Abstract
A homodyne detection system to acquire the thickness of silicon wafers is constructed and described. By harnessing the relationship between the transmission phase change of a 4.3-THz light beam and the incident angle controlled by a mechanical rotating stage， the thickness value of sample can be precisely deduced using the standard residual error method. The results indicate that the fitted thickness of the sample differs by only 2.5~3 μm from more accurate results measured by optical microscopes， achieving terahertz non-destructive thickness measurement with micron level accuracy. The experiment validates the effectiveness of terahertz quantum-cascade laser in non-contact and nondestructive measurement.
Terahertz light with a good transparency in most dielectrics can penetrate a certain thickness of material and achieve high-precision imaging. It thus can promise a wide range of applications in non-destructive testing and non-contact measurement
In this letter， a homodyne detection system at a working frequency of 4.3 THz is built by employing a terahertz QCL and a terahertz array detector. Two terahertz beam splitters and two gold coated mirrors are used to construct a terahertz coherent optical path， and a terahertz lens is utilized to form the interference fringes which are detected and clearly displayed by a terahertz array detector. By introducing a precision rotating mechanism into one branch of the optical path， the transmission phase curve used to fit the thickness of the measured sample is acquired. Finally， based on the standard residual error fitting method， the thickness fittings of multiple high-resistivity silicon wafers are realized. The fitting results are compared with the higher precision measurement results from optical microscopes， from which the accuracy of the above thickness measurements is roughly estimated.
The schematic diagram and photos of the terahertz homodyne detection system are shown in
Fig. 1 Terahertz homodyne detection system：（a） Schematic of the terahertz light path；（b） Photos of the experimental setup
图1 太赫兹零差探测系统： （a） 太赫兹光路示意图；（b） 实验装置照片
The terahertz QCL， operating at 77 K， is placed in a liquid nitrogen dewar and has the same active region and fabrication process as Ref. （11）. The size of the laser is 100 μm × 2 mm. The working center frequency and the average output power of the laser are about 4.3 THz and 0.8 mW， respectively. The terahertz array used in this experiment is a commercial detector which has an array size of 320 × 240. The pixel size of the array is 23.5 μm. The noise equivalent power （NEP） is about 100 pW/H
After the parallel light beam from QCL passes through the beam splitter B1， it is divided into two branches of reference beam and object beam. The former will enter the array along a straight line， and meet the latter which is directed by two mirrors M1 and M2 to go through the sample. In the experiments， for a high accuracy of the calculation， these three components of B1， M1， and M2 should be strictly aligned at an angle of 45° with respect to the incident beam. B2 is not strictly 45° for exhibiting the interference fringes. Specifically， the object light has a certain phase distribution on the array plane， allowing the interference fringes to be distinguishable at the detector array.
Firstly， in order to verify the coherence property of the terahertz beam used， a terahertz detector array was used to measure the terahertz interference fringes at the convergence point after L1 without the sample between M1 and M2. The measurement result is shown in
Fig. 2 The measured terahertz beam spot including both the reference light and the interference fringes
图2 测量得到的包括参考光和干涉条纹的太赫兹束斑
From
The above phase change can also be achieved by placing a transparent sample between M1 and M2， with the thickness changing. It should be noted that the sample must be a substance that can be easily penetrated by the terahertz light. In order to quickly achieve the change in sample thickness， we loaded the sample onto a rotational mechanical stage. The rotation of the sample leads to a change of terahertz light pathway inside the sample， resulting in changes in the optical path and the phase difference at the interference fringes.
By utilizing the coherence property of the light emitted from terahertz QCL， we can deduce the thickness （T） of solid samples with parallel front and rear surfaces using the aforementioned homodyne detection system.
