Eigenmode Wavefront Decoupling Algorithm for LC–DM Adaptive Optics Systems
Round 1
Reviewer 1 Report
Good work done. The symbols are rewritten to match the equations. For example, under Eq. 4, " where Mi is the ith DM eigenmode". The Mi should be changed to Mi to match the Equation. Also, the ith and others through the whole text of the paper.
Author Response
Good work done. The symbols are rewritten to match the equations. For example, under Eq. 4, " where Mi is the ith DM eigenmode". The Mi should be changed to Mi to match the Equation. Also, the ith and others through the whole text of the paper:
Author response: Thank you very much. All of the symbols are rewritten to match the equations and we carefully check the whole paper again.
Author action: The full text has been revised and marked in yellow. The action can be found in the highlighted manuscript.
Reviewer 2 Report
In this manuscript, the authors propose the eigenmode wavefront decoupling algorithm to realize the decoupling control of the dual corrector of liquid crystal-deformable mirror adaptive optics system. I recommend it for publication after minor revisions:
1. Introduction: In the second paragraph, the authors compare the LC-based SLM with deformable mirror-based SLM. A detailed analysis has been given in a recently published review paper: K. Yin, et al. Light Sci. Appl. 11, 161 (2022). This reference will help readers to better understand the advantage of LC-based SLMs.
2. In Fig. 8, the authors show the infrared band resolution plate imaging before and after correction. Can the authors provide a figure of modulation transfer function before and after correction? It can help readers to better understand the correction process.
3. Can the authors add some sentences to explain the large-amplitude wavefront correctors mentioned in the manuscript?
4. Please carefully check the grammars. For example, I found some grammatical mistakes in the 73rd and 347th lines (make the system to work—make the system work; eigenmode—eigenmodes)
Author Response
Reviewer#2, In this manuscript, the authors propose the eigenmode wavefront decoupling algorithm to realize the decoupling control of the dual corrector of liquid crystal-deformable mirror adaptive optics system. I recommend it for publication after minor revisions:
Concern # 1: Introduction: In the second paragraph, the authors compare the LC-based SLM with deformable mirror-based SLM. A detailed analysis has been given in a recently published review paper: K. Yin, et al. Light Sci. Appl. 11, 161 (2022). This reference will help readers to better understand the advantage of LC-based SLMs.
Author response: Thank you very much for your positive suggestion. I have added the reference in this paragraph.
Author action: We updated the manuscript by
Liquid crystal (LC) wavefront correctors present the advantages of high density, large amplitude, and low cost, which can compensate for the shortcomings of DMs. The principles and advantages of LC devices can be found in detail from reference [11].
Reference:
- Yin, K., Hsiang, EL., Zou, J. et al. Advanced liquid crystal devices for augmented reality and virtual reality displays: principles and applications[J]. Light Sci Appl, 2022, 11, 161.
Reviewer#2, Concern # 2: In Fig. 8, the authors show the infrared band resolution plate imaging before and after correction. Can the authors provide a figure of modulation transfer function before and after correction? It can help readers to better understand the correction process.
Author response: Thank you very much. According to common sense, we know that MTF reflects the relationship between spatial frequency and signal intensity. Usually, when there is no aberration in the system, the signal intensity corresponding to each spatial frequency is the maximum intensity. With the influence of aberration, the signal strength becomes worse, but MTF does not represent the diffraction limit resolution that can be resolved, but only reflects the degree to which the signal intensity is close to no aberration at all the spatial frequency. Generally, when the rms of residual aberration is less than 1/14, the system reaches the diffraction limit resolution. For our results, Figure 7 shows the wavefront aberration before and after correction. After correction, the wavefront residual aberration is significantly reduced, so the calculated MTF must be better and will be the same as the MTF without aberration. As the calculation of MTF is to construct the point spread function through the wavefront aberration, and then take the absolute value after Fourier transform of the point spread function of the system, so the result reflected by the MTF is the same as the wavefront aberration. Therefore, we believe that the results given in Figure 7 are consistent with the results given in MTF, and they both indirectly reflect the correction effect. The wavefront aberration is the most direct parameter in this paper, so we insist on retaining the wavefront aberration instead of MTF. We usually use to calculate the theoretical diffraction limit and experimental evaluate the resolution of the imaging system with the USAF resolution board to verify whether the corresponding line pairs can be clearly seen. We have rephrased the relevant paragraphs for clarity.
