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Abstract
Optical delay lines have wide applications in terahertz time-domain spectroscopy and optical coherence tomography. In this study, a fast-rotating optical delay line (FRODL) with 24 turntable reflection surfaces was designed. By analyzing the working principle of the FRODL, a mathematical model was established for the nonlinear parameter error of the FRODL delay time. By constructing the polarization Michelson interference system and testing the FRODL structure, the error of actual assembly parameters of the FRODL was approximately 0.015 mm, the actual delay time of the FRODL was greater than 43.5 ps, and the linearity was 99.785%.
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1. INTRODUCTION
As a device that can change the relative delay time of reference light and detection light in optical detection systems, optical delay lines have a wide range of applications in terahertz time-domain spectroscopy (THz-TDS), optical coherence tomographic imaging, ultrafast time-resolution spectroscopy, and pump detection technology [1–5]. The nonlinearity of optical delay lines refers to the degree of deviation between the actual delay time curve of the optical delay line device and the delay-time fitting line. The nonlinearity of optical delay lines directly affects the accuracy and consistency of the sampled signal. The nonlinear change in the delay time causes nonlinear changes in the pulse path of the femtosecond laser, which leads to the nonlinearity of the sampled signal. The greater the nonlinear error of the sampled signal, the more severe the distortion of the collected sampled signal and the greater the difficulty in subsequent data processing [6–8]. A typical optical delay line is a linear optical delay line (LODL), which uses a motor-driven microdisplacement device, such as a spiral lead screw, to drive a plane mirror or pyramid prism back and forth in a one-dimensional direction, resulting in time delay [9–11]. It has the advantages of good linearity, high-quality output beam, and good smoothness. However, fast high-frequency scanning cannot be achieved because of the mechanical inertia of the stepper motor. Various rotating optical delay lines (RODLs) with different operating principles have been proposed to solve the problem of low scanning frequencies in LODLs. An RODL is usually driven by a motor in a two-dimensional or three-dimensional direction to rotate multiple mirrors, polyhedral prisms, and polyhedral rotating mirrors, resulting in changes in the optical path [12–15]. The RODL was driven by a rotating motor, which significantly increased the scanning frequency of the optical delay line. The maximum scanning frequency can exceed 1 kHz but is limited by the shape of the surface. In general, the nonlinearity of the rotating cubic-type RODL is approximately 10% [16,17]. The linearity of spiral and involute RODLs can reach 99% [15,18,19]; however, mechanical structure processing is difficult, high processing accuracy is required, and processing cost is high. Moreover, the periodicity of the RODL structure is low, and it is difficult to further improve the scanning frequency [20]. To achieve widespread application of optical delay lines, it is necessary to solve the nonlinear error problem of various types of RODLs on the reflecting surface.
Therefore, in this study, a new type of fast-rotating optical delay line (FRODL) was proposed. In response to the nonlinear error problem of FRODLs, we first explored the working model, working angle, delay time, and theoretical nonlinearity of the FRODL structure from a theoretical perspective. Second, an analysis was conducted on various processing and assembly parameters, and the impact of parameter error on the nonlinearity of the delay time was obtained. Finally, based on polarizing Michelson interferometric calibration technology, the actual delay time of the FRODL structure was tested, and the actual error situation of the FRODL structure was obtained based on the trend of the delay time variation.
2. THEORETICAL MODEL
A. Working Principle
In this study, we designed an FRODL composed of a turntable reflection surface (TRS) array that operates on optical components, such as a coupling lens, turntable, focusing lens, and plane mirror. The radius of the rotary table base circle of the FRODL was $R = 80\;{\rm mm}$, the angle between the tangent of the TRS and the rotary table base circle was $\alpha = 30^\circ$, the angle between the TRS and the center of the rotary table base circle was $\theta = 15^\circ$, the focal length of the focusing lens was $f = 30\;{\rm mm}$, the diameter was $D = 25.4\;{\rm mm}$, the focal length of the coupling lens was ${f_0} = 4.6\;{\rm mm}$, and the lens diameter was ${D_0} = 3.6\;{\rm mm}$.
