Abstract

Precision wavefront sensing and interferometry are essential in many fields of industry and fundamental research. Characterization of semiconductor devices, optics in lithography systems, and biologic features of living cells all require measurement resolution at the nanometer level. The field of high-contrast imaging in space-based astronomy has pushed wavefront sensing requirements to a new regime with current and future concepts requiring sensitivity on the order of 10 pm. Techniques to achieve this level of precision have been demonstrated, but require large, expensive instrumentation with custom light sources, and therefore do not provide a solution for in-space operation. Here we demonstrate experimentally the ability to detect picometer-level wavefront errors at spatial frequencies limited only by the pixel count of the sampling detector using a simple, inexpensive method. The system is based on the Zernike wavefront sensor (ZWFS) that implements the phase-contrast technique whereby the DC portion of an optical wavefront is phase-shifted with respect to its higher spatial frequency components. In our demonstration, a highly repeatable deformable mirror is used to introduce phase variations into an optical path. We readily sense 60 pm RMS changes in wavefront errors with the ZWFS system with measurement repeatability on the order of 0.6 pm. This technique is an enabling technology for future astronomy missions; however, there are widespread applications to many other fields requiring high-precision interferometry.

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

1. INTRODUCTION

A metrology system capable of detecting and characterizing sub-wavelength surface geometry is an area of critical importance to many fields. Feature sizes for integrated chips are now at the nanometer level, requiring imaging systems with even finer resolution [13]. In biology, the desire to resolve fine features in living cells has led to the development of advanced microscopy techniques [47]. However, astronomy and astrophysics instruments often drive the development of precision optical metrology systems. Prime examples of this are the experiments designed to detect gravitational waves from astrophysical sources using laser metrology systems with sensitivities on the order of ${10^{- 17}}\text{m}$ [8,9]. These techniques are point-to-point measurements that measure the change in optical path between two end points. Multiple point-to-point gauges can be combined to form an optical truss to extract limited spatial information of an optical system, such as rigid-body motion of individual elements [10,11]. Both of these systems rely upon stabilized laser sources to perform the measurement.

More recent astronomical mission concepts aimed at the direct detection of extra-solar planets (exoplanets) through high-contrast imaging have driven requirements on optical metrology systems even further. These concepts require instruments capable of sensing changes in two-dimensional wavefronts with picometer-level sensitivity over very long integration times [1217]. Techniques to make surface metrology measurements at the sub-nanometer level have been demonstrated in the laboratory, but require dynamic excitation of the optics under test to perform the measurement [18,19]. Other methods require a combination of image plane focus diversity and long integration times [20] or specially engineered systems with sources fine-tuned for their temporal coherence [3]. Our approach differs from these in that we demonstrate a passive method of sensing that is compatible with space-based instruments and has no reliance on custom sources or instrumental diversity.

In the present study, we demonstrate the ability to measure two-dimensional wavefront changes at the picometer level. Our method is based on the Zernike wavefront sensor (ZWFS), which implements the phase-contrast concept [21,22]. The technique, initially applied to the field of microscopy modulates phase delays in an optical path to intensity variations at an exit pupil. In microscopy, this enabled direct detection of transparent cellular structures immersed in a fluid medium. Here, we adopt the technique to quantitatively assess spatially varying phase errors in an optical path. This technique was first proposed for astronomy applications in 1975 as a method to characterize atmospheric seeing errors in ground-based telescopes [23]. The technique has since been implemented in several ground-based instruments including those with adaptive optics systems [2426] and those performing high-contrast imaging [2729]. It was also shown to be the most photon efficient wavefront sensing method across all spatial frequencies [30]. These applications have resulted in ZWFS demonstrations at the single-digit nanometer level. The push to picometer-level precision and repeatability has been made for space-based applications, with laboratory-based ZWFS demonstrations at the 10 pm RMS level [31]. However, the implementation in these demonstrations requires substantial calibration and is limited to low spatial frequency measurements. In the present study, we are concerned with the precision limits of the ZWFS and its ability to measure picometer-level wavefront errors at all spatial scales, without the use of a priori calibration steps.

