Compact hand-guided 3D scanning terahertz sensor platforms with 3D-printed aspherical telecentric f-θ lens

Abstract We report on the development of a handheld three-dimensional (3D) terahertz scanning system with an aspherical telecentric 3D-printed f-θ lens using selective laser sintering. The lens covers a broader scan line of 50 mm with its larger aperture, compared to the 20 mm range in our initial work, which was presented at the European Microwave Week 2021. In order to evaluate the adaptability of the optomechanical components with different sensor units, two different integrated frequency-modulated continuous wave radar modules based on monolithic microwave integrated circuit technology, operating in W- and D-bands are tested within the measurement scheme. The optomechanical part consists of a galvanometer scanner mirror and the f-θ lens. The optical system enables B-scans perpendicular to the manual translational movement of the sensor unit by its user. An integrated guiding wheel system with rotary encoder makes it possible to correlate the measurement points to their respective locations, enabling complete 3D volumetric inspection of the corresponding structures, which is particularly useful for the inspection along cracks and welds.


Introduction
Millimeter-wave and terahertz frequency-modulated continuous wave (FMCW) radar systems have proven to be very suitable for various applications in nondestructive testing [1]. These include material characterization [2], layer thicknesses determination [3], and numerous other versatile applications in the field of imaging and inspection have emerged in this regime recently [4][5][6]. Current systems typically consist of a sensor unit, which is used in combination with a traversing system for scanning the measured object. A unique handheld imaging system solution is shown in [1]. This handheld unit contains an FMCW transceiver based on a voltage-controlled oscillator -driven conventional III-V semiconductor frequency multiplier chain device in combination with a horn antenna as shown in Fig. 1(a) [6], including a rotary encoder integrated into one of the wheels. In FMCW radar systems a chirp signal, i.e. a linearly swept frequency ramp, is mixed with its time-delayed version echo which is reflected from the sample under test. This down-mixed signal, its beat frequency, is linearly proportional to the range of the reflective point. The FMCW signal in our primary setup is focused by a highdensity polyethylene (PE) lens in the bottom plate of the unit into the target below. Depth cross sections (B-scans) can be recorded with the hand feed motion by means of guiding wheels. The drawback of such a system concept is that uneven surface structures such as cracks or plastic weld seams can only be inspected transversely to their direction of travel.
Focal plane arrays as in [7] potentially offer an interesting alternative in the form of handguided line arrays, but do not provide depth information as an FMCW radar does. Synthetic aperture radar line arrays such as in [8] potentially offer another attractive solution in the form of compact monolithic microwave integrated circuit (MMIC)-based approaches. However, this requires the availability of appropriate assemblies for the desired frequency range, whereas the quasi-optical concept presented here can be used directly with different FMCW radar modules, which do not need to be specially designed and implemented.
While the use of monostatic all-electronic FMCW radar transceivers based on semiconductor technologies in the millimeter wave allows measurements over a full waveguide band, even at frequencies beyond 300 GHz, for further integration of the handheld device the geometric size is a limiting factor. The development of ultra-wideband millimeter wave and terahertz radar MMICs in recent years now offer an interesting alternative [9][10][11].
In our previous work that was presented at the EuMW2021 Conference [12], a highly compact 80 GHz MMIC radar module [13], shown in Fig. 1(b), was used instead of the waveguide integrated components. We designed and used an aspherical telecentric f-θ lens, fabricated with a computer numerical control machine using polytetrafluoroethylene (PTFE). f-θ lenses are commonly used in laser scanning systems. Unlike a regular lens' focal plane which is curved, that of an f-θ lens is flat. Moreover, the relationship between the scan angle and the focal point is linear and proportional to the focal length of the lens. For suitable threedimensional (3D) imaging performance with these lenses, one must ensure that the shape and size of the focus along the scan range is as homogeneous as possible. Telecentricity is another aspect considered in our lens design, which keeps the focused beam along the scan range perpendicular to the image plane. More detail on this topic is presented in the Section "Telecentric aspherical f-θ lens." Finally, with a galvanometer scanner mirror we steer the beam mechanically across the f-θ lens. We showed in [14] that thanks to the very low dispersion of PTFE in the subterahertz spectrum, a single f-θ lens can be used with different FMCW radars radars at different frequency spectra.