Author action: We updated the manuscript by”
After we obtain the wavefront aberration of the system, we can simulate the point spread function (PSF) of the system and calculate the optical transfer function (OTF) of the system by Fourier transform of the PSF, then take the absolute value of the OTF, that is, the modulation transfer function (MTF) of the system. MTF reflects the relationship between spatial resolution and signal intensity. Since the rms of the wavefront residual aberration is 0.07um, which is better than 1/14 in the 0.95-1.7 um waveband, and the MTF curve will be close to the diffraction limit. In order to experimentally prove that the resolution can reach the diffraction limit, the CG-USAF-1951-0 standard resolution plate was used as the observation target, and the imaging experiment was performed. The USAF target was placed near the light source, and the image was captured with the infrared camera before and after AO correction. Figure 8(a) shows the USAF target captured using the infrared camera in the 1.5–1.7-μm waveband before and after DM correction. Before the DM correction, the camera was blurred, and the image details could not be distinguished. After the DM correction of the top 55 eigenmodes, the fifth element of the fifth group in the USAF target was distinguished, and the corresponding element spatial frequency was 50.8 line pairs/mm, that is, the resolution was 19.7 μm. The diffraction limit resolution of a optical system can be calculated by, where is the wavelength, is the pupil diameter and f is the effective focus length. In this AO system, the aperture of the entrance pupil was 20 mm, the effective focus length is 200mm and the diffraction limit at the central wavelength of 1.6 μm was 19.5 μm. Therefore, the image of 1.5–1.7-μm waveband attained diffraction-limited resolution after the DM correction. Figure 8(b) shows the 950–1500-nm waveband before and after correction. After correction, the first element of the sixth group was resolved. The corresponding spatial frequency was 64 line pairs/mm, that is, the resolution was 15.6 μm. The diffraction-limited resolution of the central wavelength at 1.2 μm was 14.6 μm. Therefore, the image in the 950–1500-nm waveband achieved diffraction-limited resolution.”
Reviewer#2, Concern # 3: Can the authors add some sentences to explain the large-amplitude wavefront correctors mentioned in the manuscript?
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Author response: Thank you very much. I have added some sentences to explain the large-amplitude wavefront correctors.
Author action: We have added some sentences in the first paragraph:
Wavefront aberrations can be divided into low-order aberrations and high-order aberrations according to spatial frequency. Usually, the peak-to-valley (PV) values of low-order aberrations are larger, which we call it large amplitude, while the PV values of high-order aberrations are relatively small, so we call it small amplitude.
Reviewer#2, Concern # 4: Please carefully check the grammars. For example, I found some grammatical mistakes in the 73rd and 347th lines (make the system to work—make the system work; eigenmode—eigenmodes)
Author response: Thank you very much. We have carefully checked the whole paper and corrected the mistakes.
Author action: The action can be found in the highlighted manuscript.
Reviewer 3 Report
Rather strange manuscript, containing a lot of wrong or at least strange statements. The Reviewer task цфs tantalized by the rather poor English of the authors, so in some case we simply did not understood their statements.
Let us begin from the statements which are obviously wrong.
As far as we understand, authors say that the big telescopes with AO systems do not work in the visible spectral range (page 1), but even the non-specialists do know that today there is a couple of dozens or so of large ground-based telescopes with AO, working in the visible range.
On the page 2 the authors say that “the working waveband of LCs (liquid crystals) is limited to 700–950 nm”. Very strange statement since there are a lot of LC devices, working in visible and larger IR ranges – see for instance numerous devices, produced by Holoeye, Hamamatsu and other companies.
We do not understand why the authors think that the LC devices can provide higher stroke than that of deformable mirrors – according to our knowledge, the situation is opposite.
The structure of the paper is rather strange. It lacks the normal introduction to the idea and method – why coupling of modes and decoupling are necessary, what is the procedure and algorithm etc. Here we also expect to find some mathematics, but the authors start with setup description, and only in the next section we see some mathematical treatment.
However, we are somewhat doubtful regarding the correctness of this mathematical description. In particular, the authors assume that the set of DM’s eigenmodes is a set of orthogonal functions. It looks strange since in the case of the round DM the orthogonal basis is represented by Zernike polynomials, and eigenmodes of the mirror are not orthogonal. Maybe we have misunderstood some of author’s ideas, but in this case it is due to their poor English and incompletely comprehensive mathematical description.
The authors give practically no details of their apparatus and first of all of the mirror and of the LC device. In consideration of DM it is especially important since we again meet some unclear positions. The authors start with discussion of limitations of number of actuators of DM, and they say that the mirrors with this number more than 200 are hardly available. However, in the paper [12], which is, seemingly, the starting point for the described research, there was used the mirror with this number over 3000. Seemingly, the authors do not distinguish the mirrors with the zonal control (where this number can be very high, but where there are no eigenmodes) and the mirrors with the modal control (bimorph and similar, where the number of electrodes is approximately equal to the number of eigenmodes and it is not too high). However, it is well known, that for the lower aberrations the stroke of modal mirrors can be rather large and then it is not clear why the people are to use the dual corrector approach, presented in [12] and in this manuscript.
