Abstract

The newest experimental validation report of the coverage for the rotationally non-symmetric departure of a freeform surface in adaptive interferometry is about 20 µm. A compact adaptive interferometer is introduced to test unknown freeform surfaces with larger departure. The cascaded DMs (woofer and tweeter) can effectively double the measurable rotationally non-symmetric departure, to ∼80 µm using current DM technology. With a constrained decoupling control algorithm, the woofer and tweeter can averagely share the aberrations without coupling. DM surface monitoring is addressed by a time-division-monitoring (TDM) technique, which avoids separate monitoring devices and configurations and thus makes a compact configuration. Measurements of two different surfaces are presented: a nearly flat freeform with ∼40 um departure, and an off-axis paraboloid with ∼50 um of asymmetric departure.

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

1. Introduction

Optical freeform surfaces have elegant performance in modern optical systems in aberrations correction, illuminating shaping and so on. However, its metrology is still a great challenge. Interferometry, which has made rapid progress in spherical and aspherical surfaces test in various precise non-contact metrology techniques, is preferred in freeform surfaces metrology as well. Many aspheric interferometry ideas, such as null test, non-null test and sub-aperture stitching, have been transplanted to freeform surfaces interferometry. But for a surface with an unknown shape, such as a freeform surface in fabrication, the flexible corrector is necessary. Several flexible correctors such as the partial null optic (PNO) [13] and the mobile refractive aspheric null lens (RANL) [4,5] have been developed to compensate aberrations of aspheric surfaces and the rotationally symmetric departure of freeform surfaces. Especially with the mobile RANL, the coverage range for rotationally symmetric departure reaches up to 230λ (λ=632.8 nm) [4]. Meanwhile, researches for rotationally non-symmetric departure compensation are advancing in exploration as well. Some variable null correctors have been developed but only afford several low order aberrations [68]. The adaptive optics such as the spatial light modulator (SLM) and deformable mirror (DM), which can generate various kinds of aberrations in theory, have been introduced as alternatives in recent years. S. Xue et al. [4,912] have done a series of excellent works in the SLM based flexible test for freeform surfaces. However, the biggest challenge is the phase control method of SLMs, as was noted in [11]. The performance now is limited in moderate test accuracy (1/30 λ rms) [11]. Moreover, the dynamic range is very limited owing to the SLMs capacity. Although R. Chaudhuri introduced a design of SLM [13] based interferometer which was capable of 150 µm coverage, it was only a simulation and no experiments have been reported so far. The latest experiment report [4,9,11] of the SLM coverage for rotationally non-symmetric departure is about 20 µm (here Holoeye LC 2012). Therefore, the performance of SLMs in accuracy and flexibility has yet to be further improved.

As an alternative, DM has elegant performance in aberration correction as well. The commercialized DM with continuous surface now has a very simple control procedure and increasing aberration correction capacity. It has been employed to perform flexible tests for aspheric and freeform surfaces in many cases [10,1417]. The latest report of the DM (ALPAO DM-88) accuracy and coverage (for rotationally non-symmetric departure) is about 0.002 λ (rms) and 40 µm (although the experiment validation was about 23 µm) [16], respectively. Obviously, it has a better performance than SLMs. However, the high accuracy comes from the assistant of the DM surface monitoring. Because the general DMs are developed for adaptive imaging systems such as astronomical telescopes, the surface control accuracy (relative 5% PV error) cannot meet the requirement of high precision optical tests. In this respect, the performance of DM is even worse than the SLM. But fortunately, the DM surface can be precisely monitored by some devices or configurations. In 2014, Fuerschbach [15] carried out the null test for a φ polynomial mirror with the assistant of a DM, in which the DM surface was measured by the Zygo interferometer in advance. Note that the off-axis testing in this method may enlarge the coverage but is limited in several low-order aberrations such as astigmatism. In 2016, Huang. employed a deflectometry system for DM deformation monitoring to assist interferometric freeform surfaces test [10]. In 2018, we integrated the DM monitoring configuration into the interferometer for freeform surfaces test with polarized design [16]. In all these cases, additional DM monitoring devices (or configurations) are inevitable, which makes the whole system complicated. It is just the drawback of DMs in adaptive interferometer compared with SLMs.

To sum up, the reported DM has larger coverage (maximum 40 µm departure) and higher accuracy (0.002 λ) in freeform surfaces interferometry than the SLM (20 µm coverage and 0.03 λ rms accuracy). The biggest drawback is the requirement of additional devices or configurations for DM surface monitoring. However, the rotationally non-symmetric departure coverage ranges of both the DM and SLM are unsatisfactory compared with those of correctors for rotationally symmetric departure. If larger coverage is wanted, more advanced DMs and SLMs are needed. More capable SLMs, such as mentioned in [11] and [13], can accommodate about 60 µm and 150 µm departure. However, no related experiments have been reported so far. More capable DMs with both large stroke (such as 100 µm) and high resolution (such as 150 actuators at 25 mm aperture) are difficult to produce. Even the larger coverage DMs are available, the monitoring approaches by general interferometers [16] and wavefront sensors [17] are ineffective due to their capacity. It means that the integrated interferometric instrument is hard to achieve because we have to rely on the deflectometry system to monitor the DM surface. Therefore, multiple elements combination would be the development thinking. The combination of SLMs is difficult because it is based on diffraction but the DMs combination is promising because it is just based on reflection and the superposition is easy to control. In fact, the DMs combination has made some progress in adaptive systems for laser atmospheric transmission [18] and ophthalmology [19]. However, the idea of multiple DMs combination in an adaptive interferometer would face two thorny questions. The one is the multiple DMs monitoring would lead to either multiple monitoring devices or complex monitoring configurations. The other is the multiple DMs aberrations coupling and sharing. The two focuses are what we work on.

In this paper, a compact adaptive interferometer is introduced to test the unknown freeform surface with a large departure (especially rotationally non-symmetric part). Two DMs are employed in this interferometer, allowing aberrations from the test surface to be split between them. The DM control algorithm was optimized to reduce the stroke coupling between the two DMs. It enlarges the rotationally non-symmetric departure coverage to a maximum of 80 µm, which is the largest coverage ever reported in the experiment (except for off-axis configuration). The complicated configuration for two DM surfaces monitoring is addressed by the tunable wave plate (TWP) enabled time-division-monitoring (TDM) technique. The monitoring for two DMs is performed by the common path with the freeform surface test path in the interferometer one by one. Therefore the separate DM monitoring devices and configurations are no need, which makes a compact configuration.

2. Principle

The whole system shown in Fig. 1(a) concludes three parts: interferometer, hybrid compensating system (HCS) and the tested freeform surface. The interferometer generally is a Fizeau or Twyman-Green configuration (here a Twyman-Green interferometer is presented). The collimated beam transmitted from beam expander is divided by the beam splitter (BS) into two parts. The one is reflected by the BS and then arrives at BS again after being reflected by the reference mirror, which acts as the reference beam. The other transmits through the BS and then arrives at the HCS. After the aberration correction by the HCS, the wavefront would be reflected by the tested surface and arrive at the HCS again, accepting another aberration correction after traveling through the HCS again. The two aberration corrections make the wavefront aberration near null. Finally, the nearly collimated beam returns back to the interferometer, acting as the test beam. The test beam and reference beam interfere with each other at the BS and the resulting interferogram is imaged on to the CCD by an imaging lens. The interferogram can characterize the tested surface figure if the system parameters are known for ray tracing. Other parameters are available except the DMs deformation in HCS, which would be monitored separately.

 

Fig. 1. System layout of the compact interferometer.

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The HCS is a compact polarized design with the combination of correctors, including cascaded DMs (called woofer and tweeter below), PNO (optional), two tunable wave plates (TWPs) and a polarized beam splitter (PBS). Generally, the woofer has a relative larger stroke which is suited to low order aberration compensation while the tweeter has high actuators density and thus high resolution for high order aberrations correction. The specific beam propagation in HCS is illustrated in Fig. 1(b). The red line refers to the incident beam while the blue line means the reflected one by the tested surface. The p-polarized beam from the interferometer is all reflected by the PBS and then arrives at the woofer after traveling through the TWP 1, which now works as λ/4 WP. After reflected by the woofer DM, the beam travels through the TWP 1 again and becomes the s-polarized beam. Therefore, it would all pass through the PBS and arrive at the tweeter after traveling through the TWP 2, which works as λ/4 WP as well. After reflected by the tweeter, the beam travels through the TWP 2 again and becomes the p-polarized beam once again. Therefore, it would all be reflected by the PBS to the PNO, which is replaceable to compensate for different rotationally symmetric departure. The p-polarized beam arrives at the tested freeform surface and is reflected back to the HCS. It shows that the HCS does not change the polarized direction of the transmitted beam (red line). Therefore, the reflected beam (blue line) would not change its polarized direction when traveling back through the HCS. That is the reflected beam would travel through the HCS along the incident path and go back to interferometer in form of a p-polarized beam. A typical commercial available DMs, such as ALPAO DMs, are capable of providing 30 µm-40 µm correction amount (according to aberrations type, except tilt), which can afford aberrations of 60 µm -80 µm PV correction by the cascaded two DMs. Note that the beam travels through the HCS twice and thus the maximum aberration coverage about 160 µm is achieved. That is the rotationally non-symmetric departure of 80 µm PV is measurable.