Fig. 3 Schematic diagram of the optical path when the terahertz beam passes through the sample
图3 太赫兹光束通过样品时的光路传播示意图
| ， | (1) |
| ， | (2) |
When the sample is rotated， the accumulated phase through the sample at the incident light with the wavelength λ will be changed by a difference of transmission phase （φ） given by
| ， | (3) |
As shown in
Fig. 4 Theoretical curves of transmission phase （φ） changing with sample rotation angle （θ） under different thickness （T） conditions
图4 不同厚度（T）条件下，传输相位（φ）随转角（θ）变化的理论计算值
In this letter， we selected three 4-inch silicon wafers with a nominal thickness of 500 ± 10 μm for thickness measurement verification. Firstly， we accurately measured the thickness of the silicon wafer using an optical microscope （Olympus SZX16 with DP27 camera）， and the measurement results are shown in
Fig. 5 Thickness measurement results of silicon wafers numbered 1 （a）， 2 （b）， and 3 （c） based on optical microscopes， as well as the measured length of a 1 mm standard microscale plate （d）
图5 基于光学显微镜的硅片厚度测量结果： （a） 晶圆编号1； （b） 晶圆编号2； （c） 晶圆编号3， 以及（d） 1mm标准微刻度尺的长度测量结果
During the measurement， we loaded the silicon wafers on a rotating mechanical stage via a circular holder and aligned the center of the wafers with the rotation axis to ensure that the terahertz beam passes through the center of the silicon wafers. As shown in
Fig. 6 Experimental results of transmission phase （φ） with the rotation angle （θ） after the terahertz light passes through the wafer （1+2） and the wafer （1+2+3）
图6 THz光透过晶圆（1+2）和晶圆（1+2+3）后传输相位（φ）随样品旋转角（θ）变化的实验测量结果
Fig. 7 Variation of the calculated RMSE with the fitting thickness：（a） wafer （1+2）；（b） wafer （1+2+3）
图7 RMSE计算值随拟合厚度的变化：（a） 晶圆（1+2）；（b） 晶圆（1+2+3）
Fig. 8 Experimental （solid circle） and fitting （solid line） transmission phase with different rotation angle
图8 传输相位随不同旋转角度变化的实验测量（实心圆）和理论拟合（实线）结果
| Wafer number | Nominal thickness （μm） | Thickness with an optical microscope （μm） | Thickness with homodyne detection （μm） |
|---|---|---|---|
| 1 | 500 ± 10 | 500.00 | / |
| 2 | 500 ± 10 | 495.34 | / |
| 3 | 500 ± 10 | 510.09 | / |
| 1， 2 | 1 000 ± 20 | 995.43 | 992.50 |
| 1， 2， 3 | 1 500 ± 30 | 1 505.52 | 1 503.00 |
Due to the fact that with the above method of observing the movement of interference fringes， one cannot directly obtain the thickness of the tested sample， we need to use a suitable method to fit the phase change curve to indirectly obtain the thickness of the tested samples. For this reason， for the curve fitting process in this letter we adopted the standard residual error fitting method. That is， for a given set of （T0， n0）， we can obtain the residual error of the transmission phase as follows，
 ， | (4) |
where i=1，2，…，m. By calculating the root mean square error （RMSE）， it is possible to determine whether the curve fitting is good or not. Through m iterations （i=m）， the minimum RMSE is obtained， resulting in （Tm， nm）. We used a value of 3.4 for both n0 and nm as the refractive index of silicon.
From
The above thickness measurement method is based on the correspondence between the number of movements （i. e.， the transmission phase change is an integer multiple of 2π） of the interference fringes and the rotation angle of the sample， and there is a certain error in determining the movement of the interference fringes. In order to obtain more accurate results， a fast detector， such as a terahertz quantum-well photodetector
In summary， we have constructed a homodyne detection system using the terahertz quantum-cascade laser as the coherent light source and the terahertz array as the coherent beam detector. By employing a precise rotating stage to drive the measured samples to rotate， periodic variation of the transmission phase was obtained. By recording this periodic variation and combining it with the standard residual error fitting method， non-destructive thickness measurements of high-resistivity silicon wafers were achieved. For nominal thicknesses with 1 500 ± 30 μm and 1 000 ± 20 μm， the measurement results are 1 503.00 μm and 992.40 μm， respectively. The results exhibit very good agreement with those from the optical microscope， with a difference of about 2.5 μm and 3.0 μm， respectively. The above micron level non-destructive thickness measurement method based on terahertz homodyne detection system has the characteristics of non-contact and non-destructive testing and can measure any position of the sample， providing an effective method for high-precision thickness measurement in terahertz applications.
