0 Introduction

Laser tracing measurement system is a high-precision and large-size measuring instrument. Laser precision tracing measurement technology plays an increasingly important role in the fields of high-precision measurement, such as aerospace, automobile, shipbuilding and machinery manufacturing^[1-4]. In 2010, a German company ETALON launched a laser tracing measurement system, which greatly improved the measurement accuracy of spatial distance by means of a laser through the patented design of the reference sphere.

High-precision machining has raised the requirement of measurement accuracy of laser tracing measurement technology. Domestic and foreign scholars extensively explored error modeling and error compensation methods of laser tracing measurement system. Lau ^[5]established the model of laser tracer's shafting perpendicularity error, dead path error of laser interferometry and laser beam alignment error and analyzed the ranging error caused by beam deviation, but they did not consider the performance of optical components operating under non-ideal conditions.

In a laser tracing measurement optical system, various factors may lead to measurement errors, such as the performance of optical components under non-ideal conditions and the orientation errors of optical components, which largely affect the interference fringe contrast of a laser tracing measurement optical system^[6-11]. The analysis of these errors can provide theoretical basis for the improvement of accuracy, reliability evaluation, optical system design and optical component selection of the laser tracing measurement system^[12-15].

A method was proposed to analyze the influences of the non-ideal spectroscopic performance of optical components and orientation errors of a laser tracing measurement optical system on the tracing performance. An analysis method for the performance optical components under non-ideal conditions and orientation errors of optical components can effectively improve the measurement accuracy of a laser tracing measurement optical system.

1 Interference Fringe Contrast Model of Laser Tracing Measurement Optical System

The schematic diagram of a laser tracing measurement optical system is shown in Fig. 1. The circularly polarized light generated by the laser source was transformed to linearly polarized light after passing through the analyzer P₁. The p-polarization light which passed through the polarizing beam splitter PBS₁ was transformed to circularly polarized light as the reference beam after transmitting through the quarter-wave plate QW₁^[16].

Meanwhile, the s-polarization light was reflected after transmitting through PBS₁. The measurement part is shown in the dashed block diagram in Fig. 1. The s-polarization transmitted through the quarter-wave plate QW₂ to obtain circularly polarized light. After being reflected by the reference sphere, the light transmitted through QW₂ again to obtain p-polarization light. Then the light transmitted through PBS₁ again and the quarter-wave plate QW₃ to obtain circularly polarized light. The light transmitted through the beam splitter BS₂ in the tracing part after being reflected by the cat's eye retroreflector, and the reflected light was split by BS₂. The reflected light was received by a position sensitive detector (PSD). At the same time the transmission light passed through QW₃ to obtain s-polarization, which was reflected by PBS₁ and then transmitted through QW₁ to obtain circularly polarized light as the measuring beam.

The interference part is also shown in the dashed block diagram in Fig. 1. The reference beam and measuring beam enter the interference part of a laser tracing measurement optical system and generate four interference signals with phase differences of 90°, which are received and processed by the four photodetectors in the interference part respectively. When the cat's eye retroreflector made random moves in the measurement space, the position of the laser spot reflected by the target was changed, resulting in changes in the optical path difference of the laser interferometer system. The changes in the position of the laser spot were detected by the position detector PSD. The change in the optical path difference was observed by the laser interferometer system.

The propagation direction of the laser beam was z-axis; y-axis was parallel to the incident surface of laser beam and perpendicular to z-axis; x-axis was perpendicular to y-z plane. The circularly polarized light beam emitted by a single-frequency laser had the frequency f, an amplitude of E₀₁, and the initial phase was φ₀. The light was polarized at an angle μ to the transmission axis of the analyzer P₁^[8].

$ \boldsymbol{E}_{x}=\boldsymbol{i}\left[\cos \mu \cdot \boldsymbol{E}_{01} \sin \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right)\right] $

(1)

$ \boldsymbol{E}_{y}=\boldsymbol{j}\left[\sin \mu \cdot \boldsymbol{E}_{01} \sin \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right)\right] $

(2)

where i is the polarization direction vector of p-polarization and parallel to the incident plane; j is the polarization direction vector of s-polarization and perpendicular to the incident plane.

