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

Round 1

Reviewer 1 Report

Summary:

The manuscript presents an innovative approach to the two-axial measurement of laser beam angular deflection using a single-axis sensor. The main contribution lies in the experimental validation of the method, which is poised to enhance precision in metrology applications. The strength of the paper is in its detailed experimental setup and the initial demonstration of the method's potential for accurate angular measurements.

General Concept Comments:

The paper introduces a potentially impactful method for angular measurement, but there are areas that require further clarification and development. The hypothesis that a single-axis sensor can effectively measure two-axial deflections is intriguing; however, the methodological execution lacks a comprehensive analysis of uncertainties (statistical vs. systematic errors), the effects of non-ideal conditions, such as imperfect polarization separation, and the estimation of angular dynamic range. 

Review Comments:

The review of the topic can be improved as it does not adequately address the comparison with differential wavefront sensing methods, which are known for better resolution and also provide laser beam tilt with respect to a reference beam, or other methods. Many examples are given in an earlier article published by the authors, however, the manuscript would benefit from a short thorough literature review to contextualize its contributions within the current state of the field. 

Specific Comments:

• Fig 6-8:The manuscript could be clearer in its presentation of the experimental results, particularly in distinguishing between Experiments 2 and 3 (and 1). The source of discrepancies between these experiments is not well explained (refer to the sections describing these experiments). If the measurements are taken for the same experimental settings, but from different measurement runs, these data should be used to quantify the repeatability and provide to some degree uncertainties of the measurements. 
• Then, environmental noise sources should be discussed to explain eventual discrepancies between the individual measurements.
• Figures 6-12 illustrate measurement results. It would be good to include uncertainties to validate the repeatability claims. Error bar estimation/approximation is needed to verify the confidence level of these measurements.
• Figures 6-12: The authors could think about combining some figures, especially Fig. 6 and 7, which already show the same data. 
• Fig. 9 and 11: Cross talk: The conclusions drawn from Experiments 4 and 5 regarding the dependency of phi and theta need a stronger empirical basis. The manuscript should explore and explain the reasons behind the observed cross-coupling in Fig 11. in theta. Why is there a significant slope of about 0.11 rad/rad, while Fig. 9 does not show a dependency on phi? Is this true or due to repeatability? What is the time between two data points? Is this cross talk be generated by the measurement, or is it driven by some external noise sources?
• Fig. 12: An average including standard deviation would be beneficial to estimate uncertainties (see also earlier comments). 
• line 377 and 379: The second theta in the sentence must be replaced with phi ?!

General Questions:

• The manuscript asserts an error of less than 2 µrad over the dynamic range of 15 µrad (13%), is this sufficient and if so, why? How can one improve? What is the limiting noise source? Repeatability = uncertainties? A clear discrimination between systematic and statistical error might be beneficial for the understanding.  
• They explain that large angular variations might cause non-linearities. Why and what is the maximum dynamic range of the presented technique? When does it become non-linear and to what degree can it be corrected using the equation 1? (It would be beneficial to add some numbers to get an impression of the limitations).

Ethics and Data Availability:

• None

Final Recommendation:

The manuscript has the potential to contribute significantly to the field of precision metrology. However, it requires minor revision to address the methodological gaps, and provide a more robust statistical analysis of the experimental data if available. Once these issues are addressed, the manuscript would be a strong candidate for publication.

Author Response

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Author Response File: Author Response.pdf

Reviewer 2 Report

The manuscript provides a way of enabling two-axial measurements of the deflection of a beam based on a single-axis sensor. The key idea is to direct a laser beam alternately into one of two arms of a measurement system. The work is interesting, but I cannot see a reason why people use such a sensor. Firstly, compared to existing two-axis autocollimators and angular interferometers, the proposed method does not show advantages, while the accuracy is much lower. Secondly, the proposed method is not simultaneously measuring two angles, but by turns. For real application like coordinate machines, that’s not what people want. Thirdly, as shown in Fig.5, the existing sensor is actually very small, the authors saved one small component but added a lot of other components, makes the system actually much more complicated and larger.

 Therefore, I do not recommend the manuscript for publication in this journal. Besides, there are several other issues：

 1.In the introduction, the authors claim that angular interferometers only measure single-axis, but there are many interferometers measuring two-axis angle.

2. What does the “Aθ₁ axis” in line 142 represent?

3. What is the range of measurements mentioned in line 315 of the manuscript? What’s the limitation of the measurement range?

4. In the experiment, the piezotranslator was used as the angle generating device and seems like also as the standard device, but the specification of the piezotranslator was not given, how accurate is it?

5. The absolute error is mentioned in the process of describing the experimental results, how to define the absolute error here?

6. There is no description of the experimental environment in the manuscript, the stability and noise of the measurement system should be test.

7. The description of experiment 2 and experiment 3 in the manuscript is unclear.

8. For the outlier in Figure 12, no explanation is given in the manuscript, but it is simply eliminated.

9. In experiment 5, with the change of angle φ, angle ϑ also changes, and the latter is about one-eighth of the former, which cannot be considered that the two do not affect each other.

10. Measurement errors cannot be visually seen in the data plots.

11. The data plots are too rough and need to be improved.

Comments for author File: Comments.pdf

Author Response

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Author Response File: Author Response.pdf

Reviewer 3 Report

Taking the angular deflection measurement of laser beams as a starting point, the manuscript proposes a new method for measuring small angular deflections in 2D using special prisms. Some improvements are still needed.

1. The wording of the formulas is not so formal, please modify the formulas according to the mathematical norms.

2. The descriptions of the 5 experiments are not clear enough, for example, add an explanation of the difference between experiment 2 and experiment 3.

3. In Figure 6-12, please add a linearly fitted residual series plot of the measurement results and change the font size of the title.

4. The description in the results section is not clear enough. For the coupling effect between the two-axes, e.g., in Figure 11, the red line shows a spike at the "step = 15", while the blue line does not. 

5. Please explain the basis for choosing a fixed value for one of the axes in Experiments 4 and 5.

The manuscript is well organized and written.

Author Response

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Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

The authors now well presented the significance of their work, it is an interesting research in the field of angular measurement. Now I believe it can be published after addressing the following issue:

1、As shown in Fig.8, the change of v is caused by the direction drift of laser source, while it cannot be seen in Fig.10. Please give an explain.

2、The used rotational stage (MCL Nano-MTA2) can achieve very low noise and high resolution, and the authors additionally calibrated its accuracy by using an autocollimator, so finally what is the accuracy of the used stage? The error of the stage is not taken into account in section 3.2 uncertainty analysis.

Author Response

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Author Response File: Author Response.pdf