2. ZWFS THEORY

The ZWFS operates under the fundamental principle of interferometry where two electric fields ${u_A} = A{e^{i{\phi _A}}}$ and ${u_B} = B{e^{i{\phi _B}}}$ are interfered at a common plane. The intensity image sampled at this plane is given by $I = {| {{u_A} + {u_B}} |^2}$ producing the following relationship for the relative phase difference between the two fields:

$${\phi _A} - {\phi _B} = \frac{\pi}{2} + \arcsin \left[\frac{{I - {I_A} - {I_B}}}{{2\sqrt {{I_A}{I_B}}}}\right],$$
where ${I_A}$ and ${I_B}$ are the independently measured intensity images of the two electric fields. This is the operating principle of a Mach–Zehnder interferometer where typically the electric field amplitude of each beam is spatially uniform. In the ZWFS concept, the measurement and reference beams are common mode at an entrance pupil, with spatially dependent amplitude and phase components,
$$u({\bf x}) = AP({\bf x}){e^{i\phi ({\bf x})}},$$
where $A$ is the average electric field amplitude and $P({\bf x})$ is the spatially varying pupil function. To generate the “reference” and “measurement” beams, the electric field is brought to a focal plane and subsequently spatially filtered, producing low- and high-pass components, respectively. Physically, this is implemented by placing a “phase plate”—an etched glass dimple—at the focal plane of the beam, as shown in Fig. 1. The spatial extent of the dimple is on the order of the diameter of the point spread function (PSF), thereby acting only over the low spatial frequency component of the field. The etched depth is defined such that it induces a $\pi /2$ phase delay over this portion of the field. Therefore, the two interfering beams are (1) the low-pass spatially filtered, and phase-shifted reference beam, ${u_B}({\bf x})$, and (2) the high-pass spatially filtered beam, ${u_A}({\bf x})$. Mathematically, this can be formulated as
$${u_B}({\bf x}) = L[u({\bf x})]{e^{i\pi /2}} = i Ab({\bf x}){e^{i\beta ({\bf x})}},$$
$${u_A}({\bf x}) = H[u({\bf x})] = AP({\bf x}){e^{i\phi ({\bf x})}} - Ab({\bf x}){e^{i\beta ({\bf x})}},$$
where $L$ and $H$ represent the low- and high-pass spatial filtering operators, respectively (see Section 1, Supplement 1). The complex low-pass portion of the beam, $Ab({\bf x}){e^{i\beta ({\bf x})}}$, is determined through a model of the phase plate interaction with the PSF. Upon recombination at a subsequent pupil plane, and by sampling an image of the interference pattern, $I({\bf x})$, the phase of the incoming beam can be determined by Eq. (1) as follows:
$$\phi ({\bf x}) - \beta ({\bf x}) = \frac{\pi}{4} + \arcsin \left[\frac{{I({\bf x}) - {I_{P({\bf x})}} - 2{I_{b({\bf x})}}}}{{2\sqrt {2{I_{P({\bf x})}}{I_{b({\bf x})}}}}}\right],$$
where ${I_{P({\bf x})}}$ is a pupil intensity measurement in the absence of spatial filtering (i.e., without the ZWFS dimple in place), and ${I_{b({\bf x})}}$ is the modeled intensity-equivalent of the low-pass field. Here, $\beta ({\bf x})$ represents the phase component of the low-pass portion of the beam. For a system with high Strehl ratio, as well as a sufficiently small dimple diameter relative to the size of the PSF, this term will be small in magnitude and low spatial frequency in nature. If the ZWFS is used in differential mode where changes in wavefront are of interest (i.e., $\Delta \phi ({\bf x})$), and the change in wavefront is small in order to preserve the existing Strehl ratio, then $\beta ({\bf x})$ can be approximated as a constant and will become negligible in the measurement.
 figure: Fig. 1.

