High-Precision Diagnosis of the Whole Process of Laser-Induced Plasma and Shock Waves Using Simultaneous Phase-Shift Interferometry

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The article suggests an interesting technique for detecting shock waves using an interferometer. The article is well written, and all the statements are confirmed in the text.

The following points can be mentioned as comments

1. The paper does not show the ultimate sensitivity of this method: what is the minimum phase shift that can be detected using the proposed scheme?
2. Despite the fact that the authors claim that their system is highly sensitive and capable of detecting weak shock waves, nevertheless, the work demonstrates the detection of waves with a phase shift of the order of a radian or more, which is a large phase shift for interferometry. And it is also unclear why such a large phase shift of the wavelength order is poorly detected by a conventional interference circuit (according to the authors). This point needs to be clarified.

Author Response

Dear Reviewer,

We truly appreciate your recognition of our paper and the valuable comments you provided. Here are our responses to your two Comments:

Comment 1: The paper does not show the ultimate sensitivity of this method: what is the minimum phase shift that can be detected using the proposed scheme?

Response 1: It is true that the minimum detectable phase shift of this method was not clearly presented in the original paper. However, the revised paper has supplemented this information. In the section "3.2 Uncertainty Discussion and Precision Analysis", the minimum detectable phase shift of the system was determined through the stability test of the background interferogram. Over fifty experiments were conducted, with an interval of over 10 minutes between each experiment to ensure that the environment returned to its previous state. The collected background interferograms were then processed. Due to the speckle noise in the light spot and environmental vibrations, the phases of the interferograms exhibited non-constant fluctuations.

After calculating the root mean square error (RMSE) of all background phases, it can be seen from the two-dimensional phase distribution histogram that the phase jitter values range from 0.01 to 0.08 radians, with an average of 0.045 radians. Based on these experiments, the minimum resolvable phase shift of the system is determined to be approximately 0.045 radians. This means that laser-induced plasma (LIP) or shock waves (SW) can be accurately detected when the absolute value of their phase shift exceeds much larger values.

Comment 2: Despite the fact that the authors claim that their system is highly sensitive and capable of detecting weak shock waves, nevertheless, the work demonstrates the detection of waves with a phase shift of the order of a radian or more, which is a large phase shift for interferometry. And it is also unclear why such a large phase shift of the wavelength order is poorly detected by a conventional interference circuit (according to the authors). This point needs to be clarified.

Response 2: The experimental results intuitively demonstrate the spatial distribution of phase shifts from "large" to "small" through the changes in brightness and gradient of the false - color images. The central region (strong perturbation) shows a strong color contrast (corresponding to a phase shift of about 1 radian), while the edge region (weak perturbation) has a smooth color transition (corresponding to a phase shift close to zero). As can be seen from Fig 6(a)-(b), at 40 mm, the outline of the shock wave can still be distinguished, and the phase shift distribution can effectively show its boundary with the background gas. At 90 mm, from Fig 7(e)-(h), the interface profile changes from a curved surface to a flat surface, clearly presenting the changes of the shock wave during propagation.

These all prove that the system can detect different phase shift situations, not just phase shifts of about 1 radian. Traditional interference methods can indeed measure large phase shifts of about 1 radian, but they have limitations in presenting the spatial distribution of shock waves. When dealing with weak perturbations, the traditional carrier - frequency interferometry is prone to error effects such as aliasing and frequency - domain leakage due to small fringe deflections. This leads to the loss of a lot of detailed information during image processing. When measuring weaker shock waves, even if there is a phase shift of about 1 radian, it is difficult for traditional methods to clearly distinguish the fringe deflection, let alone clearly present the internal structure and spatial distribution of the shock wave.

In conclusion, the revised paper clarifies the minimum detectable phase shift of the SPSI system and elaborates on the advantages of this system compared with traditional interference methods under different phase shift conditions, further improving the research content. Thank you again for your valuable comments. We hope our response satisfies you.

Best regards,

Lou Gao et al.
School of Physics, Nanjing University of Science and Technology

Reviewer 2 Report

Comments and Suggestions for Authors

A method for the diagnosis of laser induced plasma and shock waves based on phase-shift interferometry was proposed. The performances of the proposed method was verified through experiments. I think the paper can be accepted for pulbicaiton providing the following issues can be well addressed.