Fig. 1. FRODL schematic diagram. (a) FRODL two-dimensional simplified model: incident optical path (IOP), reflected optical path (ROP), coupling lens (CL), focusing lens (FL), plane mirror (PM), turntable reflection surface (TRS), and base circle (BC). (b) Physical diagram of the turntable.
As shown in Fig. 1(a), the working principle of the FRODL is as follows. A femtosecond laser is transmitted through an optical fiber, emitted from the center of the coupling lens into free space, incident on a rotating TRS. The incident beam is reflected by the TRS, focused with a focusing lens, and then reflected by a plane mirror. The reflected beam returns to the coupling lens in its original path after passing through the focusing lens and TRS and is finally coupled to the optical fiber by the coupling lens. The turntable is the rotating main body of the FRODL, and the mechanical structure of the turntable is processed using a computer numerical control milling machine. A reflective mirror film is pasted on the surface of the processed turntable to ensure that the laser reflection power of the reflective surface reaches 85% or higher. A physical image of the processed turntable is shown in Fig. 1(b), with each TRS arranged counterclockwise around the base circle of the turntable. The incident beam can be passed through the TRS twice by the focusing lens and plane mirror, which not only increases the delay time of the FRODL, but also makes the reflected beam more conveniently coupled into the fiber through the coupler [21,22].
To further verify the feasibility of the FRODL optical path, simulations were conducted on the working optical path at different angles of FRODL rotation. The typical simulation results are shown in Fig. 2.
The black and red light in the figure represent the incident and reflected beams, respectively. The simulation set the rotation-angle range of the reflected beam with a power greater than 90% after it passed through the coupling lens, representing the effective working angle of the FRODL. The results showed that when the main body of the FRODL turntable rotated between ${-}{2.5}^\circ$ and 2.5° (clockwise as positive and counterclockwise as negative), the reflected light returned to the coupling lens. When the rotation angle continued to increase to more than $\pm {2.5}^\circ$, the longitudinal displacement of the reflected beam exceeded the inlet pupil diameter of the coupling lens, resulting in the coupling lens being unable to focus the reflected beam into the fiber. Therefore, the theoretical operating range of each TRS of the FRODL was [${-}{2.5}^\circ$, 2.5°].
B. Mathematical Model
Assuming that the incident light radially enters the TRS along the base circle of the turntable, Fig. 3(a) shows the case where the incident light passes through a certain TRS center position, with the center of the turntable as the origin, the radial direction of the base circle of the turntable as the $x$-axis direction, and the tangential direction as the $y$-axis direction, establishing a two-dimensional coordinate system. The radius of the FRODL turntable base circle is $R$, the angle between the tangent of the TRS and the turntable base circle is $\alpha$, and the angle between the TRS and the center of the turntable base circle is $\theta$. The focus lens has a focal length $f$ and diameter $D$, whereas the coupling lens has a focal length ${f_0}$ and diameter ${D_0}$. $A$ and $C$ are the two ends of the TRS. ${\boldsymbol {AB}}$ is tangential to the base circle of the turntable, and the center angle of ${\boldsymbol {AB}}$ is $\theta$. The initial position of the TRS before FRODL rotation is determined using ${\boldsymbol {OA}}$ and ${\boldsymbol {OC}}$.
Fig. 3. FRODL optical path diagram. (a) Initial optical path position of FRODL. (b) Position of optical path after rotation.
Fig. 4. FRODL theoretical delay time theoretical results. (a) Delay time. (b) Nonlinear error curve.
