# Two-Axial Measurement of the Angular Microdeflection of a Laser Beam Using One Single-Axis Sensor

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Optical Element for the Rotation of Laser Beam Deflections

_{i}Y

_{i}Z

_{i}and X

_{o}Y

_{o}Z

_{o}, where the indices i and o represent the local coordinate systems of the input and output beams, respectively. Axes X

_{i}and X

_{o}are positioned along the direction of the nominal path of the beam: if the nominal path of the beam is horizontal, then axes Z

_{i}and Z

_{o}are vertical and point upwards (parallel to the Z-axis of the global coordinate system).

_{i}-axis. The deflection angle φ shown in the figure is counterclockwise, looking from above the X

_{i}Y

_{i}plane, and we assume that it has a positive value. After passing through the optical element and undergoing two reflections inside it, the output beam is deflected in the vertical plane. In the local coordinate systems, a rotation around the Z

_{i}-axis changes to a rotation around the Y

_{o}-axis of the same value but with the opposite sign (clockwise looking from above the X

_{o}Z

_{o}plane). The angle of rotation of the output beam around the Y

_{o}-axis has a value of −φ (the absolute value is the same, but the sign is negative). Hence, in the global coordinate system, a rotation around the Z-axis is changed to a rotation around the X-axis with the same absolute value but with the opposite sign.

_{i}-axis. In this case, the deflection angle of the beam before entering the optical element has a negative value (clockwise looking from above the X

_{i}Z

_{i}plane). The direction of the beam deflection was chosen for better visibility of the beam path, as the output beam is deflected in the horizontal plane. In the local coordinate systems, a rotation around the Y

_{i}-axis changes to a rotation around the Z

_{o}-axis with the same value and the same sign. An angle of rotation of the output beam around the Z

_{o}-axis has a negative value of ϑ. Hence, in the global coordinate system, a rotation around the Y-axis changes to a rotation around the Z-axis with the same absolute value and with the same sign.

_{o}-axis (|φ

_{L}|) is equal to the absolute value of the rotation of the input laser beam around the Z

_{i}-axis (|φ

_{E}|). The accuracy of this conversion was determined in [26], and the error in the conversion ΔE

_{C}(the difference between the −φ

_{L}and φ

_{E}values) can be approximated by Equation (1):

_{C}≈ 0.018 × R

_{Y}× ϑ

_{E},

_{Y}is the rotation of the optical element around the Y-axis relative to its nominal position in degrees, and ϑ

_{E}is the rotation of the entering laser beam around the Y

_{i}-axis.

_{Z}; however, these factors are less important, as their influence is smaller.

#### 2.2. Method of Two-Axial Measurement with One Single-Axis Sensor

#### 2.3. Subsystem for Altering the Beam Polarisation

#### 2.4. Single-Axis Sensor for Laser Beam Deflection

_{1}, the value of the signal from the second photodiode as I

_{2}, and so on, then the function F related to the fringe constant is given by Equation (3) [20]:

_{n}) [20]:

_{0}that is related to the centre of the measurement range. In this configuration, the sensor does not enable absolute measurements—these require the configuration with the CCD camera.

#### 2.5. Setup Used for Experimental Verification of the Proposed Method

## 3. Results

#### 3.1. Results Overwiev

^{2}value for a linear fit (calculated using Statgraphics Centurion 19 v19.1.3 software) was 99.9274%. The linear characteristic of a measurement system is important, because even if there were some systematic errors, for example, caused by an error in the determination of the piezotranslator amplification, the proportional response of the measurement system verifies the performance of the method.

^{2}values for the linear fits were 99.9277% for ϑ and 99.7965% for φ. In second run of Experiment 2, the maximum absolute error in the measurement of ϑ was 1.45 µrad, while that for φ was 1.39 µrad. The R

^{2}values for the linear fits were 99.9316% for ϑ and 99.7383% for φ. None of the errors in Experiment 2 exceeded 2 µrad.

^{2}value for a linear fit was 99.6856%. The maximum absolute error in the measurement of ϑ was 3.15 µrad. However, one measurement of ϑ was clearly an outlier. It was proved by statistical methods that the 17th observation in the data shown in Figure 8 by the red line was an evident outlier caused by unidentified random disruption and this data point was removed from the data set. After elimination of this data point, the maximum absolute error in ϑ decreased to 1.27 µrad. A drift of the measured ϑ values was observed, but the differences were within the range of the expected laser pointing angular drift. Additionally, the next experiment showed that there was no influence of φ on the measurement of ϑ. There is also no reason φ would affect the measurement of ϑ, because while ϑ is being measured, the beam is only reflected by the polarizing beamsplitter. The time between two data points in a single experiment was approximately 1 min. Taking into account all of the above, it was concluded that there was no real cross talk. The increase in ϑ, visible in Figure 8, was caused by the angular component of laser pointing instability. Alternatively, there could have been a change in the direction of the laser beam caused by the thermal drift in the setup. But in both cases, there was drift of the laser beam direction.