The transmission coefficients of the p-polarization of the polarizing beam splitters (PBS₁, PBS₂, and PBS₃) were t_p1, t_p2, and t_p3 respectively and the reflection coefficients of the s-polarization of the polarizing beam splitters were r_s1, r_s2, and r_s3 respectively. θ was the angle between the fast axis direction of the quarter-wave plate QW₁ and the light vector vibration direction of the p-polarization. α was the angle between the fast axis direction of the quarter-wave plate QW₂ and the light vector vibration direction of the p-polarization. δ was the angle between the fast axis direction of the quarter-wave plate QW₃ and the light vector vibration direction of the p-polarization. The energy ratios of the transmitted light and the reflected light of the beam splitters BS₁ and BS₂ were A₁/B₁ and A₂/B₂ respectively. τ was the angle between the fast axis direction of the half-wave plate HW and the light vector vibration direction of the p-polarization. Δφ was the variation of the optical path difference caused by the movement of the cat's eye retro-reflector.

The reference beam and the measurement beam converged and interfered at QW₁. At this position, the intensity of the reference beam was:

$ I_{R}=\left(t_{p 1} \cdot \cos \mu+\left(1-r_{s 1}\right) \cdot \sin \mu\right) \cdot E_{01} $

(3)

The intensity of the measuring beam at the same position was:

$ \begin{aligned} I_{L}= & \left\{r_{s 1} \cdot \sin \delta \cdot\left(\frac{A_{2}}{A_{2}+B_{2}}\right)^{2} \cdot \cos \delta \cdot\right. \\ & {\left[4 \cdot t_{p 1} \cdot \cos \alpha \cdot \sin \alpha+2\left(1-r_{s 1}\right)\left(\cos ^{2} \alpha-\sin ^{2} \alpha\right)\right] . } \\ & {\left[r_{s 1} \cdot \sin \mu+\left(1-t_{p 1}\right) \cdot \cos \mu\right]+} \\ & \left(1-t_{p 1}\right)\left(\cos ^{2} \delta-\sin ^{2} \delta\right)\left(\frac{A_{2}}{A_{2}+B_{2}}\right)^{2}\cdot\\ & {\left[2 \cdot t_{p 1} \cdot \cos \alpha \cdot \sin \alpha+\left(1-r_{s 1}\right)\left(\cos ^{2} \alpha-\sin ^{2} \alpha\right)\right] \cdot} \\ & \left.\left[r_{s 1} \cdot \sin \mu+\left(1-t_{p 1}\right) \cdot \cos \mu\right]\right\} \cdot E_{01} \end{aligned} $

(4)

The intensity of the interference signal received by the photoelectric detector PD₁ was^[6] :

$ \begin{aligned} I_{\mathrm{PD} 1} & \sqrt{\left(H_{1} \cdot I_{2}+H_{2} \cdot I_{1}\right)^{2}+\left(H_{2} \cdot I_{2}-H_{1} \cdot I_{1}\right)^{2}} . \\ & \sin \left(\Delta \varphi+\varphi_{1}\right) \end{aligned} $

(5)

where

$ \begin{gathered} H_{1}=t_{p 3} \cdot \cos (2 \tau) \cdot \frac{A_{1}}{A_{1}+B_{1}} \cdot \cos \theta \cdot I_{R} \\ H_{2}=\left(1-r_{s 3}\right) \cdot \cos ({\rm{ \mathsf{ π} }}+2 \tau) \cdot \frac{A_{1}}{A_{1}+B_{1}} \cdot \sin \theta \cdot I_{R} \\ I_{1}=t_{p 3} \cdot \cos (2 \tau) \cdot \frac{A_{1}}{A_{1}+B_{1}} \cdot \cos \theta \cdot I_{L} \\ I_{2}=\left(1-r_{s 3}\right) \cdot \cos ({\rm{ \mathsf{ π} }}+2 \tau) \cdot \frac{A_{1}}{A_{1}+B_{1}} \cdot \sin \theta \cdot I_{L} \\ \varphi_{1}=\arctan \left(\frac{H_{2} \cdot I_{2}-H_{1} \cdot I_{1}}{H_{1} \cdot I_{2}+H_{2} \cdot I_{1}}\right) \end{gathered} $

The intensity of the interference signal received by the photoelectric detector PD₂ was:

$ \begin{aligned} I_{\mathrm{PD} 2} \sim & \sqrt{\left(G_{1} \cdot K_{2}+G_{2} \cdot K_{1}\right)^{2}+\left(G_{1} \cdot K_{1}-G_{2} \cdot K_{2}\right)^{2}} . \\ & \sin \left(\Delta \varphi+\varphi_{2}\right) \end{aligned} $

