Interferometric polarization compensation based on one single polarization-maintaining fiber

Yibin Qian; Jiakun Li; Peizhi Jia; Qibo Feng; Jing Zhao

doi:10.1364/OE.513867

1. Introduction

Interferometry plays a unique and irreplaceable role in the field of precision measurement. It has the advantages of high precision, high sensitivity, and non-contact measurement. Among them, dual-frequency heterodyne interferometry, due to its anti-interference ability, is widely used in high-end equipment such as high-precision machine tools [1–5] and coordinate measuring machines [6,7], as well as in research in cutting-edge fields such as gravitational wave measurement [8,9]. For general laser interferometers, the heat emitted by the laser will cause errors in the measurement results, such as beam drift caused by thermal deformation of the laser. The heat also affects the thermal stability of the measurement unit. Therefore, the laser and the measurement device usually need to be kept away from each other to isolate the heat source. Furthermore, in order to minimize the influence of external environmental interference on the laser during the process from the laser to the measuring device and, at the same time, reduce the size of the measuring device, PMFs are often used to connect the laser and the measuring device. However, the introduction of PMF will also introduce new errors. These errors come from defects such as misalignment of the PMF and internal stress of the PMF. The former causes a certain crosstalk when dual-frequency beam enters the PMF. The latter causes inconsistent birefringence due to the uneven distribution of internal stress in the PMF and crosstalk occurs at certain nodes in the PMF [10–13]. These factors will cause the polarization state of the emitted light from the PMF not to be completely linearly polarized but to have a certain ellipticity, resulting in errors in subsequent measurements. For example, according to our previous research, when the PMF is misaligned, temperature changes will cause large nonlinear errors. In order to ensure that the nonlinear error introduced by the PMF is not higher than the nanometer level, the alignment error needs to be maintained below 2.87° at least [10]. Therefore, there is an urgent need to compensate for the polarization state of the emitted light from the PMF and reduce its ellipticity, thereby effectively reducing nonlinear errors and improving the measurement accuracy of the entire system. A lot of research has been conducted at home and abroad on error compensation in interferometry. Hou, W. et al. proposed dual-detector phase comparison to compensate nonlinear error in the interference optical path in real time. They used two detectors to offset the phase error to eliminate the first-order nonlinear error and reduce the error to the nanometer level [14,15]. Badami V. G. et al. reduced the first-order nonlinear error caused by installation errors by adjusting the azimuth angle of PBS and mirrors and controlling the peak-to-peak value below 0.5 nm [16]. Wu C. et al. studied nonlinear error compensation based on beam separation. By splitting dual-frequency beam to eliminate the nonlinear error caused by mixing, the periodic nonlinear error is less than 40 pm [17]. Ellis J. D. et al. used two optical fibers to couple dual-frequency beam, respectively, and then coupled into the optical fiber after forming the reference beam to eliminate the nonlinear error introduced by optical fiber transmission [18]. Dong Y. et al. proposed a modified nonlinear-error correction method for errors due to carrier phase delay and accompanied optical intensity modulation and reduced the nonlinear error to less than 1 nm. [19] The optical compensation method is the study of how to fix interferometric errors by changing the structure of the optical path or adding optical devices. It focuses on fixing factors that cause nonlinear errors in optical devices like mirrors and PBS. In addition, there is also research on algorithmic compensation of interferometric measurement errors, that is, the correction of the acquired nonlinear errors through algorithms. Heydemann P.L.M. achieved compensation by correcting the QD data, reducing the nonlinear error from 1 µm to 10 nm [20]. Xie J. et al. used an iterative algorithm to compensate for nonlinear errors and achieve nanoscale displacement measurement [21]. Kim Y. et al. proposed a new 13-sample error-compensating phase-shifting algorithm and achieved a surface shape measurement accuracy of 3 nm [22]. There are also error compensation studies based on machine learning and neural network methods [23–25]. Among the above compensation methods, algorithmic compensation is post-interference compensation, which lacks analysis of nonlinear error sources and has limited compensation potential. The majority of current optical compensation techniques primarily focus on the analysis and correction of nonlinear errors occurring in optical components, such as polarizing beam splitters and corner cube prisms. However, these methods seldom address the nonlinear error that arise from the coupling of optical fibers. In particular, the research on simultaneous compensation of dual-frequency orthogonal linearly polarized beam in the transmission of one single PMF has not yet been involved. In our previous research [26,27], it was found that the nonlinear error introduced by the PMF must not be ignored. Therefore, this paper studies the problem of polarization compensation for interferometry based on one single PMF and, for the first time, proposes a method that can simultaneously realize polarization compensation for both single-frequency interferometry and heterodyne interferometry based on one single PMF. By employing this technique, the ellipticity of the emitted light is diminished, resulting in a decrease in the nonlinear error of the system. The proposed technique employs basic optical elements, including polarizers and wave plates, to perform polarization compensation for interferometry using one single PMF. We innovatively establish a mathematical model of polarization compensation for interferometry based on one single PMF. The polarization state of the light emitted from the PMF was analyzed, and the influence of the ellipticity of the light emitted from the PMF and the angle with the optical axis of the Quarter Wave Plate (QWP) on the polarization compensation effect was simulated. Finally, an experimental verification system for polarization compensation for the interferometry of one single PMF was built. In view of the ellipticity of the light emitted from the PMF caused by the defect of the laser source itself, the defect of the PMF, and the alignment error between the PMF and the laser source, this approach effectively decreases the polarization state of the emitted light from 4.09° to 1.15° when using a PMF with an ER of 22 dB, which is equivalent to increasing the ER of the system to 33.95 dB. Following the implementation of polarization compensation, a notable reduction in the nonlinear error is observed, decreasing from an initial value of 7.22 nm to a significantly lower value of 2.02 nm. The system's nonlinear error is diminished by a significant margin of 71.99%, resulting in a substantial enhancement in the system's accuracy.