However, this configuration is faced with two difficulties. The first one is the DM surface monitoring. The whole system is a flexible compensating system but not an absolute null system. The final figure error would be extracted by a model-based ray tracing, in which the DMs surface figure would be modeled as well. In our previous work, the DM surface was monitored by an additional interferometric configuration [16]. Another set of PBS, BS, polaroid, imaging lenses and CCD is needed. If we follow this approach, the system would be too complex because there are two DMs to be monitored. Therefore, we proposed a time-division-monitoring (TDM) method using only the interferometer itself without any assistant element. The whole procedure is illustrated in Fig. 2. After the tested wavefront acquisition described above, the TWP 1 and TWP2 are turned off. Thus the optical path in the whole system is changed to a simple configuration like Fig. 2(a), in which the p-polarized beam from interferometer would not change polarized direction after traveling the TWP 1 twice. The reflected p-polarized beam by the woofer would not pass through the PBS but all be reflected back to the interferometer. The resulted interferogram realizes the woofer monitoring. Then the woofer is controlled to achieve a flat surface by all the actuators reset, as is shown in Fig. 2(b), whose rms value is less than promised 7 nm (in fact about 5 nm using Alpao DM 88-25 and DM 97-25 in Sec. 4). The flat surface figure error can also be tested and stored for removing. Then the TWP 1 is turned to λ/4 WP again as is shown in Fig. 2(c). The reflected p-polarized beam by the woofer would change to s-polarized one, and thus all passes through the PBS. The s-polarized beam travels back through the PBS after being reflected by the tweeter, passing through the TWP2 (off) back and forth. Then the beam changes to p-polarized one after the reflection by the woofer and twice crossing the TWP 1. Finally, the p-polarized beam is all reflected by the PBS and travels back to the interferometer for the tweeter monitoring. The whole process is achieved by the status switching of TWPs and no other assistant device is needed, which greatly simplifies the system structure compared with previous methods [10,1517]. Of course, this procedure costs about 1 minute and thus losses the instantaneity to some extent. The DM 88 and DM 97 employed in our experiments are equipped with the high stability option [20], which promises less than 10nm rms open-loop stability over several hours. Thus, the TDM method still accommodates high accuracy in the freeform optics metrology process.

 

Fig. 2. TDM process for DMs monitoring (a) woofer monitoring, (b) middle process, (c) tweeter monitoring.

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Another difficulty is the DM coupling and sharing. Normally, it is difficult to achieve both a large stroke and a high spatial resolution with only a single deformable mirror (DM). The DM with more actuators has a higher resolution but a lower stroke. Fewer actuators refer to relatively larger stroke but lower resolution. The cascaded DMs overcome this issue by setting actuators of woofer and tweeter at intervals [21]. The minimum spatial resolution will not be less than the tweeter even if the actuators of woofer and tweeter are set randomly. The woofer and tweeter are generally “designed” to afford the low order and high order aberrations compensation respectively but not always. In fact, the “woofer” and “tweeter” are only the nominal names and they would have cross aberrations correction in work. Just because of this issue, DMs would suffer compensation coupling, as is shown in Fig. 3(a). The only detector in the interferometer cannot identify the coupling phenomenon when DMs produce opposite phase compensations (i.e., coupling). The coupling is bound to result in a pointless waste of wavefront correction. Eventually, amounts of DM stroke volume would be completely consumed by the accumulated coupling along with the increased closed-loop time if the coupling cannot be effectively suppressed. Control algorithms are critical for such systems, which have been researched for a long time in various adaptive optics [19,2227] such as the astronomical telescope, laser beam purification, laser transmission in the atmosphere and retinal cells imaging. It is mainly divided into two categories. The one is the wavefront sensor-based decoupling (WSD) algorithm [19,22,26] and the other is the wavefront sensor-less-based decoupling (WSLD) algorithm [2325,27]. However, both do not apply to the cascaded DMs decoupling in an adaptive interferometer.

 

Fig. 3. (a) Wavefront coupling illustration, (b)-(d) traditional decoupling, (e)-(g) desired decoupling.

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Firstly, the closed-loop controlling in an adaptive interferometer is based on the resulting interferogram rather than the wavefront and image signal in traditional cases. The resulting interferogram may even be unresolvable in case of large departure surfaces. Secondly, the final task of closed-loop controlling in an adaptive interferometer is not only the decoupling but also the aberration sharing on average for DMs. However, most of the decoupling algorithms reported are designed only for aberration decomposition, which makes the woofer and tweeter afford different aberration types, as are shown in Figs. 3(b)–3(d). Most adaptive optics control algorithms are only concerned with the output of the system, and not the individual DM shapes. The proposed adaptive interferometer, however, requires knowing the DM shapes to calibrate the system. Furthermore, since the DM shapes are measured by the interferometer itself, each shape must be in the capture range of the interferometer. Therefore it is important that the adaptive interferometer DM control algorithm distributes the AO correction between the two DMs to ensure that they are both measurable by the embedded interferometer, as is shown in Figs. 3(e)–3(f). Therefore, excessive aberration commitment with only one DM is undesirable (Fig. 3(b)) because it would make the DM surface unmeasurable. Therefore, the decoupling in the adaptive interferometer is redefined to prevent the negative counteract of the same aberrations but encourage the positive superposition with balanced distribution for the DMs rather than simply aberration decomposition. It is called decoupling and sharing. Of course, the algorithm should not affect the final compensation result (Fig. 3(g)).

3. Cascaded DMs decoupling and sharing in an adaptive interferometer

The interferogram in adaptive controlling would experience two stages. Firstly the no fringe sub-regions and unresolvable sub-regions of the interferogram are transformed into resolvable regions. Secondly, global resolvable but dense fringes are relieved to sparse fringes even near null fringes. Therefore, the decoupling would also be divided into two steps, aiming at the two stages of interferogram in the adaptive process.

The first step of decoupling control is to transform the no fringe sub-regions or unresolvable sub-regions of the interferogram to resolvable sub-regions, which is a WSLD process. The WSLD algorithms to date reported in adaptive freeform surface interferometers are only two: the SPGD algorithm and genetic algorithm, which both are employing actuator voltages as control signal vectors directly [10,12]. However, the control signal of actuator voltages cannot achieve decoupling in the case of multiple adaptive optics. In this paper, we take the SPGD algorithm as an example to illustrate the decoupling process. The basic SPGD idea in an adaptive interferometer [10] can be expressed as follows.

$${J^i}\textrm{ = N}({I_{unresov}^i} )$$
$$\delta {J^i} = J({{{\boldsymbol {V}}}_{\mathop{\rm w}\nolimits}^i + \delta {{\boldsymbol {V}}}_{\mathop{\rm w}\nolimits}^i;{{\boldsymbol {V}}}_{\mathop{\rm t}\nolimits}^i + \delta {{\boldsymbol {V}}}_{\mathop{\rm t}\nolimits}^i} )- J({{{\boldsymbol {V}}}_{\mathop{\rm w}\nolimits}^i;{{\boldsymbol {V}}}_{\mathop{\rm t}\nolimits}^i} ),$$
$$\left\{ \begin{array}{l} {{\boldsymbol {V}}}_{\mathop{\rm w}\nolimits}^{i + 1} = {{\boldsymbol {V}}}_{\mathop{\rm w}\nolimits}^i - {\gamma_{\mathop{\rm w}\nolimits} }\delta J\delta {{\boldsymbol {V}}}_{\mathop{\rm w}\nolimits}^i\\ {{\boldsymbol {V}}}_{\mathop{\rm t}\nolimits}^{i + 1} = {{\boldsymbol {V}}}_{\mathop{\rm t}\nolimits}^i - {\gamma_{\mathop{\rm t}\nolimits} }\delta J\delta {{\boldsymbol {V}}}_{\mathop{\rm t}\nolimits}^i \end{array} \right.,$$
where, $J$ is the system performance metric and $\delta J$ is the variation of J; the superscript i is the iteration number; the performance metric J is defined as $\textrm{N}({I_{unresov}^i} )$, the pixel number of unresolvable regions in the interferogram (I) [12] rather than the SSD (sum of squared gray level differences between any two pixels of the interferogram) [10] because the SSD is not a monotone function. The machine vision with morphology operation can be used for regions segmentation and pixel number count [12]; subscripts $\textrm{w}$ and ${\mathop{\rm t}\nolimits}$ refer to the woofer and tweeter, respectively; ${{{\boldsymbol {V}}}_\textrm{w}} = {[{{v_{\textrm{w}1}},{v_{w2}}, \cdots ,{v_{w\textrm{n}}}} ]^{\mathop{\rm T}\nolimits} }$ and ${{{\boldsymbol {V}}}_\textrm{t}} = {[{{v_{\textrm{t}1}},{v_{\textrm{t}2}}, \cdots ,{v_{\textrm{t}m}}} ]^{\mathop{\rm T}\nolimits} }$ are the control signal vectors of actuator voltages; subscripts n and m are index numbers of the woofer and tweeter actuators, respectively; $\delta {{\boldsymbol {V}}}$ are small random perturbations having identical amplitudes in Bernoulli probability distribution, which are the optimization variables;$\gamma$ is the optimization increment. Because $J$ would be a decreasing function with iterations increasing, thus the negative $\delta J$ are preferred. Therefore, the operator in Eq. (3) should be minus.