By continuously scanning perpendicular to the motion axis as shown in Fig. 2 we measure full C-scans. The collimated radar signal incident on the galvoscanner mirror is deflected along two axes, 90 • along the main propagation axis and then rotated a second time according to the rotated angle of the mirror onto the f-θ lens. The microwave radiation is then focused on the corresponding imaging position on the scan axis. The reflected signal traverses the same path back to the transceiver. The FMCW operation of the radar module allows retrieving the depth information in each measurement point and B-scan along each scan line. A third dimension is introduced by moving the sensor unit along the sample and 3D volumetric images are obtained.
We presented the first implementation and evaluation of such a hand-guided volumetric terahertz imaging platform in [12]. In this contribution, we discuss the new improvements on the system. First, a new lens is designed with a larger aperture and is fabricated using a different method with an alternative material. The new lens is 3D printed using the selective laser sintering (SLS) printing technique, which made it possible to realize a rectangular aperture for the lens, matching the mirror's aperture for maximum coverage of the radiation, while minimizing the apodization of the beam. It has a larger aperture of 66 × 44 mm 2 , enabling us to scan a bigger area along the motion axis. The material used in this 3D printer is polyamide (PA12). PA's index of refraction varies very little over the operation bandwidth of the scanner and is seen as a suitable material for this frequency band. Moreover, due to the low dispersion of the material, the group refractive index corresponding to each radar's operation band is measured and used in the optical design. The lens is printed on 120 μm layers, which is small compared to the smallest wavelength used in the operating system, 1.7 mm. The SLS technique also gives a more homogeneous printed lens compared to techniques like fused filament additive printing, where the cooling of the material could cause inhomogeneous material distribution inside the lens. Furthermore, a second SiGe-MMIC radar [15] with different specifications, Fig. 1(c), is used in addition to the first module in order to validate the adaptability of the system to different frequency bands.

SiGe-MMIC FMCW radar
With an FMCW radar the sample is illuminated by a chirp signal with a typically linear frequency modulation. Taking B to be the modulation bandwidth and T the modulation time, the modulated frequency can be described as  where α = B/T is the frequency modulation rate and f 0 the initial frequency. The reflected echo signal, s R (t) ∝ s T (t − δt) is an attenuated and time-shifted version of the transmitted signal. The time delay is then related to the range from the radar as δt = R/ ν n with ν n being the group velocity of electromagnetic radiation in the propagation medium. The received signal of a single reflector is mixed with the local oscillator's signal resulting in: with beat frequency (f b ). The theoretical range resolution, or Rayleigh resolution, of the FMCW radar is where ν n , n, and c 0 are the group velocity, refractive index of the propagation medium, and speed of light in vacuum, respectively. The ultra-wideband SiGe-MMIC FMCW radar front-ends used in this work employ a positive linear frequency sweep in W-and D-bands of the microwave spectrum. The specifications of each radar is shown in Table 1.
The monostatic transceiver is stabilized with a fractional-N PLL chip generating a chirp signal. The ramp duration can be as short as T = 1 ms. In our setup for both radar modules we used the complete available bandwidth with T = 1.5 ms modulation time and 6.5 ms reset time between ramps. The range resolution limit for each radar in vacuum is presented in Table 1. The beam in the 80 GHz radar module is directed using an elliptical integrated dielectric lens antenna with a 3 dB beamwidth of 5°, whereas the 150 GHz radar is coupled to a commercial standard pyramidal horn antenna yielding a 3 dB beamwidth of 12°. The lens antenna is made of PTFE which has a constant refractive index in the sub-terahertz region. For the 150 GHz radar system, since the beam is quite divergent compared to the antenna lens integrated module, an additional collimating lens is used.