2. Principle

As shown in Fig. 1 [28], the emitted light of a general polarized orthogonal dual-frequency laser consists of two non-orthogonal elliptically polarized beams E₁ and E₂ with frequencies f₁ and f₂ respectively, and ellipticities ρ₁ and ρ₂ respectively. The degree of non-orthogonality is β Assuming that the x-axis of the coordinate system is parallel to E₁, the Jones matrix of E₁ is as follows:

(1)$${E_1} = {a_1}\cdot\left[ {\begin{array}{{c}} {\textrm{cos}{\rho_1}\textrm{exp}[{i({2\pi {f_1}t} )} ]}\\ {\sin {\rho_1}\textrm{exp}\left[ {i\left( {2\pi {f_1}t + \frac{\pi }{2}} \right)} \right]} \end{array}} \right]$$

To facilitate analysis, the normalized Jones matrix of E₁ is:

(2)$${E_1} = \left[ {\begin{array}{{c}} {\cos {\rho_1}}\\ {i\sin {\rho_1}} \end{array}} \right]$$

Fig. 1. Schematic diagram of the polarization state of the emitted light from a dual-frequency laser.

Download Full Size | PDF

The normalized Jones matrix of E₂ is:

(3)$$\begin{aligned} {E_2} &= \left[ {\begin{array}{{cc}} {sin\beta }&{ - cos\beta }\\ {cos\beta }&{sin\beta } \end{array}} \right]\left[ {\begin{array}{{c}} {\cos ({90^\circ{-} {\rho_2}} )}\\ {isin({90^\circ{-} {\rho_2}} )} \end{array}} \right]\\ &= \left[ {\begin{array}{{cc}} {sin\beta }&{ - cos\beta }\\ {cos\beta }&{sin\beta } \end{array}} \right]\left[ {\begin{array}{{c}} {sin{\rho_2}}\\ { - icos{\rho_2}} \end{array}} \right] = \left[ {\begin{array}{{c}} {sin\beta sin{\rho_2} + icos\beta cos{\rho_2}}\\ {cos\beta sin{\rho_2} - isin\beta cos{\rho_2}} \end{array}} \right] \end{aligned}$$

In the realm of optical systems, half-wave plates (HWPs) are commonly employed for the purpose of aligning dual-frequency lasers and PMFs. Nevertheless, in the process of aligning, it proves challenging to prevent a specific alignment error from occurring between the orthogonal linearly polarized beam output by the dual-frequency laser and the fast and slow axes of the PMF.

