The Suppression Effect of an Imaging System on the Geometric Tilt-to-Length Coupling in a Test Mass Interferometer

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This paper presents a model for geometric tilt-to-length coupling in a test mass interferometer, proposing an imaging system-based solution combined with post-processing to potentially suppress the TTL coupling. Critical findings include that TTL couplings significantly surpass LISA requirements. While first-order couplings cannot simply be eliminated by the imaging system, in-orbit post-processing could mitigate them. The optimal location of the imaging system nearly eliminates TTL coupling. Sensitivity analysis further verifies this optimal position, but potential noise interference remains a challenge, warranting further research.

1. How did you determine the optimal conjugate relation? How precise is your method?

2. How do you mitigate the first order in your in-orbit post-processing scheme?

3. Could you provide more detailed information regarding how the optimal conjugate relation was derived?

4. Are there other methods to eliminate or mitigate TTL coupling without introducing an imaging system?

5. What are the challenges in implementing this novel optical configuration in actual experiments?

6. Does your model consider potential noise sources, such as temperature pulses or mechanical movements?

7. The image is too blurry, and the authors should improve the resolution.

Comments on the Quality of English Language

The language can be polished.

Author Response

1. How did you determine the optimal conjugate relation? How precise is your method?

Response: Thank you for reviewing my manuscript. Through the theoretical derivation of the geometric TTL after introducing the imaging system in the third part of the paper, we determine the optimal conjugate relation which refers to the position of the imaging system (i.e., the conjugate relation between the TM and the detector), where the TTL coupling minimizes. Specifically, in Section 3.3, the optimal position of imaging system (i.e., the optimal conjugate relation), was theoretically obtained by setting the second-order term of the optical path length to zero. The precision of this method is primarily dependent on the accuracy of the simulation software ASAP and MATLAB. As illustrated in Figure 7, under the optimal conjugate relation, the simulation accuracy can reach 0.0004 pm. In addition, the sensitivity analysis in Figure 8 further validates the effectiveness of this optimal position, i.e., the optimal conjugate relation can fully accord with the Taiji requirement of TTL coupling within a range of ±2.33 mm.

 

2. How do you mitigate the first order in your in-orbit post-processing scheme?

Response: Thank you very much for asking the question. Through the formulas (12) and (32), the lateral displacement between the center of rotation and the point of reflection contributes the first-order term coefficient of the TTL coupling. If the initial position during the installation is less than the available movement range of the TM within the inertia sensor, the TTL coupling can be reduced by translating the TM in the vertical optical axis direction. Otherwise, it can be deducted by the in-orbit data post-processing, when the TM jitters can be measured by the DWS (differential wavefront sensing) technology and the optical path length differences can be measured via the multiple interferometers, thus the first order term coefficient can be fitted and derived by in-orbit post-processing.

 

3. Could you provide more detailed information regarding how the optimal conjugate relation was derived?

Response: We are very grateful that you have raised this critical question in this manuscript. The analytical expression of the geometric TTL with imaging system in the TM interferometer is derived in Section 3.1 of the paper. We find that the placement of imaging system (i.e., the corresponding conjugate relationship) contributes to the second-order term of the TTL coupling. Hence, in Section 3.3, we theoretically obtained the optimal imaging system position, i.e., the optimal conjugate relationship, by setting the second-order term coefficient of the OPD to zero. Furthermore, the numerical simulation verification of the theoretical expression is carried out in the Section 4 of the paper, and the simulation results are in good accordance with the theoretical model, greatly lower than the TTL requirement of the Taiji/LISA.

 

4. Are there other methods to eliminate or mitigate TTL coupling without introducing an imaging system?

Response: Thank you very much for asking the question. Several researches on how to vanish the TTL coupling have been developed, i.e., the noise can be completely eliminated when the two interfering beams meet some certain conditions. For example, there is no TTL coupling when the pivot point of the Gaussian beam coincides with the center of the radius of wavefront curvature [Ref 16]. The TTL coupling also vanishes when the two Gaussian beams have the identical parameters and interfere on a very large SEPD (single element photodiode) [Ref 18]. In terms of the interference between the Gaussian beam and flat-top beam, the noise can be removed when the waist position of the Gaussian beam coincides with the rotation point of the flat-top beam by using a large SEPD [Ref 27]. However, there are some difficulties in experimentally validating the reduction of TTL coupling based on these two methods.