However, Eq. (3) cannot complete the decoupling. According to [28], we can transform the control signal vector of actuator voltages ${{\boldsymbol {V}}}$ to Zernike coefficients control signal ${\boldsymbol A}$ by a transformation matrix ${\boldsymbol B}$. That is

$$\left\{ \begin{array}{l} {{\boldsymbol A}_{\mathop{\rm w}\nolimits} } = {{\boldsymbol B}_{\mathop{\rm w}\nolimits} } \cdot {{{\boldsymbol {V}}}_{\mathop{\rm w}\nolimits} }\\ {{\boldsymbol A}_{\mathop{\rm t}\nolimits} } = {{\boldsymbol B}_\textrm{t}} \cdot {{{\boldsymbol {V}}}_t} \end{array} \right..$$
In fact, the initial transformation matrix B has been prepared by the manufacturer and can be calibrated by measuring the influence function of each actuator. The detail method to acquire the transformation matrix ${\boldsymbol B}$ is described in [28]. ${{\boldsymbol A}_{\mathop{\rm w}\nolimits} }$ and ${{\boldsymbol A}_{\mathop{\rm t}\nolimits} }$ in Eq. (4) are the Zernike coefficients controlling signal of woofer and tweeter, which are defined to characterize lower- order and higher-order aberrations as follows.
$$\left\{ \begin{array}{l} {{\boldsymbol A}_{\mathop{\rm w}\nolimits} } = {[{{a_{{\mathop{\rm w}\nolimits} 2}},{a_{{\mathop{\rm w}\nolimits} 3}}, \cdots ,{a_{{\mathop{\rm w}\nolimits} q}},0,0, \cdots ,0} ]^{\mathop{\rm T}\nolimits} }\\ {{\boldsymbol A}_{\mathop{\rm t}\nolimits} } = {[{0,0, \cdots ,0,{a_{{\mathop{\rm t}\nolimits} (q + 1)}},{a_{{\mathop{\rm t}\nolimits} (q + 2)}}, \cdots ,{a_{{\mathop{\rm t}\nolimits} 37}}} ]^{\mathop{\rm T}\nolimits} } \end{array} \right.,$$
where the q is the artificial boundary of aberration decomposition, which determined according to the actual situation. According to Eq. (5), the optimization variables are now converted to Zernike coefficients ${{\boldsymbol A}_{\mathop{\rm w}\nolimits} }$ and ${{\boldsymbol A}_{\mathop{\rm t}\nolimits} }$. Thus Eqs. (2) and (3) change to
$$\delta {J^i} = J({{\boldsymbol A}_{\mathop{\rm w}\nolimits}^i + \delta {\boldsymbol A}_{\mathop{\rm w}\nolimits}^i;{\boldsymbol A}_{\mathop{\rm t}\nolimits}^i + \delta {\boldsymbol A}_{\mathop{\rm t}\nolimits}^i} )- J({{\boldsymbol A}_{\mathop{\rm w}\nolimits}^i;{\boldsymbol A}_{\mathop{\rm t}\nolimits}^i} ),$$
$$\left\{ \begin{array}{l} {\boldsymbol A}_{\mathop{\rm w}\nolimits}^{i + 1} = {\boldsymbol A}_{\mathop{\rm w}\nolimits}^i\textrm{ - }{\gamma_{\mathop{\rm w}\nolimits} }\delta J\delta {\boldsymbol A}_{\mathop{\rm w}\nolimits}^i\\ {\boldsymbol A}_{\mathop{\rm t}\nolimits}^{i + 1} = {\boldsymbol A}_{\mathop{\rm t}\nolimits}^i\textrm{ - }{\gamma_{\mathop{\rm t}\nolimits} }\delta J\delta {\boldsymbol A}_{\mathop{\rm t}\nolimits}^i \end{array} \right.,$$
where, $\delta {\boldsymbol A}_{\mathop{\rm w}\nolimits} ^i$ and $\delta {\boldsymbol A}_{\mathop{\rm t}\nolimits} ^i$ are small random perturbations having identical amplitudes and a Bernoulli probability distribution. The corresponding control signal vectors of the actuator voltages ${{{\boldsymbol {V}}}_{\mathop{\rm w}\nolimits} }$ and ${{{\boldsymbol {V}}}_{\mathop{\rm t}\nolimits} }$ are acquired subsequently by inverse operation of Eq. (4) to control the woofer and tweeter with few coupling.

The second step of decoupling control is to relieve global resolvable but dense fringes to sparse fringes. In this step, the Zernike coefficients are available to control the DMs directly, which refer to a WSD process. Because of the static aberrations in the system, the optimization search is not always necessary and thus the control signal vectors of the actuator voltages ${{{\boldsymbol {V}}}_{\mathop{\rm w}\nolimits} }$ and ${{{\boldsymbol {V}}}_{\mathop{\rm t}\nolimits} }$ are directly employed for DMs decoupling control with Eqs. (4) and (5). The SPGD process is no necessary, or to put it another way, the SPGD algorithm is executed with performance metric $J\textrm{ = PV}(I )$ and large optimization increment $\gamma$. Thus only several iterations are needed. After this process, the sparse fringes not necessary near null fringes are obtained. A threshold value of the PV such as 10 λ is set as a terminal condition.

However, the decoupling algorithm described above is designed for aberration decomposition. This idea is suitable for most other situations but not applicable in the adaptive interferometer because there is a practical consideration making the interferometer run with difficulty. If the woofer afforded great low order aberrations (Fig. 3(b)) but the tweeter suffered little high order aberrations, the unequal surface stroke would make woofer monitoring suffer great fringe density, what's worse, unresolvable fringes, while the tweeter would sit by. Take an example. The PV value of the coma under-compensation is 120 µm (about 60 µm departure of the tested surface). All of the coma would be compensated by woofer according to Eq. (5) when the q in Eq. (5) is larger than eight. With two reflection compensation, the woofer monitoring in Fig. 2(a) would suffer about 60 µm PV coma (about 95 fringes, λ=632.8 nm), which is an arduous task for general interferometers. This case would be relieved if the tweeter pays a fair share. But the decoupling algorithm does not allow it because tasks for the woofer and tweeter are independent.

