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

(1)$$f_{{T}}( t) = f_{0} + {Bt\over {T}} = f_{c} + \alpha t,\; $$

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:

(2)$$s_{{m}}( t) \propto \cos\left(2{\pi} {2BR\over \nu_{{n}}T}t\right) = \cos( 2{\pi}f_{{b}}t) ,\; $$

with beat frequency (f _b). The theoretical range resolution, or Rayleigh resolution, of the FMCW radar is

(3)$$\delta R = {\nu_{{n}}\over 2B} = {{c_0}\over 2nB},\; $$

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.

Table 1. SiGe-MMIC radar specifications

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 mm². 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

(4)$$L = f\theta,\; $$

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 [Reference Goldsmith16]. 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.

Figure 3. Range spectrum obtained from the reflections of each interface between air and the material showing two reflection peaks.

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 $n_{g}^{80} = 1.573$ and $n_{g}^{150} = 1.554$.

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):

(5)$$z( r) = {cr^2\over 1 + \sqrt{1-( 1 + \kappa) c^2r^2}} + \sum_{i = 1}^{N} \alpha_{2i} r^{2i},\; $$

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 mm² 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 $\Delta \theta = 40^\circ$ 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.

Figure 4. (a) 3D representation of the ray tracing of the designed aspherical telecentric f-θ lens and (b) footprint of the focused rays.

The mirror shown in Fig. 4(a) is tilted at angles of $\pm 20^\circ$, $\pm 10^\circ$, and at $0^\circ$, 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 $\mp 19.37/$ mm, $\mp 9.97$ 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.

Figure 5. On- and off-axis spot diagram in the image plane for five scan angles $\pm 20^\circ$, $\pm 10^\circ$, and $0^\circ$, without and with Airy disk for comparison.

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 well-suppressed 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.

Figure 6. Seidel coefficients on each lens surface and on the image plane.

Zernike polynomials model wavefront error compared to an ideal wavefront in the presence of optical aberrations [Reference Bass, DeCusatis, Enoch, Lakshminarayanan, Li, Macdonald, Mahajan and Van Stryland17]. 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.

Figure 7. (a) MTF and (b) PSF cross-section in the logarithmic scale in tangential (T) and sagittal (S) planes. The simulations are done on the on-axis point with $0^\circ$ deflection, and two off-axis points with $10^\circ$ and $20^\circ$ deflection of the radiation.

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.

Figure 8. Focused Gaussian beam on image plane for $\pm 20^\circ$, $\pm 10^\circ$, and $0^\circ$ scan angles.

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.

Figure 9. Handheld terahertz FMCW scanning platform.

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 mm² 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 $\pm 23,\; \, 377^\circ$ at a maximum speed of $\approx \! 1700^\circ /{\rm s}$. Optimized for our setup, it moves $\pm 15^\circ$ at $\approx \! 50^\circ /{\rm s}$. 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.

Figure 10. Front and back of the 3D printed lens made of PA.

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 [Reference Schreiner, Sauer-Greff, Urbansky, von Freymann and Friederich3], 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 [Reference Schreiner, Sauer-Greff, Urbansky, von Freymann and Friederich3].

Figure 11. Reflected signal from the focal plane in the presence of a metal reflector and in its absence with (a) 80 GHz and (b) 150 GHz radars.

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 $\pm 20^\circ$.

Figure 12. A-Scans of a reflector in the image plane at seven positions on the scan line measured with (a) 80 GHz and (c) 150 GHz radars. B-Scans of a reflector in the image plane along scan-axis, and in ± 10 mm of the focal plane measured with (b) 80 GHz and (d) 150 GHz radars.

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.

Figure 13. Custom-made stepped-wedge sample of PE material: (a) back view and (b) side view. The dimensions of features and patterns are in mm. The area distinguished within the dashed rectangle roughly shows the scanned area. The steps change from 15 to 11 mm in 1 mm steps. The diameters of the holes increase in each column linearly in steps of 4, 2, and 1 mm, whereas in each row they are halved.

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 cross-sections 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.

Figure 14. B-Scans of the step-wedge sample along the scan- and motion-axes at different positions measured with (a) 80 GHz and (b) 150 GHz radars. The measured step thicknesses, in mm, along the line x = 0 are written in white on the B-scan image. The values match the optical thicknesses according to an index of refraction of approximately 1.54 for PE.

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.

Figure 15. C-Scans of the step-wedge sample along the scan- and motion-axes acquired using (a) 80 GHz and (b) 150 GHz radars.

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.

Figure 16. 3D representation of the reflection positions in the volume from the sample's interfaces measured with (a) 80 GHz and (b) 150 GHz MMIC radar units. R ₁ represents the reflection range from the first surface facing the radar module, and R ₂ determines the reflection ranges from the stepped side of the sample.

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.

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.

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 [Reference Friederich, May, Baccouche, Matheis, Bauer, Jonuscheit, Moor, Denman, Bramble and Savage18], 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.