Fig. 1. Schematic of the ZWFS testbed; see text for details of operation. The interaction of the ZWFS dimple and point spread function (PSF) is emphasized in the detailed view. Equipment details: IFO, 4D Inc. 4020 HSPS Interferometer; DM, Boston Micromachines Multi-3.5 Deformable Mirror; ZWFS/D, Etched Fused Silica, Silios Inc.; ZWFS/C, Andor iXon Ultra 897 EMCCD; L1–L4, lenses; Newport Inc. PAC0XX (various); FM, fold mirror: Thorlabs PF2011-F01; BS, beam splitter: Thorlabs BSW27.

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In evaluation of Eq. (5), it is apparent that the resulting intensity profile produced by phase variations is sinusoidal in nature, and thus multi-valued. Therefore, as phase errors in the input pupil exceed a certain range, the interpretation of the intensity signal becomes ambiguous, which ultimately limits the dynamic range of the ZWFS. For a system incorporating a single dimple with a phase delay of $\pi /2$, this range is limited to total phase errors of ${-}\pi /4$ to $3\pi /4$. It is possible to overcome these limitations through fringe-unwrapping techniques; however, this was not considered in our study.

A distinguishing aspect of this approach to previous efforts is that it does not require a forward response of the system in order to measure the wavefront. In particular, the results of [31] are achieved by first characterizing the change in the ZWFS signal for a known change in the system aberration. Specifically, this requires measuring a change of pupil intensity for a known, controlled change in phase. These “forward” intensity measurements for a given aberration (also called the interaction matrix) are then inverted to give the control matrix. With the control matrix, the reverse process is enabled whereby intensity changes at the detector are used to estimate the magnitude of each individual mode. This approach has two primary limitations: (1) it requires an active system in order to assemble the interaction matrix, and (2) it limits the ZWFS measurements to the same number of discrete spatial modes used to generate the control matrix, and is thus incapable of measuring arbitrary phase aberrations.

The approach presented here does not have these limitations. Our reconstruction algorithm provides the flexibility to reconstruct the phase without the need for a forward model, only knowledge of the PSF size relative to that of the ZWFS dimple. Both of these parameters can be measured directly. This enables phase reconstruction on a pixel-by-pixel basis and, therefore, the ability to sense high spatial frequency modes limited only by the number of illuminated pupil pixels on the detector.

3. EXPERIMENTAL SETUP

A schematic of the ZWFS testbed is shown in Fig. 1. A commercially available Twyman–Green interferometer (IFO) is used to project a planar wavefront ($\lambda = 632.8\,\text{nm}$) into the system. The beam is resized by a pupil relay (L1 and L2) illuminating the optic under test at a conjugate plane. Upon reflection, surface figure changes on the test optic will be encoded as spatially varying phase delays in the returning wavefront, which are readily measured by the IFO using classical interferometry. A beam splitter (BS) in the returning path redirects half of the light to a separate optical path in the system where it is turned by a fold mirror (FM) and brought to a focus at a plane by L3 containing the ZWFS dimple (ZWFS/D). A camera (ZWFS/C) placed at a re-imaged pupil generated by L4 is then used to measure the intensity signal inherent to the ZWFS. The IFO and ZWFS/C are triggered using the same clock, and can thus acquire frames synchronously. The testbed, while not in vacuum, was located in a thermally controlled laboratory environment stable to 25 mK during the data acquisition. Additionally, while the experiment was performed using monochromatic light, the broadband performance of the ZWFS has been studied extensively and is shown to maintain sensing performance [27,32].

The ZWFS dimple consists of a 1 mm thick fused silica substrate with an etched circular dimple at its center. Following previous efforts [27], the diameter of the dimple was specified to $1.06\,\lambda /D$, or $8.05\,\unicode{x00B5}\text{m}$ for a ${f}/{12}$ beam and 632.8 nm wavelength source. The depth of the mask was specified to produce a $\pi /2$ phase delay in this region. As the mask works in transmission, the index of refraction of the glass must be considered. Assuming ${n} = {1.457}$ for fused silica, the depth was specified to 347 nm for the current testbed. The mask was manufactured by Silios Technologies (Peynier, France) using reactive ion etching techniques. Confocal microscopy of the dimple was performed on the as-manufactured masks confirming a diameter and depth of $8.1\,\unicode{x00B5}\text{m}$ and 350 nm, respectively.