We do not want to waste our time further, considering other numerous mistakes or unclear passages in the manuscript. Since the experimental results, described in the manuscript, seem to be rather interesting, we strongly advise the authors to prepare the new manuscript on the subject and to invite somebody with the better English as an interpreter. We do not believe that the current text can be improved – it would be much better to start again from the beginning and to prepare the brand-new manuscript. It has to be properly organized, it has to contain much more rigid mathematics and theory; it has to explain the idea, the method and the apparatus in much more details etc.
Author Response
Reviewer#3, Rather strange manuscript, containing a lot of wrong or at least strange statements. The Reviewer task цфs tantalized by the rather poor English of the authors, so in some case we simply did not understood their statements. Let us begin from the statements which are obviously wrong.
Sorry for the poor writing and we have polished the paper by an English polishing agency. We have tried our best to express our work well. Thank you for your understanding.
Concern # 1: As far as we understand, authors say that the big telescopes with AO systems do not work in the visible spectral range (page 1), but even the non-specialists do know that today there is a couple of dozens or so of large ground-based telescopes with AO, working in the visible range.
.
Author response: Thanks for your suggestion. Yes, you are right, there are indeed large-aperture telescopes in the visible wave band. This article also cites the large telescopes in the visible wave band, such as the PALM-3000 system on the 5-m Haier Telescope. What we want to express is that due to the wavelength is short, the aberration in the visible wave band has a large amplitude high spatial frequency. Hence, a high-order and large-stroke deformable mirror is required. At present, the largest AO system for large telescope visible imaging is the PALM3000 AO system that uses a 3888-unit deformable mirror, but it cannot work alone due to the small stroke. A large stroke deformable mirror is needed to work in conjunction, and the cost and complexity of the AO system is extremely high. We have modified this paragraph to express our thought well.
Author action: The first paragraph:
Adaptive optics (AO) technology is indispensable to solve atmospheric turbulence interference and restore the diffraction limit resolution of ground-based large-aperture tele-scopes [1-3] and is widely used in laser shaping, fundus imaging, laser communication, and other aberration correction fields [4-6]. Currently, most AO systems of large-aperture telescopes work in the infrared waveband and cannot provide the high-resolution images of high-radiant intensity stars in the visible waveband [7], because the wavelength of visible light is shorter than that of infrared, the intensity of atmospheric turbulence is proportional to λ−6/5 (λ is the wavelength), and atmospheric turbulence in the visible light band is strong (the atmosphere coherence length r0 is small). Although AO system imaging in the visible waveband can be achieved, the cost and complexity of the system will increase significantly.
Concern # 2: On the page 2 the authors say that “the working waveband of LCs (liquid crystals) is limited to 700–950 nm”. Very strange statement since there are a lot of LC devices, working in visible and larger IR ranges – see for instance numerous devices, produced by Holoeye, Hamamatsu and other companies.
.
Author response: Sorry for our poor expression. The working band of liquid crystal can be very wide, here we refer to the liquid crystal band that can be used for AO correction. Since the modulation principle of the liquid crystal corrector is to control the deflection of the liquid crystal molecules, the response speed of the molecules will be sharply slower with longer wavelengths. The products of companies such as Holoeye have a response time of more than 10ms, which is not allowed in the AO system. Turbulence cannot be compensated in real time. The characteristic of our LC is that the response time is as fast as 0.65ms, which is comparable to the response characteristics of the deformable mirror, so it can be applied to the AO system. We have done a lot of research work on liquid crystal AO.
Single frame overdriving method to improve the LC response to 0.65ms
Hongbin Hu, et al. Advanced single-frame overdriving for liquid-crystal spatial light modulators,” OPTICS LETTERS / Vol. 37, No. 16 / August 15, 2012.
New fast response LC materials designed by our group
Peng Z. H., et al. Fast response property of low-viscosity difluorooxymethylene-bridged liquid crystals,” Liquid Crystals, 40(1),91-96(2013).
Control bandwidth of LC AO system with PD controller
Xingyun Zhang, et al. “Improved bandwidth of open loop liquid crystal adaptive optics systems with a proportional-derivative controller,” OPTICS EXPRESS, Vol. 27, No. 8, 2019.
Author action: We have rewritten the LC paragraph:
Liquid crystal (LC) wavefront correctors present the advantages of high density, large amplitude, and low cost, which can compensate for the shortcomings of DMs. The principles and advantages of LC devices can be found in detail from reference [11]. However, due to the slow response of LCs in the long infrared waveband and severe LC dispersion in the broadened waveband, the working waveband of LCs is limited to 700–950 nm for a fast response without dispersion, which is narrower than that of DMs [12].
Concern #3: We do not understand why the authors think that the LC devices can provide higher stroke than that of deformable mirrors – according to our knowledge, the situation is opposite.
.
Author response: Thank you very much. The correction principle of the liquid crystal corrector is based on binary optical diffraction to generate a 2pi wavefront, so its correction amount depends on the pixel density of the liquid crystal corrector, and the liquid crystal can reach more than 4k*4k pixels, so it is very easy to achieve large corrections of. Our team's research field is LC AO system and starts in 2000. The relevant conclusions on LC AO systems have been verified by the given references and engineering application. For the latest progress, please pay attention to the literature: doi:10.1093/mnras/staa841(Xingyun Zhang, et al. Progress of liquid crystal adaptive optics for applications in ground-based telescopes. MNRAS 494, 3536–3540 (2020)). The open loop LC is used as cover article in 2008 of Applied Optics and we also participated in the writing of the 3rd chapter of R. K. Tyson’s Book adaptive optics progress, more than 100 papers on LC AO are published by our group.
"Applied Optics" sets the open-loop LC adaptive optics system as the cover of the August 2008 issue
We wrote the chapter of "Liquid Crystal Wavefront Corrector" Published by InTech Press in 2012
Author action: Thank you
Concern # 4: The structure of the paper is rather strange. It lacks the normal introduction to the idea and method – why coupling of modes and decoupling are necessary, what is the procedure and algorithm etc. Here we also expect to find some mathematics, but the authors start with setup description, and only in the next section we see some mathematical treatment.
However, we are somewhat doubtful regarding the correctness of this mathematical description. In particular, the authors assume that the set of DM’s eigenmodes is a set of orthogonal functions. It looks strange since in the case of the round DM the orthogonal basis is represented by Zernike polynomials, and eigenmodes of the mirror are not orthogonal. Maybe we have misunderstood some of author’s ideas, but in this case it is due to their poor English and incompletely comprehensive mathematical description.
.
Author response: Thank you very much. We will try our best to express our work well. Actually, the DM’s eigenmodes is a set of orthogonal functions, this conclusion has been verified in the literature [24], we have cited this literature. Its orthogonality can also be found by mathematical formulas (2-4).
Some other literatures are list below for you to understand.
- Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, "Distributed modal command for a two-deformable-mirror adaptive optics system," Appl. Opt. 46, 4329-4340 (2007)
Ende Li, Yun Dai, Haiying Wang, and Yudong Zhang, "Application of eigenmode in the adaptive optics system based on a micromachined membrane deformable mirror," Appl. Opt. 45, 5651-5656 (2006)
Mohammad Azizian Kalkhoran, Ann Fitzpatrick, A. Douglas Winter, Chris S. Kelley, Edmund Warrick, Mark D. Frogley, and Gianfelice Cinque "Characterization of double-deformable-mirror adaptive optics for IR beam shaping in hyperspectral imaging", Proc. SPIE 11351, Unconventional Optical Imaging II, 113511A (4 May 2020); https://doi.org/10.1117/12.2554162
Author action: Thank you.
Concern # 5: The authors give practically no details of their apparatus and first of all of the mirror and of the LC device. In consideration of DM it is especially important since we again meet some unclear positions. The authors start with discussion of limitations of number of actuators of DM, and they say that the mirrors with this number more than 200 are hardly available. However, in the paper [12], which is, seemingly, the starting point for the described research, there was used the mirror with this number over 3000. Seemingly, the authors do not distinguish the mirrors with the zonal control (where this number can be very high, but where there are no eigenmodes) and the mirrors with the modal control (bimorph and similar, where the number of electrodes is approximately equal to the number of eigenmodes and it is not too high). However, it is well known, that for the lower aberrations the stroke of modal mirrors can be rather large and then it is not clear why the people are to use the dual corrector approach, presented in [12] and in this manuscript.
We do not want to waste our time further, considering other numerous mistakes or unclear passages in the manuscript. Since the experimental results, described in the manuscript, seem to be rather interesting, we strongly advise the authors to prepare the new manuscript on the subject and to invite somebody with the better English as an interpreter. We do not believe that the current text can be improved – it would be much better to start again from the beginning and to prepare the brand-new manuscript. It has to be properly organized, it has to contain much more rigid mathematics and theory; it has to explain the idea, the method and the apparatus in much more details etc.
