Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors reported their work on hard x-ray polarization detection based on Compton scattering. The authors claimed that an alternative approach to the polarization reconstruction method based on simulated Compton scattering distribution for determining the polarization state. They also claimed the development of correction methods to account for additional “pseudo Compton events”. However, details of the reconstruction algorithm are not explicitly written in the manuscript. Therefore, the article should have major revision. My major comments on the manuscript are listed below.

1, Line 107-110: The authors mentioned “From an adjustment of the model … can be extracted.” I agree with this point. However, the method or algorithm to fit the model distribution to the experimental result is not explicitly described. The method is the core of the algorithm. The authors should describe it. Otherwise, no one can reproduce their results.

2. Figure 2: There are error-bars in those plots. The meaning of the error-bars is not described. Do those represent fitting errors? How did authors evaluate those errors?

3. Line 200-201: The authors concluded that the reconstruction algorithm yields good results compared to the expected values. The expected values are P_1,i=0.988+/-0.004 and P_2,i = 0.023+/-0.047. In this case, P_L~0.988, Kai~0.012. The reconstructed results are P_L=0.899+/-0.028, Kai=2.9+/-1.3. The expected value is not within the error of the reconstructed data even the authors used almost ideally polarized light generated by e-beams circulating in a storage ring. The accuracy of the determination of the polarization state could be worth for x-rays having arbitrary polarization states. Is it possible to use this device for determining the polarization degree of x-rays having arbitrary polarization states?

Author Response

Response to Reviewer 1 Comments

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted changes in the re-submitted files.

 

 

Comments 1: Line 107-110: The authors mentioned “From an adjustment of the model … can be extracted.” I agree with this point. However, the method or algorithm to fit the model distribution to the experimental result is not explicitly described. The method is the core of the algorithm. The authors should describe it. Otherwise, no one can reproduce their results.

Response 1: We agree with this comment. Therefore, we have added the following sentences starting on page 4 in line 115:

“For the adjustment of the model function to the data a cost function assuming Poisson distributed data is minimized using the iminiut routine [26] which is based on [27]. The cost function is constructed using the cost.poisson_chih2() function of the iminuit routine which computes an asymptotically χ²-distributed cost for Poisson-distributed data. The uncertainty of the fitting procedure is estimated by the Minuit.hesse() function of the iminuit routine. This estimates the uncertainty based on the covariance matrix and yields a one sigma uncertainty. By repeating the fitting procedure for different bin sizes of the scattering distribution, possible errors due to binning effects can be estimated. Furthermore, a bootstrapping approach is applied for each of these steps to resample the simulated distributions to assess their statistical uncertainty.”

 

Comments 2: Figure 2: There are error-bars in those plots. The meaning of the error-bars is not described. Do those represent fitting errors? How did authors evaluate those errors?

Response 2: Thank you for pointing this out. These errors represent the fit error resulting from the covariance matrix which corresponds to one standard deviation. For clarification, we have added the following sentence in the figure caption of figure 2 on page 5:

“The displayed error bars correspond to the fit error as described in section 3.”

 

Comments 3: Line 200-201: The authors concluded that the reconstruction algorithm yields good results compared to the expected values. The expected values are P_1,i=0.988+/-0.004 and P_2,i = 0.023+/-0.047. In this case, P_L~0.988, Kai~0.012. The reconstructed results are P_L=0.899+/-0.028, Kai=2.9+/-1.3. The expected value is not within the error of the reconstructed data even the authors used almost ideally polarized light generated by e-beams circulating in a storage ring. The accuracy of the determination of the polarization state could be worth for x-rays having arbitrary polarization states. Is it possible to use this device for determining the polarization degree of x-rays having arbitrary polarization states?

Response 3: It seems to be that there is a misunderstanding. The values P_1,i and P_2,i correspond to the polarization of the incident synchrotron beam. In section 6 the polarization of the photons which Compton scattered off the gold target are of interest. Therefore, the reconstructed polarization values need to be compared with the corresponding theoretical values (P_L,th and χ_th). To emphasize this, we changed and added the following sentences starting on page 6 in line 172:

“The 2D scattering distribution on the detector crystal resulting from Compton scattering events on the detector crystal where the incident photons previously also Compton scattered from the gold target within the set energy window as well as the reconstruction of the experimental data can be seen in figure 4. Note, that we have two subsequent Compton scattering events, first is the scattering off the gold target of which the scattered radiation will be analyzed on its polarization by the second scattering event being Compton scattering on the detector crystal.”