The optic under test is a microelectromechanical-systems (MEMS)-based deformable mirror (DM) with a clear aperture of 4.4 mm. The actuators of the DM modulate the height of its reflective surface by means of electrostatic actuation. Each actuator is individually addressable, and thus a variety of spatially varying figure changes can be imposed on the mirror surface. The DM in the testbed has 140 actuators arranged in a square grid (${12} \times {12}$, with absent corners). DMs of this type, but at a higher actuator count (i.e., ${48} \times {48}$), are baselined for future coronagraph designs, which establishes a strong connection to future high-contrast imaging missions [16,17]. Each actuator in the testbed has a nominal full stroke of approximately $2.4\,\unicode{x00B5}\text{m}$, and is driven by a 14 bit voltage driver, providing a least-significant-bit (LSB) of approximately 150 pm in peak-to-valley surface figure deformation. It should be noted that while this setup is applied to the millimeter size aperture of the DM, the technique only requires an image of a surface at a conjugate plane and is thus agnostic to aperture size. For example, in astronomy applications, the ZWFS can be used to detect phase variations of meter-class optics in addition to those downstream in the optical path. For microscopy, high resolution measurements can be obtained by imaging sub-millimeter specimens. Lateral resolution is therefore limited only by the wavelength of light projected into the system and the pixel count of the imaging detector.

 figure: Fig. 2.

Fig. 2. Schematic of data acquisition process. An individual measurement in either the “ON” or “OFF” command (CMD) comprises an average of ${M}$ individual frames. The differential wavefront estimate is the difference of these two measurements averaged over ${ N}$ cycles.

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As the testbed is subject to ambient air turbulence, randomly varying phase variations will corrupt the underlying measurement of the DM surface. As such, measurements were performed in a differential mode where wavefront changes are obtained by subtracting two absolute measurements. This also allows for the cancellation of non-common path errors between the ZWFS and IFO. For wavefront changes above 1–2 nm RMS, a single measurement or a small number of averages is sufficient to accurately capture the signal. To observe sub-nanometer-level effects, additional averaging steps are required, and it is imperative to minimize the time between nominally flat and actuated measurements. The data acquisition process is shown in Fig. 2. The voltage command of each actuator on the DM is commanded by a square wave. For individual actuation cycles, a collection of M intensity images is acquired by the camera in either the ON or OFF states. These intensity measurements are then averaged for the ${{n}^{\text{th}}}$ cycle as follows:

Tables Icon

Table 1. Camera and Imaging Parameters of the ZWFS Testbed

 figure: Fig. 3.

Fig. 3. Measurement validation of ZWFS measurements using a commercial Twyman–Green interferometer (IFO). Wavefront error maps are shown for the ZWFS and IFO at three discrete points and highlight the good agreement between the two techniques (colormap units in nm). Deviations occur above wavefronts errors of approximately 20 nm RMS due to intensity inversions in the ZWFS signal (highlighted with dashed circles).

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$$I_n^i = \frac{1}{M}\sum\limits_{m = 1}^M I_m^i,$$
where $i$ is an index indicating that the measurement was acquired while the DM was in the ON or OFF state. The phase of the wavefront in either state is then established using the methods described above, and the differential measurement of the ${{n}^{\text{th}}}$ cycle is as follows:
$$\Delta {\phi _n} = \phi (I_n^{\text{ON}}) - \phi (I_n^{\text{OFF}}).$$

This is repeated a total of ${N}$ times throughout the experiment to establish the average differential phase measurement,

$$\Delta {\phi _N} = \frac{1}{N}\sum\limits_{n = 1}^N \Delta {\phi _n}.$$

The rate at which measurements of this type can be performed is limited by the camera frame rate and the mechanical response time of the DM ($75\,\unicode{x00B5} \text{s}$). For the current testbed, the maximum camera frame rate, F, was 237 Hz, achieved through ${2} \times {2}$ binning of pixels resulting in an exposure time of approximately 4.2 ms. The IFO source provided a nominal photon-flux of approximately 42,200,000 photons per pixel per second at the detector during one pupil image (without the ZWFS dimple in place). Additional parameters associated with the testbed and camera can be found in Table 1.