After rotating the angle $\gamma$ (clockwise as positive and counterclockwise as negative), the working optical path is shown by the red dotted line in Fig. 3(b). The position of the TRS is determined by $A^\prime $, $B^\prime $, and $C^\prime $. The intersection points $E^\prime $, ${F^\prime _0}$, and $F^\prime $ of the reflected light with the TRS, focusing lens, and plane mirror are shown. The two-dimensional rotation matrix formula can be used to obtain the following:
Figure 3(b) shows the amplified optical path diagrams of the laser propagation before and after the rotation angle $\gamma$ of the FRODL. The figure shows that the optical path of the laser propagation at the initial position of the FRODL, denoted as $L$, is
After rotating the angle $\gamma$ of the FRODL, the optical path of the laser propagation, $L^\prime $, is given by
The optical path difference of the FRODL is mainly due to the changes in the horizontal axis position of the intersection point between the incident light and the TRS before and after rotation. By substituting and simplifying the position vectors before and after the FRODL rotation, the delay time generated by the rotation angle $\gamma$ of the FRODL, $t(\gamma)$, can be expressed as
According to the simulation results of the FRODL optical path, its working angle is [${-}{2.5}^\circ$, 2.5°], located in the paraxial region of the lens, $\sin \gamma \approx \gamma$, $\cos \gamma \approx 1$; thus, Eq. (4) can be approximated as
The nonlinearity of the FRODL can be determined from the error between the theoretical delay time and the fitting delay time at the corresponding angle:
where $\Delta {t_{{\max}}}$ is the theoretical delay time, and $t$ is the total theoretical delay time of the FRODL.The delay time of the FRODL is generated by a change in the optical path during the rotation of the turntable. The theoretical working angle of the FRODL is introduced into Eq. (5), and the theoretical delay time, which can reach 43.522 ps, is shown in the midpoint line of Fig. 4(a). Linear fitting was performed using the least-squares method to determine the relationship between the FRODL theoretical delay time and the rotation angle. After fitting, the ideal delay time was 43.465 ps and the sensitivity of the ideal delay time to the rotation angle was 8.693 ps/°. The nonlinear errors in the theoretical FRODL and the ideal delay times are shown in Fig. 4(b). Under different rotation angles, the nonlinear errors of the FRODL delay time exhibit a quadratic curve. As shown in the figure, when the rotation angle is 2.5°, the maximum nonlinear error of the FRODL is 0.132 ps, and the theoretical nonlinearity is 0.304%. The FRODL contains 24 periodic TRS structures, using a regular 3000 rpm DC brushless motor, with a theoretical scanning frequency of up to 1200 Hz and a single-waveform acquisition time of only 0.833 ms. The scanning frequency of the LODL using a high-speed galvanometer can reach 20 Hz, and the acquisition time of a single waveform is 50 ms. Compared with the LODL, the FRODL can significantly improve the scanning frequency of THz-TDS systems, providing a foundation for the real-time acquisition of terahertz signals.
Fig. 5. Influence curve of the machining parameter errors. (a) Total delay time curves for different machining parameter errors. (b) Nonlinearity degree curves for different machining parameter errors. (c) Roundness machining error curves of rotary table base circle. (d) Mirror inclination machining error curves. (e) Rotary table reflection and surface-position machining error curves.
3. ERROR ANALYSIS AND DISCUSSION
The processing and assembly accuracy of the FRODL turntable are typically the main causes of delay time errors. This study establishes a delay model for FRODL error parameters, analyzes the total delay time and nonlinear error size under the error of each processing and assembly parameter, and determines the processing and assembly errors of the FRODL.
A. Processing Error Analysis
Owing to the limitations of machining accuracy, an error always exists between the actual parameters obtained from the mechanical structure processing of the FRODL turntable and ideal parameters. According to Eq. (5), the actual processing parameters of the FRODL are primarily the radius $R$ of the base circle of the turntable, the angle $\alpha$ between the tangent of the TRS and the base circle of the turntable, and the angle $\theta$ of the TRS relative to the center of the base circle of the turntable. Assuming that the errors of the FRODL machining parameters are $\Delta R$, $\Delta \alpha$, and $\Delta \theta$, the actual machining parameters are recorded as ${R_1}$, ${\alpha _1}$, and ${\theta _1}$, respectively, with ${R_1} = R + \Delta R$, and the others are the same. When there is an error in processing, the delay time ${t_1}$ generated by the rotation angle $\gamma$ of the FRODL is
The error structure parameters of the FRODL are substituted into Eq. (7), and the effects of the processing parameter unit errors on the total delay time and nonlinearity are shown in Fig. 5. As shown in Fig. 5(a), as the processing parameter error increases in the positive direction, the total delay time of the FRODL parameter exhibits an increasing trend. The FRODL had a sensitivity of 0.544 ps/mm for the roundness error of the turntable, 1.001 ps/° for the tilt error of the TRS, and 0.207 ps/° for the position error of the TRS. In Fig. 5(b), the nonlinearity of the FRODL increases with an increase in the absolute value of the processing parameter error. Combining Eq. (6) and Fig. 5(a), it is demonstrated that the increase in the total delay time after a positive increase in the processing parameter error is smaller than the increase in the nonlinear error, resulting in an increase in the nonlinearity of the FRODL.