 

• The theoretical part is too brief, only some literal expressions were given. How the phase or amplitude isextracted, was not mentioned in this part. Please also includes some issential formulas in this part.
• The performance of the proposed method should be quantitative described in the abstract and the conclusion part.
• It was claimed as a “high-precision”method, but how precise it is? Which should be clearly presented.

Author Response

Dear Reviewer,

We sincerely appreciate your positive feedback on our paper and the insightful comments you offered. In response to your three comments, we have the following clarifications:

Comment 1: The theoretical part is too brief, only some literal expressions were given. How the phase or amplitude isextracted, was not mentioned in this part. Please also includes some issential formulas in this part.

Response 1: We understand your concern about the brevity of the theoretical part in the original paper. In the revised version, we have added a new section, "2.1 Optics diagnosis of LIP and SW", to provide a more comprehensive theoretical foundation. This section focuses on the relationship between the refractive index, electron density, and molecular density, which are crucial parameters in understanding laser - induced plasma (LIP) and shock waves (SW). We explain how obtaining accurate refractive index information is significant for deeply understanding the physical properties of LIP and SW.

At the end of the "3.1 Experimental Setup and Data Processing" section, we have added more content to clarify the data processing steps. We describe how the phase values obtained from the Carré method are initially wrapped in the range of [−π,π]. To obtain the true phase values, we use a phase unwrapping algorithm. We explain that by subtracting the initial background phase from the unwrapped phase, we can obtain the distribution of the actual phase shift. This additional information makes the data processing procedure more transparent and understandable.

Comment 2 and 3: The performance of the proposed method should be quantitative described in the abstract and the conclusion part. It was claimed as a “high-precision”method, but how precise it is? Which should be clearly presented.

Response 2 and 3: To thoroughly address your concern regarding the precision of the SPSI system, we have incorporated a new subsection titled "3.2 Uncertainty Discussion and Precision Analysis" into the experimental part of the paper. In this section, we conduct a comprehensive analysis of the uncertainty sources in the experiment, which mainly include phase shift recovery errors, systemic noise, and system stability issues. We meticulously explain the mechanisms through which these factors influence the measurement precision.

Furthermore, we experimentally determined the minimum detectable phase shift of the system. By performing 50 trials on the stability of the background interferogram, we observed that the phase jitter values vary from 0.01 to 0.08 radians, with an average value of 0.045 radians. This implies that the system's minimum resolvable phase shift is approximately 0.045 radians. We firmly believe that these quantitative values effectively illustrate the high precision of the SPSI system. The relevant results have been added to both the summary and conclusion sections, ensuring that the paper provides a more complete and accurate representation of the system's performance.

Best regards,

Lou Gao et al.
School of Physics, Nanjing University of Science and Technology

Reviewer 3 Report

Comments and Suggestions for Authors

This paper presents experimental results of synchronized phase shift interferometry applied to a laser-induced plasma. In particular, results are presented comparing the synchronized phase shift interferometry with the more standard approach using a single carrier-frequency interferogram and fast Fourier transform techniques to extract the phase.

In general, the paper is well written and clear. References are appropriately cited. The results are of interest to those working with laser-induced plasmas.

However, I have a number of concerns which need to be addressed before publication.

The first concern relates to the novelty of the work. Essentially, the paper discusses the benefits of the synchronized phase shift interferometry techniques over the standard approach. However, the experimental approach has been described in multiple papers (for example ref 18, Fukuda et al, Optics Express, 2017) and the phase-shifting analysis is a common interferometric approach (albeit using other experimental configurations to achieve the same outcome). Therefore, to proceed with publication, the authors need to be clearer on how their work extends current knowledge which, for example, could relate to novelty in the application of the technique.

A second concern is that experimental results are presented without any discussion of uncertainties. These should be quantified and included with the results.

