## Abstract

Recently, the research interest in indoor active millimeter wave (mmW) imaging by applying the synthetic aperture radar (SAR) technique is increasing. However, there is a lack of proper computer-aided design (CAD) tools at the system level, and almost all the R&D activities rely on experiments solely. The high cost of such a system stops many researchers from investigating such a fascinating research topic. Moreover, the experiment-oriented studies may blind the researchers to some details during the imaging process, since in most cases they are only interested in the readout from the receivers and do not know how the waves perform in reality. To bridge such a gap, we propose a modeling approach at mmW frequencies, which is able to simulate the physical process during SAR imaging. We are not going to discuss about advanced image reconstruction algorithms, since they have already been investigated intensively for decades. To distinguish from previous work, for the first time, we model the data acquisition process in a SAR imaging system successfully at mmW frequencies. We show how to perform some system-level studies based on such a simulator via a common PC, including the influence of reflectivity contrast between object and background, sampling step, and antenna's directivity on image quality. The simulator can serve system design purposes and it can be easily extended to THz frequencies.

©2012 Optical Society of America

## 1. Introduction

The mmW/THz frequency band, being a good trade-off between resolution and penetration ability, has become an ideal candidate for contraband detection [1–3]. Recently, there has been intensive research in the lab for mmW/THz SAR imaging systems via prototypes [4–6]. However, due to the lack of large sensor arrays at mmW/THz frequencies, typically only one pixel is applied in experiments for system studies. How to evaluate the influence of different parameters, such as topology of spatial sampling, interval between sensors, sensor selection, etc., on imaging performance, thus how to optimize a SAR imaging system, is an interesting and important question. A solid understanding of system parameters can help us to build a mmW/THz SAR imaging system in reality.

Till now, there is little work reported on system modeling and most of the research is solely based on experiments, in which few sensors are applied. Moreover, considering the current status of mmW/THz techniques, practical experiments are limited by cost, time and availability of proper components. Some researchers rely on experimental results too much and sometimes there is a lack of proper understanding of the behavior of waves during the imaging process. Undoubtedly, experimental studies are important. However, a good theoretical model can give more insight of the whole physical process, and therefore system designers would benefit from it.

In microwave research, there has been a lot of commercial software for full-wave calculations nowadays. However, the size of the problem which can be solved is typically limited to a few wavelengths. An indoor SAR imaging system aims at imaging at a standoff distance and the imaging array itself can be as large as tens of wavelengths, so the scale of the problem has been far beyond the limitation of the software above. On the other hand, due to the fast speed and ease of calculation, ray-tracing approach is very popular in optics, but the diffraction effect, which is quite obvious at mmW/THz frequencies, is not taken into account [7]. In this paper, we would like to discuss how to model a SAR imaging system at 100 GHz and we will illustrate how to apply such a system-level simulator to perform system studies, including the influence of object`s reflectivity, sampling step, and antenna's directivity on the image quality. The latter two cases will be linked to the experiments in [5,6], so as to make the discussions practical.

The paper is structured as follows: in Section Two, we first introduce the system setup and the applied sensors. How to model a SAR imaging system at high frequencies will be explained in Section Three. In Section Four, we will discuss the influence of the reflectivity of the object, sampling step, and the sensor's directivity on attained images. Conclusions follow in Section Five.

## 2. Applied sensors and system benchmark

To follow the experiments in [5,6], two kinds of antennas are taken into account, namely, a WR 10 probe and a pyramid horn antenna. In practice, they are widely used in the lab due to the mature waveguide technology at W-band, compared to microstrip antennas. The aperture fields of the two considered antennas are approximated by the empirical formulas in [8]. The far-field radiation patterns are investigated based on our in-house 3D array factor calculator, which has been successfully applied in [9] to understand the behavior of Hadamard phase patterns. In [10], the beam quality as predicted by the calculations is quite comparable in terms of half-power beamwidth (HPBW) with the measurement results reported in [11]. So the aperture field definitions are qualified to support further discussions for system studies.

A representative SAR imaging setup is shown in Fig. 1(a)
, including a transmitter, a receiver, and an object. Following the routine in [5,6,12], a monostatic radar setup is considered, in which the transmitting and receiving antennas are assumed to be at the same position. However, it is easy to expand our model to a bi-static radar setup. The transceiver is located at *(x’,y’,0),* and a general point on the object is represented by *(x,y,z _{0})*.