References
Tonouchi M. Cutting-edge terahertz technology ［J］. Nature Photonics， 2007， 1： 97-105. 10.1038/nphoton.2007.3 [Baidu Scholar]
Barry J R， Lee E A. Performance of coherent optical receivers ［J］. Proceedings of the IEEE， 1990， 78（8）： 1369-1394. 10.1109/5.58322 [Baidu Scholar]
Baldi A. Residual stress analysis of orthotropic materials using integrated digital image correlation［J］. Experimental Mechanics， 2014， 54（7）： 1279-1292. 10.1007/s11340-014-9859-1 [Baidu Scholar]
Disawal R， Suzuki T， Prakash S. Simultaneous measurement of thickness and refractive index using phase shifted Coherent Gradient Sensor ［J］. Optics and Laser Technology， 2016， 86： 85-92. 10.1016/j.optlastec.2016.06.015 [Baidu Scholar]
Köhler R， Tredicucci A， Beltram F， et al. Terahertz semiconductor-heterostructure laser ［J］. Nature， 2002， 417： 156-159. 10.1038/417156a [Baidu Scholar]
Hübers H -W， Pavlov S G， Semenov A D， et al. Terahertz quantum cascade laser as local oscillator in a heterodyne receiver ［J］. Optics Express， 2005， 13（15）： 5890-5896. 10.1364/opex.13.005890 [Baidu Scholar]
Zhou Z T， Zhou T， Zhang S Q， et al. Multicolor T-ray imaging using multispectral metamaterials ［J］. Advanced Science， 2018， 5： 1700982. 10.1002/advs.201870044 [Baidu Scholar]
Tan Z Y， Wan W J， Wang C， et al. Subwavelength resolved terahertz real-time imaging based on a compact and simplified system ［J］. Chinese Optics Letters， 2022， 20（9）： 091101. 10.3788/col202220.091101 [Baidu Scholar]
Qiu F C， You G J， Tan Z Y， et al. A terahertz near-field nanoscopy revealing edge fringes with a fast and highly sensitive quantum-well photodetector ［J］. iScience， 2022， 25（7）： 104637. 10.1016/j.isci.2022.104637 [Baidu Scholar]
Williams B S. Terahertz quantum-cascade lasers ［J］. Nature Photonics， 2007， 1： 517-525. 10.1038/nphoton.2007.166 [Baidu Scholar]
Tan Z Y， Wang H Y， Wan W J， et al. Dual-beam terahertz quantum cascade laser with >1 W effective output power ［J］. Electronics Letters， 2020， 56（22）， 1204-1206. 10.1049/el.2020.1376 [Baidu Scholar]
Hosako I， Sekine N， Oda N， et al. A real-time terahertz imaging system consisting of terahertz quantum cascade laser and uncooled microbolometer array detector ［J］. Proceedings of SPIE， 2011， 8023： 80230A. 10.1117/12.887947 [Baidu Scholar]
Liu H C， Song C Y， SpringThorpe A J， et al. Terahertz quantum-well photodetector ［J］. Applied Physics Letters， 2004， 84（20）： 4068-4070. 10.1063/1.1751620 [Baidu Scholar]
Zhang Z Z， Fu Z L， Wang C， et al. Research on terahertz quantum well photodetector ［J］. J. Infrared Millim. Waves.张真真， 符张龙， 王长， 等.太赫兹量子阱探测器研究进展［J］. 红外与毫米波学报， 2022， 41（1）： 103~109 [Baidu Scholar]
Guo X G， Cao J C， Zhang R， et al. Recent progress in terahertz quantum-well photodetector［J］. IEEE Journal of Selected Topics in Quantum Electronics， 2013， 19（1）： 8500508. 10.1109/jstqe.2012.2201136 [Baidu Scholar]