Lastly, the comparison between the techniques is currently qualitative. Figure 5 in the paper shows a comparison of the two approaches but there is no quantitative analysis of the phase nor the improvement in sensitivity. The two results show significant differences in the measured phase shift across the shock wave. It is unclear why the phase shift in the undisturbed air is significantly different between the two images (this is presumably the reference condition where phase shifts in each case should be zero). Perhaps these differences are due to the accuracies of each approach but, if so, the authors should demonstrate that this is indeed due to the technique, and not due to shot-to-shot variations.

A few further points:

Figure 1: No reference is provided for this figure. It seems to be largely identical to the figure presented by Fukuda but is reproduced without acknowledgement.

Line 92: should be Figure 2(a), not 1(a)

Figure 2: I presume the wavelength for the HWP is 532 nm

Line 182: should be Greek mu not u for micro.

Author Response

Dear Reviewer,

We are truly grateful for the perceptive comments you provided. Regarding your three comments, we offer the following clarifications:

Comment 1: The first concern relates to the novelty of the work. Essentially, the paper discusses the benefits of the synchronized phase shift interferometry techniques over the standard approach. However, the experimental approach has been described in multiple papers (for example ref 18, Fukuda et al, Optics Express, 2017) and the phase-shifting analysis is a common interferometric approach (albeit using other experimental configurations to achieve the same outcome). Therefore, to proceed with publication, the authors need to be clearer on how their work extends current knowledge which, for example, could relate to novelty in the application of the technique.

Response 1: We sincerely appreciate your concern regarding the novelty of our work. Our study does build upon the simultaneous phase shift interferometry (SPSI) technique, yet it presents several innovative aspects that contribute to expanding the existing knowledge in this field.

Firstly, considering the application aspect, while Fukuda et al. (2017) applied a similar technique for three - dimensional imaging of the refractive index distribution, our research is uniquely centered on the high - precision diagnosis of the entire process of laser - induced plasma (LIP) and shock waves (SW). We are dedicated to capturing the initial spatio-temporal evolution of LIP, a process that involves intricate physical phenomena like high - density electron generation and plasma expansion. Moreover, we meticulously track SW at different attenuation levels, from the robust shock waves close to the LIP source to the feeble shock waves on the verge of decaying into sound waves. This all - encompassing investigation of the LIP - SW system throughout its evolution represents a novel application of the SPSI technique.

Secondly, we have made significant progress in surmounting technical obstacles. In the measurement of high - electron - density LIP, the Faraday rotation effect is a formidable challenge as it introduces an indeterminate constant phase shift. Our system ingeniously integrates pump - probe technology, polarization - element coupled phase, and Carré phase - recovery methods to efficiently rectify these phase - shift errors. This technical solution is distinct from previous studies. Even if other research endeavors employ phase - shifting analysis, they might not have tackled this specific problem within the context of LIP and SW diagnosis. Our approach guarantees accurate phase measurement in the presence of the Faraday rotation effect, marking a substantial advancement in the realm of interferometric measurements for LIP and SW.

Finally, in practical applications, our SPSI system exhibits remarkable advantages in detecting transient weak perturbations. This characteristic is of utmost importance for applications in high-vacuum plasma, low - pressure shock waves, and stress waves in optical materials.

We have incorporated these elements into the Introduction section to highlight the novelty of our work. We firmly believe that these aspects vividly illustrate the originality of our research and its valuable contribution to the existing body of knowledge.

Comment 2: A second concern is that experimental results are presented without any discussion of uncertainties. These should be quantified and included with the results.

Response 2: We sincerely appreciate your second comment, and we are fully aware of the significance of discussing uncertainties in our experimental results. In the revised manuscript, we have made a dedicated effort to address this concern by adding a new section.

In the "3.2 Uncertainty Discussion and Precision Analysis" section, we have meticulously analyzed the uncertainties in our experiment. We comprehensively considered multiple factors that might influence the accuracy of our measurements. For instance, when it comes to phase shift recovery, the Carré phase recovery algorithm has its inherent error sources. In a high - electron - density plasma environment, the Faraday rotation effect further complicates the situation. When the signal - to - noise ratio is sufficiently high (SNR > 30 dB), the phase error for uniform phase shifts remains relatively small. However, for non - uniform perturbations such as shock wave fronts, additional errors are inevitably introduced.