^{2}value for a linear fit was 99.8529%. The maximum absolute error in the measurement of φ was 0.84 µrad. The maximum difference between the corresponding values of ϑ obtained from Experiments 2 (both runs) and 4 was 1.62 µrad, a value similar to those obtained in the single experiments. Therefore, it was concluded that φ had no influence on the measurement of ϑ: the measurements of ϑ gave similar results for constant φ and for increasing φ. This was expected, because the theoretical model allows only the influence of ϑ on the measurement of φ. In Figure 10, no angular drift of the laser beam is visible in the measurements of φ angle. This does not mean that there was no angular drift of the laser beam. Since the spatial direction of the laser beam is random the beam deflection might have been present at that time in the vertical plane.

#### 3.2. Uncertainty Analysis

## 4. Discussion

## 5. Patents

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Putman, C.A.J.; De Grooth, B.G.; Van Hulst, N.F.; Greve, J. A Detailed Analysis of the Optical Beam Deflection Technique for Use in Atomic Force Microscopy. J. Appl. Phys.
**1992**, 72, 6–12. [Google Scholar] [CrossRef] - Meyer, G.; Amer, N.M. Optical-beam-deflection Atomic Force Microscopy: The NaCl (001) Surface. Appl. Phys. Lett.
**1990**, 56, 2100–2101. [Google Scholar] [CrossRef] - Levesque, M.; Mailloux, A.; Morin, M.; Galarneau, P.; Champagne, Y.; Plomteux, O.; Tiedtke, M. Laser Pointing Stability Measurements. In Third International Workshop on Laser Beam and Optics Characterization; Morin, M., Giesen, A., Eds.; SPIE: Bellingham, WA, USA, 1996; Volume 2870, pp. 216–224. [Google Scholar] [CrossRef]
- Virdee, M.S. High Accuracy Form Measurement of Specularly Reflecting Surfaces by Laser Autocollimation. In Ultraprecision Machining and Automated Fabrication of Optics; SPIE: Bellingham, WA, USA, 1987; Volume 0676, pp. 66–73. [Google Scholar] [CrossRef]
- Shimizu, Y.; Matsukuma, H.; Gao, W. Optical Angle Sensor Technology Based on the Optical Frequency Comb Laser. Appl. Sci.
**2020**, 10, 4047. [Google Scholar] [CrossRef] - ISO 11670:2003; Lasers and Laser-Related Equipment—Test Methods for Laser Beam Parameters—Beam Positional Stability. ISO: Geneva, Switzerland, 2003.
- Lim, H.; Shimizu, Y. Feasible Resolution of Angular Displacement Measurement by an Optical Angle Sensor Based on Laser Autocollimation. Nanomanuf. Metrol.
**2023**, 6, 32. [Google Scholar] [CrossRef] - Huang, P.S.; Kiyono, S.; Kamada, O. Angle Measurement Based on the Internal-Reflection Effect: A New Method. Appl. Opt.
**1992**, 31, 6047–6055. [Google Scholar] [CrossRef] [PubMed] - Huang, P.S.; Ni, J. Angle Measurement Based on the Internal-Reflection Effect Using Elongated Critical-Angle Prisms. Appl. Opt.
**1996**, 35, 2239–2241. [Google Scholar] [CrossRef] [PubMed] - Zhang, S.; Kiyono, S.; Uda, Y. Nanoradian Angle Sensor and in Situ Self-Calibration. Appl. Opt.
**1998**, 37, 4154–4159. [Google Scholar] [CrossRef] [PubMed] - Villatoro, J.; García-Valenzuela, A. Measuring Optical Power Transmission near the Critical Angle for Sensing Beam Deflection. Appl. Opt.
**1998**, 37, 6648–6653. [Google Scholar] [CrossRef] [PubMed] - Gray, J.; Thomas, P.; Zhu, X.D. Laser Pointing Stability Measured by an Oblique-Incidence Optical Transmittance Difference Technique. Rev. Sci. Instrum.
**2001**, 72, 3714–3717. [Google Scholar] [CrossRef] - García-Valenzuela, A.; Diaz-Uribe, R. Detection Limits of an Internal-Reflection Sensor for the Optical Beam Deflection Method. Appl. Opt.
**1997**, 36, 4456–4462. [Google Scholar] [CrossRef] [PubMed] - Zhang, A.; Huang, P.S. Total Internal Reflection for Precision Small-Angle Measurement. Appl. Opt.
**2001**, 40, 1617–1622. [Google Scholar] [CrossRef] [PubMed] - García-Valenzuela, A.; Sandoval-Romero, G.E.; Sánchez-Pérez, C. High-Resolution Optical Angle Sensors: Approaching the Diffraction Limit to the Sensitivity. Appl. Opt.
**2004**, 43, 4311–4321. [Google Scholar] [CrossRef] [PubMed] - Garcia-Valenzuela, A.; Pena-Gomar, M.; Villatoro, J. Sensitivity Analysis of Angle-Sensitive Detectors Based on a Film Resonator. Opt. Eng.
**2003**, 42, 1084–1092. [Google Scholar] [CrossRef] - Garcia-Valenzuela, A.; Diaz-Uribe, R. Approach to Improve the Angle Sensitivity and Resolution of the Optical Beam Deflection Method Using a Passive Interferometer and a Ronchi Grating. Opt. Eng.
**1997**, 36, 1770–1778. [Google Scholar] [CrossRef] - Kwee, P.; Seifert, F.; Willke, B.; Danzmann, K. Laser Beam Quality and Pointing Measurement with an Optical Resonator. Rev. Sci. Instrum.
**2007**, 78, 73103. [Google Scholar] [CrossRef] [PubMed] - Morrison, E.; Meers, B.J.; Robertson, D.I.; Ward, H. Automatic Alignment of Optical Interferometers. Appl. Opt.
**1994**, 33, 5041–5049. [Google Scholar] [CrossRef] [PubMed] - Dobosz, M. Interference Sensor for Ultra-Precision Measurement of Laser Beam Angular Deflection. Rev. Sci. Instrum.
**2018**, 89, 115003. [Google Scholar] [CrossRef] [PubMed] - Taylor Hobson Taylor Hobson: Autocollimators. Online Resource. Available online: https://www.taylor-hobson.com/products/alignment-level/autocollimators (accessed on 4 November 2023).
- Möller Wedel Optical: Electronic Autocollimators. Online Resource. Available online: https://moeller-wedel-optical.com/en/products/electronic-autocollimators/ (accessed on 4 November 2023).
- Dobosz, M.; Jankowski, M.; Mruk, J. Application of Interference Sensor of Angular Micro-Displacement in Measurements of Machine Rotational Errors. Precis. Eng.
**2019**, 60, 12–20. [Google Scholar] [CrossRef] - Renishaw XL-80 Laser System. Online Resource. Available online: https://www.renishaw.com/en/xl-80-laser-system--8268 (accessed on 4 November 2023).
- Renishaw XM-60 and XM-600 Multi-Axis Calibrator. Online Resource. Available online: https://www.renishaw.com/en/xm-60-and-xm-600-multi-axis-calibrator--39258 (accessed on 4 November 2023).
- Nadolski, A. Badanie Wpływu Ustawienia Układu Zmieniającego Kierunek Odchyleń Wiązki Lasera Na Działanie Czujnika Mikroodchyleń Wiązki 2021. Master’s Thesis, Warsaw University of Technology, Warsaw, Poland, 2021. [Google Scholar]
- Mad City Labs Inc. Piezo Nanopositioning Systems, Vacuum Nanopositioning, Microscope Stages & Micropositioners. Atomic Force Microscopes, Near Field Scanning Optical Microscopes, Single Molecule Microscopes. Product Catalog 800A. Available online: http://www.madcitylabs.com (accessed on 4 November 2023).
- Mad City Labs, Inc. MCL Nano-Drive Technical Specification. Online Resource. Available online: http://www.madcitylabs.com/nanodrive.html (accessed on 4 November 2023).