The previously mentioned discrepancy is commonly referred to as the alignment angle error θ. Taking this error into account, the emitted light of the PMF can be expressed as:

(4)$$\begin{array}{l} E{^{\prime}_1} = \left[ {\begin{array}{{cc}} {{e^{i\Delta L}}}&0\\ 0&1 \end{array}} \right]\left[ {\begin{array}{{cc}} {\cos 2\theta }&{\sin 2\theta }\\ {\sin 2\theta }&{ - \cos 2\theta } \end{array}} \right]\left[ {\begin{array}{{c}} {\cos {\rho_1}}\\ {i\sin {\rho_1}} \end{array}} \right]\\ E{^{\prime}_2} = \left[ {\begin{array}{{cc}} {{e^{i\Delta L}}}&0\\ 0&1 \end{array}} \right]\left[ {\begin{array}{{ccc}} {\cos 2\theta }&{\sin 2\theta }\\ {\sin 2\theta }&{ - \cos 2\theta } \end{array}} \right]\left[ {\begin{array}{{c}} {\sin \beta \sin {\rho_2} + i\cos \beta \cos {\rho_2}}\\ {\cos \beta \sin {\rho_2} - i\sin \beta \cos {\rho_2}} \end{array}} \right] \end{array}$$

Let E’₁ represent the reference beam and E’₂ represent the measurement beam, as depicted in Fig. 2, then the above formula can be written as:

(5)$$\begin{array}{l} E{^{\prime}_1} = \left[ {\begin{array}{{c}} A\\ a \end{array}} \right]\\ E{^{\prime}_2} = \left[ {\begin{array}{{c}} B\\ b \end{array}} \right] \end{array}$$

In the formula, A and B are the amplitudes of the reference beam and the measurement beam, and a and b are the crosstalk of the reference beam and the measurement beam, that is, the amplitude of the beam coupling from its alignment axis to another axis.

Fig. 2. Schematic diagram of reference beam and measurement beam.

Download Full Size | PDF

After sorting, E’₁ and E’₂ can be written as:

(6)$$\begin{array}{l} E{^{\prime}_1} = \left[ {\begin{array}{{c}} A\\ a \end{array}} \right] = \left[ {\begin{array}{{c}} {{e^{i\Delta L}}({\textrm{cos}2\theta \textrm{cos}{\rho_1} + i\textrm{sin}2\theta \textrm{sin}{\rho_1}} )}\\ {\textrm{sin}2\theta \textrm{cos}{\rho_1} - i\textrm{cos}2\theta \textrm{sin}{\rho_1}} \end{array}} \right]\\ E{^{\prime}_2} = \left[ {\begin{array}{{c}} B\\ b \end{array}} \right] = \left[ {\begin{array}{{c}} {{e^{i\Delta L}}[{\sin ({2\theta + \beta } )\textrm{sin}{\rho_2} + i\cos ({2\theta + \beta } )\textrm{cos}{\rho_2}} ]}\\ { - \cos ({2\theta + \beta } )\textrm{sin}{\rho_2} + i\sin ({2\theta + \beta } )\textrm{cos}{\rho_2}} \end{array}} \right] \end{array}$$

The aforementioned formula demonstrates that in practical instances, the dual-frequency beam from the PMF is elliptically polarized, and there is a certain ellipticity introduced by the crosstalk. Hence, the reduction of crosstalk induced by the PMF can be achieved by mitigating the ellipticity of the outgoing beam, thereby compensating for the nonlinear error in the heterodyne interference system. The compensating effect is depicted in Fig. 3.

The Diagram of polarization compensation system for heterodyne interferometry based on one single PMF is shown in Fig. 4. The key component of the system to perform polarization compensation is the QWP, which can change the polarization state of the passing beam. At a specific angle, it can reduce the ellipticity of the light emitted from the PMF. In the case of orthogonal beams, it can be observed that the QWP exerts an equivalent influence on the ellipticity of both beams.

Fig. 3. Schematic diagram of polarization compensation.

Download Full Size | PDF

Fig. 4. Diagram of polarization compensation system for heterodyne interferometry based on one single PMF.