4. RESULTS

The ZWFS measurements were first compared to measurements from the IFO for wavefront error amplitudes greater than 1 nm as a validation of the measurement accuracy. A second-order approximation of Eq. (5) was used to convert normalized intensity images to phase estimates [27]. Linear phase components (i.e., tip and tilt) were also removed from the measurements, and a Gaussian image filter was applied to reduce pixel noise. Figure 3 displays this comparison whereby the DM was commanded to produce a mid-spatial frequency checkerboard pattern by applying alternating positive/negative voltages to the individual actuators. This pattern was chosen as it represents the highest spatial frequency and lowest amplitude mode that can be commanded by the DM (i.e., $\pm 1\,\,\text{bit}$). Lower order modes are readily sensed by the ZWFS; however, their reconstruction at small phase amplitudes is limited by the discretization of the DM actuators and voltage commands (see Section 3, Supplement 1). From the results presented in Fig. 3, good agreement is apparent between the IFO and ZWFS for wavefront changes below 20 nm RMS. Total wavefront errors (static $+$ change) beyond this range lead to non-linear behavior in the ZWFS signal, as identified by Eq. (5). These areas are highlighted by dashed circles in the largest stroke ZWFS map (point C) in Fig. 3. At these locations, the local phase change induced by the DM combined with static optical errors in the testbed (i.e., due to upstream optics, misalignments, etc.) exceed the ${-}\pi /4$ value (${-}{79}\;\text{nm}$), and intensity inversions begin to appear. Therefore, the ZWFS estimation begins to underestimate the overall wavefront error as shown in the trend.

As a test of the sensitivity floor of the ZWFS, the checkerboard pattern was produced through a $\pm {1}$ bit change to the individual actuators of the DM. This pattern was dithered temporally at a rate of 10 Hz, and ${ M} = {5}$ frames were acquired for each of the actuated and unactuated measurements. The frames were captured at least 12 ms before and after the actuator command, which is 2 orders of magnitude larger than the mechanical response time of the DM. Approximately 340 phase measurements were acquired in this fashion, and various levels of measurement averaging were performed (${ N} = {1}$ to 340). Figure 4(a) displays the measured wavefront error change for an increasing number of averages. The solid curve represents the spatial RMS mean of the measurement, whereas the shaded area corresponds to the standard deviation over 70 independent trials. The corresponding wavefront error maps for ${N} = {1}$, 10, 50, and 340 averages are also shown in Fig. 4(b). It is apparent that without averaging (i.e., ${N} = {1}$), the underlying checkerboard pattern on the DM is masked by atmospheric effects and the noise properties of the camera. The mean value of these fluctuations is on the order of 225 pm RMS with a standard deviation of approximately 130 pm—larger than the expected signal from the DM. As ${N}$ is increased, these stochastic effects begin to average out, and the checkerboard pattern is revealed. For ${N}\; \gt \;{100}$, the estimate converges to ${\sim}60\,\text{pm}$ RMS. When repeated over the 70 independent trials, the standard deviation in measured wavefront error reaches sub-picometer values for ${N}\; \gt \;{150}$, demonstrating the extreme precision of the ZWFS measurement technique. It should also be noted that the acquisition time required to reach this level of precision is approximately 15 s, which is 2–3 orders of magnitude faster than previous efforts [20]. This speed can be increased further if desired through the use of a higher frame rate camera.

 figure: Fig. 4.

Fig. 4. (a) Mean (solid curve) and standard deviation (shaded area) of detected wavefront over 70 independent trials. The insert highlights the sub-picometer repeatability of the system. The dashed line represents the mean determined from a numerical model of the experiment, in the absence of atmospheric effects. (b) Wavefront error maps representing one trial of a ZWFS measurement for varying levels of averaging.