(6)

where

$ \begin{gathered} G_{1}=r_{s 3} \cdot \cos ({\rm{ \mathsf{ π} }}+2 \tau) \cdot \frac{A_{1}}{A_{1}+B_{1}} \cdot \sin \theta \cdot I_{R} \\ G_{2}=\left(1-t_{p 3}\right) \cdot \cos (2 \tau) \cdot \frac{A_{1}}{A_{1}+B_{1}} \cdot \cos \theta \cdot I_{R} \\ K_{1}=r_{s 3} \cdot \cos ({\rm{ \mathsf{ π} }}+2 \tau) \cdot \frac{A_{1}}{A_{1}+B_{1}} \cdot \sin \theta \cdot I_{L} \\ K_{2}=\left(1-t_{p 3}\right) \cdot \cos (2 \tau) \cdot \frac{A_{1}}{A_{1}+B_{1}} \cdot \cos \theta \cdot I_{L} \\ \varphi_{2}=\arctan \left(\frac{G_{1} \cdot K_{1}-G_{2} \cdot K_{2}}{G_{1} \cdot K_{2}+G_{2} \cdot K_{1}}\right) \end{gathered} $

The intensity of the interference signal received by the photoelectric detector PD₃ was:

$ \begin{aligned} & I_{\mathrm{PD3}} \sim \sqrt{\left(M_{1} \cdot N_{2}+M_{2} \cdot N_{1}\right)^{2}+\left(M_{2} \cdot N_{2}-M_{1} \cdot N_{1}\right)^{2}} . \\ & \quad \sin \left(\Delta \varphi+\varphi_{3}\right) \end{aligned} $

(7)

where

$ \begin{aligned} M_{1}= & {\left[r_{s 2} \cdot \sin \theta+\left(1-t_{p 2}\right) \cdot \cos \theta\right] \cdot \frac{B_{1}}{A_{1}+B_{1}} \cdot \cos \theta \cdot I_{R} } \\ M_{2}= & {\left[r_{s 2} \cdot \cos \theta-\left(1-t_{p 2}\right) \cdot \sin \theta\right] \cdot \frac{B_{1}}{A_{1}+B_{1}} \cdot } \\ & \sin \theta \cdot I_{R} \\ N_{1}= & {\left[r_{s 2} \cdot \sin \theta+\left(1-t_{p 2}\right) \cdot \cos \theta\right] \cdot \frac{B_{1}}{A_{1}+B_{1}} . } \\ & \cos \theta \cdot I_{L}\\ N_{2}=& \left[r_{s 2} \cdot \cos \theta-\left(1-t_{p 2}\right) \cdot \sin \theta\right] \cdot \frac{B_{1}}{A_{1}+B_{1}} \cdot\\ &\sin \theta \cdot I_{L} \end{aligned} $

$ \varphi_{3}=\arctan \left(\frac{M_{2} \cdot N_{2}-M_{1} \cdot N_{1}}{M_{1} \cdot N_{2}+M_{2} \cdot N_{1}}\right) $

The intensity of the interference signal received by the photoelectric detector PD₄ was:

$ \begin{aligned} I_{\mathrm{PD} 4} \sim & \sqrt{\left(O_{1} \cdot P_{2}+O_{2} \cdot P_{1}\right)^{2}+\left(O_{2} \cdot P_{2}-O_{1} \cdot P_{1}\right)^{2}} . \\ & \sin \left(\Delta \varphi+\varphi_{4}\right) \end{aligned} $

(8)

where

$ \begin{aligned} O_{1}& =\left[t_{p 2} \cdot \cos \theta+\left(1-r_{s 2}\right) \cdot \sin \theta\right] \cdot \frac{B_{1}}{A_{1}+B_{1}} \cdot \cos \theta \cdot I_{R} \\ O_{2}& =\left[-t_{p 2} \cdot \sin \theta+\left(1-r_{s 2}\right) \cdot \cos \theta\right] \cdot \frac{B_{1}}{A_{1}+B_{1}} \cdot \sin \theta \cdot I_{R}\\ P_{1}& =\left[t_{p 2} \cdot \cos \theta+\left(1-r_{s 2}\right) \cdot \sin \theta\right] \cdot\left(\frac{B_{1}}{A_{1}+B_{1}}\right) \cdot \cos \theta \cdot I_{L}\\ P_{2}& =\left[-t_{p 2} \cdot \sin \theta+\left(1-r_{s 2}\right) \cdot \cos \theta\right] \cdot \frac{B_{1}}{A_{1}+B_{1}} \cdot\\ & \sin \theta \cdot I_{L} \end{aligned} $