**Figure 1.**Proposed optical element that rotates the deflection of a laser beam. The green arrows show the directions of rotation with a positive sign: dashed red line—the nominal path of the beam, solid red and blue lines—the deflected beams.

**Figure 2.**Setup for two-axial measurement using one mono-axial sensor: P1 and P2—polarisers, VPR—variable phase retarder, QR—quarter-wave retarder plate, PBS—polarising beamsplitter, NBS—non-polarising beamsplitter, M—mirror, SOE1—optical element for rotation of beam deflection, SOE2—optical element for shifting the beam, S—angular sensor. Black arrows show the directions of possible adjustments.

**Figure 3.**Principle of operation for the subsystem altering the beam polarisation: P1—polariser, VPR—variable phase retarder, QR—quarter-wave retarder plate.

**Figure 4.**Main principle of operation of the sensor: PB—polarising beamsplitter, CCR—corner cube reflector, P90—right-angle prism.

**Figure 5.**Setup used for experimental verification of the proposed method: P1 and P2—polarisers, VPR—variable phase retarder, QR—quarter-wave retarder plate, PBS—polarising beamsplitter, NBS—non-polarising beamsplitter, M—fixed mirror, SOE1—optical element for rotation of beam deflection, SOE2—optical element for raising the beam, MM—mirror on piezotranslator controlling the beam.

**Figure 14.**Average values of the results from 3 runs with extended uncertainties: black, dashed line—y = x line.

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**MDPI and ACS Style**

Dobosz, M.; Jankowski, M.; Mruk, J.
Two-Axial Measurement of the Angular Microdeflection of a Laser Beam Using One Single-Axis Sensor. *Sensors* **2023**, *23*, 9276.
https://doi.org/10.3390/s23229276

**AMA Style**

Dobosz M, Jankowski M, Mruk J.
Two-Axial Measurement of the Angular Microdeflection of a Laser Beam Using One Single-Axis Sensor. *Sensors*. 2023; 23(22):9276.
https://doi.org/10.3390/s23229276

**Chicago/Turabian Style**

Dobosz, Marek, Michał Jankowski, and Jakub Mruk.
2023. "Two-Axial Measurement of the Angular Microdeflection of a Laser Beam Using One Single-Axis Sensor" *Sensors* 23, no. 22: 9276.
https://doi.org/10.3390/s23229276