The galvoscanner mirror and the FMCW radar module have independent controller units which communicate with software and each other through trigger signals. After the FMCW signal of a single sweep is received the mirror moves to the next position. The data are sampled at f s = 1 MHz and transferred to a portable computer through USB interface for further processing. Simultaneously the software processes the volumetric data and presents cross-sections of the sample in real time as the sensor moves. The processed data are mapped to an image and displayed with a pixel size of 0.5 × 0.5 mm 2 . Although the lens design and simulation has been done for a line scan of 40 mm, we were able to detect features and perform valid measurements up to 54 mm scan length, with a line measurement rate of 0.97 Hz, giving a maximum translational speed of v max = 0.48 mm/s for this setting. As the limiting variable on the maximum reachable translational speed is the modulation speed of the MMIC radar, this can be further increased by using faster modulation radars. With the current available systems, however, higher raster scan rates can be achieved by using larger scanning angle steps, where higher lateral resolution limits are acceptable. The possibility of printing the optical elements in 3D provides us with more degrees of freedom to find optimal solutions, in terms of the optical design and system integration. This is especially helpful when designing elements that do not have a circular symmetrical structure, or when the designed lens' surface differs from a typical spherical surface. Moreover, 3D printing makes it possible to fully integrate the optics in the housing of the scanner.

Telecentric aspherical f-θ lens
The focus of an ideal f-θ lens moves with a constant speed on a line laying in a flat plane at a distance equal to the focal length of the lens. The displacement of the spot from the optical axis with respect to the scanning angle is where f is the effective focal length of the lens and θ is the scan angle in radians. For a scan area of ΔL = 40 mm with Δθ = 40°, the lens' focal length needs to be 57.29 mm. A standard focusing lens has a common optical aberration, called Petzval field curvature, which shows that the focal plane of a standard lens is not flat. This is an important criterion in f-θ lens design that the field curvature remains zero at non-paraxial field points. Telecentricity is another criterion included into our optimization for perpendicular incidence on the image plane. Although the range resolution of the radar imaging system is determined by the system's bandwidth, the lateral resolution is limited mainly by the center frequency and the aperture size of the system transverse to the range axis. In optical design for the microwave and terahertz spectrum, it is tried to keep the wavefront shape as little as possible distorted by geometrical aberrations and wave distortions [16]. The geometrical aberrations are defined as the displacement of rays in the image plane from an ideal paraxial image point. Ideally, the residual geometrical aberration of the designed optical system is negligible with regard to the wavelength, so that diffraction is the major effect on the wavefront. The wave distortions can be analyzed by measuring the deviation of the propagated wavefront and its ideal not-distorted shape. Diffraction, being unavoidable due to the limited aperture of the optical system and the large wavelength of the radiation compared to a value of zero assumed in ray optics, should be the dominant limiting factor in the performance of the optical system. By reducing and eliminating these aberrations in the design process and obtaining a diffraction-limited system, the wavefront will preserve its ideal profile throughout propagation.
Before designing and simulating the lens system, the group refractive index of PA at the employed frequencies is measured, since the refractive index in each applied modulation band is approximately constant and we are interested in the group velocity of each module's signal when propagating. A slab of PA is 3D printed with a thickness of 20.45 mm, with flat parallel faces. Radiating the collimated FMCW signal onto the sample, the group velocity is achieved, which gives us the group refractive index value of PA in that modulation frequency band according to v n = c 0 /n g , with n g as the group refractive index. Figure 3 shows the range spectrum of the signal's reflections from each boundary interface. The optical path differences (OPDs) between the two reflection positions for 80 and 150 GHz radars are 32.16 and 31.78 mm, respectively. This means that the group refractive indices of PA in each modulation band are n 80 g = 1.573 and n 150 g = 1.554. For the lens design an optimization merit function is defined with lens thickness and lens face radii as the initial variables. The new custom-made f-θ objective lens has aspherical faces for more optimization variables. The aspherical surface in lens design is described by the equation of the sag, z(r): with c, c = 1/R curvature of the surface at the vertex; r, surface coordinates, distance from axis; κ, conic section constant; α 2i , polynomial aspherization coefficients. Furthermore, it has a rectangular aperture of 44 × 66 mm 2 to match the rectangular aperture of the scanning mirror, made out of PA, and a central thickness of d = 20 mm. It is optimized to accept a cone angle of Du = 40 • to scan a line of ΔL = 40 mm. The design frequencies are taken to be the beginning and end frequencies of each modulation band, namely 68, 93.6, 121, and 174 GHz, corresponding to wavelengths of 4.4, 3.2, 2.48, and 1.72 mm. The schematic of the designed lens is shown in Fig. 4(a) with different scan angles for optimization and performance evaluation, whereas Fig. 4(b) shows the footprint of the focused radiation in the image plane relative to the aperture of the optical objective.