Thus decoupling algorithm must be implemented with sharing in the third step. The SPGD algorithm resume. We redefine control signal vectors of the Zernike coefficients as

$$\left\{ \begin{array}{l} {{\boldsymbol A}_{\mathop{\rm w}\nolimits} } = [{{a_{{\mathop{\rm w}\nolimits} 2}},{a_{{\mathop{\rm w}\nolimits} 3}}, \cdots ,{a_{{\mathop{\rm w}\nolimits} q}},0,0, \cdots ,0} ]\\ {{\boldsymbol A}_{\mathop{\rm t}\nolimits} } = [{{a_{{\mathop{\rm t}\nolimits} 2}},{a_{{\mathop{\rm t}\nolimits} 3}}, \cdots ,{a_{{\mathop{\rm t}\nolimits} q}},{a_{{\mathop{\rm t}\nolimits} (q + 1)}}, \cdots ,{a_{{\mathop{\rm t}\nolimits} 37}}} ]\end{array} \right..$$
Thus the tweeter would be involved in the correction of lower-order aberrations according to Eq. (8) in this step. Similarly, in case of large high order aberrations, the woofer would be involved in the correction of high order aberrations as well. To prevent the coupling, the control perturbations signal vectors $\delta {{\boldsymbol A}_{\mathop{\rm w}\nolimits} }$ and $\delta {{\boldsymbol A}_{\mathop{\rm t}\nolimits} }$ are provided as
$$\left\{ \begin{array}{l} {\boldsymbol A}_{\mathop{\rm w}\nolimits}^{i + 1} = {\boldsymbol A}_{\mathop{\rm w}\nolimits}^i\textrm{ + }{\gamma_\textrm{w}}\delta J\delta {\boldsymbol A}_{\mathop{\rm w}\nolimits}^i\\ {\boldsymbol A}_{\mathop{\rm t}\nolimits}^{i + 1} = {\boldsymbol A}_{\mathop{\rm t}\nolimits}^i - {\gamma_\textrm{t}}\delta J\delta {\boldsymbol A}_\textrm{t}^i\textrm{ = }{\boldsymbol A}_{\mathop{\rm t}\nolimits}^i - {\gamma_\textrm{w}}\delta J\delta {\boldsymbol A}_\textrm{w}^i \end{array} \right..$$
where implies the reduced $|{{{\boldsymbol A}_\textrm{w}}} |$ and increased $|{{{\boldsymbol A}_{\mathop{\rm t}\nolimits} }} |$. $\delta {{\boldsymbol A}_{\mathop{\rm t}\nolimits} }$ is set equal to $\delta {{\boldsymbol A}_\textrm{w}}$ while ${\gamma _\textrm{t}}$ is equal to ${\gamma _\textrm{w}}$, which makes the equal amount of reduced $|{{{\boldsymbol A}_\textrm{w}}} |$ and increased $|{{{\boldsymbol A}_{\mathop{\rm t}\nolimits} }} |$. It promises a relatively constant amount of total aberration correction. The performance metric J in this step is defined as
$${J^i}\textrm{ = PV}({{I^i}} )\textrm{ + }\left|{\textrm{PV}\left( {\sum {{\boldsymbol A}_{\mathop{\rm w}\nolimits}^iZ} } \right) - {\mathop{\rm PV}\nolimits} \left( {\sum {{\boldsymbol A}_\textrm{t}^iZ} } \right)} \right|,$$
which makes a relative equal aberration undertaking at the woofer and tweeter while the resulting interferogram basically remains unchanged. This method achieves aberration sharing between the woofer and tweeter while keeping the total aberration compensation capacity. In this case, the low order aberrations shared by the two DMs are positive superposition with less coupling. The whole process is illustrated in Fig. 4.

 

Fig. 4. Flow chart of cascaded DMs decoupling with sharing in adaptive interferometer.

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4. Experiment validation

Experiments were carried out to test an unknown surface at a 24 mm aperture. The large departure was produced by mechanical pressing a flat mirror, with an unknown amount of surface sag. The interferometer was a self-built Tyman-Green configuration as is shown in Fig. 1. The woofer and tweeter are DM88 (88 actuators at 25 mm aperture) and DM97 (97 actuators at 25 mm aperture) from Alpao corporation. The TWP is the LCC1421-B from Thorlabs, which can achieve 0-λ/2 phase retardance. The wavefront distortion is promised less than λ/4 PV and would be tested and calibrated in advance. In fact, a rotatable λ/4 wave plate also works. The beam expander provided a collimated beam with a 26 mm aperture, which matches the DMs’ aperture. The PNO was not employed in the case of testing a plane based mirror because we focused on the compensation performance of the rotationally non-symmetric departure. The DMs posture calibration was implemented in advance by the method proposed in our previous work [16].

The complete experiment process is presented in Fig. 5. The uncompensated interferogram is presented in Fig. 5(a), in which the no fringe area refers to the large departure. According to the flowchart shown in Fig. 4, the first step is pixel segmentation. The machine vision with morphology operation was employed to segment the no fringes sub-regions and unresolvable sub-regions from resolvable areas and the result is shown in Fig. 5(b), which has about 8022 pixels. The process from Figs. 5(b) to 5(d) is to transform the no fringes sub-regions and unresolvable sub-regions to resolvable areas, with the SPGD based decoupling algorithm. It is obvious that the algorithm worked well because Fig. 5(d) shows resolvable fringes at full aperture. The second step (Figs. 5(d)–5(f)) was to relieve global resolvable but dense fringes to sparse fringes. In this step, Zernike coefficients were available to control the DMs directly with decoupling operation Eqs. (4) and (5). This process should be performed two or three times iteratively. Of course, the SPGD optimization was effective as well by several iterations with a large $\gamma$. 5 λ PV was set as the terminal condition. The final sparse fringes were obtained as Fig. 5(f). After completing the second step in Fig. 5, the departure compensation for the tested surface was achieved. However, we did not promise measurable surfaces of woofer and tweeter.

 

Fig. 5. Interferograms in the whole experiment process, (a) uncompensated interferogram, (b) interferogram segmentation, (c)-(f) are interferograms in the process of SPGD decoupling, (h) and (j) are the fringes characterizing the woofer and tweeter surfaces after decoupling, (i) and (k) are those after aberration sharing step, which come from (h) and (j), respectively, (g) is the final interferogram characterizing tested freeform surface.

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The third step, DMs were measured by the TDM method as the process of Fig. 2. The resulting interferograms characterizing the woofer and tweeter surfaces are presented in Fig. 5(h) and 5(j). We found dense fringes even dark areas in the interferogram characterizing the woofer and sparse those for the tweeter. Obviously, the woofer undertook large aberrations while the tweeter shared little due to the inherent characteristic of the decoupling algorithm. Measurement of the woofer surface would be in trouble due to the unresolvable fringes in Fig. 5(h). Therefore, the third step was carried out for relative average commitment. The constrained SPGD based decoupling and sharing algorithm was executed until the PV value difference of the two resulted interferograms was less than 5 λ. The revised interferograms characterizing the woofer and tweeter surfaces are presented in Fig. 5(i) and 5(k), with a relatively similar density of fringes. The resulted interferogram characterizing the tested freeform surface is shown in Fig. 5(g), which has little variations with Fig. 5(f). That is the third step is only to share the compensations between the DMs with decoupling, not affecting the total compensation ability. Note that the PV value of the interferogram in Fig. 5(i) and 5(k) are 30.2 λ and 26.5 λ, respectively. The total compensated aberration by twice reflections of the DMs is about 105.1 λ (PV).

For comparison, the aberration correction performance of the woofer only, tweeter only, and cascaded woofer-tweeter without decoupling in the first step is illustrated in Fig. 6. Figure 6(a) provides the performance in the four cases in the first step. The residual pixel numbers of no fringe sub-regions and unresolvable sub-regions, after being compensated by the standalone woofer and tweeter (the yellow and blue line), were 5112 and 3806, respectively. Neither woofer nor tweeter can undertake complete departure compensation by itself. Without decoupling, the cascaded DMs were unable to accomplish the task as well due to the wasteful consumption between the DMs. There were still 2851 residual pixels unresolvable (the green line). With decoupling, the cascaded DMs allowed full play and the residual pixel numbers reduced to zero. The coupling coefficients [29], characterizing the extent of linearly dependent of the correction of woofer and tweeter, is illustrated in the right part of Fig. 6(a). The resulting interferograms in the four cases are presented in Fig. 6(b) as well, which implies the best aberration correction performance in decoupling.

 

Fig. 6. The performance comparison in the first step between woofer only, tweeter only, woofer-tweeter without decoupling and woofer-tweeter with decoupling, (a) the residual pixel number of unresolvable regions in the mentioned four cases with iterations, (b) the resulted interferograms in the four cases.

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In the third step described above, the DMs surfaces were measured by the TDM method as the process of Fig. 2. The TWP1 and TWP2 were turned off and Fig. 5(i) was obtained to describe the woofer surface figure. Then the woofer was turned to rest shape (flat surface), which is produced by sending the so-called “offset” command to the DM actuators. The “offset” commands are provided by the manufacturer. Note that the rest shape would not be flat surface immediately after the “offset” command was sent due to the thermal effect [30] if the preceding work time was long. In our experiment, the SPGD algorithm running time was less than 40s and thus the thermal effect can be ignored. Even so, the woofer surface after the “offset” commands sent was not the absolute flat and thus the slight surface error would be stored for subsequent calibration (as is shown in Fig. 2(b)). Then the TWP1 was turned to λ/4 WP again and Fig. 5(k) was obtained to describe the tweeter surface figure. As we mentioned in Sec. 2, this procedure cost about 1 minute and thus lost the instantaneity to some extent. The DM 88 and DM 97 used in our experiment was equipped with the high stability option, which promised less than 10nm rms open-loop stability over several hours [20]. For validating the DMs surface stability in the TDM method, we monitored woofer and tweeter surfaces in 5 minutes at temperature 25 °C in a thermostatic chamber with different initial strokes. Three independent experiments were carried out for the woofer and tweeter. The results of woofer monitoring are shown in Figs. 7(a)–7(c) and those of tweeter are shown in Figs. 7(d)–7(f). The rms uncertainty of each DM surface in 5 minutes was less than 4nm (0.0063 λ, λ=632.8 nm) and thus the induced error in the final result was less than 5 nm (0.0075 λ), which still accommodated high accuracy for freeform optics metrology. Note that the error of the TWP working in “on” and “off” cases were slightly different, which would be considered and modeled in the system with a virtual error surface.