$ \varphi_{4}=\arctan \left(\frac{O_{2} \cdot P_{2}-O_{1} \cdot P_{1}}{O_{1} \cdot P_{2}+O_{2} \cdot P}\right) $

The interference fringe contrast indicates the clarity of the fringes in the interference field. The total fringe contrast in the laser tracing measurement optical system and the fringe contrast of the four-way interference signal were^[7] :

$ K_{\mathrm{TOTAL}}=\frac{2\left(E_{r}^{\prime} / E_{l}^{\prime}\right)}{1+\left(E_{r}^{\prime} / E_{l}^{\prime}\right)^{2}} $

(9)

where

$ \begin{aligned} E_{r}^{\prime}= & I_{R} \cdot\left[\cos \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right)-\sin \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right)\right] \\ E_{l}^{\prime}= & -I_{L}\left[\cos \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}+\Delta \varphi\right)+\sin (2 {\rm{ \mathsf{ π} }} f t+\right. \\ & \left.\left.\varphi_{0}+\Delta \varphi\right)\right] \end{aligned} $

and

$ K_{\mathrm{PD} 1}=\frac{2\left(E_{r 1}^{\prime} / E_{l 1}^{\prime}\right)}{1+\left(E_{r 1}^{\prime} / E_{l 1}^{\prime}\right)^{2}} $

(10)

where

$ \begin{aligned} E_{r 1}^{\prime}= & I_{1} \cdot \sin \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right)-I_{2} \cdot \cos \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right) \\ E_{l 1}^{\prime}= & -H_{1} \cos \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}+\Delta \varphi\right)-H_{2} \sin (2 {\rm{ \mathsf{ π} }} f t+ \\ & \left.\varphi_{0}+\Delta \varphi\right) \end{aligned} $

and

$ K_{\mathrm{PD} 2}=\frac{2\left(E_{r 2}^{\prime} / E_{l 2}^{\prime}\right)}{1+\left(E_{r 2}^{\prime} / E_{l 2}^{\prime}\right)^{2}} $

(11)

where $E_{r 2}^{\prime}=-G_{1} \cos \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right)+G_{2} \sin \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right)$, $E_{l 2}^{\prime}=-K_{1} \cos \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}+\Delta \varphi\right)-K_{2} \sin (2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}+\Delta \varphi)$

and

$ K_{\mathrm{PD} 3} =\frac{2\left(E_{r 3}^{\prime} / E_{l 3}^{\prime}\right)}{1+\left(E_{r 3}^{\prime} / E_{l 3}^{\prime}\right)^{2}} $

(12)

where

$ \begin{aligned} E_{r 3}^{\prime}= & M_{1} \sin \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right)-M_{2} \cos \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right) \\ E_{l 3}^{\prime}= & -N_{1} \sin \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}+\Delta \varphi\right)-N_{2} \cos (2 {\rm{ \mathsf{ π} }} f t+ \\ & \left.\varphi_{0}+\Delta \varphi\right) \end{aligned} $

and

$ K_{\mathrm{PD} 4}=\frac{2\left(E_{r 4}^{\prime} / E_{l 4}^{\prime}\right)}{1+\left(E_{r 4}^{\prime} / E_{l 4}^{\prime}\right)^{2}} $

(13)

where

$ \begin{aligned} E_{r 4}^{\prime}= & O_{1} \sin \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right)-O_{2} \cos \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}\right) \\ E_{l 4}^{\prime}= & -P_{1} \sin \left(2 {\rm{ \mathsf{ π} }} f t+\varphi_{0}+\Delta \varphi\right)-P_{2} \cos (2 {\rm{ \mathsf{ π} }} f t+ \\ & \left.\varphi_{0}+\Delta \varphi\right) \end{aligned} $

Eqs.(9)-(13) indicate the interference fringe contrast model for a laser tracing measurement optical system.

2 Influences of Various Error Sources on Interference Fringe Contrast Model

In a laser tracing measurement optical system, the main optical components include an analyzer, beam splitters, polarizing beam splitters, quarter-wave plates, and a half-wave plate. Based on Eqs. (9)-(13), the influences of the performance of optical components under non-ideal conditions and orientation errors on the interference fringe contrast of a laser tracing measurement optical system were investigated below.