Download Full Size | PDF

The beam passing through the optical axis of the QWP can be expressed as:

(7)$$\begin{aligned} {{E^{\prime}}_{1QWP}} &= R({0^\circ } )\left[ {\begin{array}{{cc}} 1&0\\ 0&{ - i} \end{array}} \right]R{({0^\circ } )^{ - 1}} \cdot E{^{\prime}_1}\\ &= \left[ {\begin{array}{{cc}} {{{\cos }^2}\alpha - \textrm{i}{{\sin }^2}\alpha }&{\sin \alpha \cos \alpha + \textrm{i}\sin \alpha \cos a}\\ {\sin \alpha \cos \alpha + \textrm{i}\sin \alpha \cos a}&{{{\sin }^2}\alpha - \textrm{i}{{\cos }^2}\alpha } \end{array}} \right] \cdot E{^{\prime}_1} \end{aligned}$$

Its orthogonal beam can be expressed as:

(8)$$\begin{aligned} {{E^{\prime}}_{2QWP}} &= R({90^\circ } )\left[ {\begin{array}{{cc}} 1&0\\ 0&{ - i} \end{array}} \right]R{({90^\circ } )^{ - 1}} \cdot E{^{\prime}_2}\\ &= \left[ {\begin{array}{{cc}} {{{\cos }^2}\alpha - \textrm{i}{{\sin }^2}\alpha }&{\sin \alpha \cos \alpha + \textrm{i}\sin \alpha \cos a}\\ {\sin \alpha \cos \alpha + \textrm{i}\sin \alpha \cos a}&{{{\sin }^2}\alpha - \textrm{i}{{\cos }^2}\alpha } \end{array}} \right] \cdot{-} E{^{\prime}_2} \end{aligned}$$

A polarimeter can be used to determine the angle of the fast axis of the QWP to ensure the best polarization compensation effect.

Given that the polarimeter is limited to analyzing the polarization state of beam with a single frequency, it becomes necessary to employ a polarizer to fully block one frequency of dual-frequency light. Once the angle of the QWP has been determined, the polarizer can be subsequently removed.

The process is as follows: After the beam passes through the HWP and is aligned with the fast and slow axes of the PMF, the beam is coupled into the PMF through the coupler. At this time, the ellipticity introduced by the alignment is the least. The emitted light of the PMF is compensated by the QWP, and the polarization state is confirmed by a polarimeter. The QWP fast axis is identified when the polarimeter finds the least ellipticity.

3. Simulation

Matlab was used to simulate the polarization state of the outgoing beam from a PMF with ellipticity after passing through the QWP as the angle between the outgoing beam and the optical axis of the QWP changes.

The ER of PMF can be expressed as [29]:

(9)$$ER \approx{-} 20\log ({\tan (\varphi + |\delta |)} )$$

In the formula, φ is the incremental ellipticity introduced by the alignment error of the PMF, and δ is the ellipticity introduced by the defects and external stress of the PMF.

At the wavelength of 633 nm, the ER of currently used optical fibers with better polarization-maintaining effects is usually 20 dB. Taking a certain model PMF as an example (whose ER is 22 dB). From formula (10), it can be seen that under ideal alignment conditions for a PMF with an ER of 22 dB, the emitted light has an ellipticity of up to 4.54°.

Due to the structural defects of the PMF, the azimuth angle of the light emitted from the PMF will still change to a certain extent after alignment. Considering that the optical axis will not change after the QWP is set, it becomes imperative to ascertain the extent of the azimuth angle within which compensation can be effectively attained.

Set the ellipticity of the incident light to 4.54°, and the angle between the optical axis of the QWP and the azimuth angle of the emergent light is θ. Figure 5 illustrates the variations in ellipticity of the light emitted from the PMF (Left vertical axis) and its impact on nonlinear error (Right vertical axis) as the angle θ transitions from -15° to 15° (Horizontal axis). Polarization compensation can be effectively achieved through simulation, with a notable range of ±4.60° observed between the optical axis of the QWP and the azimuth angle. This range surpasses the subsequent experimental findings, which confirmed a narrower range of ±1.13° for azimuth angle changes. Consequently, the simulation results indicate the suitability of the compensation scheme. The simulation results also indicate that within the actual azimuth angle change range of ±1.13°, the polarization compensation can reduce the ellipticity to 1.14°. After obtaining the mathematical expression of the compensation of the emergent light of the PMF, according to the heterodyne interference error model of a single PMF, the reduction of the nonlinear error before and after compensation can be obtained. According to the heterodyne interference error model from Ref. [30]:

(10)$$\Delta {L_{non}} = \frac{\lambda }{{4\pi }}\Delta {\varphi _{non}} ={-} \frac{\lambda }{{4\pi }}(\frac{{Ab + Ba}}{{AB}}\sin \mathrm{\Delta }\varphi + \frac{{ab}}{{AB}}\sin 2\mathrm{\Delta }\varphi )$$

Fig. 5. The influence of the angle between the optical axis of the QWP and the azimuth angle of the emitted light on ellipticity and nonlinear error compensation.

Download Full Size | PDF

In the formula, $\Delta {L_{non}}$ is the nonlinear error, $\Delta {\varphi _{non}}$ is the nonlinear error term and $\Delta \varphi $ is the phase difference. Using the formula (7), (8) and (10), the heterodyne interference error model of a single PMF is established and the specific impact on the nonlinear error of interference measurement can be calculated through the ellipticity of the beam before and after compensation, so as to effectively evaluate the compensation effect. It was seen that the nonlinear error decreased from 7.88 nm to 2.01 nm, so demonstrating the viability of this proposed approach.

4. Experiments and results

In order to verify the effectiveness of the proposed compensation method, a polarization compensation system for the interferometry based on one single PMF was built for experimental verification.

4.1 Experimental system and conditions

In order to enhance the quality of experimental results, it is imperative to initially establish an excellent level of alignment between the optical axis of the laser source and the PMF. In this experiment, a HWP and a coupling lens are employed to achieve alignment between the laser source and the PMF. Moreover, it is required to first remove the QWP and apply external loading to the PMF. At this time, the birefringence of the PMF changes and thereby affects the beat length. The polarization state of the light emitted from the PMF also undergoes variations within a certain range. A polarimeter is used to observe and track alterations in the polarization state of the light emitted from the PMF. When the change in polarization state is the smallest, it is considered to be aligned.

The experiment was carried out under normal circumstances, with the room temperature maintained at 18 ± 1°C. The doors and windows of the laboratory were closed, and an optical platform with a foundation was used to minimize the impact of air disturbance and solid vibration. The experimental device is shown in Fig. 6. The adjustment of the polarizer and wave plates is facilitated through the utilization of electric rotary stages. The polarizer is controlled by Aerotech's ANT95R-360 rotary stage with a resolution of 0.01″, the HWP is controlled by Jiangyun Optoelectronics Y200RA100 rotary stage with a resolution of 1.8”, and the QWP is controlled by PI Company's M-116 rotary stage with a resolution of about 0.5”. The rotary stages used in the experiments can all meet their respective accuracy requirements. In our case, the PMF is a certain model PMF, with an ER of 22 dB and a length of 1 m.

Fig. 6. The experimental device of the polarization compensation system.

Download Full Size | PDF

4.2 Experimental process and result analysis

A polarizer with an extinction ratio of >100000:1 at a wavelength of 633 nm is used to shield one beam of dual-frequency orthogonal polarized light. We use a photodetector to observe the beat signal of the emitted light while rotating the polarizer. When the beat signal is minimum (the peak value is about 0.016 V, which is equal to the noise signal of the photodetector when no light enters), it is considered that one beam of the dual-frequency orthogonal polarized lights is completely shielded. When using a motorized rotary stage to control the HWP, the alignment accuracy mainly depends on the resolution of the polarimeter. When the maximum absolute value of the polarization ellipticity of the emitted light, Emax, reaches its minimum value, it indicates that the alignment is deemed to be complete. The polarization state of the emitted light can be confirmed by monitoring the change in ellipticity. During one revolution of the wave plate, the ellipticity will change according to the angle between the laser azimuth angle and the optical axis of the wave plate. The goal of the experiment is to reduce the ellipticity of the outgoing light through alignment and polarization compensation, thereby minimizing the nonlinear error of the system. The polarimeter utilized in this paper is PAX1000VIS/M from Thorlabs. The measurement dynamic range is -60 dBm to +10 dBm. The maximum sampling rate is 400 samples per second. The resolution for measuring the azimuth angle and ellipticity is ±0.25° for both parameters.

