# Off-Axis Diffractive Optics for Compact Terahertz Detection Setup
## 5 Shorter Focal Lengths  
As a proposed design method has proved to be successful in designing off-axis lens-like structures with the focal length equal to 100 mm, the additional simulations have been performed. These numerical calculations allowed determination of how short the focal length can be while maintaining good geometrical parameters of the focal spot. The diameters of lenses have been cropped to 50 mm in all these simulations. However, it must be noted that here as well as in previously described cases the reconstructing beam is described with flat phase and Gaussian intensity distribution. It means that the introduced 50 mm aperture has not limited active structure region significantly. The phase distributions of four lens-structures with focal lengths varying from 50 mm to 10 mm together with the corresponding focal spots are presented in Figure 7.  
The longest focal length chosen to be shown in this figure is 50 mm, as no significant changes can be observed in the intensity distribution for longer focal lengths (like the 100 mm focal spot shown in Figures 3 and 5). On the other hand, distortions from the circularity of the focal spot become more and more visible for smaller focal lengths. For \(f=25\) mm the focal spot is already significantly elongated, while for \(f=10\) mm it resembles a stripe not a dot.  
The additional study on the shape of focal spots is presented in Figure 8.  
Figure 6: Off-axis focal spots, obtained from (**a**). PA12 lens, (**b**). transparent paraffin lens, (**c**). white paraffin lens and (**d**). reused paraffin lens. All images are normalized to unity, while the maximum power detected for each structure in respect to the PA12 lens is denoted as \(P_{max}\).  
Figure 7: The comparison of phase distributions (**top** panels) and calculated focal spots (**bottom** panels) of the off-axis lenses with varying focal lengths: (**a**). f = 50 mm, (**b**). f = 40 mm, (**c**). f = 25 mm and (**d**). f = 10 mm. The same two parts of all lenses have been magnified to make fine changes in the phase maps more visible.  
The orange line denotes circularity of the focal spots, calculated as the ratio of the diameters of the spots measured along horizontal (a) and vertical (b) axes. The violet line corresponds to the degree of the deflection of the incident beam--in this case it is simply a tangent of the angle between the lens surface and the deflected beam. The circularity equal to one would denote the perfect circle, but the smaller focal length the closer it gets to zero. It means that the shape is more elliptical. The dependence of the spot circularity on the focal length is in the shape of two lines with different slopes, crossing around focal length which equals to 60 mm. As it can be seen in the Figure 8, for longer focal lengths the dependence is almost flat while for shorter ones the curve is almost parallel to the beam deflection line (up to the value of circularity equal to 1). Mathematically, this curve can be described by two asymptotes, as in Equation (2):  
\[\begin{cases}\frac{a}{b}=1&\text{for }f\rightarrow\infty,\\ \frac{a}{b}=N=\frac{f}{y_{d}}&\text{for }f\to 0.\end{cases} \tag{2}\]  
The long-focal-length-limit behavior of the curve is easily understandable, as the shift from the main optical axis to the center of the structure becomes much smaller than the focal length. In this case the structure starts to neglect off-axis requirements and it is focusing almost on-axis with the perfect circular focal spot. The interesting behavior is observed for a short-focal-length-limit as the curve approaches tangent of the deflection angle of the THz beam. It seems to be the fundamental limit of off-axis focal spot circularity, at least in this design method. The breakthrough point lies at 60 mm, where the focal length is equal to the shift from the optical axis and two asymptotes intersect.