And modified the sentence starting on page 8 in line 216:

“From our new algorithm we reconstructed a degree of polarization of P_L = 0.899 ± 0.028 and a polarization angle of χ = 2.9 ± 1.3 for the photon beam which Compton scattered off the gold foil at an observation angle of 63.4°.”

It is possible to determine the degree of polarization for arbitrary polarization states with the presented device. That our reconstruction algorithm is able to determine the degree of polarization is shown in figure 2 on page 5 and discussed in section 4.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear authors,

this work is very well written and a great contribution to the scientific community. I recommend it for publication as it is. 

There is a typo in line 34: missing ")" and line 137: missing space. 

Concerning Fig. 4b, could you please comment on the factor of almost 10 more counts in 4b compared to 4a, which is not there in the corresponding data of Fig. 5?

 

Please add a sentence at the end of the conclusion for an outlook. What will your nice work be used for / applied to?

 

Thanks and all the best.

Author Response

Response to Reviewer 2 Comments

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted changes in the re-submitted files.

 

 

Comments 1: There is a typo in line 34: missing ")" and line 137: missing space.

Response 1: Thank you for pointing this out. The missing “)” on page 1 in line 34 has been added as well as the missing space on page 5 in line 149.

 

Comments 2: Concerning Fig. 4b, could you please comment on the factor of almost 10 more counts in 4b compared to 4a, which is not there in the corresponding data of Fig. 5?

Response 2: In Fig.5 is the reconstruction normalized to the magnitude of the experimental data for a better visualization, in contrast to Fig. 4b where this is not done, as the focus in this figure is on the pattern. Therefore, we added the following sentence in the image caption of Fig. 5 on page 7 to the following:

“The reconstructed distribution is normalized to the magnitude of the experimental distribution.”

And modified the following sentence starting on page 8 in line 212:

“Furthermore, figure 4b and the red shaded curve in figure 5 show the reconstruction of the experimental scattering distribution, which is normalized to the magnitude of the experimental distribution, displayed in figure 4a and figure 5, respectively, using the model function in Equation (4).”

 

Comments 3: Please add a sentence at the end of the conclusion for an outlook. What will your nice work be used for / applied to?

Response 3: To emphasize this, we added the following sentence on page 8 in line 226:

“This kind of detector can be applied for an efficient determination of the linear polarization of hard x-rays as are emitted in a lot of fundamental interaction processes.”

And the sentences starting on page 8 in line 252:

“In a recent experiment, hard x-rays were scattered off a gold target [29]. The presented reconstruction algorithm is used to determine the linear polarization of the scattered radiation. The publication of the polarization analysis is pending. Additionally, this routine will be used in future experiments to analyze the polarization of atomic emission and scattering processes for energies beyond 200 keV. This will be possible with our novel Compton telescope, which is in the commissioning phase [15].”

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

1. Please provide the name of the software used for Monte Carlo simulations or related information.
2. The introduction does not list recent (post-2020) literature on Compton polarimetry.
3. In Section 5, Background Correction, please analyze the causes of pseudo-Compton events.
4. How was the threshold value of 1.5 determined, and what is the theoretical basis for it? It is recommended to verify the effect of different threshold values on polarization reconstruction accuracy through Monte Carlo simulations.
5. Is the threshold value selection related to the energy of the X-ray? Please clarify.
6. What are the experimental conditions for the data in Figure 3? Please provide this information before the results.
7. It is suggested to add error bars in Figure 5.
8. Is the silicon detector model mentioned in Section 2 the one used in subsequent experiments? Please clarify and provide the detector parameters.
9. Please describe how the energy spectrum broadening was handled.

Author Response

Response to Reviewer 3 Comments

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted changes in the re-submitted files.

 

 

Comments 1: Please provide the name of the software used for Monte Carlo simulations or related information.

Response 1: The code of the Monte Carlo simulations is written in EGS5 (reference [25]) and described in reference [24] (on page 4 see line 108). The self-written code has no name and we can only reference these papers.

 

Comments 2: The introduction does not list recent (post-2020) literature on Compton polarimetry.