 

Fig. 7. DM surfaces stability in different initial stroke in five minutes at 25 °C,(a)-(c) are three repetitive experimental results of the woofer, (d)-(f) are three repetitive experimental results of the tweeter.

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With the final interferogram characterizing DMs shown in Figs. 5(i) and 5(k), the corresponding DM surface figures were extracted by ray tracing with the reverse optimization reconstruction (ROR) method [33]. Then the DM surface figures were modeled in the ray tracing model. With the final interferogram characterizing the freeform surface, which is illustrated in Fig. 5(g), the final figure of the tested surface was extracted by system based ray tracing with the ROR method. Figure 8(a) shows the final figure map of the tested freeform surface. The total surface sag with 64.46 λ (40.8 µm) PV and 7.821 λ (4.95 µm) rms value at 24 mm aperture was reconstructed. Due to the lack of instruments for full-aperture measurement as cross validation, the Taylor Hobson profilometer was employed to measure the two contour lines of marked cross sections. The results compensation of the two contour lines between the Taylor Hobson profilometer and our interferometer is illustrated in Fig. 8(b). The other two repetitive experiments were carried out and the results are shown in Figs. 8(c) and 8(d), which show about 0.01 λ rms uncertainty. Note that Fig. 8(b) only provides a visualized but loose comparison due to the low registration accuracy of the contour lines. A rigorous cross validation would be presented by another experiment below.

 

Fig. 8. Final surface figure map and contour line, (a) surface figure map by our adaptive interferometer, (b) contour lines comparison with Taylor Hobson profilometer. (c) and (d) are surface figure maps tested in the other two repetitive experiments.

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For more rigorous cross validation, another experiment testing an off-axis paraboloidal mirror was carried out. The tested mirror was designed for laser collimation in a Doppler velocimeter with elliptical aperture (18.8 mm and 24.6 mm in x and y direction) and 101 mm reflected focal length at 90° off-axis angle. The parent focal length is 77.5 mm. A PNO was designed to compensate for the vast majority of spherical aberrations. The first picture in Fig. 9(a) presents the resulted interferogram compensated by the PNO, with unresolvable sub-regions around. Interferograms variation in DMs compensation process is shown in Fig. 9(a). Figure 9(b) illustrates the interferogram and corresponding surface sag of the woofer while Fig. 9(c) presents those of the tweeter. The final tested surface figure map is shown in Fig. 9(d), which is the departure from the best fit sphere. The total PV value of the Fig. 9(d) is 149.02 λ (94.30 µm) PV, in which about 68.29 λ (43.21 µm) is rotationally symmetric departure corrected by the PNO. The residual rotationally non-symmetric departure of about 80.73 λ (51.09 µm) PV value was covered by cascaded DMs. Note that the inherent aberration induced by the rotationally non-symmetric departure is about 150 λ, which is just about 4 times the sum of the surface sags of the woofer and tweeter (respective two reflections). Figure 9(e) refers to the surface figure error, which is the difference between Fig. 9(d) and the known designed shape. Figure 9(f) is the surface figure error by Zygo interferometer with the aberration free method. Note that the exit pupil of the tested configuration in aberration free method is a circle. Thus, the map difference of Figs. 9(e) and 9(f) should be extracted with mapping, which is presented in Fig. 9(g), with 0.0074 λ rms error.

 

Fig. 9. Test result of the off-axis paraboloidal mirror, (a) the interferogram variation process, (b) the interferograms and corresponding surface sag of the woofer, (c) the interferograms and corresponding surface sag of the tweeter, (d) the surface figure map by the adaptive interferometer, (e) surface figure error, (f) surface figure error by Zygo interferometer, (g) the difference of (e) and (f) with mapping.

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5. Discussion

5.1 Measurement accuracy

The test accuracy of these model-based testing methods [4,3133] is mainly determined by inaccurate element structure parameters and element position inconsistency with an actual system. The general modeling errors for element parameters and misalignments in a non-null interferometer had been reported in our previous works [3133], which induced less than 10−3 λ residual rms error in the final result after calibration. In the adaptive interferometer with cascaded DMs in this work, additional errors would be introduced by the DMs and TWPs. Of course, these errors were generally tested and store in advance for removing. The transmission and reflection errors of TWP are easily measured by the interferometer. The DM misalignments can be measured in the case of its flat surface [16]. Although the TDM method loss real-time monitoring to some extent, the rms uncertainty of the DM surface in 5 minutes would be less than 4 nm (0.006315 λ, λ=632.8 nm, temperature 25 °C) as mentioned above. The specific error values are listed in Table 1. The whole measurement accuracy is higher than SLM based interferometer because the phase control accuracy of the SLM now is about 0.033 λ [11].

Tables Icon

Table 1. The specific error values

5.2 System compaction

Currently, the most popular adaptive interferometer for freeform surfaces metrology is mainly based on DMs and SLMs. Compared with SLM based interferometers, the biggest weakness of DM based interferometer is that assistant monitoring devices [10,15,17] or configurations [16,17] are needed to promise the DM aberration compensation accuracy. Especially for cascade DMs in this work, the system would be more complicated due to two monitoring configurations or devices. In this paper, the problem is solved by the TWP based TDM method. No other configurations and devices but the interferometer self are necessary for DMs monitoring. The cascade DMs monitoring is afforded by the same interferometer in the time-division procedure. With this technique, the DMs based adaptive interferometer is free from the complexity of the system from now on. A compact configuration is thus achieved.

5.3 Coverage

Because the rotationally symmetric departure can be addressed by the RANL and PNO, we focus only on rotationally non-symmetric departure. The combination of SLMs is difficult because it is based on diffraction but the DMs combination is promising because it is just based on reflection and the superposition is easy to control. The commercial available DM (such as Alpao DM) has a better coverage performance than the SLM according to the reported Refs. [11,16]. The polarized design allowed twice reflections of the same DM surface and the aberration coverage is thus roughly doubled. Of course, the posture of the DM surface in the system is influential. The 45° reflection setup [10] would loss 10%-20% coverage compared with 90°reflection according to aberration forms. The off-axis testing configuration [15] may enlarge the coverage but is limited in several low-order aberrations such as astigmatism. Table 2 presents the specific aberration correction capacity between cascaded DMs and SLM in different aberrations forms. The data of SLM capacity comes from Fig. 2 in [4]. It is obvious that the cascaded DMs perform far better in coverage. In this interferometer, two cascaded DMs can generated maximum 160 µm (252 λ) aberrations, which enabled the large departure coverage (80 µm) with high actuators density (about 150 actuators at 25 mm valid aperture). It is hard to achieve with only one DM of the same amount of actuators and the same aperture because the DMs with high density actuators are generally designed for the higher-order aberrations correction but with smaller stroke. Even if there is a DM of the same capacity, the cost would be too high and the general interferometer cannot monitor the DM surface with such a large stroke as well. The cascaded DMs have addressed this issue because the cascaded DMs share the aberration compensation and the interferometer just needs to monitor the DM one by one with relative little stroke. It means that the cascade DMs technique is a preferred choice in pure interferometric configuration to test large departure.

Tables Icon

Table 2. Correction capacity comparison in different aberrations form

Note that our previous work [16] did not apply to commercial interferometers due to the DM monitoring configuration. In this work, this issue is addressed. The dual cascaded DMs (DCD) configuration with the polarized design would not change the beam polarized direction from the interferometer. It would support multiple DCD configurations, which would further expand the adaptive interferometer coverage in a simple supplement framework for general commercial interferometers, such as Zygo interferometers as shown in Fig. 10. Of course, the PNO or RANL would be easy to fit in, for rotationally symmetric departure compensation.

 

Fig. 10. Multiple DCD design for large departure test with general commercial interferometers.

