Abstract
The focal shift effect of terahertz (THz) beam focusing when using a wide-aperture refractive lens has been investigated. The deviation of focus position caused by the focal shift effect can adversely affect the imaging or measurement quality of a THz system. In this study, reference values of relative focal shifts and combinations of different lens apertures, focal lengths, and working frequencies were analyzed and discussed through theoretical calculation and finite element analysis simulation. When using the commercial lens, the actual focus was determined based on the focal shift effect to ensure the working efficiency of a terahertz system. Concerning the customized lens design, the focal shift distance was compensated in the focal length optimization according to the working frequency. These two approaches guaranteed the good performance of a THz system.
The performance of a terahertz (THz) imaging or spectroscopy system is closely related to the quality of the THz bea
The focal shift theory was first derived from the Huygens-Fresnel principle by Li and Wol
. | (1) |

Fig. 1 The schematic of THz beam focusing and focal shift effect
图1 太赫兹波束聚焦及焦移效应示意图
, | (2) |
where a and f denote the semi-aperture and focal length of THz WARL, respectively. The intensity of point P is:
, | (3a) |
, | (3b) |
where indicates the intensity of geometrical focus, and represents the Fresnel number.

Fig. 2 Theoretical calculations of focal shift effect (a) intensity distributions along the optical axis at different Fresnel numbers, (b) relative focal shift and peak intensity with Fresnel number
图2 焦移效应的理论计算(a)不同菲涅尔数下沿光轴方向的强度分布曲线,(b)相对焦移和峰值强度与菲涅尔数的关系曲线
In the simplification of
Electromagnetic simulation based on finite element analysis (FEA) was performed to analyze the intensity distribution of THz beam focusing, so as to explore the focal shift effect of THz WARLs. Considering broadband THz research and applications, the focusing characteristics of the same THz WARL at different working frequencies were first studied. The incident wave was chosen to be a Gaussian plane wave. The boundary was set to be an absorbing condition to simulate the actual infinite space. The specific methods for THz WARL design and optimization were analyzed in our previous wor

Fig. 3 Focal shift analysis of the THz WARL with a 20 mm aperture and focal length (a) to (c) FEA simulations at 0.1, 0.3, and 0.5 THz, respectively, (d) intensity distributions along the optical axis, (e) relative focal shift and peak intensity with working frequency and Fresnel number
图3 孔径和焦距均为20 mm的太赫兹WARL的焦移特性分析 (a)至(c)0.1、0.3和0.5 THz下的FEA仿真结果,(d)沿光轴方向的强度分布曲线,(e)相对焦移和峰值强度与工作频率和菲涅尔数的关系
Similarly, FEA simulations were conducted using different apertures (2a), focal lengths (f), and working frequencies (ν), with N values ranging from 2 to 32. The results are rendered in

Fig. 4 FEA simulation results obtained using different lens apertures, focal lengths, and working frequencies
图4 不同透镜孔径、焦距和工作频率下的FEA仿真结果
According to the FEA simulations, the focus position of a THz WARL is closer to the theoretical position when N is large. With respect to a customized THz WARL, the aperture and focal length can be freely designed following the working frequency and desired N value. Although N is relatively small, the focal shift can be compensated by optimizing the surface parameters. For example, the THz WARL with a 20 mm aperture and the focal length was discussed in section 3. When the working frequency is 0.1 THz, the THz WARL has an N of 1.67 and of -23.4%, and the physical focal length is calculated to be 15.32 mm. If the physical focal length is set to 20 mm, the geometrical focal length is calculated to be 28.75 mm. When the working frequency is 0.1 THz, the physical focal length is 14.7 mm, and the focal shift value is 5.3 mm (= -26.5%), as demonstrated in

Fig. 5 FEA simulations at 0.1 THz of the THz WARL with a 20 mm aperture and focal length, (a) without and (b) with focal shift compensation
图5 孔径和焦距均为20 mm的太赫兹WARL在0.1 THz频率下的FEA仿真,(a)有焦移补偿,(b)无焦移补偿
However, the choice of aperture and focal length is relatively limited for commercial THz WARLs. The commercial THz WARLs with plano-convex surfaces of three companies were investigated. The parameters and analyses of some commercial lenses are listed in
Brand | Surface type | Diameter (mm) | Focal length (mm) | (mm) |
---|---|---|---|---|
Batop | Aspherical | 25.4 | 10-67 | 1.9-13.0 |
50.8 | 35-100 | 5.2-14.9 | ||
Spherical | 25.4 | 100-150 | 0.9-1.3 | |
50.8 | 200-250 | 2.1-2.6 | ||
Thorlabs | Spherical | 50.8 | 75-500 | 1.0-7.0 |
76.2 | 115-150 | 7.8-10.2 | ||
101.6 | 151.5-200 | 10.5-13.8 | ||
Tydex | Spherical | 25.4 | 50-250 | 0.5-2.6 |
50.8 | 75-2000 | 0.3-7.0 | ||
76.2 | 100-450 | 2.6-11.8 | ||
101.6 | 200-5 000 | 0.4-10.5 |

Fig. 6 Focal shift () in relation to lens parameters () and working frequency, the area corresponding to the situation where commercial THz WARLs could be used is marked with a light-yellow background
图6 焦移(Δf/f)与透镜参数(a2/f)和工作频率的关系,浅黄色背景标记了使用商用太赫兹WARL时涉及的参数所对应的区域
In conclusion, the focal shift effect of THz beam focusing when using THz WARLs was studied. As revealed by the calculation of the theoretical expression, the actual focus position always shifted towards the lens direction. The relative focal shift is inversely proportional to the Fresnel number of a THz WARL. FEA simulations of different lens apertures, focal lengths, and working frequencies were conducted. The focal shift values were consistent with the theoretical calculation. The inaccurate focus position will impact the working efficiency of the components of a THz system, leading to curtailed imaging or measurement quality. Hence, the focal shift effect should be minimized in the practical use of THz WARLs. Concerning commercial THz WARLs, especially the configurations of focal length greater than the aperture, the actual focus position should be determined under the focal shift effect. For customized THz WARLs, if there are specific requirements for the lens size and working frequency, the focal shift distance can be offset by increasing the theoretical focal length in the design. These two approaches can guarantee the good performance of a THz system, such as the high spatial resolution of a THz image and large dynamic range of the THz spectrum.
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