The mirror shown in Fig. 4(a) is tilted at angles of +20 • , +10 • , and at 0 • , deflecting the beam at twice its tilting angle onto the f-θ lens. The footprint diagram shown in Fig. 4(b) shows the chief ray's incident position for the simulated angles to be +19.37/ mm, +9.97 mm, and 0 mm, respectively.
Besides being diffraction-limited, telecentricity and flat image plane criteria in the new design are evaluated. It is taken into consideration that the beam diameter at all angles is kept constant in different ranges from the radar. The effective and back focal length are EFL = 58.01 mm and BFL = 50 mm, respectively. For performance evaluation and optimization the Rayleigh criteria in optics are taken into account here. Figure 5 shows the spot diagram on the optical focal plane at five different off-and on-axis scan angles and for all the wavelengths, for a geometrical optics evaluation of the system. The incident ray distributions show signs of other aberrations for the off-axis field points in addition to the spherical aberration on all points.
To compare the aberrations magnitude to limiting effect of diffraction on the lens' performance, we compare the geometrical spot radius to the Airy radius, the radius of the optical system's response to a point source after diffraction. The maximum and root-mean-square spot radii are R max = 1.032 mm and R rms = 310 μm, respectively, where the Airy radius is R Airy = 5.66 mm. As a result, the geometrical aberrations are negligible and wellsuppressed compared to the diffraction.
Of the different numerical methods for modeling optical aberrations, we look at two of them, e.g. the Seidel coefficients and Zernike polynomials. Figure 6 shows the contribution of each lens surface to the aberration coefficients and the accumulated effect of them on the optical image plane. The remaining aberrations are well negligible to the smallest wavelength of 1.7 mm at 174 GHz.
Zernike polynomials model wavefront error compared to an ideal wavefront in the presence of optical aberrations [17]. In a well-designed system W pv < λ/4, where W pv is the peak-to-valley wavefront error in wavelengths, or more strictly W rms < λ/14, with W rms as RMS wavefront error in wavelengths. For the designed f-θ lens the wavefront error at the highest off-axis point gives W pv = 0.019 ≪ λ/4 and W rms = 0.0054 ≪ λ/14,   confirming the system to be aberration-free. Lastly, we look at the Strehl ratio, the ratio between peak value of the aberrated image of a point source and the maximum diffraction intensity. When S > 0.8 an optical system is considered to have a good performance. Strehl ratio is calculated from the Zernike polynomials for our lens to be S = 0.999. All the investigated parameters ensure a diffraction-free optical system.
The lateral resolution of the optical system can be found by looking at the cut-off spatial frequency of modulation of the complex optical transfer function (MTF). According to Fig. 7(a), presenting the accumulative MTF functions for all the four for the positive scan angles, the cut-off frequency is ν max ≈ 0.208 cycles/ mm, yielding a theoretical lateral resolution of R lateral = 4.8 mm. Moreover, the simulated point spread function (PSF) on the image plane for all wavelengths and over different scan points, Fig. 7(b), shows that the focused beam profile is preserved along the scan line. Here for the sake of simplicity, it is assumed that the signal power is equally distributed over the opening angle for all the rays. It is seen in the PSFs that the signals distribution is kept over the scan range.
