.
The last study focuses on the distance of the FBS lens to the horn antenna ($L$). The FBS lens under investigation is designed for the distance of $L$=5cm. At this distance, the collimator face of the FBS lens compensates the spherical phase distribution of the incident beam and makes the incident rays parallel. If the FBS lens is located at a slightly different distance from the horn antenna, the collimator surface cannot compensate the spherical phase of the incident beam completely. Using the paraxial approximation, the resultant residual phase distribution can be expressed as:
in which the axial offset of the lens is denoted by $\Delta L$. This equation suggests that if the lens is displaced farther away from the horn antenna, i.e. $\Delta L>0$, the transmittance of the lens includes a negative quadratic phase distribution which is equivalent to imposing a focusing lens on the transmitted beam. Hence, for $\Delta L>0$ we expect a narrower output flat-top beam. And, if the lens is located closer to the horn, i.e. $\Delta L<0$, we expect the output beam to become wider. This phenomenon is very interesting in practice since it proposes a simple mechanism for tuning the output beam radius or the field-of-view. By minor displacement of the FBS lens with respect to the horn antenna one can change the radius of the flat-top beam. By the simulation of the designed lens this hypothesis is proved. Figure 5(c) shows the achieved flat-top beam for $\Delta L$=0, -5mm, and +5mm. It is clear that the beam radius is changed considerably without degradation of the flat-top beam quality. The relative displacement is $\Delta L/L$=$\pm$10% and the resultant beam radius change is about $\pm$20%.4. Fabrication and measurement
The optimized axially symmetric FBS lens shown in Fig. 3(b) was fabricated by precision computer numerical control (CNC) machining of a PTFE block. The fabricated lens was then inserted in a circular frame and mounted on a pair of rails in front of the IMPATT diode source. Figures 6(a) and 6(b) show a photograph of the fabricated lens and the overall source assembly. As mentioned before, the IMPATT diode source is a product of Terasense Inc. [21]. This source radiates coherent continuous wave (CW) signal at around 100GHz. The source is coupled to the free-space via a mm-wave conical horn antenna and produces about 100mW average power.
Fig. 6. (a) Fabricated FBS lens, (b) Fabricated FBS lens installed in front of the IMPATT diode source, (c) Detector array (camera), (d) Measurement setup.
To measure the intensity of the wave radiated from the introduced source and FBS lens assembly, we used an array detector (camera) provided by the same company which is shown in Fig. 6(c). The detector pixel size is 3mm and is based on high-mobility GaAs semiconductor technology. The detector is a square law direct detector which measures the incident field intensity. The array detector was placed on a low reflection stand at the distance of $d$=3m from the source assembly as shown in Fig. 6(d) and is moved manually on a transverse line in the E- and H-planes to record the beam intensity in two orthogonal directions. At each measurement step, 32 samples with a pitch size of 3mm were recorded. Then the detector array was displaced laterally to record another part of the beam profile. Finally, the recorded sections were concatenated to obtain the whole shape of the beam profile.
As a first experiment, the lens is located at its nominal distance from the horn antenna which is denoted by zero axial displacement ($\Delta L$=0). Then, the lens is displaced $\Delta L$=$\pm$5mm from its nominal location. At each lens location, the generated output beam was measured. This process was done for both the E- and H-planes. The normalized measured beam intensities along with the simulation results are presented in Fig. 7. The left-hand side figures are H-plane measurements for various displacements of the FBS lens and the right-hand side figures are E-plane measurements. The output beam is slightly broader in the H-plane as a result of the minor deviation of the conical horn antenna beam in the E- and H-planes.
Fig. 7. Measured flat-top beam intensity (solid line) at the distance of $d$=3m from the 100GHz source and FBS lens assembly for various locations of the FBS lens with respect to the horn antenna ($\Delta L$=0, -5mm, and +5mm) in the E-plane (right-hand side) and in the H-plane (left-hand side) with the simulation results (dotted line) superimposed. The vertical lines define the nominal flat-top beamwidth for $\Delta L$=0.
The measured flat-top beam profiles are in good agreement with the simulations. Specifically, the measured beams show a sharp roll-off as expected. The width of the flat-top beam is changed by displacement of the lens with respect to the horn antenna in a similar way as explained in the previous section. The measured beams show noise-like fluctuations which can be attributed to the limited accuracy of our measurement setup and slight differences in the responsivity of the pixels of the detector array. But, the overall quality of the measured beams complies with our expectations and validates our proposed design method.
Since the radius of the flat-top beam in almost four times the radius of the FBS lens, and as inferred from the geometry of Fig. 2, the radiated beam from the FBS lens aperture diverges to cover the intended region in the output plane. The radiated beam has a flat-top shape not only in the output plane but also in other planes normal to the axis of the FBS lens. To study this issue, the electric field intensity at various distances from the FBS lens aperture is measured and illustrated in Fig. 8. The FBS lens is originally designed to generate a flat-top beam of radius 40cm at $d$=3m. So, the measurements were conducted at different distances of $d$=1, 2, 3 and 3.5m. The simulated results at the mentioned distances are also shown which are in agreement with the measurements. It is evident that the intensity distribution is flat-top at all distances but the beamwidth is increasing with distance as expected. It is also noteworthy that the skirt of the intensity distribution is sharper at the shorter distances and the flat-top beam disperses with the distance as a result of diffraction. It should also be highlighted that the Fraunhofer region of the designed FBS lens is at distances longer than 2$D_l^2/\lambda \approx 6.7$m. Hence, all the mentioned distances lie within the Fresnel region of the FBS lens.
Fig. 8. Measured electric field intensity (solid line) at different distances of $d$=1, 2, 3 and 3.5m from the 100GHz source and FBS lens assembly in the H-plane with the simulation results (dashed line) superimposed.
5. Conclusions
In this paper, a straightforward design procedure for the shaping of the radiated beam of a mm-wave conical horn antenna into a flat-top beam in the Fresnel region is proposed. A single element double curved refractive lens is proposed for this purpose. The lens profile is initially estimated based on the principles of geometrical optics and is optimized and simulated by a full-wave integral equation method. The proposed design technique is used for shaping the radiated beam of a 100GHz IMPATT diode oscillator which is integrated with a conical horn antenna. A prototype FBS lens for the mentioned mm-wave source is fabricated and measured. It is proved that flat-top intensity distribution with small ripples and a sharp roll-off can be generated. It is also shown that the radius of the generated beam can be tuned by changing the relative distance of the FBS lens and the horn antenna. Moreover, it is verified that the radiated beam keeps its flat-top shape at various distances from the lens aperture. The proposed FBS lens is promising for stand-off imaging systems at THz and mm waves in which uniform illumination of the field-of-view with negligible spillover is important.
Acknowledgments
The authors would like to thank Mr. Saeed Babanezhad for the fabrication of the prototype lens. All the reported measurements were conducted in Terahertz and Far-Infrared Lab., Department of Electrical Engineering, Sharif University of Technology.
Disclosures
The authors declare no conflicts of interest.
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