The alignment experiment, as depicted in Fig. 7, demonstrates that the presence of defects in the PMF leads to ellipticity in the emitted light. Moreover, variations in external environment induce changes in the azimuth angle of the emitted light from the PMF. The change in azimuth angle is also related to the misalignment between the laser source and the PMF. A decrease in the magnitude of the alignment error corresponds to a decrease in the magnitude of the change in azimuth angle. When the HWP is properly aligned by an electric rotating stage, the minimum value of Emax is 4.09°. At this time, it is considered that the laser source is aligned with the optical axis of the PMF, and the ER of the system is 22.97 dB. The experiment also demonstrated that the alteration in azimuth angle, when optimally aligned, was ±1.13°. The QWP is positioned at the exit end of the PMF through an electric rotating stage. Considering that the QWP will no longer rotate after the compensation system is settled, the angle of its optical axis has a certain value. Hence, the variation in the angle between the azimuth angle of the light emitted by the PMF and the optimal angle of the optical axis of the QWP falls within the interval of ± 1.13°. Figure 5 shows that the angle between the azimuth angle of the light emitted from the PMF and the optical axis of the QWP can reduce the ellipticity within the range of ± 1.13°, which proves the feasibility of the experiment. An electric rotation stage is utilized to regulate the rotation of the QWP in order to determine the optimal angle for achieving polarization correction. The impact of polarization adjustment is illustrated in Fig. 8, as observed through experimental investigations.

Fig. 7. Schematic diagram of the light emitted from the PMF before polarization compensation (a) is the polarization state of the measuring beam and (b) is the polarization state of the reference beam.

Download Full Size | PDF

Fig. 8. Actual polarization compensation results measured by polarimeter (a) is the polarization state of the measuring beam and (b) is the polarization state of the reference beam.

Download Full Size | PDF

At the optimal angle of the QWP optical axis, the ellipticity of the light emitted from the PMF is reduced from 4.09° to 1.15°. The ellipticity of the orthogonal polarized light also decreased from 4.07° to 1.05°, proving that the compensation effect is applicable to the orthogonal polarized light simultaneously. At this time, the ER of the system has experienced a rise from 22.97 dB to 33.95 dB. The experimental results indicate that the nonlinear error resulting from the laser source to the PMF in the single PMF heterodyne interferometry system, as described by formula (10), has been calculated to be 7.22 nm. After polarization compensation, the nonlinear error is effectively mitigated to 2.02 nm. The reduction in nonlinear error led to a significant improvement of 71.99%, resulting in an enhanced accuracy of the measuring system. The above experiment is a combination of polarization compensation experiments and model simulation verification. If sufficient conditions are provided, it is more convincing to directly prove the feasibility of this method through nanometer or sub-nanometer scale interferometric measurement experiments. We are actively raising funds and optimizing experimental conditions to complete verification experiments in the near future.

4.3 Stability test

The stability experiments were conducted on the system before and after compensation, respectively. The measurement device is fixed on the optical platform. A computer is used to connect the polarimeter for sampling, with data points being recorded at regular intervals of 1 second. The stability experiment continued until the polarization state changed for 25 cycles, and the ellipticity and stability data of the system before and after compensation were obtained. The comparison of ellipticity and stability before and after compensation is shown in Fig. 9, where V is the variance. The upper and lower limits are the maximum and minimum values of the ellipticity in each cycle before and after compensation of the emergent light. The dots are the average value of the ellipticity in each cycle before and after compensation. It can be seen that the ellipticity before compensation varies between 2.94° and 4.09°, with a range of 1.15° and a fluctuation variance of 0.2140. The ellipticity after compensation varies between 0.98° and 1.15°, with a range of 0.17° and a fluctuation variance of 0.0043. The application of polarization compensation techniques is observed to significantly diminish the ellipticity exhibited by the emitted light, hence enhancing the ER of the system. Furthermore, it significantly enhances the measurement's stability.

Fig. 9. Comparison of ellipticity and stability before and after compensation.