The other method to subtract TTL coupling is via the time-delay interferometry (TDI) postprocessing algorithm. This solution relies on noise minimization and uses the differential wavefront sensing (DWS) measurements to estimate the TTL contribution on the current design configuration of LISA, then use these estimates to subtract the TTL noise, ensuring that any remaining TTL noise is below the current estimate of the other noise sources [Postprocessing subtraction of tilt-to-length noise in LISA]. However, this approach only works as long as the TTL noise is the dominating noise source in the TDI output in a given frequency range, and the model of the expected noise without TTL coupling to whiten the data during the estimation process.

 

5. What are the challenges in implementing this novel optical configuration in actual experiments?

Response: Thank you very much for asking the question. In the actual experiments, it necessitates the execution of precise positional adjustments and alignments pertaining to the imaging systems in conjunction with the detectors. At the same time, through the sensitivity analysis of the optimal configuration, it's determined that an alignment accuracy of 2.33 mm is adequate, implying the realization is rather feasible considering the present technical level.

 

6. Does your model consider potential noise sources, such as temperature pulses or mechanical movements?

Response: Thank you very much for asking the question. Our theoretical model does not take these two noise sources into account, and merely focused on the discussion of TTL coupling noise. During actual in-orbit detection, there is indeed the presence of noise emerging from thermal pulses or mechanical movements, which subsequently couples into the final interference signals, potentially influencing the extraction of gravitational wave signals. However, these two types of noise have little coupling with the TTL noise, so the noise sources are not considered in the paper.

 

7. The image is too blurry, and the authors should improve the resolution?

Response: Thank you very much for the reminder concerning the image quality. The resolution of the images in the paper has been enhanced.

Reviewer 2 Report

Comments and Suggestions for Authors

In the manuscript, authors derived analytical expressions and gave numerical simulations for the geometric tilt-to-length (TTL) coupling in a test mass interferometer, present in a space born interferometers.  These results were used to investigate influence of an imaging system on TTL noise suppression.

The model and the calculations provide theoretical bases for design, adjustments of the optical set up in the LISA and Taiji programs. By playing w/o optical imaging system, and with different arrengments of optics in the imaging system, using parameters from the Taiji program, they found optimal design of the system that improves TTL below the requirements of the Taiji program. Despite using parameters from one program only, they claum that that the model described in the manuscript applys to both LISA and Taiji program. This can only be justified if the differences between optical systems of the two programs are minor or such that results can indeed are applicable to both programs. In the introduction, only history of improved systems for LISA was discussed.

Does the sketch of the optical setup given in Fig. 2, within the black dotted line, present the true schematic of the Taiji optics?  If it is so, are there previous theoretical calculations and publications of the TTL coupling, for this particular simple system? Is the imaging system, like this one considered in Fig. 2, within the green dotted line, truly new as an approach to lower TTL noise?

 Although the results shown in Figure 4 (b) are obtained with parameters characteristic for the Taiji program, they show that the TTL coupling significantly exceeds the Taiji requirement,. How is that? If what is called requirements are the actual specs of this program, if the optical system is the copy of that in Taiji, and if the parameters are from this system, one expects that theory and actual performance of the program should agree. Or the theory is wrong? I expect authors to comment on this.

As discussed about optical schematics in Fig. 2, both measured and reference beam are from the same laser. It is evident that the two paths of the interferometer, for the two beams, are of different lengths. This difference should be within the coherence length of the laser used in the program. Adding bunch of optics elements for the imaging system, will increase the length of one path vs the other path. It is expected to take this, the actual coherence length of the laser used in Taiji program, into consideration when redesigning the optical system!

There is technical mess on pages 14 and 15, with figures not in place and repeated.

After this comments and suggestions are taken into account by authors, I can recommend this manuscript for publications in Photonics.

 

Comments on the Quality of English Language

Just a minor correction. 

Author Response

1. The model and the calculations provide theoretical bases for design, adjustments of the optical set up in the LISA and Taiji programs. By playing w/o optical imaging system, and with different arrengments of optics in the imaging system, using parameters from the Taiji program, they found optimal design of the system that improves TTL below the requirements of the Taiji program. Despite using parameters from one program only, they claum that that the model described in the manuscript applys to both LISA and Taiji program. This can only be justified if the differences between optical systems of the two programs are minor or such that results can indeed are applicable to both programs. In the introduction, only history of improved systems for LISA was discussed.