Response 2: You are correct that we did include literature from the last 5 years. Therefore, we revised the references in the introduction. We substituted on page 1 in line 17 the reference previously at number [8] with [8, 9] and added in line 19 the paper [15] as recent publications (even though [8] is not recent). We do not have more recent publications in this field and we also do not know different recent papers from other groups. If you know more recent papers, please let us know. To implement the modifications, we changed the two following sentences starting on page 1 in line 19:

“In the last decades efficient Compton polarimeters were developed based on large-volume position-sensitive semiconductor crystals [10–15]. Within the Stored Particles Atomic Physics Research Collaboration (SPARC) [16], these detectors enabled a wide variety of novel experiments dedicated to the polarization features in atomic interaction processes as radiative recombination [5], Bremsstrahlung [6], elastic scattering [17] or the transitions within highly charged heavy ions [18].”

 

Comments 3: In Section 5, Background Correction, please analyze the causes of pseudo-Compton events.

Response 3: Pseudo Compton events can be caused by two independent photons hitting the detector at the same time when these photons fulfill the energy condition. The origin of these photons varies. Possible origins for example are scattered photons off the shielding/experimental setup or from the target, e.g. K_α + tail of the Compton peak, background radiation from arbitrary sources,… While it definitively is important to understand that such events can occur, from our perspective, a discussion of the origin of these events is beyond the scope of this work, especially since the origin varies depending on the experimental setup and investigated energy window. Since the applied random correction does not include information on the origin of the occurring random events, we also think, that it does not have to be discussed further in the manuscript.

 

Comments 4: How was the threshold value of 1.5 determined, and what is the theoretical basis for it? It is recommended to verify the effect of different threshold values on polarization reconstruction accuracy through Monte Carlo simulations.

Response 4: The threshold value is based on the trade-off between high statistics and a good signal-to-noise ratio and determined by trial and error. We set as a lower boundary that we have more real Compton events than random double hits. Therefore, the threshold value is not based on theoretical model, it is based on the above-mentioned criteria, and we changed and added the following sentences starting on page 5 in line 164:

“For this, only events within a distance, where the ratio of real and pseudo Compton events lies under a set threshold (here 1.5), are considered in the analysis. The threshold value was chosen based on a trade-off between high statistics and a good signal-to-noise ratio.”

Your suggestion is very interesting but unfortunately, this is out of scope for this work.

 

Comments 5: Is the threshold value selection related to the energy of the X-ray? Please clarify.

Response 5: This value should be independent of the photon energy. The threshold is used to suppress a strong influence from random pseudo Compton events. The above-mentioned criteria in Response 4 are not dependent on energy and therefore, also the threshold. As it is a set value, we think, that the question does not have to be discussed in the manuscript.

 

Comments 6: What are the experimental conditions for the data in Figure 3? Please provide this information before the results.

Response 6: We understand your point. The decision on this arrangement is based on the premise to first explain the methodology before introducing the experiment for validation. Unfortunately, to visualize the background correction experimental data are necessary. Moving the plot after section 6 would make it more difficult for the reader. We hope you can now understand our choice to arrange the text.

 

Comments 7: It is suggested to add error bars in Figure 5.

Response 7: Thank you for pointing this out. We agree with your suggestion and have added error bars in Figure 5 and the following sentence in the figure caption on page 7:

“For the experimental data, the statistical uncertainty is given.”

 

Comments 8: Is the silicon detector model mentioned in Section 2 the one used in subsequent experiments? Please clarify and provide the detector parameters.

Response 8: Yes, the detector described in Section 2 is the one used in subsequent experiments. To further clarify we modified the following sentences in Section 2 on page 2 line 61:

“In figure 1a the detector which is relevant for this work is presented and described in the following.”

And on page 3 in line 67:

“This novel Compton polarimeter allows for an efficient use in an energy range of up to 200 keV with an energy resolution of 1 keV FWHM measured at 60 keV [14]. A more detailed description can be found in [14].”

 

Comments 9: Please describe how the energy spectrum broadening was handled.