Download Full Size | PDF

5. Conclusion

This paper studies the issue of polarization compensation for interferometry based on one single PMF. 1) For the first time, a polarization compensation method for interferometry based on one single PMF is proposed. It is equally applicable to both heterodyne interferometry and single-frequency interferometry based on one single PMF. The proposed solution employs a simple structure and exclusively relies on basic optical components, including polarizers and wave plates. The implementation of this approach is both cost-effective and straightforward. 2) A mathematical model for interferometry polarization compensation based on one single PMF is proposed. According to the model, the compensation of the light emitted from the PMF was simulated. A typical PMF with an ER of 22 dB was used as an example to demonstrate the compensation scheme is suitable. 3) A polarization compensation experiment was conducted, and the experimental results show that when using a PMF with an ER of 22 dB and an ellipticity of the exiting light of 4.09°, the polarization compensation system can reduce the ellipticity of the exiting light of the PMF from 4.09 to 1.15°, the range is reduced from 1.15° to 0.17°, and the fluctuation variance is reduced from 0.2140 to 0.0043. At this time, the ER of the system has increased from 22.97 dB to 33.95 dB. After compensation, the nonlinear error is reduced from 7.22 nm to 2.02 nm. The reduction in nonlinear error amounted to a significant decrease of 71.99%. The measurement accuracy of the interferometry system based on one single PMF is significantly improved. Simultaneously, given that the approach proposed in this paper is able to compensates for the light emitted from the PMF, it is theoretically well-suited for interferometric measurement systems employing either single or multiple PMFs, hence possessing a broad spectrum of potential applications.

Funding

Fundamental Research Funds for the Central Universities (No. 2023JBZX004); National Natural Science Foundation of China (No. 52375523); Ministry of Science and Technology of the People's Republic of China (No. 2022XAGG0200).

Disclosures

The authors declare that there are 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.

References

1. M. Deng, H. Li, S. Xiang, et al., “Geometric errors identification considering rigid-body motion constraint for rotary axis of multi-axis machine tool using a tracking interferometer,” International Journal of Machine Tools and Manufacture 158, 103625 (2020). [CrossRef]

2. Z. Geng, “Review of geometric error measurement and compensation techniques of ultra-precision machine tools,” Light: Advanced Manufacturing 2, 211–227 (2021). [CrossRef]

3. H. Haitjema, “Calibration of displacement laser interferometer systems for industrial metrology,” Sensors 19(19), 4100 (2019). [CrossRef]

4. S. Ibaraki and M. Hiruya, “A novel scheme to measure 2D error motions of linear axes by regulating the direction of a laser interferometer,” Precis. Eng. 67, 152–159 (2021). [CrossRef]

5. C. J. Evans, “Precision engineering: an evolutionary perspective,” Philosophical Transactions of the Royal Society A 370, 3835–3851 (2012). [CrossRef]

6. Y. Shang, J. Lin, L. Yang, et al., “Precision improvement in frequency scanning interferometry based on suppression of the magnification effect,” Opt. Express 28(4), 5822–5834 (2020). [CrossRef]

7. D. Zheng, C. Du, Y. Hu, et al., “Research on optimal measurement area of flexible coordinate measuring machines,” Measurement 45(3), 250–254 (2012). [CrossRef]

8. Z. Luo, “Space laser interferometry gravitational wave detection,” Advances in Mechanics 43, 415–447 (2013).

9. H. Grote and Y. Stadnik, “Novel signatures of dark matter in laser-interferometric gravitational-wave detectors,” Phys. Rev. Res. 1(3), 033187 (2019). [CrossRef]

10. Y. Qian, J. Li, and Q. Feng, “Error analysis of heterodyne interferometry based on one single-mode polarization-maintaining fiber,” Sensors 23(8), 4108 (2023). [CrossRef]

11. G. Walker and N. Walker, “Alignment of polarisation-maintaining fibres by temperature modulation,” Electron. Lett. 23(13), 689–691 (1987). [CrossRef]

12. T. T. Aalto, M. Harjanne, K. Markku, et al., “Method for the rotational alignment of polarization-maintaining optical fibers and waveguides,” Opt. Eng. 42(10), 2861–2867 (2003). [CrossRef]