## 3. Millimeter wave SAR imaging system modeling

The procedures for modeling a SAR imaging system are illustrated in Fig. 1(b), in which oblique words correspond to the functions in the model. First, waves are radiated by the transmitting antenna and propagate to illuminate the object of interest. Incident waves interact with the object and reflected waves propagate backwards to the receiving antenna. Complex coupling happens between incident waves and the receiver, including both amplitude and phase. In practical measurements, these correspond to the readout of coherent receivers. All these above form one cycle of data sampling. Next, the transceiver moves on the sampling plane by repeating previous procedures. After finishing all the data sampling steps, image reconstruction is implemented. Essentially, this model tracks the physical process of wave radiation, propagation, reflection and detection sequentially. For numerical computations, all the components are discretized into uniform grids. Typically, we define ten grids per wavelength in the simulations. Better accuracy can be obtained at the cost of higer computational load. In what follows, we will explain the five functions in the model one by one.

#### 3.1 Wave propagation

Wave propagation is a basic problem in any model of an imaging system. As discussed previously, full-wave calculations are impossible at system level at mmW frequencies. By making a good trade-off between computation load and accuracy, the Rayleigh-Sommerfield diffraction is implemented, as shown in Eq. (1), by which *P _{0}* and

*P*stand for the points of interest on the source and destination planes, respectively. The distance between the two points is represented by

_{1}*r*,

*θ*is the angle folded by the normal direction of the source plane of the vector formed by

*P*and

_{0}*P*,

_{1}*S*is the aperture of the source,

*λ*is the wavelength in the medium and

*k*is the wavenumber.

This approach has also been applied to study focal plane imaging systems at mmW frequencies [13]. In [9], due to the introduction of fast Fourier transform (FFT), the integral calculation of Eq. (1) has been speeded up significantly, so that we can excute system simulations on common personal computers nowadays taking only a few seconds for each sampling point. This is necessary for simulating imaging systems at mmW/THz frequencies.

#### 3.2 Interaction between waves and object

The interaction between incident waves and the object determines the scattered waves, which is described by the reflectivity function *f(x,y)*. This function is simply the ratio between reflected fields and incident fields for a given material. Material properties, including dielectric constant and conductivity at mmW frequencies, are not as well-known as at microwave frequencies. Some measurement results can be found in [14,15]. Material properties at THz frequencies are being investigated and results are published in [16], including more than 200 kinds of materials. Moreover, as the interests in mmW systems increase, dedicated material characterization has become a hotspot nowadays. So in the near future, more accurate modeling of objects can be expected. Since the object has been defined on grids, *f(x,y)* can be described as a 2D matrix consequently. In the simulation, the complex response is determined by both reflection coefficients regarding different materials and phase shifts corresponding to the shape of the object. This method originates from wave optics and it has also been applied in mmW imaging studies [7,9,13].

From the electromagnetics point of view, the reflection coefficient depends on the polarization state of the waves. Unfortunately, there is little work quantising the cross-polarization features of materials at the moment. Moreover, multiple reflections may play an important role during reflection, also Fabry-Perot effects may introduce certain resonant frequencies. We will investigate these features via both full-wave calculations and measurements at mmW frequencies in our future work.

#### 3.3 Coupling between incident waves and antennas

By implementing wave propagation calculations, the field distribution on the aperture of receiving antennas can be obtained. However, due to the finite aperture of a practical antenna or probe, what is obtained from measurements is not exactly the field distribution at an ideal point. The detected signal is influenced by both the dimension and the aperture field of the applied antenna. In [17], this problem is formulated in spatial frequency domain. As a special case, the distance between the antennas and the plane of interest with a dedicated field distribution is zero in the considered case, since the coupling happens between antennas and the fields just over their apertures. Consequently, the field coupling mechanism can be described by a complex coefficient *C _{ab}* defined in Eq. (2), corresponding to the coherent response of receivers.

*E*and

_{a}*E*are the aperture fields of the antenna and incident fields over its aperture, respectively.

_{b}*C*is a normalized overlap integral, whose amplitude is between 0 and 1. In what follows, we will approximate the detector response or measurement data in practical experiments in this way.