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 figure: Fig. 5.

Fig. 5. Standard deviation of RMS wavefront difference over 70 independent trials. The blue line is experimental data, and the orange line is from numerical simulation. The measurement precision increases with the number of averages, ${N}$. Because the experiment was performed in air, over small averages, the precision is dominated by the time-varying atmosphere. Note that over large averages, the slope of the experimental variance more closely matches that of the numerical simulations.

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The inherent performance limits of the ZWFS in our experiment can be estimated based upon the random noise sources in the testbed. Here, we neglect the effects of external sources such as thermal variations, testbed vibrations, or atmospheric turbulence, and instead focus on the random processes of the photodetection process. In particular, shot (photon) noise, ${\sigma _s}$, detector dark current, ${\sigma _d}$, and detector read noise, ${\sigma _r}$, are included. These noise sources are irreducible for fixed integration times, and therefore set a lower limit for the phase reconstruction process. Their respective contribution to the total error, ${\sigma _t}$, can be expressed as a sum of variances as follows:

$${\sigma _t} = \sqrt {\sigma _s^2 + \sigma _d^2 + \sigma _r^2} .$$

These effects were included in a numerical model simulating the wavefront propagation through the ZWFS testbed (see Section 2, Supplement 1). The parameters of the model were matched to those in the experiments, and a 60 pm RMS checkerboard pattern of the same spatial frequency was simulated in the data. The mean wavefront error estimate predicted by this method is plotted against the experimental results in Fig. 4. These predicted results represent the quadrature sum of the simulated phase with the modeled error sources. While it is evident that the error sources dominate for low values of ${N}$, the accuracy of the technique is increased simply by averaging more frames. After approximately 100 averaged data sets, the numerical prediction is within 1 pm of the input wavefront.

 figure: Fig. 6.

Fig. 6. Numerical simulation demonstrating the dynamic range of the ZWFS for varying levels of (a) tilt and (b) focus. Top row, phase reconstruction estimate, with errors apparent for amplitudes greater than 160 nm PV. The corresponding motion of the PSF relative to the ZWFS dimple is also noted. Bottom row, differential phase estimate when a 240 pm PV checkerboard pattern is applied on top of the dominate tilt/focus terms. Errors in the reconstruction are highlighted by dashed circles. All colormap units in nm.

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The measurement repeatability, represented here as the standard deviation of the wavefront change between the actuated/unactuated states for 70 independent trials, is plotted in Fig. 5. In the absence of noise sources other than those associated with the photon-detection process, the numerical model predicts the sub-picometer repeatability for a single measurement (${N} = {1}$). Upon averaging, the standard deviation in the numerical estimate follows a power law with an exponent of approximately 0.45, approaching the fundamental limit of 0.5 set by photon noise. In comparison, the repeatability of a single experimental measurement is approximately ${130} \times$ greater in magnitude than that of the numerical results, indicating that the external noise sources such as atmospheric effects dominate in this region. The effect of averaging reduces these effects drastically for low values of ${N}$, as evident by the initial slope of the line in Fig. 5. However, as the number of averages is increased (${N}\; \gt \;{150}$), the nature of the experimental data begins to more closely match that of the numerical simulations. In this regime, the experiment repeatability is now limited by the photon-detection process.