Systemic noise also has an impact. The misalignment of the micro - polarizer array, fluctuations in laser energy, and the readout noise of the CMOS sensor all contribute to the overall uncertainty. We carefully took all these factors into account and accurately calculated the total uncertainty in the measured phase shift.

To precisely determine the minimum detectable phase shift of the system, we conducted 50 trials on the stability of the background interferogram. Through these trials, we observed that the phase jitter values fell within a certain range, and the average value obtained therefrom indicates the system's minimum resolvable phase shift.

We have integrated these uncertainty - related results into the relevant parts of the experimental results section. This not only enriches the content of our experimental analysis but also provides a more comprehensive and reliable basis for the interpretation of our research findings.

Comment 3: Lastly, the comparison between the techniques is currently qualitative. Figure 5 in the paper shows a comparison of the two approaches but there is no quantitative analysis of the phase nor the improvement in sensitivity. The two results show significant differences in the measured phase shift across the shock wave. It is unclear why the phase shift in the undisturbed air is significantly different between the two images (this is presumably the reference condition where phase shifts in each case should be zero). Perhaps these differences are due to the accuracies of each approach but, if so, the authors should demonstrate that this is indeed due to the technique, and not due to shot-to-shot variations.

Response 3:

We sincerely appreciate the reviewer’s insightful comments. Below, we address the concerns raised by providing quantitative analysis and clarifying the observed differences in phase shifts between the SPSI and conventional methods.

 

1. Quantitative Analysis of Sensitivity Improvement

In the revised manuscript (Section 3.2, Uncertainty Discussion and Precision Analysis), we have added rigorous uncertainty quantification for the SPSI system. Key quantitative results include: Total phase shift uncertainty: Controlled within ±0.008 radians, corresponding to a refractive index uncertainty of Δn ≈ ±1.52 × 10⁻⁶ (for λ = 532 nm). Minimum resolvable phase shift: 0.045 radians, equivalent to a refractive index accuracy of Δn ≈ ±8.55 × 10⁻⁶. Comparison with conventional carrier-frequency interferometry (SCFI): SCFI’s noise-equivalent phase error under low fringe contrast (C < 0.2) exceeds 0.5 radians, an order of magnitude higher than SPSI (Section 3.3, Fig. 6d).

These values directly demonstrate the quantitative superiority of SPSI in resolving weak perturbations, such as low-energy shock waves.

2. Explanation of Phase Shift Differences in Undisturbed Air

The apparent non-zero phase shift in the "undisturbed air" regions of Fig. 6 (formerly Fig. 5) arises from fundamental limitations of the conventional method, not shot-to-shot variations. We clarify as follows:

1. Noise and Fringe Contrast Limitations

In SCFI (Fig. 6c-d), the low fringe contrast (<0.2) and high-frequency noise (aliasing, frequency-domain leakage) distort phase recovery. The FFT-based bandpass filtering used in SCFI removes high-frequency details, leading to baseline phase drift (non-zero offsets in undisturbed regions).

In contrast, SPSI (Fig. 6b) uses the Carre algorithm to recover phase without relying on carrier frequencies or filtering, eliminating these artifacts.

1. Experimental Validation of Stability

To exclude shot-to-shot variations, we conducted 50 repeated trials of background interferograms (Section 3.2, Fig. 4). The results confirmed:

Phase jitter in SPSI: 0.01–0.08 radians (average 0.045 rad), consistent with the system’s minimum resolvable phase shift.

SCFI’s instability: Phase errors >0.5 radians under identical conditions, as shown in Fig. 6d.

This confirms that the observed differences stem from systematic limitations of SCFI, not experimental fluctuations.

3. Enhanced Figures for Clarity

In Visualization 2 (supplementary material), we provide additional data comparing SPSI and SCFI measurements of the same shock wave (D = 40 mm). These results reinforce that SPSI achieves sub-radian phase resolution, while SCFI fails to resolve perturbations below 0.5 radians.

4. Conclusion

The revised manuscript now includes quantitative metrics for sensitivity and uncertainty, and the observed phase differences are conclusively attributed to the superior noise immunity and accuracy of SPSI. We thank the reviewer for highlighting this critical point, which has strengthened the clarity and rigor of our analysis.