_{ab}#### 3.4 Mechanical scanning of the transceiver

In SAR imaging systems, sampling needs to be implemented over a certain plane. It can be realized either by scanning a single receiver, or by using a receiver array. In most experiments, data sampling is realized by mechanical scanning of antennas due to the lack of large coherent receiver arrays at such high frequencies. The scanning process is defined by an interval and the position of the transceiver changes step by step in the *x-y* plane, in terms of coordinates. Since the measurement facilities are usually bulky, it is common to move the object instead of the transceivers, as in [5,6]. However, relative motion between the object and the transceiver can still be described in the same way.

#### 3.5 Image reconstruction

Image reconstruction is an old topic and there has been a lot of work on it. Since the purpose of this paper is not to look for a better image reconstruction algorithm, we simply choose the same algorithm as Sheen [12]. Essentially, it is a backward-wave reconstruction process. The amplitude decay with range is ignored since in small-scale indoor systems, it will have little impact on focusing the image. By decomposing spherical waves into a superposition of plane-wave components, image reconstruction can be expressed by Eq. (3).

*s(x,y)*is the detector response and the wavenumber

*k*is divided into three components

*k*,

_{x}*k*, and

_{y}*k*in the Cartesian coordinate system.

_{z}*f’(x,y)*is the reconstructed reflectivity function, which is expected to be similar to

*f(x,y).*

## 4. Some applications of the system-level simulator

For ease of studies and comparison, we use similar system parameters as in [5,6], in which a pyramide horn antenna is applied for illumination and a WR 10 probe is used for detection. The dimensions of the scanned aperture and the gun are 0.42m by 0.42m and 0.17m by 0.15m, respectively, which are almost completely the same as in the experiments. The working frequency is 100 GHz. As in [9], a dummy gun model is investigated, including several features, such as grating (upper-right part), slope (bottom-left part) and some random scatters (bottom-right part).

#### 4.1 Influence of object's reflectivity

First, a simple case is investigated, so as to validate this simulator. The gun-shape object is assumed to be located on top of a sheet with a certain reflectivity. The contrast between the reflectivities of the object and the background would influence the imaging result. In case of different contrast values, simulation results are shown in Fig. 2 . According to the upper row of Fig. 2, as the contrast between object and background decreases, the amplitude image of the sampling data becomes more and more confusing due to the stronger reflected waves from the background. In Fig. 2(a), the background almost disappears in the reconstructed image because of the quite large contrast. In Fig. 2(b), the moderate contrast between object and background illustrate both parties quite clearly. The object and the background almost merge together in the reconstructed image in Fig. 2(c), in which only the grating, slope and randomly-rough patterns are visible. In practice, we can approximate the resolving ability of a SAR imaging system by taking objects of different materials into account once the noise and SNR performance of the receivers are known.

#### 4.2 Influence of sampling step

The example above is quite simple, but it has proved that such a simulator can serve system studies. More advanced topics are possible. Sampling intensity is an interesting question, since it decides the number of transceivers, thus the cost of the system in practice. In [5], based on a lot of measurement results, it is commented that the sampling step does not need to strictly follow the Nyquist sampling criteria when a high-gain antenna is applied. We have simulated the experiments by the simulator. Since experiments are implemented in the anechoic chamber, the reflectivity of the background is simply defined to be zero in the simulations. Various sampling steps have been studied, from half a wavelength to ten wavelengths, with half a wavelength as the increment.

Figure 3 illustrates how the image quality changes as the sampling step becomes larger. In Fig. 3(a), the Nyquist sampling criteria is satisfied and the object is very clear. When the sampling step is one wavelength, the Nyquist sampling criteria is violated. However, as shown in Fig. 3(b), the difference with the reconstructed image in Fig. 3(a) is not visible. When the sampling step becomes as large as 3.5 wavelengths, the reconstructed image becomes blurred, but the reconstructed image of the object is still quite acceptable in Fig. 3(c), although no interpolation is implemented. The bright lines around the central region of the gun correspond to aliasing due to the spectrum overlap introduced by high spatial-frequency components. Since the shape of the object in [5] is quite simple, the reconstructed image is not very sensitive to the sampling step. So by using a high-gain antenna, a relatively large sampling step is not that fatal. Consequently, sampling criteria can be loosened. So the conclusions that we draw from our simulation-based studies and the results obtained from the experimental studies converge.

#### 4.3 Influence of antenna's directivity

In [6], according to experimental studies, it is commented that high-gain antennas can give better images compared to low-gain antennas. In what follows, we are going to put it into question. Regarding different setups, we have studied the following cases based on two types of sensors introduced in Section Two: horn-horn, horn-probe, and probe-probe, with the former one and the latter one as the transmitting antenna and the receiving antenna, respectively. Simulation results for these three cases are shown in Fig. 4 .