Response: Thank you very much for asking the question. Firstly, the LISA and Taiji programs share similar structural parameters in relation to their TM, with half of the side length being 23 mm, and equipollent adjustment and processing levels. Secondly, the slight differences in the Taiji and LISA indicators come from the disparities in the control level of the TM jitter between the two programs, without any differences in order of magnitude of the TTL requirement. Therefore, the method introduced in this paper can be applied to both LISA and Taiji programs. The reason why the progress of Taiji has not been referenced is because there have not been any publicly published discussions about the TTL coupling of the TM interferometer in the Taiji program.

 

2. Does the sketch of the optical setup given in Fig. 2, within the black dotted line, present the true schematic of the Taiji optics?  If it is so, are there previous theoretical calculations and publications of the TTL coupling, for this particular simple system? Is the imaging system, like this one considered in Fig. 2, within the green dotted line, truly new as an approach to lower TTL noise?

Response: Thank you very much for asking the question. Figure 2 is a schematic diagram of the TTL coupling in the TM interferometer of the Taiji program, which contains key optical components (not all optical components in the TM interferometer), but is not the actual optical path diagram of the TM interferometer used in the Taiji program (please refer to this paper for more details and we have added this reference in our revised manuscript). There have been the theoretical calculations and publications of the TTL coupling which is cited in the Introduction, and our theoretical calculation results are the same. However, we mainly proposed the optimal conjugate relationship through the theoretical and numerical means. The imaging system is not a new approach to lower TTL noise, on the contrary, it was proposed early and the suppression effects of different imaging systems have been verified through both numerical simulations and experiments, as cited in the references (15-20) in the paper. However, the relevant experiments do not consider the complete structural parameters and alignment errors of the TM, which is also one of the main innovation points of this paper.

 

3. Although the results shown in Figure 4 (b) are obtained with parameters characteristic for the Taiji program, they show that the TTL coupling significantly exceeds the Taiji requirement. How is that? If what is called requirements are the actual specs of this program, if the optical system is the copy of that in Taiji, and if the parameters are from this system, one expects that theory and actual performance of the program should agree. Or the theory is wrong? I expect authors to comment on this?

Response: Thank you for raising this significant question in the manuscript. In Figure 4(b), the TTL coupling that significantly exceeds the requirement of the Taiji program is a result of a configuration without an imaging system. This is just for comparing the magnitude of the TTL coupling with and without the imaging system, in order to demonstrate the capability of the imaging system in suppressing the TTL noise. Indeed, the imaging system will be introduced in the space-borne gravitational wave detection programs, whatever LISA or Taiji. But rest assured, the theory is correct.

 

4. As discussed about optical schematics in Fig. 2, both measured and reference beam are from the same laser. It is evident that the two paths of the interferometer, for the two beams, are of different lengths. This difference should be within the coherence length of the laser used in the program. Adding bunch of optics elements for the imaging system, will increase the length of one path vs the other path. It is expected to take this, the actual coherence length of the laser used in Taiji program, into consideration when redesigning the optical system!

Response: Thank you very much for asking the question. Figure 2 is a schematic diagram corresponding to the simplified theoretical model, rather than being the comprehensive representation of the Taiji TM interferometer optical path diagram. The introduction of black and green frames serves to intuitively exhibit the suppressing effect of the imaging system on TTL coupling. In the actual TM interferometer of the Taiji program, both the measurement beam and the reference beam will pass through the imaging system simultaneously. The difference is that the measurement beam will be reflected by the TM, but the reference beam will not. Therefore, the optical path difference between the reference beam and the measurement beam is very small in actual interferometric measurement, less than 1 m. Indeed, an ultra-stable frequency laser is employed in the space-borne gravitational wave detection programs and the ultimate linewidth is better than 0.1Hz, thus the corresponding coherent length is calculated 3*10^9m(c/Δυ), which can meet the requirement of the arm length difference with 3 million km, the distance of a desktop experiment is surely satisfied. Even in the ground experiments, the stable frequency lasers in use also exhibit a linewidth level of 100 Hz, granting a substantial coherent length of 3*10^6 m.