13. P. Arora, A. Agarwal, A. S. Gupta, et al., “Simple alignment technique for polarisation maintaining fibres,” Rev. Sci. Instrum. 82(12), 1 (2011). [CrossRef]

14. W. Hou, “Optical parts and the nonlinearity in heterodyne interferometers,” Precis. Eng. 30(3), 337–346 (2006). [CrossRef]

15. W. Hou, Y. Zhang, H. Hu, et al., “A simple technique for eliminating the nonlinearity of a heterodyne interferometer,” Meas. Sci. Technol. 20(10), 105303 (2009). [CrossRef]

16. V. Badami and S. Patterson, “A frequency domain method for the measurement of nonlinearity in heterodyne interferometry,” Precis. Eng. 24(1), 41–49 (2000). [CrossRef]

17. C.-M. Wu, “Heterodyne interferometric system with subnanometer accuracy for measurement of straightness,” Appl. Opt. 43(19), 3812–3816 (2004). [CrossRef]

18. J. D. Ellis, A. J. H. Meskers, J. W. Spronck, et al., “Fiber-coupled displacement interferometry without periodic nonlinearity,” Opt. Lett. 36(18), 3584–3586 (2011). [CrossRef]

19. Y. Dong, P. Hu, M. Ran, et al., “Correction of nonlinear errors from PGC carrier phase delay and AOIM in fiber-optic interferometers for nanoscale displacement measurement,” Opt. Express 28(2), 2611–2624 (2020). [CrossRef]

20. P. L. Heydemann, “Determination and correction of quadrature fringe measurement errors in interferometers,” Appl. Opt. 20(19), 3382–3384 (1981). [CrossRef]

21. J. Xie, L. Yan, B. Chen, et al., “Iterative compensation of nonlinear error of heterodyne interferometer,” Opt. Express 25(4), 4470–4482 (2017). [CrossRef]

22. Y. Kim, K. Hibino, N. Sugita, et al., “Error-compensating phase-shifting algorithm for surface shape measurement of transparent plate using wavelength-tuning Fizeau interferometer,” Optics and Lasers in Engineering 86, 309–316 (2016). [CrossRef]

23. C. Wang, E. D. Burnham-Fay, J. D. Ellis, et al., “Real-time FPGA-based Kalman filter for constant and non-constant velocity periodic error correction,” Precis. Eng. 48, 133–143 (2017). [CrossRef]

24. Z. Li, K. Herrmann, F. Pohlenz, et al., “A neural network approach to correcting nonlinearity in optical interferometers,” Meas. Sci. Technol. 14(3), 376–381 (2003). [CrossRef]

25. T. B. Eom, J. A. Kim, C.-S. Kang, et al., “A simple phase-encoding electronics for reducing the nonlinearity error of a heterodyne interferometer,” Meas. Sci. Technol. 19(7), 075302 (2008). [CrossRef]

26. F. Zheng, Z. Liu, F. Long, et al., “High-precision method for simultaneously measuring the six-degree-of-freedom relative position and pose deformation of satellites,” Opt. Express 31(8), 13195–13210 (2023). [CrossRef]

27. J. Li, Q. .Feng, C. Bao, et al., “Method for simultaneously and directly measuring all six-DOF motion errors of a rotary axis,” Chin. Opt. Lett. 17, 011203 (2019). [CrossRef]

28. J. Tan, H. Fu, P. Hu, et al., “A laser polarization state measurement method based on the beat amplitude characteristic,” Meas. Sci. Technol. 22(8), 085302 (2011). [CrossRef]

29. A. Méndez and T. F. Morse, Specialty Optical Fibers Handbook (Elsevier, 2011).

30. H. Fu, Y. Wang, P. Hu, et al., “Real-time compensation of nonlinearity in heterodyne interferometers based on quadrature demodulation and extremum operation,” Opt. Eng. 59(04), 1 (2020). [CrossRef]

Interferometric polarization compensation based on one single polarization-maintaining fiber

Abstract

1. Introduction

2. Principle

3. Simulation

4. Experiments and results

4.1 Experimental system and conditions

4.2 Experimental process and result analysis

4.3 Stability test

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (10)

Optics Express