2.1 Influences of Placement Angle of Analyzer

In the absence of other error sources, the incident light was circularly polarized at an azimuth of μ to the transmission axis of the analyzer. According to the established models (Eqs. (9)-(13)), the values of interference fringe contrast were obtained (as shown in Table 1). The simulation results based on the interference fringe contrast model shows that the placement angle of the analyzer had an influence on the interference fringe contrast. If light was polarized at an angle μ of 65° to the transmission axis of the analyzer and passed through the analyzer, the contrast of the four-way interference fringe reached 0.9996. In this case, the interference fringes of the system were the clearest, thus improving the measurement precision. When the placement angle of the analyzer was not equal to 65°, the interference fringe contrast of the system decreased significantly, and when μ=25°, the interference fringe contrast (0.4263) decreased by 57%. When μ=85°, the interference fringe contrast (0.3528) is decreased by 65%. Therefore, when the placement angle of the analyzer was 65°, a laser tracing measurement optical system achieved the highest quality beam with the highest interference fringe contrast and stability.

2.2 Influences of BS Split Ratio

The change in the split ratio of the beam splitter BS₁ in the interference part and the beam splitter BS₂ in the tracing part affected the energy of the interference signal, thus changing the contrast value of interference fringe.

Table 2 shows the interference fringe contrast values of a laser tracing measurement optical system with a split ratio of BS₂ of 7∶3 and different split ratios of BS₁. Under the fixed split ratio of BS₂ of 7∶3 and different split ratios of BS₁, the interference fringe contrast value of the resulting interference signal remained unchanged.

Fig. 2 shows the change in the contrast of the resulting interference signal of the laser tracing measurement system with a split ratio of BS₁ of 5∶5 and different split ratios of BS₂. Under the fixed split ratio of BS₁, the changing split ratio of BS₂ shows the significant influence on the fringe contrast of the resulting interference signal. When the split ratio of BS₂ was changed from 8∶2 to 2∶8, the contrast of interference fringes decreased by 87% from 0.9996 to 0.1281. However, in order to obtain interference fringes with high quality and meet the requirement of the tracing measurement system, the reflected light of beam splitter in the tracing part should be applied to the position detector PSD to achieve the tracing measurement of the cat's eye retro-reflector. Under the split ratio of beam splitters in the interference part (BS₁) of 5∶5, when the splitting ratio of BS₂ changed from 2∶8 to 8∶2, the fringe contrast of the interference signals received by the photodetectors increased, but the injection light intensity onto the PSD reflected by BS₂ decreased.

2.3 Influences of Non-ideal Spectroscopic Performance of PBS for Laser Tracing Measurement Optical System

With a split ratio of BS₁ of 5∶5 and the split ratio of BS₂ of 7∶3, the contrast of the interference fringe of a laser tracing measurement optical system was optimum. An effect of the PBS with non-ideal spectroscopic performance on the contrast of interference fringes were explored with the above conditions. The transmittance and the reflectance of polarizing beam splitter were respectively T_p and R_s.

The values of interference fringe contrast with ideal transmittance and different reflectivities are plotted in Fig. 3(a). With an ideal transmittance, the value of interference fringe contrast decreased with the decrease in the reflectivity and the interference fringe contrast of the signals obtained by the four receivers PD₁, PD₂, PD₃, and PD₄ were the same. When the reflectivity of the PBS₁ decreased from 1.0000 to 0.9400, the values of interference fringe contrast decreased by about 1.5% from 0.9996 to 0.9846.

Fig. 3(b) shows interference fringe contrast with ideal reflectivity and different transmittances. With ideal reflectivity, the value of interference fringe contrast decreased with a decrease in the transmittance. The contrast values of the interference fringes received by the photoelectric receivers PD₁, PD₂ and PD₃ were lower than that of the interference fringe received by PD₄. When the transmittance of PBS₁ decreased from 1.0000 to 0.9400, the value of interference fringe contrast decreased by about 0.76% from 0.9996 to 0.9920.

According to the models (Eqs. (9)-(13)), the effects of PBS₂ and PBS₃ with non-ideal spectroscopic performance on the interference fringe contrast in laser tracing optical system were analyzed. Table 3 shows the interference fringe contrast of PBS₂ under the ideal transmittance and different reflectivities. Table 4 shows the interference fringes contrast value of PBS₂ under the ideal reflectivity and different transmittances. Table 5 shows the interference fringes contrast value of PBS₃ under the ideal transmittance and different reflectivities. Table 6 shows the interference fringes contrast value of PBS₃ under the ideal reflectivity and different transmittances. It can be seen that PBS₂ and PBS₃ with non-ideal spectroscopic performance had less influence on the interference fringe contrast in laser tracing optical systems. What's more, when PBS₁, PBS₂ and PBS₃ were with non-ideal spectroscopic performance at the same time, the interference fringe contrast in a laser tracing optical system was seldom changed.