Author response: Sorry for not expressing clearly, we do not mean that there are no deformable mirrors with more than 200 actuators or large strokes, we mean that there are no deformable mirrors with both large strokes and large number of actuators, which are actually beyond ability of modern mechanical machining. There are many famous deformable mirror companies, including French ALPAO, Dutch OKO, American CILAS Xinetics and other companies, various types of deformable mirrors have performance advantages and disadvantages, you can go to the company website to refer to their performance parameters. It is because of this design and manufacturing limitation that people began to study the AO system of double deformable mirrors, and a lot of related literature can be found in google scholar, search keywords: woofer-tweeter adaptive optics system.
Finally, we are very grateful for your valuable time and comments. We will seriously revise our paper to make our expression clearer so that everyone can read it as clearly as possible.
Author action: Thank you.
Author Response File: 
Author Response.pdf
Round 2
Reviewer 3 Report
The authors have coped only with part of our notes. They have corrected the definitely wrong statements about the lack of AO systems in visible and about the narrow spectral band of LC performance. They have not properly introduced their method and apparatus. We do not see the details of the LC device and of DM. We do not see the proper explanation of the decoupling procedure. When speaking about the orthogonality of DM modes, they, seemingly, simply do not understand the mathematical sence of orthogonality. How can we treat the eigenmodes of the zonal DM as orthogonal, if their are simply the hills or depletions? If they do not agree, they should have proven it mathematically. They also do not understand that in the mirrors with the modal control, which can be also used for astronomy AO, especially in IR, the lower is the mode, the higher is the stroke, and for the lowest aberrations it can exceed the stroke of LC device in several times.
I am not against publishing these results, but I am staying on my previous position - this text is to be thrown away, and the authors are to write another one, starting with explanation of their method, proving its advantages, then discussing the decoupling algorithm and its necessity, then introducing their apparatus in details (parameters of DM; parameters of the LC device; their responce times, the role of dispersion etc.) and only then shift to the simulation, experiment and results.
Author Response
The authors have coped only with part of our notes. They have corrected the definitely wrong statements about the lack of AO systems in visible and about the narrow spectral band of LC performance.
Concern # 1: They have not properly introduced their method and apparatus. We do not see the details of the LC device and of DM.
Author response: Thank you very much. Compared with others, our proposed method is mainly improved in two aspects. The first is to construct a deformable mirror eigenmode to replace the traditional Zernike mode, thereby improving the wavefront fitting accuracy. The second is the decoupling of the two correctors when they work together. Since the two correctors jointly correct the same wavefront, the wavefront must be properly distributed without affecting each other to ensure cooperative work.
This article uses a 256×256 pixels reflective nematic liquid crystal wavefront corrector manufactured by Meadowlark optics Company in the United States. The liquid crystal material is synthesized by us and manufactured by Meadowlark optics Company, with a diameter of 6.14 mm, a pixel size of 24 um and a response time of 0.65ms, as shown in the figure below. The most important characteristic is the fast response, Liquid Crystal Adaptive Optics Pioneer, Professor G. D. Love has cited our result in his research: Gregor Thalhammer, Richard W. Bowman, Gordon D. Love, Miles J. Padgett, and Monika Ritsch-Marte, "Speeding up liquid crystal SLMs using overdrive with phase change reduction," Opt. Express 21, 1779-1797 (2013). [6-8] are our published results.
The DM 145 is circle distribution brought from ALPAO (DM145) with aperture of 30 mm, the spacing between the actuators d is 2.5mm. The parameters in the simulation model are normalized values. For example, the parameter d in the model is 2.5/30=0.083.
We have made modifications in the Introduction section of the paper to re-describe the method proposed in this paper and give detailed parameters of the apparatus used in the simulation and experiment sections.
Author action: We updated the manuscript by
Introduction part:
In this paper, we proposed a wavefront decoupling method based on DM eigenmodes, which used the DM response matrix to construct the eigenmodes, thereby preventing the fitting error of DM with the Zernike modes. Then we discuss the wavefront decomposition method based on eigenmodes, the aberration in the infrared waveband were decomposed and corrected using the DM. The DM was controlled with eigenmodes to improve the compensation accuracy. The aberration in the visible waveband was compensated by DM and LC. In order to make the aberration generated by LC not include the aberration that has been corrected by DM, the decoupling control method of LC and DM must be studied. The reason for cross coupling between the two correctors was analyzed, and we calculated the projection of the LC response matrix on the DM eigenmode response matrix to obtain the coupling term of the two matrices when correcting the same wavefront, then, the control signal of LC was reset with the coupling term to realize aberration decoupling between the two correctors thereby achieving high-precision correction in the visible waveband. Finally, a setup of LC–DM AO system was developed for a 2-m telescope. The experimental results showed that the proposed method can effectively realize dual corrector decoupling. Compared with the correction accuracy of traditional Zernike decomposition method, the correction accuracy of the proposed method was considerably improved.