Figures 5(c)–5(e) represent the nonlinear errors in the processing parameters for different errors. By comparison, it can be observed that the nonlinear error of the FRODL time delay increases with the absolute value of the processing parameter error. The processing error of the TRS inclination had the greatest impact on the total delay time of the FRODL, whereas the processing error of the TRS position had the smallest impact on the total delay time of the FRODL, which is consistent with the results shown in Figs. 5(a) and 5(b). Because of the higher machining error and cost of the FRODL structural parameters, considering the nonlinear error of the FRODL itself, accuracy of the delay line encoder, and processing and assembly costs of the turntable, this study selected the maximum delay time error caused by the error of each FRODL machining parameter to not exceed 0.05 ps. As shown in Fig. 5(a), the machining error of the roundness of the rotary table base should be within $\pm {0.092}\;{\rm mm}$, machining error of the TRS inclination angle should be within $\pm {0.050}^\circ$, and machining error of the corresponding center angle should be within $\pm {0.242}^\circ$.
B. Assembly Error Analysis
In most cases, during the assembly process of the turntable, the assembly error is manifested as the eccentricity error of the turntable, which can be divided into lateral eccentricity error $\Delta x$ and longitudinal eccentricity error $\Delta y$. The working optical paths of the FRODL for these two types of eccentricity errors are shown in Fig. 6.
Fig. 6. Schematic diagram of the working optical path of single wedge reflector during eccentric installation. (a) Transverse eccentricity. (b) Longitudinal eccentricity.
Fig. 7. Influence curve of assembly parameter errors. (a) Delay time curves with different installation errors. (b) Nonlinearity of delay time for different installation errors. (c) Lateral eccentricity error. (d) Longitudinal eccentricity error.
The solid line represents the position of the turntable without eccentricity, the dotted line represents the actual installed position of the turntable, $\Delta x$ represents the lateral displacement of the turntable axis, and $\Delta y$ represents the longitudinal displacement of the turntable axis. The impact of the lateral offset error on the FRODL was mainly due to the change in the distance between the reflecting surface of the turntable and the reflecting mirror, whereas the vertical offset error was mainly due to the change in the position of the laser incident on the reflecting surface of the turntable. When there is both a lateral eccentricity error $\Delta x$ and longitudinal eccentricity error $\Delta y$ during the turntable installation process, the delay time ${t_2}$ of the FRODL can be expressed as
Figure 7 shows the influence curves of the installation error of the turntable. As shown in Fig. 7(a), the sensitivity of the FRODL to the lateral eccentricity error is 0.506 ps/mm, and the sensitivity to the longitudinal eccentricity error is 0.292 ps/mm. The nonlinearity of the delay time in Fig. 7(b) increases with the absolute value of the installation error. By comparing Figs. 7(c) and 7(d), it can be observed that when both lateral and longitudinal eccentricity errors exist in the FRODL, the impact of the lateral eccentricity error on the nonlinearity of the delay time is greater. Based on the requirement that the total delay time error does not exceed 0.05 ps, the lateral eccentricity of the FRODL assembly should be controlled within $\pm {0.099}\;{\rm mm}$, and the longitudinal offset error should be less than $\pm {0.171}\;{\rm mm}$.