Response 9: Thank you for pointing this out. The energy resolution of the detector is considered in the simulations. For this the energy resolution of each strip was determined by analyzing the monoenergetic Rayleigh peak. The resulting energy resolution is fed into the simulations for the polarization analysis. To clarify this in the manuscript we added the sentence starting on page 4 in line 115:

“Both, the experimental and the simulated azimuthal scattering distributions need to be sorted into histograms with the same bin size, i.e. all scattering events with azimuthal scattering angles in a certain range will be sorted into the same bin of the histogram.”

We hope that we correctly understand your comment and can give you a satisfying answer. If we misunderstood your comment, please let us know.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

The manuscript provides a solid presentation of the practical reconstruction of X-ray polarization in a SPARC Compton polarimeter. 

The first two sections provide a well-written introduction and basic overview of Compton Polarimetry. The section is written at the correct level and efficiently presents the key details necessary for understanding the rest of the manuscript. 

Line 34 is missing a right parenthesis after "keV" that should be added. 

 

Section three provides the theoretical background and reconstruction technique used by the analysis. The reconstruction procedure is well described, and the literature references are appropriate. Equation 4 needs to be broken up into separate lines as it is too long in its present form. 

 

Section four shows Monte Carlo results from the reconstruction algorithm, and Figure 2 nicely illustrates the strengths and weaknesses of the algorithm. Section five provides a well-written summary of background corrections due to pseudo-Compton events. Section 6  provides a validation of the reconstruction algorithm using experimental data recorded on a Gold film beamline scattering measurement. The resulting polarization measurements indicate a measurement of the degree of polarization of 3% and a measurement of the polarization angle with an accuracy of 1-4 degrees. 

The conclusion section only briefly summarizes the structure of the paper. The paper could be strengthened by comparing the accuracy of this reconstruction technique with other techniques already published in the literature. Although it is beyond the scope of this presentaion, I note that a maximum likelihood or Bayesian approach could seamlessly include both the polarization measurement and the background corrections into a single analysis chain, thereby simplifying the analysis. The conclusion also fails to discuss the broader implications of the work. For example, does the measured reconstruction accuracy provide the necessary resolution to enable new science capabilities or discoveries that are the focus of the use of the instrument? A description of how this work provides improvements that lead to discoveries of capability would advance the interest of the general reader.

 

As it stands, this is a solid piece of work that is technically accurate. The manuscript provides a well-written description of the reconstruction algorithm used to analyze data from the detector. The paper does not extend to describe the larger implications of the accomplished research. 

 

 

Author Response

Response to Reviewer 4 Comments

Thank you very much for taking the time to review this manuscript. Please find the detailed responses below and the corresponding revisions/corrections highlighted changes in the re-submitted files

 

Comments 1: Line 34 is missing a right parenthesis after "keV" that should be added.

Response 1: Thank you for pointing this out. The missing “)” on page 1 in line 34 has been added.

 

Comments 2: Section three provides the theoretical background and reconstruction technique used by the analysis. The reconstruction procedure is well described, and the literature references are appropriate. Equation 4 needs to be broken up into separate lines as it is too long in its present form. 

Response 2: We agree with this comment. Therefore, we have split equation 4 on page 4 into two lines.

 

Comments 3: The conclusion section only briefly summarizes the structure of the paper. The paper could be strengthened by comparing the accuracy of this reconstruction technique with other techniques already published in the literature.

Response 3: We added the following sentences to highlight the strengths starting on page 8 from line 234:

“We expect this polarization reconstruction to be a good alternative to the already existing method which is based on an adjustment of the Klein-Nishina formula to an experimental data set and relies on additional correction by simulated isotropic scattering distribution [19,21]. As the Klein-Nishina formula treats the electrons as free, binding effects (especially the electron momentum distribution) are not considered in the evaluation of the experimental scattering distribution in the previous reconstruction method. On the other hand, the simulation generating the model azimuthal scattering distribution treats the Compton scattering on the detector crystal within the impulse approximation [32] and thus explicitly considers the scattering off bound electrons. Additionally, the previously necessary pre- and post-processing corrections due to detector properties is now no longer required as we directly simulate our detector. Moreover, we plan to further improve this reconstruction algorithm. Based on the simulated scattering distributions, not only an adjustment to the azimuthal scattering distributions is possible, but also an evaluation of the 3D-resolved θ- ϕ scattering distribution can be performed. This will probably improve the accuracy of the polarization reconstruction as first, more information is available based on including both the polar and azimuthal scattering angle and secondly, the accepted scattering events do not need to be reduced to polar scattering angles of θ = (90° ± 15°), providing more usable scattering events.”