Finally, we evaluate the system's performance with physical optics, in which the wavefront propagation throughout the optical system is observed, whereas in ray optics the system model is based on rays. We take the signal to be of Gaussian nature to be focused after reflection and refraction through the optical path. Figure 8 shows the Gaussian beam propagated and detected in the image plane for different scan angles, for the minimum and maximum frequencies of 68 and 174 GHz.
A dynamic range of 10 dB for each row is selected for a high contrast concentrating mainly on the Gaussian beam's mainlobe. In Gaussian optics the beam size is defined as the distance from the axial point where the power drops to 1/e 2 = 0.135 of its maximum value on the axis. This translates into −8.686 dB. This is the reason behind showing a dynamic range of 10 dB here.
We see a relatively well-preserved wavefront and beam waist in the image plane for all the angles. The slight deviation from an ideal case is however due to the smaller aperture of the mirror mounted on the galvoscanner compared to the Gaussian beam size of the radiation. This could be further improved by fixing the aperture matching to avoid apodization of the radiation as much as possible.

System architecture
The preliminary setup for the mobile terahertz FMCW telecentric f-θ scanning system is shown in Fig. 9.
The required space in the system when using the MMIC radar is decreased significantly in comparison to the bigger dimensions the split-block FMCW unit would need. The mirror has an aperture of 45.5 × 32 mm 2 and including the galvanometer mount an overall height of ≈130 mm. Its amplifier board is controlled via an field-programmable gate array-based platform. The mirror can rotate in between +23, 377 • at a maximum speed of ≈ 1700 • /s. Optimized for our setup, it moves +15 • at ≈ 50 • /s. The 3D printed lens is shown in Fig. 10. The printed layers have a thickness of 120 μ m and since this stepped structure of the lens surface is very small compared to the used wavelength, we do not see a relevant influence on the radiation.  A preliminary inspection of PA for terahertz and sub-terahertz applications showed us that the absorption of PA at frequencies above 450 GHz is significant and therefore, is not suitable for systems with higher operation frequencies.

System evaluation and results
As depicted in Fig. 2, we call the axis perpendicular to the sensor's translational movement, along which the beam is being steered, the scan-axis, and the one on which the sensor unit moves, the motion-axis. To eliminate effects of possible misalignment on the optical path, signal apodization, lens surface roughness, and systematic noise we perform a calibration on the received signal at all the scan points [3], so that we can achieve homogeneous B-and C-scans. This calibration technique, dividing the sample's signal to a reference signal, normalizes the amplitude of the signal, and the phase center of the IF signal is moved to the reference position as well. Furthermore internal reflections from the optical setup are compensated within the measurement signal by this calibration procedure.
In order to compare the performance of the radars, we first look at the range-gated signal of a metal reflector in the focal plane, and the signal in its absence for noise measurement. Figure 11 shows the amplitude envelope's distribution with frequency for each radar. The effects of the system noise and signal sensitivity drop at the lower and higher parts of the signal's frequency are compensated for with the three-term calibration mentioned in [3].
The first evaluation made on the scanner is its performance consistency along the scan-and range-axis. To evaluate the telecentricity performance of the f-θ objective in combination with the MMIC FMCW radar, a test measurement with a metal    reflector in the back focal plane is performed. The metal plate is then moved 10 mm toward and farther away from the sensor unit. Figures 12(a) and 12(c) show that the f-θ lens keeps the A-scan profile on the on-and off-axis points in the focal plane consistent to a very good degree, even for angles larger than the optimized value of +20 • .
Moreover, the optical path difference (OPD) to the reflecting plane is equal for all the tilted positions, suggesting a flat focal plane where the scan-line lies. At the edges of the scan axis, however, distortions become dominant and negatively affects the performance, which is shown better in Figs 12(b) and 12(d).
