# Off-Axis Diffractive Optics for Compact Terahertz Detection Setup
## 2 Design and Modeling  
The concept of a compact in-reflection setup for skin cancer diagnosis is presented in Figure 1. The two designed diffractive optical elements are working in transmission and additionally are redirecting the incident radiation off-axis and focus the beam at a particular distance. In this approach, two point-to-point lenses are used (marked as DOE). Due to the symmetry of the setup, both lenses can actually be considered to be the same diffractive structure, rotated by 180\({}^{\circ}\). The structure needs to be thin and light with short focal length, which allows focusing of THz radiation into a small focal spot located outside the contours of the device. These restrictions practically exclude the application of refractive lenses; as such, lenses would be too bulky and heavy [9]. Using a refractive lens in the described geometry would require either partial illumination of a larger lens or mounting the lens at some angle to the structure [16]. In the first case, the f-number of the lens would be very small (below 1). Such lenses are extremely thick and challenging to manufacture as refractive  
Figure 1: The configuration of the in-reflection setup for the sample examination using two off-axis diffractive structures (DOEs). The radiation from the divergent source illuminates the diffractive element focusing it at some angle and at some distance after the structure. Such solution can help in minimizing the distance between the element and the sample, thus the off-axis focusing is required. Then, the radiation reflected or scattered on the sample is collected by the element with the same shape and geometry to focus this radiation on the detector. The concept is general, allowing for the arbitrary choice of the source and detector of radiation as well as sample under test. Here, VDI Schottky diodes with frequency multipliers have been used. structures. In the second case, the focal spot would be very elongated (a sample would be placed at an angle to the focal plane of the lens). On the other hand, redirection of the terahertz radiation with use of mirrors is also possible and commonly used [17]. However, once again, in the short-focal-length and off-axis focusing design, mirrors would result in unnecessary bulkiness of the system. The redirection and focusing of the radiation with parabolic mirrors at an angle higher than 90\({}^{\circ}\) is also challenging. Therefore, our research has been focused on diffractive structures with thickness of the order of several millimeters. Light propagation in the Fresnel region and modified convolution method have been used in numerical simulations [18]. All calculations have been performed on 4096 \(\times\) 4096 pixels square matrix with 117 \(\upmu\)m sampling. The design frequency was equal to 520 GHz which corresponds to 0.58 mm design wavelength. Three different design approaches were verified: cutting only part of off-axis lens described by analytical equation, the back-propagation of diffraction-limited focal spot, and the iterative algorithm. In the first method, the diffractive lens is designed according to the lens equation in the non-paraxial approach, and its off-axis part is taken. The second method relies on the definition of the target intensity distribution (ideal focal spot) and propagating it at the negative distance to the lens plane. The third approach, in the simplified view, is based on the iterative implementation of previous two methods. The phase distribution of the structure obtained using the first method can be seen in the top left part of Figure 2.  
A diffractive off-axis lens coded in the form of kinoform with the focal length of 100 mm has been designed. The phase distribution is described by the following equation after applying modulo 2\(\pi\) operation:  
\[\phi(r)=-\frac{2\pi}{\lambda}\sqrt{f^{2}+r^{2}}, \tag{1}\]  
where \(\phi\) is the phase delay introduced by the lens, \(r\) is radial coordinate (calculated from the main optical axis), \(\lambda\) is the design wavelength and \(f\) is the focal length.