 

Comments 4: Although it is beyond the scope of this presentaion, I note that a maximum likelihood or Bayesian approach could seamlessly include both the polarization measurement and the background corrections into a single analysis chain, thereby simplifying the analysis.

Response 4: Your suggestion sounds very interesting. Our fit routine was originally build with a maximum likelihood estimator based on a Poisson distribution. Where the negative logarithmic likelihood function was minimized with iminuit. As this was not stable, we use now the iminuit cost.poisson_chih2() function. So far, we have not had experience with the Bayesian approach, but it sounds very promising. Therefore, we are interested in getting more input hereto from you as well on how we could improve our maximum likelihood approach to simplify the analysis chain.

 

Comments 5: The conclusion also fails to discuss the broader implications of the work. For example, does the measured reconstruction accuracy provide the necessary resolution to enable new science capabilities or discoveries that are the focus of the use of the instrument? A description of how this work provides improvements that lead to discoveries of capability would advance the interest of the general reader.

Response 5: We added the improvements of our reconstruction technique in Response 3. To further include the capabilities of our work we included the following sentences starting on page 8 in line 252:

“In a recent experiment, hard x-rays were scattered off a gold target [29]. The presented reconstruction algorithm is used to determine the linear polarization of the scattered radiation. The publication of the polarization analysis is pending. Additionally, this routine will be used in future experiments to analyze the polarization of atomic emission and scattering processes for energies beyond 200 keV. This will be possible with our novel Compton telescope, which is in the commissioning phase [15].”

 

Comments 6: As it stands, this is a solid piece of work that is technically accurate. The manuscript provides a well-written description of the reconstruction algorithm used to analyze data from the detector. The paper does not extend to describe the larger implications of the accomplished research. 

Response 6: We hope that our changes in the manuscript have clarified your concern.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

The authors answered all of my comments and I have no more comment on this manuscript.

Author Response

Response to Reviewer 1 Comments – Round 2

Thank you very much for taking the time to review our comments and the manuscript. We are pleased to hear that we have answered all your comments.

Reviewer 3 Report

Comments and Suggestions for Authors

The author clearly acknowledges that the threshold of 1.5 was chosen based on a trade-off between high statistics and good performance, but does not specify the exact criteria for "high statistics" and "good signal-to-noise ratio," nor does it reference similar studies to support the empirical threshold range. It is still recommended to validate the impact of different thresholds on reconstruction accuracy through Monte Carlo simulations.

Author Response

Response to Reviewer 3 Comments – Round 2

Thank you very much for taking the time to review our responses. Please find the detailed response to your new comment below and the corresponding correction highlighted changes in the re-submitted files.

 

 

Comments 1: The author clearly acknowledges that the threshold of 1.5 was chosen based on a trade-off between high statistics and good performance, but does not specify the exact criteria for "high statistics" and "good signal-to-noise ratio," nor does it reference similar studies to support the empirical threshold range. It is still recommended to validate the impact of different thresholds on reconstruction accuracy through Monte Carlo simulations.

Response 1: To further clarify how we set this threshold, we modified and added the following sentences in section 5 starting on page 5 in line 161:

“Additionally, the contribution of pseudo Compton events becomes especially strong compared to real Compton events for large distances between scatterer and absorber. This can be corrected by limiting both the experimental data and simulated distributions to events with a maximum distance between scatterer and absorber. For this, the distance distribution of the measured scattering distribution shown in Figure 3 is analyzed on the ratio of all measured events compared to estimated pseudo Compton events in a certain distance between scatterer and absorber. If the ratio drops under a chosable value, this sets the maximum accepted distance. All events (both for the experimental and simulated distributions) with larger distances are discarded in the analysis. In this work, the ratio is set to 1.5, which was chosen heuristically as a trade-off between not discarding too many real Compton events (good statistics) and not allowing too many pseudo Compton events (good signal to noise ratio).”

We are still convinced that your suggestion is beyond the realm of this work, as the background correction is only a minor correction. For measurements with dominant pseudo Compton events, this correction is of importance, which was not the case in the experiments analyzed so far. In these cases, we would investigate the threshold value in more depth.

Author Response File: Author Response.pdf