• A few further points:

Response: Regarding Figure 1, we apologize for the oversight. We have now added a proper reference in the figure caption to clarify its origin.

Regarding the error in line 92, we have corrected it. Because of the addition of the picture, we have cited the pictures in the correct order.

In Figure 2, we have amended the wavelength information for the HWP in the figure caption, specifying that it is 532 nm, as you presumed.

Finally, in Line 182, we have replaced the incorrect “u” with the Greek symbol “μ” for micro in revised manuscript.

Best regards,

Lou Gao et al.
School of Physics, Nanjing University of Science and Technology

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

Since the authors have thoroughly revised the manuscript, and all my concerns have been well adressed, I would pleasure to support its acceptance in tis current form.

Author Response

Dear Reviewer,
Thank you for your valuable suggestions that have significantly improved the manuscript. We greatly appreciate your contribution to enhancing the quality of our work.   Sincerely,
Lou Gao et al.

Reviewer 3 Report

Comments and Suggestions for Authors

I thank the authors for responding to my comments. However, there are still some outstanding issues.

1. Novelty

Some, but not all, of my concerns have been addressed. The authors indicate that the novelty lies in the area of application which I accept. However, this needs to be further highlighted with references to past studies of such flows to demonstrate the advances made.

The authors also claim that there is novelty in addressing Faraday rotation. A paper by Lehmann and Spatschek is referenced but seems to bear no relation to the current work (Faraday rotation is not mentioned in that paper). The phase shift due to this Faraday rotation is then included with the applied phase shift between the arms and subsequently ignored without justification. If this is a novel approach, then the paper needs to provide further explanation of the method and its novelty.

Lastly, in the revised version of the paper, the authors claim to “introduce” SPSI. I would argue that the authors are applying the technique – the current wording (in a number of places) gives the impression that this is a new technique.

Note that references 8, 12 and 22 are missing the journal details.

2. Uncertainties

I thank the author for quantifying the uncertainties. However, there is a claimed uncertainty of 0.008 radians but then they identify a “phase jitter” of 0.045 radians. Surely the latter is present in all measurements and is thus a better indication of the uncertainty of each measurement.

3. Background phase in SCFI

The authors argue that the FFT process has introduced a phase drift. This may be the case but has no meaning in analysing the final flow field. What is of interest is how the phase changes from the undisturbed region through into the shock wave. Thus, to compare the two methods, a final step of removing the phase drift should be applied to the FFT method so that the phase in the air is zero for both methods.

I indicated that a quantitative comparison of phases measured by each approach would enhance the quality of the paper. By this I meant a plot of phase along a common line. For example, this could be along a horizontal line at y vale of 5mm (as seen in Figure 6).

Comments on the Quality of English Language

In some places, the quality of the English could be improved - some proof-reading is needed

Author Response

Dear Reviewer,

We are truly grateful that you have taken the trouble to offer comments again. Regarding your three comments, we provide the following clarifications:

Comment 1:

Novelty

Some, but not all, of my concerns have been addressed. The authors indicate that the novelty lies in the area of application which I accept. However, this needs to be further highlighted with references to past studies of such flows to demonstrate the advances made.

The authors also claim that there is novelty in addressing Faraday rotation. A paper by Lehmann and Spatschek is referenced but seems to bear no relation to the current work (Faraday rotation is not mentioned in that paper). The phase shift due to this Faraday rotation is then included with the applied phase shift between the arms and subsequently ignored without justification. If this is a novel approach, then the paper needs to provide further explanation of the method and its novelty.

Lastly, in the revised version of the paper, the authors claim to “introduce” SPSI. I would argue that the authors are applying the technique – the current wording (in a number of places) gives the impression that this is a new technique.

Note that references 8, 12 and 22 are missing the journal details.