4. EXPERIMENT AND RESULTS ANALYSIS
A. Actual Work Angle Testing
The optical power of the FRODL will affect the signal-to-noise ratio of terahertz signals emitted by photoconductive antennas. To ensure a high signal-to-noise ratio for stimulating the THz signal, a fiber-coupled FRODL commonly used in practical engineering applications was used to test the simulation results of the FRODL working-angle range.
The working principle of the FRODL working-angle testing system is shown in Fig. 8(a), where the propagation path of the femtosecond laser is from the FLP to port 1, port 2, CL, TRS, FL, PM, FL, TRS, CL, port 2, port 3, and OPM. Owing to the constant output optical power of the femtosecond laser, the relationship between the working angle of the FRODL and the coupled optical power can be characterized by recording the optical power corresponding to the port-3 end at different angles of rotation of the FRODL. Figure 8(b) shows the proportion of the coupled optical power of the FRODL relative to the normal incidence coupled optical power at different rotation angles. The actual output power of the FRODL at a normal incidence angle is up to 20 mw, and the coupled optical output power is relatively stable at [${-}{2.5}^\circ$, 2.5°] rotation angles, ranging from 18.5 to 20 mw. The proportion of coupled optical power relative to the normal incidence coupled optical power exceeds 90%, and it meets the condition of a high signal-to-noise ratio of excited terahertz signals.
Fig. 8. FRODL system coupling power testing. (a) FRODL system coupling power-testing system: femtosecond laser pulse (FLP), optical circulator (OC), CL, TRS, PM, and optical power meter (OPM). (b) Proportion of FRODL coupling optical power relative to the normal incidence coupling optical power at different rotation angles.
B. Actual Delay Time Testing
This study constructed a polarization Michelson interferometer measurement system, as shown in Fig. 9, and tested the actual delay time corresponding to different deflection angles of the FRODL. The reference arm was a planar reflector, and the measurement arm was the FRODL optical path. Among them, the illustration in Fig. 9 shows the spatial interference fringes obtained by the interference between the reflection surface of the turntable and the plane mirror in the Michelson reference optical path. Using the change in the measurement arm optical path relative to the reference arm optical path during rotation, the grayscale value of the black box area in the illustration was calculated, and the number of changes in the interference fringes during turntable rotation was counted to obtain the actual delay time of the TRS [23].
Using the Michelson polarization interferometric system, the actual delay time of each TRS on the FRODL turntable was calibrated multiple times; the average actual total delay times of the turntable are shown in Fig. 10(a). The average delay times of the FRODL were not significantly different from the ideal delay times. The maximum nonlinear error of the average delay time was ${-}{0.071}\;{\rm ps}$, and the nonlinearity of the average delay time of the FRODL was only 0.163%. From the overall delay time results of the TRS, as shown in Fig. 10(b), the ideal delay time of the TRS was 43.522 ps and the average delay time was 43.504 ps, with a small difference between the two. Regarding the delay time of a single TRS, TRS 21 had the highest total delay time of 43.554 ps, which exceeded the ideal delay time of 0.032 ps. TRS 7 had the shortest total delay time of 43.433 ps, which is lower than the ideal delay time of 0.089 ps. Moreover, the total delay time between the TRS and turntable shows a trend of first decreasing and then increasing.
Fig. 10. FRODL test results. (a) Comparison of assembly error fitting results for TRS delay time error. (b) Comparison of actual delay time results for the TRS.
Because the positions of the incident laser, lens, and plane mirror during the calibration process are the same relative to the conditions of each TRS, the errors caused by the three are significant for all TRSs. The machining errors between each TRS of the turntable are independent, as well as independent of the machining errors of other TRSs. There is a correlation between the assembly errors, determined by the lateral eccentricity error $\Delta x$ and longitudinal eccentricity error $\Delta y$ of the rotary table center relative to the motor shaft. From this, it can be inferred that the error trend of the total delay time of this FRODL was mainly caused by the assembly error of the TRS. During the operation of the FRODL, if the center of the turntable moves on a circle with a radius of $R^\prime $ ($R^\prime = \sqrt {\Delta {x^2} + \Delta {y^2}}$), the lateral eccentricity error $\Delta {x_i}$ and longitudinal eccentricity error $\Delta {y_i}$ of the $i$th TRS of the turntable are as follows:
The actual total delay time error $\Delta {t_i}$ of the $i$th TRS can be expressed as
The nonlinear least-squares method was used to fit the error trend of the total delay time of the FRODL. The results show that the lateral eccentricity error $\Delta x$ of the center of the rotary table is 0.015 mm, and the longitudinal eccentricity error $\Delta y$ is ${-}{0.011}\;{\rm mm}$, both of which do not exceed the required accuracy of the rotary table assembly parameters and are consistent with the calibration results.