The same numerical model was used to demonstrate the dynamic range of the ZWFS and its effect on the reconstructed phase. Here noise sources are neglected as the input phase aberrations are at the nanometer level. In particular, the ability to sense linear and quadratic phase terms (i.e., tip/tilt and focus) was studied. These represent either phase components of the input electric field upstream of the ZWFS dimple, or a misalignment of the dimple relative to the PSF location in the lateral or longitudinal directions, respectively. Figure 6 displays the result of this study. The input phase, either pure tilt or focus [Fig. 6(a) and Fig. 6(b), respectively], is depicted in the first column. The top row of columns 2–4 of each sub-figure depicts the reconstructed phase using Eq. (5) at increasing phase amplitudes. At 80 nm peak-to-valley ($\pm {40}\;\text{nm}$), the ZWFS is able to reconstruct the input phase terms. However, as the phase is increased in amplitude, approaching the nominal ${-}\pi /4$ (${-}{79}\;\text{nm}$) dynamic range of the ZWFS, errors in the reconstruction become evident. For tilt, an apparent fringe begins to appear on the left hand portion of the aperture. For focus, a spherical term is introduced in the reconstruction as evident by the phase estimate in the central portion of the aperture. We note that these errors occur primarily on the portion of the field exhibiting negative phase relative to the mean. The reconstruction of the positive phase regions are generally unaffected due to the asymmetric nature of Eq. (5); however, small errors are introduced due to the interaction of the PSF with the ZWFS dimple (see Section 1, Supplement 1).

For each phase amplitude, the shift of the PSF relative to the dimple is shown for an optical setup with parameters matching our experiment. Micron-level alignment in the lateral direction is required to ensure that the linear phase terms are within the dynamic range of the ZWFS. However, this is within the capability of commercially available linear stages (as was performed in the experiment). The sensitivity to misalignment in the longitudinal direction is less stringent by a factor of approximately 100.

The ability to detect small perturbations in phase in addition to these large static aberrations was also studied. This was performed by incorporating a small checkerboard phase pattern of magnitude comparable to that generated in the experiments, in addition to the baseline tilt/focus. The differential measurement of the perturbed (i.e., tilt/focus + checkerboard) and unperturbed (i.e., tilt/focus only) fields is shown in the bottom row of Figs. 6(a) and 6(b). It is apparent that when the total phase amplitude is within the dynamic range of the ZWFS, the checkerboard pattern is well reconstructed. However, as the amplitude of tilt/focus is increased, errors in the differential phase estimate are introduced. These errors are highlighted for a single region on the checkerboard pattern by a dashed circle. In particular, considering the results shown in Fig. 6(a), a region of true negative phase transitions to a coma-like shape in the reconstruction when 160 nm of tilt is introduced. Once increased further to 240 nm, the phase reconstruction erroneously estimates this region to be positive in differential phase. This behavior matches that shown in Fig. 3 for our experiments.

5. DISCUSSION

In this study, we have demonstrated the ability to detect changes in optical wavefronts with sensitivity at the 60 pm RMS level and 0.6 pm repeatability. This was achieved through the use of a ZWFS, which implements the phase-contrast technique. We believe the combination of measurement sensitivity, in-plane spatial resolution, and acquisition speed demonstrated in this study is the first of its kind. The technique was verified through a simultaneous measurement performed by a commercial interferometer, showing strong agreement between the two methods for nanometer-level wavefront errors.

The ZWFS has been demonstrated as a capable wavefront sensing technique in prior efforts; however, the implementation presented here differs as it is agnostic to the spatial content of the phase aberrations to be measured. In particular, we note that previous ZWFS demonstrations [31] required the use of a “forward approach” where phase errors of known magnitude and spatial frequency are injected into the system and the resulting intensity variation recorded. This process allows for the characterization of those specific modes used in the calibration, but precludes the measurement of phase aberrations beyond its spatial cutoff. The method presented here, using the full reconstruction algorithm discussed above, does not depend on this process. In that sense, this technique is a true passive sensor, capable of measuring arbitrary phase aberrations and is limited only by the pixel count of the imaging detector.

Furthermore, this study demonstrates that the fundamental precision of the ZWFS technique is set only by irreducible noise sources, namely those associated with the photon-detection process. We note that this property of the ZWFS has been predicted through numerical models [30], whereas here it is demonstrated experimentally. This was achieved through a series of averaging steps to eliminate external stochastic effects such as air turbulence and testbed disturbances. Once reduced, the precision and repeatability of the ZWFS measurements matched that of numerical predictions, which captured the noise properties of the imaging camera.