Response 1: Thank you for your constructive feedback. We have carefully addressed the concerns regarding the novelty and references as outlined below:

1. Novelty in Application Area

We appreciate your acceptance of the novelty in the application of SPSI to LIP and SW diagnostics. To further highlight the advances, we have supplemented the second paragraph of the Introduction section with research progress of predecessors, elaborating on the use of interferometry for studying LIP and SW. Studies by Breitling et al., Tao et al., Sobra et al., Mao et al., and Zhang et al., which employed interferometry to investigate LIP and SW, collectively show that their work focused on early-stage strong perturbation processes, such as the initial evolution phases of LIP and SW with significant beam perturbations.

2. Faraday Rotation Handling and Algorithm Correction

We acknowledge the error in referencing Lehmann and Spatschek’s paper, which does not address Faraday rotation. This has been corrected by citing recent relevant studies (Jiang et al. 2023; Swadling et al. 2014) that directly link Faraday rotation to plasma electron density. In Section 3.2 (Data Processing) and the Introduction, we now elaborate on how the Carré algorithm mitigates phase shift deviations caused by Faraday rotation, treating the polarization error as a constant to enable robust phase recovery (see Equations 9–11).

3. Clarification on SPSI Terminology

We have revised all instances of "introduce" to "employ" (e.g., Abstract, Introduction) to clarify that we are applying and adapting the SPSI technique rather than presenting it as a new invention. The focus of the work is on its novel integration with polarization correction and Abel inversion for transient plasma and shock wave diagnostics.

Comment 2: 2.    

Uncertainties

I thank the author for quantifying the uncertainties. However, there is a claimed uncertainty of 0.008 radians but then they identify a “phase jitter” of 0.045 radians. Surely the latter is present in all measurements and is thus a better indication of the uncertainty of each measurement.

Response 2:

Thank you for your careful review and valuable suggestion regarding the uncertainty quantification.

In the revised manuscript, we have indeed addressed the distinction between the two uncertainty values to avoid confusion. The 0.008 radians represents the inherent uncertainty derived from algorithmic errors (Carré method), optical component tolerances (e.g., polarizer alignment), and detector noise, which are systematically analyzed in Section 3.3. This value reflects the precision of the measurement chain under ideal conditions, excluding environmental fluctuations.

However, as you correctly noted, the 0.045 radians phase jitter is a more comprehensive metric for practical measurements. This value accounts for total uncertainty in real-world scenarios, including uncontrollable factors like speckle noise, environmental vibrations, and laser energy stability. As stated in the manuscript: "To determine the detection system’s accuracy by accounting for all uncertainties, background phase jitter is used for quantification. The phase shifts of LIP and SW are derived relative to the phase of the background interferogram," emphasizing that the 0.045 radians represents the minimum resolvable phase shift when all error sources are integrated.

This distinction is now highlighted in Section 3.3 to clarify that the 0.008 radians is a component-wise uncertainty, while the 0.045 radians reflects the system’s operational accuracy in dynamic environments. Thank you for prompting this clarification, which strengthens the rigor of our uncertainty analysis.

Comment 3:

Background phase in SCFI

The authors argue that the FFT process has introduced a phase drift. This may be the case but has no meaning in analysing the final flow field. What is of interest is how the phase changes from the undisturbed region through into the shock wave. Thus, to compare the two methods, a final step of removing the phase drift should be applied to the FFT method so that the phase in the air is zero for both methods.

I indicated that a quantitative comparison of phases measured by each approach would enhance the quality of the paper. By this I meant a plot of phase along a common line. For example, this could be along a horizontal line at y vale of 5mm (as seen in Figure 6).

Response 3:

To address your suggestion regarding the phase transition from the undisturbed region to the shock wave, we have taken the following changes :

To clarify the phase transition from the undisturbed region to the shock wave, we selected a profile at Y=7mm for both methods. The SCPS data were subjected to 1 D phase drift correction by enforcing a zero phase reference in the undisturbed air region. The comparative results are presented in Fig. 6(e), which clearly demonstrates the phase behavior across the shock interface for both techniques. 

This modification ensures a direct visualization of the phase change through the shock wave while eliminating background discrepancies, enhancing the clarity of method performance comparison.

Finally, thank you for your feedback on the English language. We have carefully proofread the manuscript and revised many details to improve clarity and accuracy, ensuring the language meets academic standards.

Best regards,

Lou Gao et al.

School of Physics, Nanjing University of Science and Technology