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Abstract
We propose a simulation method for a multireflector terahertz imaging system. The description and verification of the method are based on an existing active bifocal terahertz imaging system at 0.22 THz. Using the phase conversion factor and angular spectrum propagation, the computation of the incident and received fields requires only a simple matrix operation. The phase angle is used to calculate the ray tracking direction, and the total optical path is used to calculate the scattering field of defective foams. Compared with the measurements and simulations of aluminum disks and defective foams, the validity of the simulation method is confirmed in the field of view of 50 cm × 90 cm at 8 m. This work aims to develop better imaging systems by predicting their imaging behavior for different targets before manufacturing.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Most theoretical and experimental studies of terahertz (THz) technology focus on THz wave generation [1], detection [2], spectroscopy [3], communication [4], imaging [5], and so on. Existing and potential applications for THz imaging are numerous. For the challenging problem of long-range observation, active THz imaging systems are able to penetrate opaque materials, protect the structure of the detected target, and achieve good image resolution [6]. To improve and optimize the THz imaging system prior to manufacturing, simulation methods that consider antenna radiation, wave propagation, and scattering of the measured object are needed.
Many physically small objects cannot be considered electrically small objects as frequencies enter THz. Physics-based models can be used to accurately determine the geometric properties of anomalies and distinguish the variations due to their presence, but the necessary computational resources for full-wave simulation is very difficult. The scattering simulation of foam density change is carried out under the condition of vertical plane wave incidence without considering the focusing characteristics of the multireflector system [7]. In vertical specular reflection imaging, the point source synthesis model can obtain the electric field distribution; however, it is still necessary to use the equal-scaled reduced electromagnetic scattering model to obtain the cross-sectional THz intensity distribution of foam void defects [8]. Bistatic imaging radars can reduce some computational resources by coupling radiation and scattering fields with conjugate fields, but the calculations of the radiation and scattering fields are still needed [9]. The raster scanning THz imaging model can be regarded as the convolution of the objective function and point diffusion function [10], and the transmission characteristics of THz waves caused by the incident angle should also be considered.
Physical optics methods have been used to simulate the incident and received electric fields at different incidence angles of multireflector THz systems [11], but the simulation is limited to measured objects with smooth surfaces [12]. Ray-tracing techniques have been shown to characterize THz images of mannequins with rough surfaces [13]. It is necessary to select a suitable scattering function to characterize the scattering characteristics of the measured object by using the ray tracing technique. The system response is simplified as a function of the incidence angle of the main ray. However, THz images of objects (such as foam voids) whose scattering properties cannot be expressed by the angle function are difficult to obtain by this main ray tracing method. In this case, it is necessary to consider the calculation of the incident field, received field and scattering field at the same time.
Therefore, a method based on angular spectrum theory is proposed to simulate the images of the multireflector THz imaging system. The phase conversion factor of each reflector is determined by using main ray tracing, and the incident electric field distribution of each scanning point is calculated by introducing a field curvature. Then, the phase conversion factor and aperture function are used to calculate the received electric field of the detector and obtain the system response. The scattering field of foam embedded defects is calculated by using point-by-point tracking. In Section 2, the THz imaging simulation of the system is introduced in detail, and the images and simulations of aluminum disks are shown. In Section 3, the scattering field calculated by the ray tracing method for embedded defects of foam is described, and images and simulations of foam defects are presented. Finally, some brief conclusions are provided in Section 4.
2. Modeling imaging system
An active bifocal THz imaging system [14] is used to validate the proposed simulation method. Figure 1(a) shows a photograph of the system. The focal length (f) of the system is 7.7 m, and the operating frequency is 0.22 THz. The 0.22 THz radiation source model is AMC666. The 0.22 THz detector is homemade with a NEP of 10−10 W/Hz1/2. The imaging distance is 8 m, and the field of view (FoV) is approximately 50 cm × 90 cm at 0.22 THz [15]. We rotated the imaging plane so that the ${z_t}$ axis was parallel to the line between the center of the main reflector and the point (0, 0, 8 m) with an angle of ${\theta _p}$ to the z axis.
The expressions of the ellipsoidal main reflector (${Z_m}$) and parabolic sub reflector (${Z_s}$) are given as Eqs. (1) and (2).
where A is the semimajor axis of the ellipsoid; B is the semiminor axis of the ellipsoid, and ${f_s}$ is the focal length of the sub reflector.It is assumed that the astigmatism of the optical system is zero and the field curvature is not zero; specifically, there is Petzval field curvature. At this time, the field curvature of the sub reflector ($Z_s^{\prime}$) and the system ($Z_m^{\prime}$) are described as follows:
Figure 1(b) shows the ray tracing process of the imaging system. The source is located at the focus of the tertiary feed reflector. THz waves are collimated by the tertiary feed reflector and emitted in the direction of ${n_{tr}}$. The incident wave (${U_{tf}}$) and the outgoing wave (${U_{tb}}$) of the tertiary feed reflector are calculated as follows:
According to angular spectrum diffraction theory, after the electric field ${U_0}$ propagates Z distance, it is expressed as follows:
where $\lambda $ is the working wavelength; ${f_x}$ and ${f_y}$ are the spatial frequencies in the x and y directions, respectively; FFT is the Fourier transform; and IFT stands for the inverse Fourier transform. The field distribution in the ${x_t}{y_t}$ plane was obtained through multiplication of the conversion factor ${T_i}$, which is calculated as follows:Based on Eqs. (7)–(13) and combined with the aperture functions of the tertiary feed reflector (${A_t}$), rotating mirror (${A_r}$), sub reflector (${A_s}$), and main reflector (${A_m}$), the calculation process of the proposed simulation method is summarized in the flow chart of Fig. 2. The transfer functions and phase conversion factors vary with rotation scanning. The incident field (${U_i}$) and received field (${U_{detector}}$) of each scanning point are obtained.
We glued aluminum disks with a diameter of 1 cm to the foam at intervals of 10 cm, as shown in Fig. 3(a). Because there is no equipment to measure refractive index in the reflection configuration, the dielectric properties of the aluminum disk are ignored in the simulation. The aluminum disks and foam are regarded as the total reflection and full transmission areas, respectively; then, the mathematical model of the measured object can be expressed as follows:
The reflection field of the measured object is ${U_r} = {U_i} \cdot test$. According to the simulation process in Fig. 2, the field distribution at the detector can be calculated to obtain the system response. Figures 3(b) and (c) show the THz image and simulation of aluminum disks. The scanning range of the system is larger than the FoV. Considering the calculation time, imaging simulation is only carried out within the FoV. The vertical coordinate ranges of Figs. 3(b) and (c) are different. The simulation do not consider the reflection of foam. The actual imaging results include the foam reflection intensity. An aluminum disk is added for positioning to determine the position of the foam board. Due to the error of manual rotation of the imaging plane, the incident angle at the center of the FoV is not zero, which causes the maximum intensity point to move down. Since the measured intensity is related to the magnification, the imaging results are uniformly normalized using the maximum intensity value. Although the dielectric properties of materials affect the field amplitude, they cannot change the normalized imaging results. From the point of the maximum intensity to the edge of the FoV, the variation trend of the intensity and radius of the aluminum disks have similar results in 0.22 THz imaging and simulation.
Fig. 3. (a) Photograph of the aluminum disks. 0.22 THz imaging (b) and simulation (c) results of the 1 cm diameter aluminum disks.
3. Calculation of the scattering field
We conducted THz images and simulations of defective foam boards with a size of 60 cm × 120 cm and a thickness of 2 cm. The foam board is ethylene-vinyl acetate copolymer (EVA), and its refractive index (${n_{EVA}} = {n_r} - i \cdot {n_i}$) is measured with the fiber-based THz TDS system [16]. At 0.22 THz, ${n_r}$ = 1.0144, and ${n_i}$ = 1.16 × 10−4. Machined defects are distributed symmetrically along the central axis of EVA boards. Figures 4(a, b) and 4(c, d) show EVA boards with embedded horizontal and vertical strip defects, respectively. The widths of the strip defect are 0.6 cm in Figs. 4(a, c) and 3 cm in Figs. 4(b, d). We tested defects with depths of 0.5, 1, and 1.5 cm. The edge of the defects is relatively smooth; therefore, we consider it as an ideal rectangular defect. Figure 4(e) shows the defect-free side of the EVA board. The defective side of the EVA board is adhered to the iron plate, which is fixed on the blackboard, as shown in Fig. 4(f).
Fig. 4. Photographs of the foam board with embedded horizontal strip defects with widths of 0.6 cm (a) and 3 cm (b). Photographs of the foam board with embedded vertical strip defects with widths of 0.6 cm (c) and 3 cm (d). Photographs of the defect-free EVA board (e) and the blackboard (f).
To minimize computational resources, the incident field is treated as a series of discrete points for geometric optical tracing. Figure 5 illustrates the discrete model. Since the incident field is represented as the product of amplitude and phase, the propagation direction of each point $({\alpha ,\beta ,\gamma } )$ is determined by the derivative of the phase angle of the incident field.
where $ANGLE$ represents the phase angle calculation. Each point must satisfy ${\cos ^2}(\alpha )+ {\cos ^2}(\beta )< 1$; otherwise, it is disregarded. Points meeting the propagation conditions are tracked to obtain the transmission optical path L, which is related to the refractive index of the medium. The electric field at each discrete incident point (${U_{trans}}$) in the transmission plane can be expressed as follows:The continuous transmitted field is obtained by scattered interpolation. According to the above method, simulation images of defective foams need to calculate the incident scattering field and the reflected scattering field, respectively. By substituting the reflected scattering field of the defective foam into the calculation process shown in Fig. 2, THz images of the defective foam can be obtained. Since THz images of the defective foam obtained by this system have both specular reflection imaging and diffuse reflection imaging, the intensity gap within the FoV is too large. We use a logarithmic scale to show the images of the defective foam.
Figures 6 and 7 show THz images and simulations of EVA boards with embedded horizontal and vertical strip defects, respectively. There are three features of the defects in the THz images: (1) the imaging characteristics of horizontal strip defects change from specular reflection to diffuse reflection with increasing incidence angle; (2) vertical strip defects are not detected outside the light spot; and (3) the change in defect intensity becomes more drastic with increasing depth. The simulation satisfies the above features. The difference between the simulation and imaging results is reflected in the light spots, which are related to the phase conversion factor. We only consider the calculation of the phase factor under the Petzval field curvature, and other aberrations are difficult to disregard in the actual measurement. The similarity between the simulation and experimental results of defective foams shows the feasibility of the simulation method, but the calculation of the phase conversion factor still needs further improvement.
Fig. 6. 0.22 THz images of the EVA boards with 3-cm-wide (a1-a3) and 0.6-cm-wide (b1-b3) embedded horizontal strip defects. (c1-c3) and (d1-d3) are the simulations of (a1-a3) and (b1-b3), respectively. The defect depths of X1, X2, and X3 are 0.5, 1, and 1.5 cm, respectively.
Fig. 7. 0.22 THz images of the EVA boards with 3-cm-wide (a1-a3) and 0.6-cm-wide (b1-b3) embedded vertical strip defects. (c1-c3) and (d1-d3) are the simulations of (a1-a3) and (b1-b3), respectively. The defect depths of X1, X2, and X3 are 0.5, 1, and 1.5 cm, respectively.
The complete simulation was executed by using a MATLAB 2016b environment on a conventional PC based on a Dell ChengMing3977 motherboard with an Intel i7-7700 CPU@3.6 GHz processor with 32 GB of RAM. To make FFT and IFT more accurate, the calculation space is –100 to 100 cm, with 0.05 cm sampling intervals. Specifically, the calculation of incident and received fields requires operation on a 4000 × 4000 matrix. Scattering field calculation requires point-by-point tracking and multiple interpolation. Not using any accelerative measures is the main reason for the long calculation time. Each pixel takes 15 s to compute.
4. Conclusion
A simulation method for multireflector THz imaging systems based on angular spectrum theory has been presented. By introducing phase conversion factors and using the main ray tracing, this method can simulate the incident field generated by the focusing system, and calculate the received electric field returned by the target scattering field. The phase angle distribution is used to calculate the direction of ray tracing to obtain the scattering field of the defective foam. The simulations were compared with the measurements to validate the simulations, and good agreement was obtained. Calculation of the phase conversion factor by using the Petzval field curvature represents an ideal case. Future work will consider the phase conversion factor of astigmatism to obtain a more accurate field.
Funding
National Natural Science Foundation of China (62004093, 62227820, 62288101, 62035014); Fundamental Research Funds for the Central Universities; Jiangsu Provincial Key Laboratory of Advanced Manipulating Technique of Electromagnetic Wave.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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