While the precision of the ZWFS technique is evident, it is limited in dynamic range. As the intensity profile inherent to the ZWFS signal is multi-valued, ambiguity in the wavefront reconstruction process can occur once local phase errors surpass the $\pi /4$ range. In fact, the reconstruction algorithm presented here will invert the sign of the estimated phase if the magnitude is sufficiently large. This was demonstrated in the experimental measurements as well as in numerical simulations. Therefore, it is necessary to ensure that the total wavefront error directly upstream of the ZWFS dimple is below the dynamic range of the system. It is also required to ensure adequate placement of the ZWFS dimple with respect to the PSF as linear and quadratic phase terms can cause errors in subsequent measurements. We note that it is possible to extend the dynamic range by introducing phase diversity at the ZWFS dimple plane. This can be achieved by utilizing two dimples with unique phase shifts (i.e., ${\pm}\pi /2$), or through the use of polarization-selective devices [33]; however, these methods present implementation challenges.

Finally, we note that the performance of the ZWFS technique shown in this study has direct applications to the field of high-contrast imaging for exoplanet detection. Future astronomy missions with coronagraph instruments require differential wavefront sensors with 10 pm RMS precision over a broad range of spatial frequencies. While further studies are required, the combination of measurement precision and operational simplicity ensures that the ZWFS technique is a compelling method to achieve this goal. However, the technique can be applied to a wide variety of additional fields requiring high-precision interferometric measurements.

Funding

Jet Propulsion Laboratory.

Acknowledgment

We thank Frank Loya for his help in establishing the ZWFS testbed and David Redding for technical advice throughout the effort. This research was carried out at the Jet Propulsion Laboratory, California Institute of Technology under a contract with the National Aeronautics and Space Administration (NASA).

Disclosures

The authors declare no conflicts of interest.

 

See Supplement 1 for supporting content.

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6. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30, 468–470 (2005). [CrossRef]  

7. A. Canales-Benavides, Y. Zhuo, A. M. Amitrano, M. Kim, R. I. Hernandez-Aranda, P. S. Carney, and M. Schnell, “Accessible quantitative phase imaging in confocal microscopy with sinusoidal-phase synthetic optical holography,” Appl. Opt. 58, A55–A64 (2019). [CrossRef]  

8. A. Abramovici, W. Althouse, J. Camp, D. Durance, J. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. Raab, R. L. Savage Jr., D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996). [CrossRef]  

9. T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the LISA gravitational reference sensor,” Class. Quantum Gravity 26, 085008 (2009). [CrossRef]  

10. F. Zhao, “Development of high-precision laser heterodyne metrology gauges,” Proc. SPIE 5634, 247–259 (2005). [CrossRef]  

11. P. G. Halverson, D. R. Johnson, A. C. Kuhnert, S. B. Shaklan, and R. Spero, “Multichannel averaging phasemeter for picometer precision laser metrology,” Proc. SPIE 3740, 646–649 (1999). [CrossRef]  

12. S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 62651I (2006). [CrossRef]  

13. G. Ruane, D. Mawet, B. Mennesson, J. B. Jewell, and S. B. Shaklan, “Vortex coronagraphs for the habitable exoplanet imaging mission concept: theoretical performance and telescope requirements,” J. Astron. Telesc. Instrum. Syst. 4, 015004 (2018). [CrossRef]  

14. B. Mennesson, S. Gaudi, S. Seager, K. Cahoy, S. Domagal-Goldman, L. Feinberg, O. Guyon, J. Kasdin, C. Marois, D. Mawet, M. Tamura, D. Mouillet, T. Prusti, A. Quirrenbach, T. Robinson, L. Rogers, P. Scowen, R. Somerville, K. Stapelfeldt, D. Stern, M. Still, M. Turnbull, J. Booth, A. Kiessling, G. Kuan, and K. Warfield, “The habitable exoplanet (HabEx) imaging mission: preliminary science drivers and technical requirements,” Proc. SPIE 9904, 99040L (2016). [CrossRef]  

15. M. R. Bolcar, S. Aloezos, V. T. Bly, C. Collins, J. Crooke, C. D. Dressing, L. Fantano, L. D. Feinberg, K. France, G. Gochar, