# Terahertz Spiral Spatial Filtering Imaging
## 2 Theory of SSF Imaging  
Figure 1 illustrates the schematic of a THz SSF imaging system, which is based on a 4\(f\) imaging system consisting of two convex lenses with the same focal length \(f\). The target is located at the front focal plane of the first lens, and the detector is located at the back focal plane of the second lens. As the SPPs have the advantages of high purity and high conversion efficiency compared to the computational holograms (CHGs) and the metamaterials, a SPP is chosen and placed at the Fourier plane for generating the spiral phase.  
Assume that a target \(O(x_{1},y_{1})\) is illuminated by a collimated THz beam, where (\(x_{1}\), \(y_{1}\)) denotes the Cartesian coordinate of the target plane. After passing through the target, the THz electric field located at the back surface of the target can be expressed as \(E_{1}(x_{1},y_{1})\), which can also be called the THz image of the target.  
The lenses in the 4\(f\) imaging system play the role of Fourier transform, and the SPP acts as a filter that changes the spatial distribution of THz beam at the Fourier plane. In the Fourier plane, the THz electric field \(E_{2}\) (\(x_{2},y_{2}\)) can be written as  
\[E_{2}(x_{2},y_{2})=\mathbb{F}\{E_{1}(x_{1},y_{1})\}H(\rho) \tag{1}\]  
where the symbol \(\mathbb{F}\{\}\) represents the Fourier transform, and (\(x_{2},y_{2}\)) denotes the Cartesian coordinate of the Fourier plane. \(H(\rho)\) is the modulation function of the SPP, and it can be expressed as  
\[H(\rho)=\text{circ}\Big{(}\frac{\rho}{R}\Big{)}e^{i\theta} \tag{2}\]  
where \(\rho\) is the polar radius in the Fourier plane, \(\text{circ}(\rho\mathbb{R})\) is the circular function describing an SPP with a radius of \(R\), and \(\theta\) is the azimuth in the Fourier plane.  
The THz electric field can be obtained at the THz detector, which reads  
\[E_{3}(x_{3},y_{3})=\mathbb{F}\{E_{2}(x_{2},y_{2})\}=E_{1}(x_{3},y_{3})\big{|} _{x_{3}=-x_{1},y_{3}=-y_{1}}*\mathbb{F}\{H(\rho)\} \tag{3}\]  
where \(E_{1}(x_{3},y_{3})\big{|}_{x_{3}=-x_{1},y_{3}=-y_{1}}\) reverses the THz image \(E_{1}\) (\(x_{1},y_{1}\)), and \(\mathbb{F}\{H(\rho)\}\) represents the point spread function (PSF) of the SPP, which can be expressed as [36]  
\[h(r)=\mathbb{F}\{H(\rho)\}=\frac{\pi R}{2r}\bigg{[}J_{0}\Big{(}\frac{kRr}{f} \Big{)}H_{1}\bigg{(}\frac{kRr}{f}\Big{)}-J_{1}\bigg{(}\frac{kRr}{f}\Big{)}H_{ 0}\bigg{(}\frac{kRr}{f}\bigg{)}\bigg{]}e^{i\varphi} \tag{4}\]  
where \(k\) is the wave vector, \(f\) is the focal length of the lens, \(J_{m}()\) is \(m\) th-order Bessel function of the first kind, and \(H_{m}()\) is \(m\) th-order Struve function. (\(x_{3}\), \(y_{3}\)) denotes the Cartesian coordinate of the THz detection plane, \(r\) is the polar radius, \(\varphi\) is the azimuth.  
Equation (3) suggests that the final THz image \(E_{3}(x_{3},y_{3})\) received by the THz detector is the convolution of the THz image \(E_{1}(x_{1},y_{1})\) and the PSF of the SPP. Since the PSF of the SPP decays quickly with the radius \(r\), the effective convolution is restricted to nearby surrounding area of each image point. According to the convolution theorem, if the nearby surrounding area of the image point is a uniform area, convolution with \(\exp(i\varphi)\) will make the image point 0. Therefore, the uniform background can be filtered out after the convolution. Instead, the edges between different intensities or phases will be highlighted, and the intensity distribution is proportional to the intensity gradient of the original THz image of the target. That is, the image contrast can be enhanced as the image background is suppressed and the edges stand out through the convolution.  
Figure 1: Schematic of the terahertz (THz) spiral spatial filtering (SSF) imaging system. A collimated THz beam illustrates the target, then it enters into the 4\(f\) imaging system. After passing through the spiral phase plate (SPP) in the Fourier plane, the filtered THz beam is received by the THz detector finally.