2.4 Influences of Quarter-wave Plate Orientation Errors

In a laser tracing measurement optical system, when the angle between the incident beam and quarter-wave plate fast axis was 45°, the quarter-wave plate can convert the linearly polarized incident beam into one with circular polarization and convert the circularly polarized incident beam into the linearly polarized beam with the polarization vector orthogonal to that of the incident beam. When other error sources did not exist, the incident light was polarized at an azimuth of 45°+e to the fast axis of the quarter-wave plate, where e represents the orientation errors of the quarter-wave plate QW₁. According to the models (Eqs. (9)-(13)) the values of interference fringe contrast of a laser tracing measurement optical system are shown in Table 7.

Table 7 shows the influences of the orientation errors (e) of the quarter-wave plate QW₁ on interference fringe contrast. It can be seen that orientation errors of QW₁ have little effect on the interference fringe contrast.

Fig. 4 shows the influences of the orientation errors (e) of the quarter-wave plate QW₂ on interference fringe contrast. If no other error source existed, the interference fringe contrast decreased with an increase in e and the interference fringe contrasts of the photoelectric receivers PD₁ to PD₄ were the same. When e of QW₂ varied from 0° to ±5°, the value of interference fringe contrast decreased by about 0.18% from 0.9996 to 0.9978.

Fig. 5 demonstrates the influence of the orientation errors (e) of the quarter-wave plate QW₃ on interference fringe contrast. The contrast value increased with an increase in the e. The interference fringe contrasts of the photoelectric receivers PD₁ to PD₄ were the same. As shown in Fig. 5, the e of the analyzer varied from 0° to 5°, whereas the value of interference fringe contrast increased by about 0.03% from 0.9996 to 0.9999.

2.5 Influences of half-wave Plate Orientation Error

The behavior of a half-wave plate HW on a beam of plane-polarized light was completely different from that of a quarter-wave plate. If the incident plane of vibration was at an azimuth 22.5° with respect to the fast axis of the half-wave plate, the transmitted beam would rotate 45° relative to the azimuth of the incident beam. The beam was emitted by the half-wave plate and split by PBS₂ to achieve uniform beam splitting. It is assumed that the angle between the fast axis and the incident plane of vibration was 22.5°+ζ when HW was oriented, where ζ represents the orientation errors of the half-wave plate. According to the models(Eqs. (9)-(13)), supposing that other error sources did not exist, when ζ of the half-wave plate varied from 0° to 5°, the interference fringe contrast of a laser tracing measurement optical system remained unchanged (Table 8).

3 Zemax-Based Simulation Experiment of Laser Tracing Measurement Optical System

To validate this theoretical model of the interference fringe contrast, the optical system simulation experiments of the laser tracing measurement were conducted based on ZEMAX. In a laser tracing measurement optical system, the polarizing beam splitter and beam splitter were realized by the multiple structural parameters and the analyzer, the quarter-wave plate, and the half-wave plate were realized by the Jones matrix in ZEMAX to optimize the design and evaluate the overall performance of the imaging optics^[17].The ZEMAX-based laser tracing measurement optical system simulation model well-established in this study is shown in Fig. 6. The basis of the simulation conditions is shown in Table 9.

In ZEMAX, the parameters were set as follows. The angle between the transmission axis of the analyzer was 65°; the angle between the polarization direction of linearly polarized beam and the fast axis of the quarter-wave plate was 45°; the angle between the fast axis of the half-wave plate and the vibration direction of linearly polarized incident beam was 22.5°; The transmittance and reflectance of the polarizing beam splitters PBS₁, PBS₂, and PBS₃ were equal to 1; the split ratio of BS₁ is 5∶5; the split ratio of BS₂ was 7∶3. The interferograms received by photodetectors PD₁, PD₂, PD₃ and PD₄ in a laser tracing measurement optical system are shown in Fig. 7.

3.1 Influences of Placement Angle of Analyzer

According to the ZEMAX simulation model established in Fig. 6, supposing that no other error sources exist, the simulation results based on ZEMAX show that the placement angle of the analyzer has a great influence on the interference fringe contrast. If light was polarized at an angle of 65° to the transmission axis of the analyzer and passed through the analyzer, the interference fringe contrasts were above 0.9900 and the interference fringes of the system were the clearest as shown in Fig. 8. These conclusions are consistent with those in Section 2.1.