Simulation Part:
Figure 1 presents the block diagram of the LC–DM cascade AO system. This system is mainly composed of an LC wavefront corrector, DM, a tip-tilt mirror (TTM), a Hartmann WFS, a wavefront processing computer, and an imaging camera. The density of the LC is 256 × 256 pixels with a pixel size of 24 μm, and the aperture of LC is 6.14 mm. The number of actuators of the DM is 145, the pitch of the DM was 2.5 mm, and its aperture is 30 mm. The aperture of TTM is 25 mm and the number of sub-apertures of the WFS is 20 × 20, and the full aperture is 5.8 mm, these devices are placed in the conjugate position by lens pairs in the optical path for high-spatial-frequency small-amplitude aberration correction, low-spatial-frequency large-aberration correction, tilt aberration correction, and wavefront measurements, respectively. The wavefront processing computer is used for wavefront reconstruction, wavefront decoupling, and the calculation of the drive signal of each corrector. The wavefront obtained using WFS is decomposed and decoupled and sent to the controller of each corrector. The controller is used to calculate the driving signal and to send this signal to each corrector. The imaging camera is employed to image the corrected light beam. In the design, the compactness and complexity of the system are comprehensively considered, and the device is selected by ensuring energy utilization, resolution, the field of view, and the zoom beam and is completed with the smallest volume and fewest devices.
In most AO systems with dual (multi) DMs, the Zernike mode method is used to decompose wavefront aberrations. Then, DMs are used to produce the conjugated Zernike mode to correct distortion and aberrations. However, due to their parameters and other factors, DMs cannot achieve a perfect fit to the Zernike mode. The ith response function of the DM was described as a Gaussian function in formula (1). The LC was described by 209 Zernike modes, the dimension of the wavefront is 256 × 256 pixels.
(1)
where , , and are the Gaussian exponent, the coupling value, and the spacing between the actuators in the DM, respectively. In this model, was 1.73, was 0.23, and was set to 0.083. These parameters are normalized values from the DM parameters that was purchased from ALPAO. Figure 2 presents the residual aberration histogram of the top 54 Zernike modes with the 145 units DM, when the DM actuator stroke is not considered. Correction residual aberrations exhibit an overall upward trend with an increase in the number of modes mainly because when the number of modes increases, the spatial frequency of the wavefront is high, and the wavefront is considerably complicated. DM cannot satisfy the requirements of high spatial resolution; the correction effect can be poor. Therefore, when the Zernike mode is used for wavefront decomposition and correction, fitting errors occur inevitably.
Experiment part:
The setup of LC–DM cascade AO system was constructed in the laboratory for a 2-m telescope (Figure 6). The system mainly constitutes two parts, namely AO and imaging parts. The AO part mainly includes TTM, DM, LC, and WFS. The DM is a 145-unit continuous surface DM produced by ALPAO, with a aperture and stroke of 30 mm and approximately 3 μm, respectively. LC is manufactured by Meadowlark optics Company with the fast LC materials synthesized by us. The number of pixels is 256 × 256, the response time is 0.65 ms, and the effective aperture is 6.14 mm. WFS is an SH wavefront sensor manufactured by FIRST-LIGHT company. The camera frame rate is approximately 1.67 kHz, the pixel size is 24 μm, and the number of micro lenses is 20 × 20. The receiving aperture is approximately 5.8 mm. The imaging part includes visible and infrared cameras. The visible camera is an iXon ultra 897 model camera produced by ANDOR, with a pixel size of 16 μm, a number of pixels of 512 × 512, and a spectral response range of 400–900 nm, The infrared camera is a Cheetah-640CLshort-wave infrared camera produced by the Xenics company, with the pixel size of 20 μm, number of pixels of 320 × 256, and spectral response range of 900–1700 nm.
Concern # 2: We do not see the proper explanation of the decoupling procedure.
Author response: Sorry for the unclear expression. The explanation of the decoupling procedure is given in Section 2.3. The decoupling procedure is divided into three steps. The first step is to decompose the aberration corrected by the LC corrector, which is realized by the steps of equations from 11 to 15. The number of modes for decomposing is analyzed in section 2.4. The second step is to calculate the projection of the LC response matrix on the DM eigenmodes matrix, thereby calculating the coupling term of the LC response to the DM response, the procedure is given by Equations from 16 to 18. The third step is to reset the control signal of the LC corrector and subtract the coupling term from the LC control signal to achieve decoupling control. We have rewritten this part and kindly hope to meet your requirement, thank you.