To verify the system's overall performance we performed a 3D imaging on a sample. The flat-bottom hole step-wedged sample made of PE, shown in Fig. 13, has holes of different diameters on each step and its thickness varies from 11 to 15 mm with 1 mm step size. The sample is illuminated from its flat side to look for those hidden features on its back side.
The scanned area is roughly shown in Fig. 13(a). B-scans along the scan-axis and motion-axis are shown in Fig. 14 at different locations. The discontinuities and steps are visible in the crosssections of the 3D volumetric data of the sample. The flat surface that is the reflection closer to the radar module. We observe that the smaller holes are better resolved in B-scans obtained with the 150 GHz radar. For example in the B-scan on the line y = 88 mm,  where three holes of 20, 10, and 5 mm diameter lie, the structure is much better resolved both in transverse and range, with the 150 GHz radar.
By looking at the C-scans at different depths in Fig. 15, the defects and steps are visible from the top-view as well. Internal hidden defects down to 1 mm are detected with the telecentric f-θ sensor system. Here, telecentricity has kept the location of features constant in different depths of the C-scan.
By finding the reflection positions from each boundary interface, R 1 and R 2 , a 3D model of each surface of the sample is obtained, shown in Fig. 16.
Due to the higher bandwidth of the 150 GHz radar, the peaks are better separated, therefore, the steps and holes structures are defined more clear. Also with its radiation at shorter wavelengths compared to the 80 GHz radar, smaller features and holes are better resolved. The optical distance between the two reflections shown in Fig. 17, is equal to the thickness of the sample at each point times material's index of refraction. The change in the sample's thickness due to the stepped structure and holes is distinguished sharper with the 150 GHz module, due to its smaller spot size.  Here, we see a very well integration of both radars with different carrier frequencies and bands with a single f-θ lens. This provides us with the flexibility to select the proper FMCW radar module based on the sample under test's optical and physical properties, such as refractive index, absorption, thickness, etc. In [18], for example, the trade-off of better signal penetration in the W-band versus better resolution in the D-band are exemplified with thick glass fiber reinforced composite structures.

Conclusion
A novel hand-guided mobile terahertz 3D imaging system with a new aspherical telecentric f-θ lens is integrated into a mobile FMCW terahertz imaging platform. In order to allow for a broader scan range, a new f-θ lens with a larger aperture has been designed. To match the aperture of the lens to that of the scanning mirror, the new lens has a rectangular aperture. The lens fabrication is made possible with the SLS 3D printing, where the used material is PA. Two ultra-wideband SiGe-MMIC FMCW radars operating at 80 and 150 GHz are used as transceivers to prove the adaptability of the system to different sub-terahertz frequency bands. We showed here, that the new opto-mechanical scanning system operates well in a wide frequency range of more than 100 GHz, thanks to PA's nearly constant refractive index throughout this frequency range. With the sensor movement along a line, the scanning system acquires a B-scan of the sample on its so-called scan-axis, which is perpendicular to the motion-axis. The lateral resolution of the f-θ lens is kept constant on each point of the scanned line, by preserving the focused beam's size and shape. Depending on the application and the test objects, the platform can easily be adapted to suitable sensors with the appropriate operation frequencies, bandwidths, measurements rates, etc. for any specific imaging scenario.
In future works, we will improve the system to have a faster imaging rate with a proper housing for in-field applications. In Figure 17. OPD between the two interfaces of the step-wedge sample calculated with the data from (a) 80 GHz and (b) 150 GHz radars. order to decrease and eliminate possible chromatic aberrations we intend to inspect the change of refractive index of the lens material throughout an even larger frequency range. Furthermore, it is intended to implement machine learning techniques in our imaging scheme to support automatic detection of anomalies [19,20].
Financial support. This work was partly funded by the Federal Ministry of Education and Research (grant numbers 16KIS1404 K, 16KIS1405, 16KIS1406); and the Fraunhofer Internal Programmes (grant number SME840104).
Competing interests. The authors report no conflict of interest.