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5.4 Limitation

The first limitation is the SPGD based algorithm. This optimization algorithm is not perfect in global search and thus leads to a local minimum [12]. Once the SPGD based algorithm sinks into a local minimum, the interferogram may keep containing unresolvable fringes. Therefore, the DMs would not be able to adequately share the aberrations. A more intelligent algorithm is needed such as the genetic algorithm [12]. In addition, the large surface deformation of DM would make the resulting interferogram deviating from the circular aperture, which would make the final ray tracing inaccuracy because the DM sags in the model are characterizing with Zernike polynomials. New orthogonal polynomials should be developed. Besides, the final coverage is not entirely dependent on the DMs’ capacity. Both the DM and the interferometer capacity limit the final departure coverage. In a traditional interferometer, the biggest limitation of the final test coverage is just the capacity of the interferometer detector. In our system, the final coverage is about four times the average range of DMs’ stroke. The interferometer only needs to cover the DMs’ stroke, which is about a quarter of the departure of the tested surface. That is the final coverage is thus enlarged about four times.

6 Conclusion

We report a compact adaptive interferometer to test the unknown freeform surface with a large departure. The cascaded DMs (woofer-tweeter) are employed to afford the aberrations. With the constrained decoupling control algorithm, the woofer and tweeter can averagely share the aberrations without coupling. Two DM surface monitoring is addressed by the TDM technique, which avoids separate monitoring devices and configurations and thus makes a compact configuration. It enlarges the rotationally non-symmetric departure coverage to maximum 120 µm, which is the largest coverage ever reported. Experiments validated the test for a freeform surface with 64.47 λ (40.8 µm) rotationally non-symmetric departure. Another experiment was designed for accuracy validation, which tested an off-axis paraboloid mirror with 80.73 λ (51.09 µm) rotationally non-symmetric departure and less than 0.01 λ rms error. It is larger than any tested rotationally non-symmetric departure ever reported in on-axis interferometric experiments. The cascaded DMs would be the competitive candidate for the huge departure test, only with a compact design.

Funding

National Natural Science Foundation of China (61705002, 61905001, 41875158); Natural Science Foundation of Anhui Province (1808085QF198, 1908085QF276); Opening Project of the Anhui Province Key Laboratory of Non-Destructive Evaluation (CGHBMWSJC05); Opening Project of the Key Laboratory of Astronomical Optics and Technology in Nanjing Institute of Astronomical Optics and Technology of the Chinese Academy of Sciences (CAS-KLAOT-KF201704); Doctoral Start-up Foundation of the Anhui University of Science and Technology (J01003208); ); National Program on Key Research and Development Project of China (2016YFC0301900, 2016YFC0302202).

Disclosures

The authors declare no conflicts of interest.

References

1. H. Liu, Q. Zhu, and Q. Hao, “Design of novel part-compensating lens used in aspheric testing,” Proc. SPIE 5253, 480–484 (2003). [CrossRef]  

2. J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007). [CrossRef]  

3. D. Liu, C. Tian, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009). [CrossRef]  

4. S. Xue, S. Chen, and G. Tie, “Flexible interferometric null testing for concave free-form surfaces using a hybrid refractive and diffractive variable null,” Opt. Lett. 44(9), 2294–2297 (2019). [CrossRef]  

5. S. Xue, S. Chen, and G. Tie, “Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres,” Opt. Express 26(24), 31172–31189 (2018). [CrossRef]  

6. M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010). [CrossRef]  

7. S. Chen, X. Shuai, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017). [CrossRef]  

8. J. Peng, D. Chen, and H. Guo, “Variable optical null based on a yawing CGH for measuring steep acylindrical surface,” Opt. Express 26(16), 20306–20318 (2018). [CrossRef]  

9. S. Xue, S. Chen, and Z. Fan, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018). [CrossRef]  

10. L. Huang, D. W. Kim, H. Choi, A. Anderson, L. R. Graves, and W. Zhao, “Adaptive interferometric null testing for unknown freeform optics metrology,” Opt. Lett. 41(23), 5539–5542 (2016). [CrossRef]  

11. S. Xue, S. Chen, and G. Tie, “Adaptive null interferometric test using spatial light modulator for free-form surfaces,” Opt. Express 27(6), 8414–8428 (2019). [CrossRef]  

12. S. Xue, W. Deng, and S. Chen, “Intelligence enhancement of the adaptive wavefront interferometer,” Opt. Express 27(8), 11084–11102 (2019). [CrossRef]  

13. R. Chaudhuri, J. Papa, and J. P. Rolland, “System design of a single-shot reconfigurable null test using a spatial light modulator for freeform metrology,” Opt. Lett. 44(8), 2000–2003 (2019). [CrossRef]  

14. C. Pruss and A. H. J. Tiziani, “Dynamic null lens for aspheric testing using a membrane mirror,” Opt. Commun. 233(1-3), 15–19 (2004). [CrossRef]  

15. K. Fuerschbach, K. P. Thompson, and J. P. Rolland, “Interferometric measurement of a concave, φ-polynomial, Zernike mirror,” Opt. Lett. 39(1), 18 (2014). [CrossRef]  

16. L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888 (2018). [CrossRef]  

17. L. Zhang, S. Zhou, D. Li, J. Li, and B. Yu, “Model-based adaptive non-null interferometry for freeform surface metrology,” Chin. Opt. Lett. 16(8), 081203 (2018). [CrossRef]  

18. C. Li, N. Sredar, K. M. Ivers, H. Queener, and J. Porter, “A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system,” Opt. Express 18(16), 16671–16684 (2010). [CrossRef]  

19. W. Zou and S. A. Burns, “High-accuracy wavefront control for retinal imaging with Adaptive-Influence-Matrix Adaptive Optics,” Opt. Express 17(22), 20167 (2009). [CrossRef]  

20. Alpao corporation, “Deformable Mirror datasheet,” https://www.alpao.com/adaptive-optics/deformable-mirrors.html (2019).

21. G. Liu, H. Yang, C. Rao, Y. Zhang, and W. Jiang, “Experimental verification of combinational-deformable-mirror for phase correction,” Chin. Opt. Lett. 5(10), 559–562 (2007).

22. R. J. Zawadzki, S. S. Choi, S. M. Jones, S. S. Oliver, and J. S. Werner, “Adaptive optics-optical coherence tomography: optimizing visualization of microscopic retinal structures in three dimensions,” J. Opt. Soc. Am. A 24(5), 1373 (2007). [CrossRef]  

23. X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express 20(20), 22143–22157 (2012). [CrossRef]  

24. M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15(10), 2745–2758 (1998). [CrossRef]  

25. H. Hofer, N. Sredar, H. Queener, C. Li, and J. Porter, “Wavefront sensorless adaptive optics ophthalmoscopy in the human eye,” Opt. Express 19(15), 14160–14171 (2011). [CrossRef]  

26. W. Zou, X. Qi, and S. A. Burns, “Wavefront-aberration sorting and correction for a dual-deformable-mirror adaptive-optics system,” Opt. Lett. 33(22), 2602 (2008). [CrossRef]  

27. M. Vorontsov, J. Riker, G. Carhart, V. S. R. Gudimetla, L. Beresnev, T. Weyrauch, and L. C. J. Roberts, “Deep turbulence effects compensation experiments with a cascaded adaptive optics system using a 3.63 m telescope,” Appl. Opt. 48(1), A47–A57 (2009). [CrossRef]  

28. M. Booth, T. Wilson, H. B. Sun, T. Ota, and S. Kawata, “Methods for the characterization of deformable membrane mirrors,” Appl. Opt. 44(24), 5131–5139 (2005). [CrossRef]  

29. W. Liu, L. Dong, P. Yang, X. Lei, H. Yan, and B. Xu, “A Zernike mode decomposition decoupling control algorithm for dual deformable mirrors adaptive optics system,” Opt. Express 21(20), 23885–23895 (2013). [CrossRef]  

30. U. Bitenc, “Software compensation method for achieving high stability of Alpao deformable mirrors,” Opt. Express 25(4), 4368–4381 (2017). [CrossRef]  

31. L. Zhang, D. Liu, T. Shi, Y. Yang, S. Chong, B. Ge, Y. Shen, and J. Bai, “Aspheric subaperture stitching based on system modeling,” Opt. Express 23(15), 19176–19188 (2015). [CrossRef]  

32. L. Zhang, D. Liu, T. Shi, Y. Yang, and Y. Shen, “Practical and accurate method for aspheric misalignment aberrations calibration in non-null interferometric testing,” Appl. Opt. 52(35), 8501–8511 (2013). [CrossRef]  

33. D. Liu, T. Shi, L. Zhang, Y. Yang, S. Chong, and Y. Shen, “Reverse optimization reconstruction of aspheric figure error in a non-null interferometer,” Appl. Opt. 53(24), 5538–5546 (2014). [CrossRef]  

References

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  • |

  1. H. Liu, Q. Zhu, and Q. Hao, “Design of novel part-compensating lens used in aspheric testing,” Proc. SPIE 5253, 480–484 (2003).
    [Crossref]
  2. J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007).
    [Crossref]
  3. D. Liu, C. Tian, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
    [Crossref]
  4. S. Xue, S. Chen, and G. Tie, “Flexible interferometric null testing for concave free-form surfaces using a hybrid refractive and diffractive variable null,” Opt. Lett. 44(9), 2294–2297 (2019).
    [Crossref]
  5. S. Xue, S. Chen, and G. Tie, “Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres,” Opt. Express 26(24), 31172–31189 (2018).
    [Crossref]
  6. M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
    [Crossref]
  7. S. Chen, X. Shuai, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
    [Crossref]
  8. J. Peng, D. Chen, and H. Guo, “Variable optical null based on a yawing CGH for measuring steep acylindrical surface,” Opt. Express 26(16), 20306–20318 (2018).
    [Crossref]
  9. S. Xue, S. Chen, and Z. Fan, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018).
    [Crossref]
  10. L. Huang, D. W. Kim, H. Choi, A. Anderson, L. R. Graves, and W. Zhao, “Adaptive interferometric null testing for unknown freeform optics metrology,” Opt. Lett. 41(23), 5539–5542 (2016).
    [Crossref]
  11. S. Xue, S. Chen, and G. Tie, “Adaptive null interferometric test using spatial light modulator for free-form surfaces,” Opt. Express 27(6), 8414–8428 (2019).
    [Crossref]
  12. S. Xue, W. Deng, and S. Chen, “Intelligence enhancement of the adaptive wavefront interferometer,” Opt. Express 27(8), 11084–11102 (2019).
    [Crossref]
  13. R. Chaudhuri, J. Papa, and J. P. Rolland, “System design of a single-shot reconfigurable null test using a spatial light modulator for freeform metrology,” Opt. Lett. 44(8), 2000–2003 (2019).
    [Crossref]
  14. C. Pruss and A. H. J. Tiziani, “Dynamic null lens for aspheric testing using a membrane mirror,” Opt. Commun. 233(1-3), 15–19 (2004).
    [Crossref]
  15. K. Fuerschbach, K. P. Thompson, and J. P. Rolland, “Interferometric measurement of a concave, φ-polynomial, Zernike mirror,” Opt. Lett. 39(1), 18 (2014).
    [Crossref]
  16. L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888 (2018).
    [Crossref]
  17. L. Zhang, S. Zhou, D. Li, J. Li, and B. Yu, “Model-based adaptive non-null interferometry for freeform surface metrology,” Chin. Opt. Lett. 16(8), 081203 (2018).
    [Crossref]
  18. C. Li, N. Sredar, K. M. Ivers, H. Queener, and J. Porter, “A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system,” Opt. Express 18(16), 16671–16684 (2010).
    [Crossref]
  19. W. Zou and S. A. Burns, “High-accuracy wavefront control for retinal imaging with Adaptive-Influence-Matrix Adaptive Optics,” Opt. Express 17(22), 20167 (2009).
    [Crossref]
  20. Alpao corporation, “Deformable Mirror datasheet,” https://www.alpao.com/adaptive-optics/deformable-mirrors.html (2019).
  21. G. Liu, H. Yang, C. Rao, Y. Zhang, and W. Jiang, “Experimental verification of combinational-deformable-mirror for phase correction,” Chin. Opt. Lett. 5(10), 559–562 (2007).
  22. R. J. Zawadzki, S. S. Choi, S. M. Jones, S. S. Oliver, and J. S. Werner, “Adaptive optics-optical coherence tomography: optimizing visualization of microscopic retinal structures in three dimensions,” J. Opt. Soc. Am. A 24(5), 1373 (2007).
    [Crossref]
  23. X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express 20(20), 22143–22157 (2012).
    [Crossref]
  24. M. A. Vorontsov and V. P. Sivokon, “Stochastic parallel-gradient-descent technique for high-resolution wave-front phase-distortion correction,” J. Opt. Soc. Am. A 15(10), 2745–2758 (1998).
    [Crossref]
  25. H. Hofer, N. Sredar, H. Queener, C. Li, and J. Porter, “Wavefront sensorless adaptive optics ophthalmoscopy in the human eye,” Opt. Express 19(15), 14160–14171 (2011).
    [Crossref]
  26. W. Zou, X. Qi, and S. A. Burns, “Wavefront-aberration sorting and correction for a dual-deformable-mirror adaptive-optics system,” Opt. Lett. 33(22), 2602 (2008).
    [Crossref]
  27. M. Vorontsov, J. Riker, G. Carhart, V. S. R. Gudimetla, L. Beresnev, T. Weyrauch, and L. C. J. Roberts, “Deep turbulence effects compensation experiments with a cascaded adaptive optics system using a 3.63 m telescope,” Appl. Opt. 48(1), A47–A57 (2009).
    [Crossref]
  28. M. Booth, T. Wilson, H. B. Sun, T. Ota, and S. Kawata, “Methods for the characterization of deformable membrane mirrors,” Appl. Opt. 44(24), 5131–5139 (2005).
    [Crossref]
  29. W. Liu, L. Dong, P. Yang, X. Lei, H. Yan, and B. Xu, “A Zernike mode decomposition decoupling control algorithm for dual deformable mirrors adaptive optics system,” Opt. Express 21(20), 23885–23895 (2013).
    [Crossref]
  30. U. Bitenc, “Software compensation method for achieving high stability of Alpao deformable mirrors,” Opt. Express 25(4), 4368–4381 (2017).
    [Crossref]
  31. L. Zhang, D. Liu, T. Shi, Y. Yang, S. Chong, B. Ge, Y. Shen, and J. Bai, “Aspheric subaperture stitching based on system modeling,” Opt. Express 23(15), 19176–19188 (2015).
    [Crossref]
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2019 (4)

2018 (5)

2017 (2)

S. Chen, X. Shuai, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

U. Bitenc, “Software compensation method for achieving high stability of Alpao deformable mirrors,” Opt. Express 25(4), 4368–4381 (2017).
[Crossref]

2016 (1)

2015 (1)

2014 (2)

2013 (2)

2012 (1)

2011 (1)

2010 (2)

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

C. Li, N. Sredar, K. M. Ivers, H. Queener, and J. Porter, “A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system,” Opt. Express 18(16), 16671–16684 (2010).
[Crossref]

2009 (3)

2008 (1)

2007 (3)

2005 (1)

2004 (1)

C. Pruss and A. H. J. Tiziani, “Dynamic null lens for aspheric testing using a membrane mirror,” Opt. Commun. 233(1-3), 15–19 (2004).
[Crossref]

2003 (1)

H. Liu, Q. Zhu, and Q. Hao, “Design of novel part-compensating lens used in aspheric testing,” Proc. SPIE 5253, 480–484 (2003).
[Crossref]

1998 (1)

Anderson, A.

Bai, J.

Bauer, M.

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Beresnev, L.

Bitenc, U.

Booth, M.

Burns, S. A.

Carhart, G.

Chaudhuri, R.

Chen, D.

Chen, S.

Choi, H.

Choi, S. S.

Chong, S.

Dai, Y.

S. Chen, X. Shuai, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

Deng, W.

Devries, G.

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Dong, L.

Fan, Z.

Fleig, J.

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Forbes, G.

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Fuerschbach, K.

Ge, B.

Graves, L. R.

Greivenkamp, J. E.

J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007).
[Crossref]

Gudimetla, V. S. R.

Guo, H.

Hao, Q.

H. Liu, Q. Zhu, and Q. Hao, “Design of novel part-compensating lens used in aspheric testing,” Proc. SPIE 5253, 480–484 (2003).
[Crossref]

He, T.

Hofer, H.

Huang, L.

Ivers, K. M.

Jiang, W.

Jones, S. M.

Kawata, S.

Kim, D. W.

Kulawiec, A.

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Lei, X.

Li, C.

Li, D.

Li, J.

Li, S.

S. Chen, X. Shuai, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

Liu, D.

Liu, G.

Liu, H.

H. Liu, Q. Zhu, and Q. Hao, “Design of novel part-compensating lens used in aspheric testing,” Proc. SPIE 5253, 480–484 (2003).
[Crossref]

Liu, W.

Liu, Y.

Miladinovich, D.

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Murphy, P.

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

Oliver, S. S.

Ota, T.

Papa, J.

Peng, J.

Porter, J.

Pruss, C.

C. Pruss and A. H. J. Tiziani, “Dynamic null lens for aspheric testing using a membrane mirror,” Opt. Commun. 233(1-3), 15–19 (2004).
[Crossref]

Qi, X.

Queener, H.

Rao, C.

Riker, J.

Roberts, L. C. J.

Rolland, J. P.

Shen, Y.

Shi, T.

Shuai, X.

S. Chen, X. Shuai, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

Sivokon, V. P.

Sredar, N.

Sullivan, J. J.

J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007).
[Crossref]

Sun, H. B.

Thompson, K. P.

Tian, C.

D. Liu, C. Tian, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[Crossref]

Tie, G.

Tiziani, A. H. J.

C. Pruss and A. H. J. Tiziani, “Dynamic null lens for aspheric testing using a membrane mirror,” Opt. Commun. 233(1-3), 15–19 (2004).
[Crossref]

Tricard, M.

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
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D. Liu, C. Tian, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
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Appl. Opt. (4)

Chin. Opt. Lett. (2)

CIRP Ann. (1)

M. Tricard, A. Kulawiec, M. Bauer, G. Devries, J. Fleig, G. Forbes, D. Miladinovich, and P. Murphy, “Subaperture stitching interferometry of high-departure aspheres by incorporating a variable optical null,” CIRP Ann. 59(1), 547–550 (2010).
[Crossref]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

C. Pruss and A. H. J. Tiziani, “Dynamic null lens for aspheric testing using a membrane mirror,” Opt. Commun. 233(1-3), 15–19 (2004).
[Crossref]

Opt. Express (13)

S. Xue, S. Chen, and G. Tie, “Adaptive null interferometric test using spatial light modulator for free-form surfaces,” Opt. Express 27(6), 8414–8428 (2019).
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S. Xue, W. Deng, and S. Chen, “Intelligence enhancement of the adaptive wavefront interferometer,” Opt. Express 27(8), 11084–11102 (2019).
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C. Li, N. Sredar, K. M. Ivers, H. Queener, and J. Porter, “A correction algorithm to simultaneously control dual deformable mirrors in a woofer-tweeter adaptive optics system,” Opt. Express 18(16), 16671–16684 (2010).
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W. Zou and S. A. Burns, “High-accuracy wavefront control for retinal imaging with Adaptive-Influence-Matrix Adaptive Optics,” Opt. Express 17(22), 20167 (2009).
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J. Peng, D. Chen, and H. Guo, “Variable optical null based on a yawing CGH for measuring steep acylindrical surface,” Opt. Express 26(16), 20306–20318 (2018).
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S. Xue, S. Chen, and Z. Fan, “Adaptive wavefront interferometry for unknown free-form surfaces,” Opt. Express 26(17), 21910–21928 (2018).
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S. Xue, S. Chen, and G. Tie, “Near-null interferometry using an aspheric null lens generating a broad range of variable spherical aberration for flexible test of aspheres,” Opt. Express 26(24), 31172–31189 (2018).
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H. Hofer, N. Sredar, H. Queener, C. Li, and J. Porter, “Wavefront sensorless adaptive optics ophthalmoscopy in the human eye,” Opt. Express 19(15), 14160–14171 (2011).
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X. Lei, S. Wang, H. Yan, W. Liu, L. Dong, P. Yang, and B. Xu, “Double-deformable-mirror adaptive optics system for laser beam cleanup using blind optimization,” Opt. Express 20(20), 22143–22157 (2012).
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W. Liu, L. Dong, P. Yang, X. Lei, H. Yan, and B. Xu, “A Zernike mode decomposition decoupling control algorithm for dual deformable mirrors adaptive optics system,” Opt. Express 21(20), 23885–23895 (2013).
[Crossref]

U. Bitenc, “Software compensation method for achieving high stability of Alpao deformable mirrors,” Opt. Express 25(4), 4368–4381 (2017).
[Crossref]

L. Zhang, D. Liu, T. Shi, Y. Yang, S. Chong, B. Ge, Y. Shen, and J. Bai, “Aspheric subaperture stitching based on system modeling,” Opt. Express 23(15), 19176–19188 (2015).
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L. Zhang, S. Zhou, D. Li, Y. Liu, T. He, B. Yu, and J. Li, “Pure adaptive interferometer for free form surfaces metrology,” Opt. Express 26(7), 7888 (2018).
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Opt. Laser Technol. (1)

S. Chen, X. Shuai, Y. Dai, and S. Li, “Subaperture stitching test of convex aspheres by using the reconfigurable optical null,” Opt. Laser Technol. 91, 175–184 (2017).
[Crossref]

Opt. Lett. (5)

Proc. SPIE (3)

H. Liu, Q. Zhu, and Q. Hao, “Design of novel part-compensating lens used in aspheric testing,” Proc. SPIE 5253, 480–484 (2003).
[Crossref]

J. J. Sullivan and J. E. Greivenkamp, “Design of partial nulls for testing of fast aspheric surfaces,” Proc. SPIE 6671, 66710W (2007).
[Crossref]

D. Liu, C. Tian, and Y. Zhuo, “Non-null interferometric aspheric testing with partial null lens and reverse optimization,” Proc. SPIE 7426, 74260M (2009).
[Crossref]

Other (1)

Alpao corporation, “Deformable Mirror datasheet,” https://www.alpao.com/adaptive-optics/deformable-mirrors.html (2019).

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Figures (10)

Fig. 1.
Fig. 1. System layout of the compact interferometer.
Fig. 2.
Fig. 2. TDM process for DMs monitoring (a) woofer monitoring, (b) middle process, (c) tweeter monitoring.
Fig. 3.
Fig. 3. (a) Wavefront coupling illustration, (b)-(d) traditional decoupling, (e)-(g) desired decoupling.
Fig. 4.
Fig. 4. Flow chart of cascaded DMs decoupling with sharing in adaptive interferometer.
Fig. 5.
Fig. 5. Interferograms in the whole experiment process, (a) uncompensated interferogram, (b) interferogram segmentation, (c)-(f) are interferograms in the process of SPGD decoupling, (h) and (j) are the fringes characterizing the woofer and tweeter surfaces after decoupling, (i) and (k) are those after aberration sharing step, which come from (h) and (j), respectively, (g) is the final interferogram characterizing tested freeform surface.
Fig. 6.
Fig. 6. The performance comparison in the first step between woofer only, tweeter only, woofer-tweeter without decoupling and woofer-tweeter with decoupling, (a) the residual pixel number of unresolvable regions in the mentioned four cases with iterations, (b) the resulted interferograms in the four cases.
Fig. 7.
Fig. 7. DM surfaces stability in different initial stroke in five minutes at 25 °C,(a)-(c) are three repetitive experimental results of the woofer, (d)-(f) are three repetitive experimental results of the tweeter.
Fig. 8.
Fig. 8. Final surface figure map and contour line, (a) surface figure map by our adaptive interferometer, (b) contour lines comparison with Taylor Hobson profilometer. (c) and (d) are surface figure maps tested in the other two repetitive experiments.
Fig. 9.
Fig. 9. Test result of the off-axis paraboloidal mirror, (a) the interferogram variation process, (b) the interferograms and corresponding surface sag of the woofer, (c) the interferograms and corresponding surface sag of the tweeter, (d) the surface figure map by the adaptive interferometer, (e) surface figure error, (f) surface figure error by Zygo interferometer, (g) the difference of (e) and (f) with mapping.
Fig. 10.
Fig. 10. Multiple DCD design for large departure test with general commercial interferometers.

Tables (2)

Tables Icon

Table 1. The specific error values

Tables Icon

Table 2. Correction capacity comparison in different aberrations form

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

J i  = N ( I u n r e s o v i )
δ J i = J ( V w i + δ V w i ; V t i + δ V t i ) J ( V w i ; V t i ) ,
{ V w i + 1 = V w i γ w δ J δ V w i V t i + 1 = V t i γ t δ J δ V t i ,
{ A w = B w V w A t = B t V t .
{ A w = [ a w 2 , a w 3 , , a w q , 0 , 0 , , 0 ] T A t = [ 0 , 0 , , 0 , a t ( q + 1 ) , a t ( q + 2 ) , , a t 37 ] T ,
δ J i = J ( A w i + δ A w i ; A t i + δ A t i ) J ( A w i ; A t i ) ,
{ A w i + 1 = A w i  -  γ w δ J δ A w i A t i + 1 = A t i  -  γ t δ J δ A t i ,
{ A w = [ a w 2 , a w 3 , , a w q , 0 , 0 , , 0 ] A t = [ a t 2 , a t 3 , , a t q , a t ( q + 1 ) , , a t 37 ] .
{ A w i + 1 = A w i  +  γ w δ J δ A w i A t i + 1 = A t i γ t δ J δ A t i  =  A t i γ w δ J δ A w i .
J i  = PV ( I i )  +  | PV ( A w i Z ) PV ( A t i Z ) | ,

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