In the upper row of Fig. 4, the amplitude images of the echoes are illustrated, corresponding to the readout signals from the sensor in the experiments. The object is completely blurred in the echoes. In case of the probe-probe combination, the profile information of the gun is almost lost. While near the border of the scanning aperture, unlike the other two cases, we can still sense some weak echoes successfully by using the probe-probe combination. Essentially, high-spatial frequency components are linked to these weak signals, which help to visualize the details of the object. Reconstructed images are shown in the bottom row of Fig. 4. Compared to Fig. 4(a), Fig. 4(b) is clearer. The improvement is limited, since the high-gain antenna itself eliminates the high spatial-frequency components. Consequently, it becomes the bottleneck towards high resolution. This agrees with the discussions in Section 4.2. Basically, as the number of probes increases, the image becomes sharper.

Due to the assigned features of the gun, large diffraction angles are introduced by the grating structure and random scatterers. Considering the size of the scanning aperture and the distance between the aperture and the object, the intersection angle is around 34 degrees. From the geometrical optics point of view, due to the slope structure, reflected waves are guided to the direction, which is 106 degrees deviated from the direction of the incident waves. It means that we cannot reconstruct this part even if we do coherent detection perfectly over a infinitely large plane. This part can only be reconstructed by using cylindrical or spherical scanning. That is why we cannot see the slope structure in all the three reconstructed images. For the grating part, the unit size is one wavelength, and a *pi* phase shift is introduced. According to the antenna array theory, the main beam is in the direction with an elevation angle of 30 degrees in this case. Consequently, most of the scattered power spills over the sampling aperture already. The information is totally lost by using either the horn-horn or the horn-probe combination. However, we can still sense the structure marginally when the probe-probe combination is applied, as shown in Fig. 5
. Moreover, we can even see some diffraction fringes, which may help us to conjecture the existence of the grating structure. For the random-phase part, only the probe-probe combination gives the complete description and some blank holes appear in the other two cases, implying that some information of the object is lost. Resolution ability in each case can also be checked from the trigger part. The probe-probe combination distinguishes the trigger well, apart from the bottom line. The horn-horn combination gives the most blurred image, thus lowest resolution.

To understand the phenomenon that high resolution is attained by using the low-gain antenna in Fig. 4, exact illuminations on the object are investigated. In the worst case, the transceiver is located at the corner of the scanning aperture; while in the best case, it is in the central point. In Fig. 6 , in the worst case by using horn antenna, only the left part of the gun is illuminated and most power escapes elsewhere without shining the gun. Even in the best case in Fig. 6(b), the incident waves on the surface of the gun differ by more than 3 dB in amplitude. On the other hand, by using the probe antenna, the object is almost uniformly illuminated, with an amplitude difference of 0.5 dB in the worst case. So by using high-gain antennas, the transceiver only senses local information of the object during each scanning; while low-gain antennas may always give a global description of the object. In [6], the degradation in effective illumination by using horn antenna is ignored and practical illuminations during scanning do not gain enough attention in the experimental studies. Actually, it is assumed that the object is always inside HPBW of the high-gain horn. However, the simulator gives us more details about the imaging process compared to experimental studies, in which generally only the readout from receivers are recorded and no information about field distributions during wave propagations can be attained.

## 5. Conclusions

In this paper, for the first time, a system-level simulator has been developed, which can support various mmW/THz SAR imaging system studies. We have explained the modeling approach, which is to track wave radiation, propagation, interaction with objects and detection of electromagnetic waves sequentially. We have shown how to use this simulator to study practical systems, including influence of reflectivity of the object, sampling step of transceivers, and the relationship between sensor’s directivity and resolution. We conclude that by using high-gain antennas, requirement on sampling steps can be weakened. Moreover, high-gain antennas can be the bottleneck towards high resolution. Compared to experimental studies, we can draw some similar conclusions and the behavior of waves during the imaging process can be clearly understood. We believe that such a simulation tool can serve system design and imaging studies in a much wider range, for example, for investigating the influence of positioning error of the transceiver, array topology, sparse sampling, compressive imaging, influence of crosstalk between adjacent pixels, bi-static radar setup, etc. To sum up, the proposed simulator is quite versatile and it may become a complementary approach to current experimental studies for mmW/THz SAR imaging systems.

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