Abstract

The PISTIL interferometry has been recently developed for the wavefront sensing of phase delays (pistons) and tilts of segmented surfaces, used in many domains such as astronomy, high-power lasers or ophthalmology. In this paper, we propose a two-wavelength version of this interferometer developed to bypass the dynamic range limitation of the ambiguous 2π phase wrapping. Principle of the technique is presented, along with experimental results obtained with a demonstration deformable mirror PTT-111 from Iris AO. Above wavelength pistons are measured with a precision and accuracy below λ/100, making the two-wavelength PISTIL interferometry a high-dynamic range technique. To prove these performances, we successfully compare the results in terms of precision and accuracy with those of a reference phase-shifting Interferometer, from a blind experimentation.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Today, there exists a large panel of projects relying on segmented wavefronts tailoring, either in coherent combination of elementary fiber lasers [13], astronomy [4], or ophthalmology [5].

In those domains, innovative phase analyzers have been developed to provide a solution to measure accurately and / or co-phase the segmented wavefronts [611]. In a general way, a co-phasing system must be able of measuring accurately the phase delay (piston) difference between each element in a range from tenths to tens units of wavelength λ, with an error at least below λ/20 and often of the order of λ/100 [24].

We present a new Two-Wavelength PISTIL (Two Wavelength PISton and TILt) interferometry concept. It is a member of lateral-shearing interferometers family and evolution of the recent PISTIL concept [12]. This phase analyzer is very handy because it consists of few optical elements easily packable and does not require an external reference wavefront. This quality makes it particularly suitable for on-site measurements (outside the laboratory).

It has the ability for piston/tip/tilt analysis of high-dynamic range segmented wavefronts, thanks to the implementation of a second wavelength: the piston range limit can be set by the interferometer user, from 10$\lambda $ to 100$\lambda $ depending on the application. Above wavelength piston differences can be estimated with residual error of λ/100 or better.

To prove the performances of this interferometer, and its interest as a segmented wavefront sensor, we choose to compare its performance relatively to Phase-shifting interferometry (PSI) [13], which is a reference wavefront sensing technique. The aim is to achieve close experimental precision and accuracy with the PSI measurements: order of magnitude of 1-10nm, ${\sim} $λ/100 at 633nm.

The demonstration device chosen to provide our segmented wavefront stimuli is a PTT-111 deformable mirror (DM) from the company Iris AO [14]. This device is an array of 37 single crystalline silicon hexagonal mirrors with a pitch of 606µm, able to move in piston with a stroke around 1.5µm. This class of DM is routinely used at the LAM (Laboratoire d’Astrophysique de Marseille) and has been extensively characterized, in particular by PSI [15].

In this paper, we firstly recall the PISTIL concept and detail the method to reach high-dynamic range, in Section 2. We present the PSI technique in Section 3. Then, experimental setup for the two-wavelength PISTIL is presented in Section 4, along with a first experimental demonstration of the method, and a comparison with the independent analysis made by PSI.

2. Two-wavelength PISTIL interferometry

The PISTIL interferometry aims to analyze the relative phase in the neighborhood of each element of the segmented wavefront. Observation of the PISTIL interference pattern has been conducted in [12] and we briefly sum up here its main lines: It relies on the combined use of three elements: a mask which extracts the central part of each segment defining elementary beams; a diffractive element creating replicas of the beams; a focal plane array to observe superimposition of pairs of neighbor beams, ultimately interfering and creating a fringes pattern (drawn in Fig. 1, left). This fringe pattern is modeled according to the relative phase difference that can exist between the adjacent segmented elements.

 

Fig. 1. Side-view layout of the PISTIL interferometer. We focus here on two collimated beams that propagate through the system and ultimately generate the interference pattern on the detector. The fringe pattern is related to the relative phase difference between segments (denoted 1 and 2 here).

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We can observe three different behaviors (Fig. 1, right) on the two wave fringe pattern depending on the phase default between adjacent elements:

  • • a shift in the fringe pattern (Fig. 1(b)), is due to a piston difference $\mathrm{\Delta }P = {P_1} - {P_2}$ (mainly expressed in nm) between the segmented elements;
  • • a frequency change in x-direction (Fig. 1(c)), is due to an inclination difference in the x-axis here, $\mathrm{\Delta }{t_X} = {t_X}_1 - {t_{{X_2}}}$ (in radian);
  • • a frequency change in y-direction, also described as a rotation (Fig. 1(d)), is due to an inclination difference in the y-axis here, $\mathrm{\Delta }{t_Y} = {t_Y}_1 - {t_{{Y_2}}}$ (in radian).
We derive the mathematical model of the PISTIL fringes pattern between two adjacent segments, to highlight the influence of each phase default as described:
$$I({x,y} )\propto 1 + \cos \left( {2\pi \left( {\frac{{2x}}{{{p_g}}} + \frac{{\mathrm{\Delta }{t_X}x}}{\lambda } + \frac{{\mathrm{\Delta }{t_Y}y}}{\lambda } + \frac{{\mathrm{\Delta }P}}{\lambda }} \right)} \right),\; $$
where $\lambda $ is the wavelength and ${p_g}$the diffractive element period, or grating period.

The global interference figure recorded (Fig. 2(a)), which takes into account all the elements of the segmented surface, is then composed of a series of small two wave interference sub-patterns as described above whose orientation is dictated by the direction defined by two neighboring surface elements. The corresponding spectrum (Fig. 2(b)), after Fourier transform, shows harmonics of interest in the directions corresponding to the two-wave fringes, which are processed [16] to obtain the relative phase map between pairs of adjacent elements. From this relative phase map, a least-square inversion algorithm [17] is applied to estimate the original segmented wavefront map. Complete derivation of the process has been led in [18], along with the first metrology analysis of a segmented mirror for small pistons.

 

Fig. 2. A pistilogram of a hexagonal meshed segmented wavefront (2a), and its corresponding Fourier plane (represented in 2D powerspectrum) (2b). Dotted lines circle different fringes orientations, which are all transformed into the corresponding Fourier harmonics (matching dots or color). A top-hat filter (circling as represented) is then used for the inverse Fourier extraction process.

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2.1. Extension of the piston difference dynamic with a two-wavelength interferometry algorithm

An issue still needs to be addressed: the fringe pattern is periodic by its definition (Eq. (1)). Whenever the phase term of the cosine,$\; \mathrm{\Delta }P/\lambda $ is outbound of the main determination interval$\; ({ - \pi , + \pi } )$ and the true value of the phase is wrapped by a relative integer,$\; n$ multiple of 2π. It translates into a λ-wrapping, within the main determination interval$\; ({ - \lambda /2\; , + \lambda /2\; } )$. The dynamic range of the interferometer is therefore limited in piston sensing to a maximum of a $\lambda /2$ piston difference between neighbor segments. In a practical reflective case (study of a mirror), this limit is even reduced to$\; \lambda /4$, as the sensed piston difference is doubled. For a 633nm wavelength, only relative pistons up to ±316nm (±158nm for a mirror) can be properly sensed before phase wrapping. A piston difference $\mathrm{\Delta }P$ is then estimated by Eq. (2), in which ${\widehat {\mathrm{\Delta }P}^\lambda }$ is the wrapped piston difference estimate.

$$\mathrm{\Delta }P = \; {\widehat {\mathrm{\Delta }P}^\lambda } + n\lambda .\; $$
This issue is typically solved by the addition of another wavelength, $\lambda + \mathrm{\Delta }\lambda $ (with$\; \mathrm{\Delta }\lambda $ the spectral gap,$\; |{\mathrm{\Delta }\lambda } |\ll \lambda $), in order to perform Two-Wavelength Interferometry (TWI), which is applied in a wide range of techniques, using a single wavelength demodulation [1921] or a combination of the wavelengths (in pre or post-processing) [2127]. Therefore, we investigated here an innovative Two-Wavelength PISTIL interferometer which performs a Vernier-like comparison between each wrapped piston difference estimate ${\widehat {\mathrm{\Delta }P}^\lambda }$ and ${\widehat {\mathrm{\Delta }P}^{\lambda + \mathrm{\Delta }\lambda }}$, and we propose the concept shown in Fig. 3 and described hereafter. We take the example of only one piston difference between two adjacent segments for demonstration purpose. This reasoning remains valid for all other segment pairs. A pistilogram of the segmented wavefront is recorded for each wavelength, one at a time, and it derives two fringe patterns of interest:
$$\left\{ {\begin{array}{{c}} {{I_\lambda }({x,y} )\propto 1 + \cos \left( {2\pi \left( {\frac{{2x}}{{{p_g}}} + \frac{{\mathrm{\Delta }P}}{\lambda }} \right)} \right)}\\ {{I_{\lambda + \mathrm{\Delta }\lambda }}({x,y} )\propto 1 + \cos \left( {2\pi \left( {\frac{{2x}}{{{p_g}}} + \frac{{\mathrm{\Delta }P}}{{\lambda + \mathrm{\Delta }\lambda }}} \right)} \right)} \end{array}.\; \; } \right.$$

 

Fig. 3. Illustration of the Vernier comparison used in the two-wavelength PISTIL interferometer. From top to bottom: A segmented wavefront with a high piston is measured with a PISTIL interferometer at two wavelengths, obtaining two slightly different pistilograms. Those figures are computed each by a Fourier-transform based algorithm to estimate the wrapped relative phases (here pistons only). The values obtained are then virtually demodulated and compared in a Vernier-like algorithm which determines the correct (n,m) couple that restores the correct relative piston input. Then the PISTIL least-square algorithm is applied to a single wavelength demodulated pistilogram to reconstruct the initial wavefront.

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By the use of the Fourier transform based algorithm [16], the piston difference $\mathrm{\Delta }P$ (hence wrapped by the process), is estimated for both wavelength separately:

$$\left\{ {\begin{array}{{c}} {\Delta P = \; {{\widehat {\mathrm{\Delta }P}}^\lambda } + n\lambda }\\ {\; \; \Delta P = \; {{\widehat {\mathrm{\Delta }P}}^{\lambda + \mathrm{\Delta }\lambda }} + m({\lambda + \mathrm{\Delta }\lambda } )} \end{array}\; .\; } \right.$$
Virtual unwrapped values are compared in both wavelength scales (adding $n\lambda $ and $m({\lambda + \mathrm{\Delta }\lambda } )$ respectively to the wrapped values,$\; n,m \in {\mathbb{\textrm N}}$), and at the perfect match, the $({n,m} )$ couple is the solver for the unwrapping process.

Once this step is done, we choose a single wavelength, either the $\lambda $ or $\lambda + \mathrm{\Delta }\lambda $ pistilogram maximizing fringes contrast, and unwrap each ${\widehat {\mathrm{\Delta }P}^\lambda }$ with the corresponding $n\; or\; m$ found. The least-square reconstruction algorithm described in the PISTIL classical process [17,18] is then applied to estimate the input piston(s) of the segmented wavefront (process described in Fig. 3).

The process finds its equivalent in the virtual wavelength analysis: indeed, from the two wavelengths we used in our Vernier-like method, one can write an equivalent wavelength:

$${\lambda _v} = \frac{{\lambda ({\lambda + \mathrm{\Delta }\lambda } )}}{{|{\mathrm{\Delta }\lambda } |}}.\; $$
It shows that the piston difference command $\mathrm{\Delta }P$ is extended to the main determination interval$\; ({ - {\lambda_v}/2, + {\lambda_v}/2} )$. This insures existence of a unique $({n,m} )$ couple that properly unwrap the piston difference$\; \mathrm{\Delta }P$. For example, a spectral gap of tens of nm between two visible wavelengths (633nm and 594nm) will introduce an equivalent wavelength of ${\sim} $ 9.6µm, almost 15 times the initial wavelengths. The suitable $\mathrm{\Delta }\lambda $ can be fixed on the application requirement, and results from a trade-off between the piston amplitude to measure and the hardware constraints.

We summarize here the Two-Wavelength PISTIL procedure:

  • 1. Pistilograms are recorded and computed by the Fourier-transform based algorithm [16,18] separately (Fig. 2, Fig. 3). The wrapped relative phases maps, wrapped, are then obtained for each pairs of segments.
  • 2. The relative piston at one wavelength is compared with the value measured at the other wavelength, being virtually demodulated until the Vernier-like match is reached (Fig. 3). The solver couple ($n$,$\; m$) is found and the step is repeated for each$\; \mathrm{\Delta }P$.
  • 3. One single wavelength is then selected and the relative piston is demodulated using the corresponding n (or$\; m$). Then the least-square reconstruction algorithm [17,18] is used to compute the estimated wavefront.

3. Phase shifting interferometry (PSI)

Our PSI bench was developed for the characterization of micro-optical components requires dedicated setups, especially for small-scale deformation characterization on their surface from room temperature down to cryogenic temperatures. The measurement of the shape and the deformation parameters of the MOEMS devices are made thanks to this interferometric characterization bench, fully developed in-house (hardware and software). Characterizations are done in static or dynamic modes, including measurements of optical surface quality at different scales, actuator stroke, maximum mirror deformation and cut-off frequency.

The bench we used is a high-resolution and low-coherence Twyman-Green interferometer (Fig. 4). It is a modular set-up allowing measurements at two scales on the MOEMS device, from a small field of view (typically 1mm) to the whole device (up to 40mm). Out-of-plane measurements are performed with PSI showing very high precision (standard deviation<1nm) [13,21]. It has the ability to use a compact cryo-vacuum chamber designed for reaching 10-6 mbar and 160K, in front of our custom interferometer. This set-up is able to measure performance of a deformable mirror at actuator/segment level and at the entire mirror level, with a lateral resolution of 2µm and a sub-nanometer z-resolution [28].

 

Fig. 4. Schematic description of PSI interferometric measurement set-up with the cryo-vacuum chamber, located at LAM. Image bottom right: PTT111 device mounted in the cryogenic chamber for characterization in space environment. Image upper left: Front window is closed and the cryogenic chamber is installed in front of the interferometric setup. The segmented deformable mirror could be successfully actuated before, during and after cryogenic cooling at 160 K.

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PSI allows the shape of the segmented surface to be restored with very fine spatial resolution, thus showing the internal deformation within the segments. PISTIL concentrates on the differential phase between each segment, meaning it is sensitive to variations of relative piston, tip and tilt, with high-dynamic range in the case of Two-Wavelength PISTIL. Those two techniques differ by the need of a reference surface in the PSI measure and temporal stability, not being necessary for PISTIL, calibrated beforehand. Both techniques could reach a nanometer accuracy as the measurement is calibrated with respect to a laser wavelength. Then, the feasibility and the truthfulness of the Two-Wavelength PISTIL interferometer for accurate and precise measurement of a wide range of segmented pistons will be validated by comparison to the reference interferometric technique.

4. Experimental demonstration of the two-wavelength PISTIL setup and blind comparison with PSI analysis on a segmented mirror

4.1. Two wavelength PISTIL experimental setup

We built an optical bench (Fig. 5) to probe the PTT-111 demonstration device with PISTIL.

 

Fig. 5. Experimental setup built at the ONERA laboratory. The PISTIL interferometer is in the upper green frame and the second wavelength addition is in the bottom. Each wavelength is recorded alternatively.

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Two HeNe lasers (${\lambda _1} = 633$nm and$\; {\lambda _2} = 594$nm) have been installed on the bench. The equivalent wavelength thus generated is ${\lambda _v} \approx 9.6$µm, which is enough for measurements of above-wavelength pistons with the DM (max stroke of 1.5µm). The segmented wavefront is conjugated by two afocal systems. The PISTIL interferometer is made of a hole mask having a pitch of $D\; = 606$µm with a hole diameter of 300µm; and of an intensity diffraction grating having a period of ${p_g} = 97$µm. The special position of the grating is explained in [16]. The camera is a JAI SP-5000-CXP4 having an array of 2580${\times} $2048 pixel² with a pixel pitch of 5µm.

4.2. Two-wavelength PISTIL interferometry demonstration

Different configurations of high relative pistons have been set on the PTT-111 DM. For each wavelength, we recorded a reference pistilogram with a flattened DM (obtained through the IRIS-AO “flat” pipeline). Then for each configuration and each wavelength, the measurement procedure was to record ten images to average them. It is a safe step that ensures no slight shift of the fringes has happened. Each resulted pistilogram is processed according to the Fourier and TWI algorithms (Fig. 2, Fig. 3). Global tip/tilt due to residual misalignment is removed before practical data processing by applying a check command.

Measurements are shown in Fig. 6. For this experiment, both high and small pistons have been set aside to test our PISTIL Vernier-like algorithm. The wavefront has been accurately estimated with no tip/tilt corruption, with RMS standard error of 4nm (see Table 1). The RMS value is obtained by computing the standard deviation of the difference between estimate and input coefficients.

 

Fig. 6. Several pistons are set on the PTT-111 DM (left) and the segmented wavefront reconstruction by the Two-Wavelength PISTIL is shown (right). The “X” marks the non-functioning segment. The RMS of the estimated wavefront is of 4nm, and is defined by the standard deviation between estimates and commands.

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Tables Icon

Table 1. Results from the Two-Wavelength PISTIL analysis of wavefront set by the PTT-111 DM and comparison with the PSI measurements.

4.3. Comparison with the PSI measurements and discussion

Once the two-wavelength PISTIL setup is demonstrated at the metrology laboratory, we compare its performance with respect to the PSI reference analysis. The interferometric fringes patterns are shown for both techniques in Fig. 7. This comparison again highlights difference of measurement between the two methods: PISTIL fringes are coding phase difference between segments while PSI is a full map sensitive technique (with inner segment deformations).

 

Fig. 7. Overview of the interferometric patterns PISTIL (left) and PSI (right). PISTIL is relative-phase sensitive (the fringes are correlated to the phase difference). The reference PSI is full map sensitive (including residual deformation on the segments).

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On Table 1, comparison of the two-wavelength PISTIL measurements is done with both the theoretical input commands and the PSI measurements. The DM has an uncertainty range for each segment and each command as given by the IRIS-AO company and related to the DM architecture. Range comparison is validated for the two-wavelength PISTIL with respect to the PSI, regarding the input commands. Precision of both techniques, in term of RMS standard error (deviation of measurements versus input values) lies in the same order of magnitude (0-10nm). As for the non-used segments (8 to 37), measurements are also accurate with RMS down to 1.5nm for the PISTIL process. The maps of the wavefront reconstruction by two-wavelength PISTIL and PSI are displayed in Fig. 8. It shows no difference, uncertainty of the demonstration device sets aside.

 

Fig. 8. Maps of pistons measured by Two-Wavelength PISTIL and PSI techniques with identical scales (values listed in Table 1). RMS PISTIL = 2 nm, RMS PSI = 4 nm. The general wavefront reconstruction is similar for both methods and both are close to the DM input commands.

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In addition, influence of one high piston command on the external segments, has been probed by comparing results for both two-wavelength PISTIL and PSI, as shown in Fig. 9. The central piston has an input command of 1µm, which is properly measured by both techniques (996.6nm for PISTIL and 986.5nm for PSI). The standard errors for this experiment are of 2nm for PISTIL and 4nm for PSI. In both wavefront map (rescaled to ±λ/20 range), three pistons in neighbor segments are clearly discernable in both PISTIL and PSI. The complex architecture underneath each segment might exhibit some non-uniformities revealed by this experiment on the highlighted segments. They are all marked clearly with both techniques.

 

Fig. 9. A piston of 1 µm is set on the PTT-111 DM and is well resolved by both methods. If we scan finely (30 nm ≈ λ/20) the estimated wavefront, one can see influence of one segment command onto the neighbor segments of the mirror, in both piston measurement of the PISTIL interferometer and the full map analysis accessible by the PSI.

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The Two-Wavelength PISTIL interferometer has matched the requirements (RMS order of magnitude of 1-10nm, ${\sim} $λ/100 at 633nm). With a different measurement sensitivity than PSI, that is to say piston difference spatial resolution instead of full segment spatial resolution, the two-wavelength PISTIL technique is able to reconstruct the input piston commands on the DM with the same accuracy. This method does not provide topographic details; however, this is not a problem in the context of intense lasers obtained by recombination of fiber lasers because it is known that the defects to be measured are limited to pistons and tilts [13], or in specific cophasing issues [4]. Moreover, the technique does not need a local reference at the time of measurement, which leads to decoupling between calibration step and measurement step.

5. Conclusion

In this work, we developed the Two-Wavelength PISTIL interferometry to achieve high dynamic piston sensing on segmented wavefront. This phase analyzer is a plug-and-play setup of few optical elements easily packable and is self-referenced, which is of great advantage for on site measurements. Extension of the piston range is achieved with almost no loss in precision compared to classical PISTIL, below$\; \lambda /100$. Moreover, we compared successfully two-wavelength PISTIL’s performance in terms of accuracy and precision with measurements from PSI.

Continuation of this research splits in two main topics: on one side, we will continue to improve the Two-Wavelength PISTIL process by conducting very high dynamic segmented wavefront analysis (up to 50-100$\lambda $) on a femtosecond coherently combined laser array having 61 channels [29]. On another side, the comparison work between two-wavelength PISTIL and PSI might continue with analysis of the DM and a passive stepped measurement standard.

Funding

LabEx PALM (ANR-10-LABX-0039-PALM).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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References

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  1. G. A. Mourou, D. Hulin, and A. Galvanauskas, “The road to high peak power and high average power lasers: Coherent amplification network (can),” AIP Conf. Proc. 827(1), 152–163 (2006).
    [Crossref]
  2. J. Bourderionnet, C. Bellanger, J. Primot, and A. Brignon, “Collective coherent phase combining of 64 fibers,” Opt. Express 19(18), 17053–17058 (2011).
    [Crossref]
  3. A. Heilmann, J. Le Dortz, L. Daniault, I. Fsaifes, S. Bellanger, J. Bourderionnet, C. Larat, E. Lallier, M. Antier, E. Durand, C. Simon-Boisson, A. Brignon, and J.-C. Chanteloup, “Coherent beam combining of seven fiber chirped-pulse amplifiers using an interferometric phase measurement,” Opt. Express 26(24), 31542–31553 (2018).
    [Crossref]
  4. I. Trumper, P. Hallibert, J. W. Arenberg, H. Kunieda, O. Guyon, H. Philip Stahl, and D. W. Kim, “Optics technology for large-aperture space telescopes: from fabrication to final acceptance tests,” Adv. Opt. Photonics 10(3), 644–702 (2018).
    [Crossref]
  5. N. Doble, D. T. Miller, G. Yoon, and D. R. Williams, “Requirements for discrete actuator and segmented wavefront correctors for aberration compensation in two large populations of human eyes,” Appl. Opt. 46(20), 4501–4514 (2007).
    [Crossref]
  6. G. Chanan, M. Troy, F. Dekens, S. Michaels, J. Nelson, T. Mast, and D. Kirkman, “Phasing the mirror segments of the Keck telescopes: the broadband phasing algorithm,” Appl. Opt. 37(1), 140–155 (1998).
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  7. S. Esposito, E. Pinna, A. Puglisi, A. Tozzi, and P. Stefanini, “Pyramid sensor for segmented mirror alignment,” Opt. Lett. 30(19), 2572–2574 (2005).
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  8. B. Benna, L. Lombard, V. Jolivet, C. Delezoide, E. Pourtal, P. Bourdon, G. Canat, O. Vasseur, and Y. Jaoun, “Brightness scaling based on 1.55 µm fiber amplifiers coherent combining,” Fiber Integr. Opt. 27(5), 355–369 (2008).
    [Crossref]
  9. C. Bellanger, B. Toulon, J. Primot, L. Lombard, J. Bourderionnet, and A. Brignon, “Collective phase measurement of an array of fiber lasers by quadriwave lateral shearing interferometry for coherent beam combining,” Opt. Lett. 35(23), 3931–3933 (2010).
    [Crossref]
  10. M. Antier, J. Bourderionnet, C. Larat, E. Lallier, E. Lenormand, J. Primot, and A. Brignon, “kHz Closed Loop Interferometric Technique for Coherent Fiber Beam Combining,” IEEE J. Sel. Top. Quantum Electron. 20(5), 182–187 (2014).
    [Crossref]
  11. P. Janin-Potiron, M. N’Diaye, P. Martinez, A. Vigan, K. Dohlen, and M. Carbillet, “Fine cophasing of segmented aperture telescopes with ZELDA, a Zernike wavefront sensor in the diffraction-limited regime,” Astron. Astrophys. 603, A23 (2017).
    [Crossref]
  12. M. Deprez, B. Wattellier, C. Bellanger, L. Lombard, and J. Primot, “Piston and tilt interferometry for segmented wavefront sensing,” Opt. Lett. 41(6), 1078–1081 (2016).
    [Crossref]
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  14. M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
    [Crossref]
  15. F. Zamkotsian, P. Lanzoni, R. Barette, M. A. Helmbrecht, F. Marchis, and A. Teichman, “Operation of a MOEMS deformable mirror in cryo: Challenges and results,” Micromachines 8(8), 233 (2017).
    [Crossref]
  16. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
    [Crossref]
  17. R. Penrose and J. A. Todd, “A generalized inverse for matrices,” Math. Proc. Cambridge Philos. Soc. 51(3), 406–413 (1955).
    [Crossref]
  18. M. Deprez, B. Wattellier, C. Bellanger, L. Lombard, and J. Primot, “Phase measurement of a segmented wave front using PISton and TILt interferometry (PISTIL),” Opt. Express 26(5), 5212–5224 (2018).
    [Crossref]
  19. K. Houairi and F. Cassaing, “Two-wavelength interferometry: extended range and accurate optical path difference analytical estimator,” J. Opt. Soc. Am. A 26(12), 2503–2511 (2009).
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  20. M. G. Löfdahl and H. Eriksson, “Algorithm for resolving 2-pi ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40(6), 984–990 (2001).
    [Crossref]
  21. A. Liotard and F. Zamkotsian, “Static and dynamic micro-deformable mirror characterization by phase-shifting and time-averaged interferometry,” Proc. SPIE 5494, 207 (2004).
    [Crossref]
  22. C. Polhemus, “Two-wavelength interferometry,” Appl. Opt. 12(9), 2071–2074 (1973).
    [Crossref]
  23. B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
    [Crossref]
  24. K. Creath, “Step height measurement using two-wavelength phase-shifting interferometry,” Appl. Opt. 26(14), 2810–2816 (1987).
    [Crossref]
  25. K. S. Mustafin and V. A. Seleznev, “Methods for increasing the sensitivity of holographic interferometry,” Sov. Phys. Usp. 13(3), 416 (1970).
    [Crossref]
  26. M. Servin, M. Padilla, and G. Garnica, “Synthesis of multi-wavelength temporal phase-shifting algorithms optimized for high signal-to-noise ratio and high detuning robustness using the frequency transfer function,” Opt. Express 24(9), 9766–9780 (2016).
    [Crossref]
  27. M. Servin, M. Padilla, and G. Garnica, “Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping,” Opt. Lasers Eng. 106, 68–74 (2018).
    [Crossref]
  28. F. Zamkotsian, E. Grassi, S. Waldis, R. Barette, P. Lanzoni, C. Fabron, W. Noell, and N. de Rooij, “Interferometric characterization of MOEMS devices in cryogenic environment for astronomical instrumentation,” Proc. SPIE 6884, 68840D (2008).
    [Crossref]
  29. I. Fsaifes, L. Daniault, S. Bellanger, M. Veinhard, J. Bourderionnet, C. Larat, E. Lallier, E. Durand, A. Brignon, and J.-C. Chanteloup, “Coherent beam combining of 61 femtosecond fiber amplifiers,” Opt. Express 28(14), 20152–20161 (2020).
    [Crossref]

2020 (1)

2018 (4)

M. Servin, M. Padilla, and G. Garnica, “Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping,” Opt. Lasers Eng. 106, 68–74 (2018).
[Crossref]

A. Heilmann, J. Le Dortz, L. Daniault, I. Fsaifes, S. Bellanger, J. Bourderionnet, C. Larat, E. Lallier, M. Antier, E. Durand, C. Simon-Boisson, A. Brignon, and J.-C. Chanteloup, “Coherent beam combining of seven fiber chirped-pulse amplifiers using an interferometric phase measurement,” Opt. Express 26(24), 31542–31553 (2018).
[Crossref]

I. Trumper, P. Hallibert, J. W. Arenberg, H. Kunieda, O. Guyon, H. Philip Stahl, and D. W. Kim, “Optics technology for large-aperture space telescopes: from fabrication to final acceptance tests,” Adv. Opt. Photonics 10(3), 644–702 (2018).
[Crossref]

M. Deprez, B. Wattellier, C. Bellanger, L. Lombard, and J. Primot, “Phase measurement of a segmented wave front using PISton and TILt interferometry (PISTIL),” Opt. Express 26(5), 5212–5224 (2018).
[Crossref]

2017 (2)

F. Zamkotsian, P. Lanzoni, R. Barette, M. A. Helmbrecht, F. Marchis, and A. Teichman, “Operation of a MOEMS deformable mirror in cryo: Challenges and results,” Micromachines 8(8), 233 (2017).
[Crossref]

P. Janin-Potiron, M. N’Diaye, P. Martinez, A. Vigan, K. Dohlen, and M. Carbillet, “Fine cophasing of segmented aperture telescopes with ZELDA, a Zernike wavefront sensor in the diffraction-limited regime,” Astron. Astrophys. 603, A23 (2017).
[Crossref]

2016 (3)

2014 (1)

M. Antier, J. Bourderionnet, C. Larat, E. Lallier, E. Lenormand, J. Primot, and A. Brignon, “kHz Closed Loop Interferometric Technique for Coherent Fiber Beam Combining,” IEEE J. Sel. Top. Quantum Electron. 20(5), 182–187 (2014).
[Crossref]

2011 (1)

2010 (1)

2009 (1)

2008 (2)

B. Benna, L. Lombard, V. Jolivet, C. Delezoide, E. Pourtal, P. Bourdon, G. Canat, O. Vasseur, and Y. Jaoun, “Brightness scaling based on 1.55 µm fiber amplifiers coherent combining,” Fiber Integr. Opt. 27(5), 355–369 (2008).
[Crossref]

F. Zamkotsian, E. Grassi, S. Waldis, R. Barette, P. Lanzoni, C. Fabron, W. Noell, and N. de Rooij, “Interferometric characterization of MOEMS devices in cryogenic environment for astronomical instrumentation,” Proc. SPIE 6884, 68840D (2008).
[Crossref]

2007 (2)

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

N. Doble, D. T. Miller, G. Yoon, and D. R. Williams, “Requirements for discrete actuator and segmented wavefront correctors for aberration compensation in two large populations of human eyes,” Appl. Opt. 46(20), 4501–4514 (2007).
[Crossref]

2006 (1)

G. A. Mourou, D. Hulin, and A. Galvanauskas, “The road to high peak power and high average power lasers: Coherent amplification network (can),” AIP Conf. Proc. 827(1), 152–163 (2006).
[Crossref]

2005 (1)

2004 (1)

A. Liotard and F. Zamkotsian, “Static and dynamic micro-deformable mirror characterization by phase-shifting and time-averaged interferometry,” Proc. SPIE 5494, 207 (2004).
[Crossref]

2001 (1)

M. G. Löfdahl and H. Eriksson, “Algorithm for resolving 2-pi ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40(6), 984–990 (2001).
[Crossref]

1998 (1)

1987 (1)

1982 (1)

1973 (1)

1970 (1)

K. S. Mustafin and V. A. Seleznev, “Methods for increasing the sensitivity of holographic interferometry,” Sov. Phys. Usp. 13(3), 416 (1970).
[Crossref]

1955 (1)

R. Penrose and J. A. Todd, “A generalized inverse for matrices,” Math. Proc. Cambridge Philos. Soc. 51(3), 406–413 (1955).
[Crossref]

Antier, M.

A. Heilmann, J. Le Dortz, L. Daniault, I. Fsaifes, S. Bellanger, J. Bourderionnet, C. Larat, E. Lallier, M. Antier, E. Durand, C. Simon-Boisson, A. Brignon, and J.-C. Chanteloup, “Coherent beam combining of seven fiber chirped-pulse amplifiers using an interferometric phase measurement,” Opt. Express 26(24), 31542–31553 (2018).
[Crossref]

M. Antier, J. Bourderionnet, C. Larat, E. Lallier, E. Lenormand, J. Primot, and A. Brignon, “kHz Closed Loop Interferometric Technique for Coherent Fiber Beam Combining,” IEEE J. Sel. Top. Quantum Electron. 20(5), 182–187 (2014).
[Crossref]

Arenberg, J. W.

I. Trumper, P. Hallibert, J. W. Arenberg, H. Kunieda, O. Guyon, H. Philip Stahl, and D. W. Kim, “Optics technology for large-aperture space telescopes: from fabrication to final acceptance tests,” Adv. Opt. Photonics 10(3), 644–702 (2018).
[Crossref]

Barette, R.

F. Zamkotsian, P. Lanzoni, R. Barette, M. A. Helmbrecht, F. Marchis, and A. Teichman, “Operation of a MOEMS deformable mirror in cryo: Challenges and results,” Micromachines 8(8), 233 (2017).
[Crossref]

F. Zamkotsian, E. Grassi, S. Waldis, R. Barette, P. Lanzoni, C. Fabron, W. Noell, and N. de Rooij, “Interferometric characterization of MOEMS devices in cryogenic environment for astronomical instrumentation,” Proc. SPIE 6884, 68840D (2008).
[Crossref]

Bellanger, C.

Bellanger, S.

Benna, B.

B. Benna, L. Lombard, V. Jolivet, C. Delezoide, E. Pourtal, P. Bourdon, G. Canat, O. Vasseur, and Y. Jaoun, “Brightness scaling based on 1.55 µm fiber amplifiers coherent combining,” Fiber Integr. Opt. 27(5), 355–369 (2008).
[Crossref]

Bourderionnet, J.

Bourdon, P.

B. Benna, L. Lombard, V. Jolivet, C. Delezoide, E. Pourtal, P. Bourdon, G. Canat, O. Vasseur, and Y. Jaoun, “Brightness scaling based on 1.55 µm fiber amplifiers coherent combining,” Fiber Integr. Opt. 27(5), 355–369 (2008).
[Crossref]

Brignon, A.

Canat, G.

B. Benna, L. Lombard, V. Jolivet, C. Delezoide, E. Pourtal, P. Bourdon, G. Canat, O. Vasseur, and Y. Jaoun, “Brightness scaling based on 1.55 µm fiber amplifiers coherent combining,” Fiber Integr. Opt. 27(5), 355–369 (2008).
[Crossref]

Carbillet, M.

P. Janin-Potiron, M. N’Diaye, P. Martinez, A. Vigan, K. Dohlen, and M. Carbillet, “Fine cophasing of segmented aperture telescopes with ZELDA, a Zernike wavefront sensor in the diffraction-limited regime,” Astron. Astrophys. 603, A23 (2017).
[Crossref]

Cassaing, F.

Chanan, G.

Chanteloup, J.-C.

Creath, K.

Daniault, L.

de Rooij, N.

F. Zamkotsian, E. Grassi, S. Waldis, R. Barette, P. Lanzoni, C. Fabron, W. Noell, and N. de Rooij, “Interferometric characterization of MOEMS devices in cryogenic environment for astronomical instrumentation,” Proc. SPIE 6884, 68840D (2008).
[Crossref]

Dekens, F.

Delezoide, C.

B. Benna, L. Lombard, V. Jolivet, C. Delezoide, E. Pourtal, P. Bourdon, G. Canat, O. Vasseur, and Y. Jaoun, “Brightness scaling based on 1.55 µm fiber amplifiers coherent combining,” Fiber Integr. Opt. 27(5), 355–369 (2008).
[Crossref]

Deprez, M.

Doble, N.

Dohlen, K.

P. Janin-Potiron, M. N’Diaye, P. Martinez, A. Vigan, K. Dohlen, and M. Carbillet, “Fine cophasing of segmented aperture telescopes with ZELDA, a Zernike wavefront sensor in the diffraction-limited regime,” Astron. Astrophys. 603, A23 (2017).
[Crossref]

Durand, E.

Eriksson, H.

M. G. Löfdahl and H. Eriksson, “Algorithm for resolving 2-pi ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40(6), 984–990 (2001).
[Crossref]

Esposito, S.

Fabron, C.

F. Zamkotsian, E. Grassi, S. Waldis, R. Barette, P. Lanzoni, C. Fabron, W. Noell, and N. de Rooij, “Interferometric characterization of MOEMS devices in cryogenic environment for astronomical instrumentation,” Proc. SPIE 6884, 68840D (2008).
[Crossref]

Fsaifes, I.

Galvanauskas, A.

G. A. Mourou, D. Hulin, and A. Galvanauskas, “The road to high peak power and high average power lasers: Coherent amplification network (can),” AIP Conf. Proc. 827(1), 152–163 (2006).
[Crossref]

Garnica, G.

M. Servin, M. Padilla, and G. Garnica, “Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping,” Opt. Lasers Eng. 106, 68–74 (2018).
[Crossref]

M. Servin, M. Padilla, and G. Garnica, “Synthesis of multi-wavelength temporal phase-shifting algorithms optimized for high signal-to-noise ratio and high detuning robustness using the frequency transfer function,” Opt. Express 24(9), 9766–9780 (2016).
[Crossref]

Grassi, E.

F. Zamkotsian, E. Grassi, S. Waldis, R. Barette, P. Lanzoni, C. Fabron, W. Noell, and N. de Rooij, “Interferometric characterization of MOEMS devices in cryogenic environment for astronomical instrumentation,” Proc. SPIE 6884, 68840D (2008).
[Crossref]

Guérineau, N.

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

Guyon, O.

I. Trumper, P. Hallibert, J. W. Arenberg, H. Kunieda, O. Guyon, H. Philip Stahl, and D. W. Kim, “Optics technology for large-aperture space telescopes: from fabrication to final acceptance tests,” Adv. Opt. Photonics 10(3), 644–702 (2018).
[Crossref]

Haïdar, R.

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

Hallibert, P.

I. Trumper, P. Hallibert, J. W. Arenberg, H. Kunieda, O. Guyon, H. Philip Stahl, and D. W. Kim, “Optics technology for large-aperture space telescopes: from fabrication to final acceptance tests,” Adv. Opt. Photonics 10(3), 644–702 (2018).
[Crossref]

He, M.

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

Heilmann, A.

Helmbrecht, M. A.

F. Zamkotsian, P. Lanzoni, R. Barette, M. A. Helmbrecht, F. Marchis, and A. Teichman, “Operation of a MOEMS deformable mirror in cryo: Challenges and results,” Micromachines 8(8), 233 (2017).
[Crossref]

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

Houairi, K.

Hulin, D.

G. A. Mourou, D. Hulin, and A. Galvanauskas, “The road to high peak power and high average power lasers: Coherent amplification network (can),” AIP Conf. Proc. 827(1), 152–163 (2006).
[Crossref]

Ina, H.

Janin-Potiron, P.

P. Janin-Potiron, M. N’Diaye, P. Martinez, A. Vigan, K. Dohlen, and M. Carbillet, “Fine cophasing of segmented aperture telescopes with ZELDA, a Zernike wavefront sensor in the diffraction-limited regime,” Astron. Astrophys. 603, A23 (2017).
[Crossref]

Jaoun, Y.

B. Benna, L. Lombard, V. Jolivet, C. Delezoide, E. Pourtal, P. Bourdon, G. Canat, O. Vasseur, and Y. Jaoun, “Brightness scaling based on 1.55 µm fiber amplifiers coherent combining,” Fiber Integr. Opt. 27(5), 355–369 (2008).
[Crossref]

Jolivet, V.

B. Benna, L. Lombard, V. Jolivet, C. Delezoide, E. Pourtal, P. Bourdon, G. Canat, O. Vasseur, and Y. Jaoun, “Brightness scaling based on 1.55 µm fiber amplifiers coherent combining,” Fiber Integr. Opt. 27(5), 355–369 (2008).
[Crossref]

Kempf, C. J.

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

Kim, D. W.

I. Trumper, P. Hallibert, J. W. Arenberg, H. Kunieda, O. Guyon, H. Philip Stahl, and D. W. Kim, “Optics technology for large-aperture space telescopes: from fabrication to final acceptance tests,” Adv. Opt. Photonics 10(3), 644–702 (2018).
[Crossref]

Kirkman, D.

Kobayashi, S.

Kunieda, H.

I. Trumper, P. Hallibert, J. W. Arenberg, H. Kunieda, O. Guyon, H. Philip Stahl, and D. W. Kim, “Optics technology for large-aperture space telescopes: from fabrication to final acceptance tests,” Adv. Opt. Photonics 10(3), 644–702 (2018).
[Crossref]

Lallier, E.

Lanzoni, P.

F. Zamkotsian, P. Lanzoni, R. Barette, M. A. Helmbrecht, F. Marchis, and A. Teichman, “Operation of a MOEMS deformable mirror in cryo: Challenges and results,” Micromachines 8(8), 233 (2017).
[Crossref]

F. Zamkotsian, E. Grassi, S. Waldis, R. Barette, P. Lanzoni, C. Fabron, W. Noell, and N. de Rooij, “Interferometric characterization of MOEMS devices in cryogenic environment for astronomical instrumentation,” Proc. SPIE 6884, 68840D (2008).
[Crossref]

Larat, C.

Le Dortz, J.

Lenormand, E.

M. Antier, J. Bourderionnet, C. Larat, E. Lallier, E. Lenormand, J. Primot, and A. Brignon, “kHz Closed Loop Interferometric Technique for Coherent Fiber Beam Combining,” IEEE J. Sel. Top. Quantum Electron. 20(5), 182–187 (2014).
[Crossref]

Liotard, A.

A. Liotard and F. Zamkotsian, “Static and dynamic micro-deformable mirror characterization by phase-shifting and time-averaged interferometry,” Proc. SPIE 5494, 207 (2004).
[Crossref]

Löfdahl, M. G.

M. G. Löfdahl and H. Eriksson, “Algorithm for resolving 2-pi ambiguities in interferometric measurements by use of multiple wavelengths,” Opt. Eng. 40(6), 984–990 (2001).
[Crossref]

Lombard, L.

Malacara, D.

D. Malacara, M. Servin, and Z. Malacara, “Interferogram analysis for optical testing”, (Marcel Dekker, New York, 1998).

Malacara, Z.

D. Malacara, M. Servin, and Z. Malacara, “Interferogram analysis for optical testing”, (Marcel Dekker, New York, 1998).

Marchis, F.

F. Zamkotsian, P. Lanzoni, R. Barette, M. A. Helmbrecht, F. Marchis, and A. Teichman, “Operation of a MOEMS deformable mirror in cryo: Challenges and results,” Micromachines 8(8), 233 (2017).
[Crossref]

M. A. Helmbrecht, M. He, C. J. Kempf, and F. Marchis, “Long-term stability and temperature variability of Iris AO segmented MEMS deformable mirrors,” Proc. SPIE 9909, 990981 (2016).
[Crossref]

Martinez, P.

P. Janin-Potiron, M. N’Diaye, P. Martinez, A. Vigan, K. Dohlen, and M. Carbillet, “Fine cophasing of segmented aperture telescopes with ZELDA, a Zernike wavefront sensor in the diffraction-limited regime,” Astron. Astrophys. 603, A23 (2017).
[Crossref]

Mast, T.

Mercier, R.

B. Toulon, J. Primot, N. Guérineau, R. Haïdar, S. Velghe, and R. Mercier, “Step-selective measurement by grating-based lateral shearing interferometry for segmented telescopes,” Opt. Commun. 279(2), 240–243 (2007).
[Crossref]

Michaels, S.

Miller, D. T.

Mourou, G. A.

G. A. Mourou, D. Hulin, and A. Galvanauskas, “The road to high peak power and high average power lasers: Coherent amplification network (can),” AIP Conf. Proc. 827(1), 152–163 (2006).
[Crossref]

Mustafin, K. S.

K. S. Mustafin and V. A. Seleznev, “Methods for increasing the sensitivity of holographic interferometry,” Sov. Phys. Usp. 13(3), 416 (1970).
[Crossref]

N’Diaye, M.

P. Janin-Potiron, M. N’Diaye, P. Martinez, A. Vigan, K. Dohlen, and M. Carbillet, “Fine cophasing of segmented aperture telescopes with ZELDA, a Zernike wavefront sensor in the diffraction-limited regime,” Astron. Astrophys. 603, A23 (2017).
[Crossref]

Nelson, J.

Noell, W.

F. Zamkotsian, E. Grassi, S. Waldis, R. Barette, P. Lanzoni, C. Fabron, W. Noell, and N. de Rooij, “Interferometric characterization of MOEMS devices in cryogenic environment for astronomical instrumentation,” Proc. SPIE 6884, 68840D (2008).
[Crossref]

Padilla, M.

M. Servin, M. Padilla, and G. Garnica, “Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping,” Opt. Lasers Eng. 106, 68–74 (2018).
[Crossref]

M. Servin, M. Padilla, and G. Garnica, “Synthesis of multi-wavelength temporal phase-shifting algorithms optimized for high signal-to-noise ratio and high detuning robustness using the frequency transfer function,” Opt. Express 24(9), 9766–9780 (2016).
[Crossref]

Penrose, R.

R. Penrose and J. A. Todd, “A generalized inverse for matrices,” Math. Proc. Cambridge Philos. Soc. 51(3), 406–413 (1955).
[Crossref]

Philip Stahl, H.

I. Trumper, P. Hallibert, J. W. Arenberg, H. Kunieda, O. Guyon, H. Philip Stahl, and D. W. Kim, “Optics technology for large-aperture space telescopes: from fabrication to final acceptance tests,” Adv. Opt. Photonics 10(3), 644–702 (2018).
[Crossref]

Pinna, E.

Polhemus, C.

Pourtal, E.

B. Benna, L. Lombard, V. Jolivet, C. Delezoide, E. Pourtal, P. Bourdon, G. Canat, O. Vasseur, and Y. Jaoun, “Brightness scaling based on 1.55 µm fiber amplifiers coherent combining,” Fiber Integr. Opt. 27(5), 355–369 (2008).
[Crossref]

Primot, J.

Puglisi, A.

Seleznev, V. A.

K. S. Mustafin and V. A. Seleznev, “Methods for increasing the sensitivity of holographic interferometry,” Sov. Phys. Usp. 13(3), 416 (1970).
[Crossref]

Servin, M.

M. Servin, M. Padilla, and G. Garnica, “Super-sensitive two-wavelength fringe projection profilometry with 2-sensitivities temporal unwrapping,” Opt. Lasers Eng. 106, 68–74 (2018).
[Crossref]

M. Servin, M. Padilla, and G. Garnica, “Synthesis of multi-wavelength temporal phase-shifting algorithms optimized for high signal-to-noise ratio and high detuning robustness using the frequency transfer function,” Opt. Express 24(9), 9766–9780 (2016).
[Crossref]

D. Malacara, M. Servin, and Z. Malacara, “Interferogram analysis for optical testing”, (Marcel Dekker, New York, 1998).

Simon-Boisson, C.

Stefanini, P.

Takeda, M.

Teichman, A.

F. Zamkotsian, P. Lanzoni, R. Barette, M. A. Helmbrecht, F. Marchis, and A. Teichman, “Operation of a MOEMS deformable mirror in cryo: Challenges and results,” Micromachines 8(8), 233 (2017).
[Crossref]

Todd, J. A.

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Figures (9)

Fig. 1.
Fig. 1. Side-view layout of the PISTIL interferometer. We focus here on two collimated beams that propagate through the system and ultimately generate the interference pattern on the detector. The fringe pattern is related to the relative phase difference between segments (denoted 1 and 2 here).
Fig. 2.
Fig. 2. A pistilogram of a hexagonal meshed segmented wavefront (2a), and its corresponding Fourier plane (represented in 2D powerspectrum) (2b). Dotted lines circle different fringes orientations, which are all transformed into the corresponding Fourier harmonics (matching dots or color). A top-hat filter (circling as represented) is then used for the inverse Fourier extraction process.
Fig. 3.
Fig. 3. Illustration of the Vernier comparison used in the two-wavelength PISTIL interferometer. From top to bottom: A segmented wavefront with a high piston is measured with a PISTIL interferometer at two wavelengths, obtaining two slightly different pistilograms. Those figures are computed each by a Fourier-transform based algorithm to estimate the wrapped relative phases (here pistons only). The values obtained are then virtually demodulated and compared in a Vernier-like algorithm which determines the correct (n,m) couple that restores the correct relative piston input. Then the PISTIL least-square algorithm is applied to a single wavelength demodulated pistilogram to reconstruct the initial wavefront.
Fig. 4.
Fig. 4. Schematic description of PSI interferometric measurement set-up with the cryo-vacuum chamber, located at LAM. Image bottom right: PTT111 device mounted in the cryogenic chamber for characterization in space environment. Image upper left: Front window is closed and the cryogenic chamber is installed in front of the interferometric setup. The segmented deformable mirror could be successfully actuated before, during and after cryogenic cooling at 160 K.
Fig. 5.
Fig. 5. Experimental setup built at the ONERA laboratory. The PISTIL interferometer is in the upper green frame and the second wavelength addition is in the bottom. Each wavelength is recorded alternatively.
Fig. 6.
Fig. 6. Several pistons are set on the PTT-111 DM (left) and the segmented wavefront reconstruction by the Two-Wavelength PISTIL is shown (right). The “X” marks the non-functioning segment. The RMS of the estimated wavefront is of 4nm, and is defined by the standard deviation between estimates and commands.
Fig. 7.
Fig. 7. Overview of the interferometric patterns PISTIL (left) and PSI (right). PISTIL is relative-phase sensitive (the fringes are correlated to the phase difference). The reference PSI is full map sensitive (including residual deformation on the segments).
Fig. 8.
Fig. 8. Maps of pistons measured by Two-Wavelength PISTIL and PSI techniques with identical scales (values listed in Table 1). RMS PISTIL = 2 nm, RMS PSI = 4 nm. The general wavefront reconstruction is similar for both methods and both are close to the DM input commands.
Fig. 9.
Fig. 9. A piston of 1 µm is set on the PTT-111 DM and is well resolved by both methods. If we scan finely (30 nm ≈ λ/20) the estimated wavefront, one can see influence of one segment command onto the neighbor segments of the mirror, in both piston measurement of the PISTIL interferometer and the full map analysis accessible by the PSI.

Tables (1)

Tables Icon

Table 1. Results from the Two-Wavelength PISTIL analysis of wavefront set by the PTT-111 DM and comparison with the PSI measurements.

Equations (5)

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I ( x , y ) 1 + cos ( 2 π ( 2 x p g + Δ t X x λ + Δ t Y y λ + Δ P λ ) ) ,
Δ P = Δ P ^ λ + n λ .
{ I λ ( x , y ) 1 + cos ( 2 π ( 2 x p g + Δ P λ ) ) I λ + Δ λ ( x , y ) 1 + cos ( 2 π ( 2 x p g + Δ P λ + Δ λ ) ) .
{ Δ P = Δ P ^ λ + n λ Δ P = Δ P ^ λ + Δ λ + m ( λ + Δ λ ) .
λ v = λ ( λ + Δ λ ) | Δ λ | .

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