In the THz-TDS system, the FRODL acquires time-domain waveforms at different terahertz positions by changing the optical path difference between the pump and detection lights. When reconstructing the waveform of the terahertz signal sampling, the nonlinear error of each TRS delay affects the accuracy of the system signal sampling. Therefore, before practical application, it is necessary to calibrate the sampling interval of each TRS on the FRODL turntable. The average nonlinear error results of multiple calibrations for each TRS are shown in Fig. 11. The figure shows that the nonlinear error trend of the FRODL target sampling interval was consistent with the theoretical nonlinear error trend. There are 52.724% sampling intervals with a nonlinear error less than $\pm {0.05}\;{\rm ps}$, 47.276% sampling intervals with a nonlinear error exceeding $\pm {0.05}\;{\rm ps}$, and all target sampling intervals with a nonlinear error less than $\pm {0.1ps}$. The maximum nonlinear error was 0.094 ps, and the actual nonlinearity of the FRODL was 0.215%, which proves that the FRODL structure has good linearity.
5. CONCLUSION
The actual delay time of the FRODL designed in this study was greater than 43.5 ps, and the linearity exceeded 99%. To address the error problem in the FRODL and assembly processes, a mathematical model was constructed for the TRS tilt error, turntable base circle roundness error, positional error, and lateral and longitudinal eccentricity errors during the installation. The nonlinearity of the delay time of the delay line was analyzed, and it was determined that the processing and lateral eccentricity errors of the reflector inclination angle had the greatest impact on the nonlinearity of the FRODL delay time. Under a delay time error of no more than 0.05 ps, the machining error of the roundness of the rotary table base should be within $\pm {0.092}\;{\rm mm}$, the machining error of the TRS inclination angle should be within $\pm {0.050}^\circ$, and the machining error of the corresponding center angle should be within $\pm {0.242}^\circ$. The lateral eccentricity of the FRODL assembly should be controlled within $\pm {0.099}\;{\rm mm}$, and the longitudinal deviation error should be within $\pm {0.171}\;{\rm mm}$. In addition, based on the Michelson interferometric calibration technology, the actual delay time of the FRODL was tested, and the error trend of the total delay time of the FRODL was fitted using the nonlinear least-squares method. The results showed that the lateral eccentricity error $\Delta x$ of the rotary table center was 0.015 mm, and the longitudinal eccentricity error $\Delta y$ was ${-}{0.011}\;{\rm mm}$. The test results showed that the nonlinear error of the 47.276% sampling interval in the actual FRODL exceeded 0.05 ps, with a maximum nonlinear error of 0.094 ps, and that the actual linearity of the FRODL was 99.785%.
Although the FRODL delay time is slightly shorter, it is not suitable for the terahertz nondestructive testing of ordinary samples. However, in the application of thin-coating thickness detection, because the thickness of the thin-coating to be detected generally does not exceed 500 µm, the delay time generated by the reflective THz-TDS system for detection is approximately 3 ps. The time resolution of the FRODL target sampling interval was 0.1 ps, and the thickness resolution was 30 µm, which can be used to detect the thin-coating thickness. In addition, preliminary experiments have shown that by changing the design parameters of the TRS, such as increasing the radius of the turntable base circle and inclination angle of the TRS, the delay time of the FRODL can be further increased. This aspect of the study will be investigated in the future.
Funding
The ninth Zhongshan Scientific Innovative Research Teams (GXTD2022010); The second batch of social public welfare and basic research projects in Zhongshan City (2022B2012).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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