Author action: We updated the manuscript by
There are three steps in the decoupling procedure. The first step is wavefront decomposition. For the proposed LC–DM cascaded AO system, DM is required to correct low-order large-amplitude aberrations to meet the imaging requirements of the infrared waveband. To make DM correct only the first N eigenmodes, the coefficient matrix m is selected as follows:
(11)
where is an n × n identity matrix, the first N diagonal elements are 1, and the rest diagonal elements are 0, that is
(12)
Substituting Formulas (11) and (12) into Formula (10), the control voltage vector can be used to calculate for the correction of the first N eigenmodes:
(13)
After voltage calculation, the actual surface shape of the DM is obtained by employing the voltage and DM response matrix. After the subtraction of the actual DM surface shape from aberration to be corrected, the residual wave surface is sent to LC:
(14)
where is the wave surface to be corrected, which is sent to the LC and is the surface shape generated with DM. Because the LC has thousands of pixels, if individual pixels are driven one by one, the calculation will be highly complicated and time-consuming, hence, LC is driven with the Zernike mode surface. The wavefront correction process of the LC is as follows [12,13]
(15)
where C and T are the control signal and response matrix of the LC, respectively. So far, we have obtained the wavefront that needs to be corrected by LC. This part of the wavefront is the residual wavefront after DM correction.
The second step is to obtain the coupling term between the LC corrector and the DM In the LC-DM cascade AO system, the DM uses the eigenmodes method for wavefront decomposition and reconstruction, which is different from LC with Zernike modes. The DM eigenmodes are obtained from DM response, thus the DM can accurately fit its own eigenmodes without affecting the LC correction. The DM control voltage calculated using Formula (13). For LC corrector, the residual aberration is corrected using Zernike modes. As there is no orthogonal relationship between the Zernike modes and the eigenmodes, the LC will generate the eigenmodes that has been corrected by the DM. That is to say, there are aberrations that have been corrected by the DM, which causes a reduction in the final correction performance. This part of the DM corrected aberration is the coupling term to be calculated and eliminated.
We first calculate the wavefront generated by LC without decoupling. The control matrix C for the LC can be calculated using Formulas (13)-(15). The LC response wavefront can be expressed as follows:
(16)
There are aberrations that have been corrected by the DM in . To avoid the repetitive generation the coupling aberration by the LC, the DM eigenmodes M is used to reconstruct the LC wavefront
(17)
where are the eigenmodes coefficients corresponding to . Because the DM corrects the aberrations of the first N eigenmodes, the wavefront reconstructed with the LC should not contain the first N eigenmodes, thus the first N eigenmodes coefficients of is the coupling aberration that must be filtered out. Therefore, the coupling aberration can be calculated as follows:
(18)
So far, the coupling term is obtain with Formula (18), the coefficient can be calculated with Formula (15). The third step is to subtract the coupling term to reset the control signal sent to the LC,thus the control signal sent to LC after decoupling is as follows:
(19)
At this time, the coupling aberration is filtered out, and the aberration decoupling between the two correctors is realized. This is a type of the control signal reset method. The basic principle is to remove the coupling term from the original signal.
Concern # 3: When speaking about the orthogonality of DM modes, they, seemingly, simply do not understand the mathematical sence of orthogonality. How can we treat the eigenmodes of the zonal DM as orthogonal, if their are simply the hills or depletions? If they do not agree, they should have proven it mathematically.
Author response: I'm very sorry, our lack of knowledge leads to not understanding what the meaning of the hills or depletions are. But I don't think it matters, I want to talk about our understanding of orthogonality. "Orthogonality" is a term borrowed from geometry. If two lines intersect at right angles, they are orthogonal. In vector terms, the two lines are independent of each other. Move along a line, and the position of the line projected onto another line does not change. In a space vector, the scalar product of two vectors is zero, that is, the two vectors are orthogonal. A necessary and sufficient condition for a vector group to be an orthogonal vector group is , where and are non-zero vectors. Now, words to our eigenmode derivation, we first calculate the coupling matrix C(i,j) between the actuators according to the response of the actuators, which is an n*n square matrix. We obtained the unitary matrix U and the diagonal matrix S by decomposing the singular value of C(i,j), the U matrix is the eigenvector of each actuators, and it is a common knowledge that the eigenvectors are orthogonal From formula (5), the eigenmodes are constituted using the eigenvectors, so it can be proved that the eigenmodes are also orthogonal. In addition, you can refer to the expressions in other literatures, and the orthogonality of eigenmodes is a consensus in the field.