3.2 Influences of BS Split Ratio

According to the ZEMAX simulation model depicted in Fig. 6, the contrast values under the different split ratios of BS₁ and BS₂ were obtained (Table 10 and Table 11). Table 10 shows the interference fringe contrast values of a laser tracing measurement optical system. Under the fixed split ratio of BS₂ (7∶3), when the split ratio of BS₁ varied from 8∶2 to 2∶8, the interference fringe contrast value was not changed. Similarly, for a split ratio of BS₁ (5∶5), when the split ratio of BS₂ varied from 8∶2 to 8∶2, the interference fringe contrast decreased by 44% from 0.9975 to 0.5613(Table 11).

3.3 Influences of Non-ideal Spectroscopic Performance of PBS for Laser Tracing Measurement Optical System

According to the ZEMAX simulation model established in Fig. 6, the influence of PBS with the non-ideal spectroscopic performance of a laser tracing measurement optical system on interference fringe contrast can be discussed. Without considering error sources, interference signals received by PD₁, PD₂, PD₃ and PD₄ were the same. As shown in Table 12, when the transmittance was not ideal, the interference fringe contrast decreased with the decrease in the reflectivity. The reflectivity of the PBS₁ shows limited effect on the interference fringe contrast. However, when PBS₂ or PBS₃ shows the non-ideal spectroscopic performance, the interference fringe contrast remained 0.9975, indicating that interference fringe contrast was seldom changed, as shown in Table 13 and Table 14. Furthermore, all of the polarizing beam splitters with the non-ideal spectroscopic performance at the same time in the entire laser tracing measurement optical system had little effect on the interference fringe contrast value. The result is consistent with the analysis conclusion in Section 2.3.

3.4 Influences of Quarter-wave Plate Orientation Errors

According to the ZEMAX simulation model established in Fig. 6, assuming that there is no other error source of interference fringe contrast, the influences of the orientation errors of QW₁ and QW₂ on interference fringe contrast were discussed (Table 15 and Table 16). The interference fringe contrast decreased with the increase in e. When e increased from 0° to 5°, interference fringe contrast of QW₁ decreased from 0.9986 to 0.9949 (Table 15). However, the interference fringe contrast of QW₂ remained 0.9974 (Table 16). As shown in Table 17, when e increased from 0° to 5°, interference fringe contrast of QW₃ increased by 0.09% from 0.9975 to 0.9985. The interference fringe contrasts of the photoelectric receivers PD₁ to PD₄ were the same. According to the results, orientation errors (e) of QW₁, QW₂ and QW₃ had different effects on the interference fringe contrast. This is consistent with the conclusion in Section 2.4.

3.5 Influences of Half-wave Plate Orientation Error

According to the ZEMAX simulation model established in Fig. 6, assuming that there was no other source of interference fringe contrast, when ζ of the half-wave plate varied from 0° to 5°, the interference fringe contrast of a laser tracing measurement optical system remained to be 0.9975 (Table 18). This result verified the conclusion in Section 2.5.

In conclusion, the simulation results obtained with the ZEMAX-based laser tracing measurement optical system model are consistent with the analysis results of the interference fringe contrast model. It demonstrates the correctness of the interference fringe contrast model of a laser tracing measurement optical system proposed in this study.

4 Experiment

Based on the comprehensive fringe contrast model of the laser tracing measurement system established in this paper, the experiment about the influence on the tracing performance brought by the fringe contrast of the laser tracing measurement optical system was carried out.

The laser tracing measurement optical system is a position-following closed-loop control system. When the tracing target is displaced, the position of the laser spot reflected by the cat's eye will shift^[13]. Its tracing process was achieved by a built-in PSD. When the target was not displaced, the incident beam split by the beam splitter BS₂ injected onto the center of the PSD. The photocurrent generated by the PSD was zero. When the position of the laser spot changed, the PSD generated different photocurrent signals according to the spot position. After the amplification and calculation of the photocurrent, the displacement of the incident beam can be determined, which provided key data for the tracing control of the laser tracing measurement optical system.

The experimental process is consistent with the tracing process of the laser tracing measurement optical system. The experimental setup is shown in Fig. 9. A Renishaw Laser Interferometer XL-80 was used as laser source. The laser beam was injected onto the center of the PSD. The incident light intensity was 0.315 mW. The model of the PSD was Hamamatsu S1880. The photo sensitivity of the PSD was 0.6A/W, and the saturation current was 0.5 mA. A lens was added to the light path, which was connected with the probe of the coordinate measuring machine (CMM) by a rod. The lens can focus the laser beam. An attenuator was placed between the laser source and the lens. It can be known from '3.2 Influences of BS split ratio' that when the splitting ratio of BS₂ changed from 2∶8 to 8∶2, the fringe contrast of the interference signals received by the photodetectors PD₁ to PD₄ increased, but the injection light intensity onto the PSD reflected by BS₂ decreased. By adjusting the attenuator, the intensity of the incident beam of the PSD can be changed.