 

5. There is technical mess on pages 14 and 15, with figures not in place and repeated.

Response: Thank you for the feedback, the positions corresponding to our simulation images can indeed be confusing. Figure 6 (on page 14) and Figure 7 (on page 15) correspond to the simulations of the traditional optical conjugate configurations, and the optimal conjugate relationship derived theoretically in the paper, respectively. Moreover, Figure 8 (on page 15) presents an in-depth sensitivity analysis, corresponding to the optimal conjugate relationship. This is aimed at validating the effectiveness and general applicability of the conjugate relationship, as derived in the manuscript.

Reviewer 3 Report

Comments and Suggestions for Authors

This is an interesting work that provides insights in superior high-precision laser interferometry design of space-borne gravitational wave detection, imaging systems and perhaps microgravitational measurements. The authors have (i) analysed the mechanism of imaging system to suppress the geometric TTL coupling in TM interferometer with the alignment errors, and proposed the (ii) targeted method where the second-order term is suppressed, and (iii) the first-order term is subtracted by in-orbit calibration. The theoretical model,  numerical simulation, optimization conjugate relationships in TM interferometer design and its corresponding positions, have been done extensively along with recent literature studies and are acceptable in technical merit.

However:

11. The images provided in both simulation/experimental design and results are very low quality, not clearly readable, hence not acceptable.

22. The literature review is adequate, including experimental inferences from recent and relevant missions like LISA or Taiji etc. However, comparative results with current proposed design to establish the superiority of current design may be useful.  

33. Some additional references may be included related to possible experimental realization of proposed design:

-Appl. Sci. 2021, 11(17), 7872; https://doi.org/10.3390/app11177872

-Phys. Rev. Applied 2020, 14, 014030; DOI: https://doi.org/10.1103/PhysRevApplied.14.014030

-Class. Quantum Grav. 2018, 35 105001DOI 10.1088/1361-6382/aab86c

-J. Opt. 2023 , 25 055601, DOI 10.1088/2040-8986/acc3ac

-J. Opt.2022, 24 065601DOI 10.1088/2040-8986/ac675e

 4.     English language may be improved for better readability of the article.

 

Comments on the Quality of English Language

English language may be improved for better readability of the article.

Author Response

This is an interesting work that provides insights in superior high-precision laser interferometry design of space-borne gravitational wave detection, imaging systems and perhaps microgravitational measurements. The authors have (i) analysed the mechanism of imaging system to suppress the geometric TTL coupling in TM interferometer with the alignment errors, and proposed the (ii) targeted method where the second-order term is suppressed, and (iii) the first-order term is subtracted by in-orbit calibration. The theoretical model, numerical simulation, optimization conjugate relationships in TM interferometer design and its corresponding positions, have been done extensively along with recent literature studies and are acceptable in technical merit. However:

1. The images provided in both simulation/experimental design and results are very low quality, not clearly readable, hence not acceptable.

Response: Thank you very much for your help in this detail. In the revised manuscript, the quality of the images has been improved.

 

2. The literature review is adequate, including experimental inferences from recent and relevant missions like LISA or Taiji etc. However, comparative results with current proposed design to establish the superiority of current design may be useful.

Response: Thank you very much for asking the question. No one has specifically considered this issue until now, and all the designs of imaging system have been purposed for the scientific interferometers. Even if for the TM interferometer, a beam rotating around a fixed point was implemented where the test bed does not feature a TM.

 

3. Some additional references may be included related to possible experimental realization of proposed design:

-Appl. Sci. 2021, 11(17), 7872; https://doi.org/10.3390/app11177872

-Phys. Rev. Applied 2020, 14, 014030; DOI: https://doi.org/10.1103/PhysRevApplied.14.014030

-Class. Quantum Grav. 2018, 35 105001DOI 10.1088/1361-6382/aab86c—

-J. Opt. 2023 , 25 055601, DOI 10.1088/2040-8986/acc3ac-

-J. Opt.2022, 24 065601DOI 10.1088/2040-8986/ac675e—

Response: Thank you very much for your help in this detail. The first reference (-Appl. Sci. 2021, 11(17), 7872; https://doi.org/10.3390/app11177872) has been incorporated into the Introduction section by us. The last three references have already been cited in the initial manuscript, namely from the previous documents 12, 24, and 25.

 

4. English language may be improved for better readability of the article.

Response: Thank you very much for your help in this detail. In the revised manuscript, the quality of the English language has been improved.