- Conan, C. Bradley, P. Hampton, O. Keskin, A. Hilton, and C. Blain, "Distributed modal command for a two-deformable-mirror adaptive optics system," Appl. Opt. 46, 4329-4340 (2007)
Ende Li, Yun Dai, Haiying Wang, and Yudong Zhang, "Application of eigenmode in the adaptive optics system based on a micromachined membrane deformable mirror," Appl. Opt. 45, 5651-5656 (2006)
Mohammad Azizian Kalkhoran, Ann Fitzpatrick, A. Douglas Winter, Chris S. Kelley, Edmund Warrick, Mark D. Frogley, and Gianfelice Cinque "Characterization of double-deformable-mirror adaptive optics for IR beam shaping in hyperspectral imaging", Proc. SPIE 11351, Unconventional Optical Imaging II, 113511A (4 May 2020); https://doi.org/10.1117/12.2554162
We have tried our best to make it clear, the modification is given in section 2.1. We hope that it will meet your requirement, thank you.
Author action:
Next, we give the derivation procedure of the DM eigenmodes. The response matrix, a two-dimensional matrix, is measured using WFS. The response function of each actuator represents the response surface of each actuator. The final surface shape of DM can be expressed through a linear combination of the response functions of each actuator:
(2)
where, n represents the number of actuators, denotes the voltage applied using the ith actuator. The coupling relationship between the response functions can be expressed with a coupling matrix C where the coupling values C(i,j) of the ith and jth actuators are calculated as follows:
(3)
where D is the DM aperture. C describes the correlation of the influence functions of different actuators and may be decomposed using the singular value decomposition method:
(4)
where S is the diagonal matrix whose diagonal elements are the eigenvalue of matrix C, U is the unitary matrix comprising the eigenvector of C, and U−1 = UT. The linear combination of the response function of each actuator and the coupling matrix eigenvector constitutes a new two-dimensional matrix, the DM eigenmodes:
(5)
where is the ith DM eigenmodes. As U(i,:) is the eigenvector of the ith actuator, the eigenvectors are orthogonal, so the eigenmodes are also orthogonal. There are n eigenmodes, which are the same as the number of actuators. A random wavefront may be described with the following eigenmodes:
(6)
where is the ith coefficient of . Considering a 145-element DM from ALPAO in the laboratory as a prototype, Figure 3 shows the surface shapes of the top 27 eigenmodes.
Fig.3 Top 27 eigenmodes profiles of the 145-unit DM
From the derivation procedure of the DM eigenmodes, it can be seen that the eigenmodes have the following properties:1) The number of eigenmodes of an DM is equal to the number of its actuators.2) The eigenmodes of the DM are orthogonal to each other. 3) As the number of modes increases, the spatial frequency of the eigenmodes increases gradually. In addition, since the eigenmodes are obtained by constructing the response function of the DM, which reflects the inherent characteristics of the DM, there is no fitting error when using the DM to generate the eigenmodes. Compared with Zernike modes presented in Figure 2, no fitting error is observed for eigenmodes in theory. Hence, the DM eigenmodes can effectively lead to a decrease in fitting errors and improvement of the final correction accuracy.
Concern # 4: They also do not understand that in the mirrors with the modal control, which can be also used for astronomy AO, especially in IR, the lower is the mode, the higher is the stroke, and for the lowest aberrations it can exceed the stroke of LC device in several times.
Author response: Sorry for our unclear expression, we seem to have not said that DM cannot use modal control for astronomical observations in the infrared band. We can definitely say that deformable mirrors are used in both infrared and visible bands in astronomy, and the control methods cover modal control and direct slope control, etc. We also accept that the stroke of DM can be many times higher than that of LC at low frequencies. What we have been expressing in this paper is that DM is hardly to achieve high driving density and large driving stroke at the same time, so the multi-correctors AO system appears. The feature of our system is that LC is used as a high-order corrector, which is limited by the working band of LC, it is mainly used to compensate the aberration in the visible band.
Author action: Thank you for your understanding.
Concern # 5: I am not against publishing these results, but I am staying on my previous position - this text is to be thrown away, and the authors are to write another one, starting with explanation of their method, proving its advantages, then discussing the decoupling algorithm and its necessity, then introducing their apparatus in details (parameters of DM; parameters of the LC device; their response times, the role of dispersion etc.) and only then shift to the simulation, experiment and results.
Author response: Thank you for your precious time and sincere comments, and I hope you can accept our efforts and work, thank you for your kind tolerance and understanding. Finally, I wish you a happy life and work.
Author action: We have updated the whole manuscript. The changes can be found in the highlighted version.
Author Response File: Author Response.pdf