A coordinate system was established with the origin as the center of the PSD. The x direction was the same as the z direction of the CMM. The y direction was the same as the y direction of the CMM. When the lens was in the initial position, the x and y position signals output by the PSD were both 0 V. When the CMM controlled the lens to move in the y direction from the initial position, the position of the laser beam on the PSD was shifted in the y direction. The response time of the y-direction position signal of the PSD rising to 4 V was recorded under different incident light intensities. The attenuator was used to adjust the incident light intensity to 0.315mW, 0.284mW, 0.252mW and 0.221mW respectively. Under the same incident light intensity, the response time of the tracing system was measured 10 times repeatedly. The measurement results are shown in Table 19. P in Table 19 represents the light intensity measurement of incident light. Δt_r represents the response time measurement of the tracing system.

The standard deviation of the mean value was obtained to evaluate the reliability of the multi-measurement average of the PSD response time. The standard deviation of the mean value can be expressed as:

$ \mathop \sigma \nolimits_{\overline {\Delta {t_r}} } = {{\sqrt {{{\sum\limits_{i = 1}^{10} {\left( {\Delta {t_r} - \overline {\Delta {t_r}} } \right)} } \over {10 - 1}}} } \over {\sqrt {10} }} $

(14)

where $\overline {\Delta {t_r}}$ indicates the mean value of each group of measurements, i indicates the number of measurements. Then the final result is expressed as:

$ \Delta t_{r}=\overline{\Delta t}_{r} \pm 2.26 \cdot \sigma_{\Delta t_{r}} $

(15)

The experimental results show that when the incident light intensity was 0.315 mW, the response time of the system was △t_r=371.5±2.28 ms. When the incident light intensity changed to 0.284 mW, the response time of the system was △t_r=389.9±3.46 ms. When the incident light intensity changed to 0.252 mW, the response time of the system was △t_r=391.2±2.40 ms. When the incident light intensity changed to 0.221 mW (i.e., when the BS₂ split ratio become 8∶2), the response time of the system was △t_r=395.2±2.28 ms. When splitting ratio of BS₂ increased, the contrast of the interference fringes increased. But at the same time, due to the weakening of the incident light intensity of the PSD caused by the change of BS₂ splitting ratio, the response time of the tracing system increased by 23.7 ms. As a result, the tracing performance of the laser tracing measurement optical system was degraded.

5 Conclusions

The influence of the non-ideal spectroscopic performance of optical components and orientation errors of a laser tracing measurement optical system on the tracing performance was explored using the model proposed in this study. The experimental results verified the proposed model. The following conclusions can be drawn:

1) The placement angle of the analyzer has a great influence on the interference fringe contrast. When the angle of the polarized light to the analyzer's transmission axis increased from 65° to 85°, each contrast of the four-way interference fringes decreased by 65% from 0.9996 to 0.3528. In a laser tracing measurement optical system, if beams were polarized at an angle of 65° to the transmission axis of the analyzer and passed through the analyzer, the interference fringe of the system were the clearest.

2) For the fixed split ratio of BS₁, the split ratio of BS₂ showed a significant influence on the fringe contrast of the resulting interference signal and light intensity onto the PSD reflected by BS₂. The measurement experiments verified that the influence was significant. When splitting ratio of BS₂ increased, the contrast of the interference fringes increased. Due to the weakening of the incident light intensity of the PSD caused by the change of the BS₂ splitting ratio, the response time of the tracing system increased by 23.7 ms. As a result, the tracing performance of the laser tracing measurement optical system was degraded.

3) In the laser tracing measurement optical system, the orientation errors of the half-wave plate or the quarter-wave plate in the interference part had little effect on the interference fringe contrast. In addition, when the performance of the polarizing beam splitter in the interference part and tracing part in an entire optical system were under non-ideal conditions, the interference fringe contrast in a laser tracing optical system was seldom changed. Under the fixed split ratio of the beam splitter in the tracing part and different split ratios of the beam splitter in the interference part, the interference fringe contrast showed minor changes.

This study provides an important theoretical basis for evaluating and improving the accuracy and reliability of laser tracing measurement systems.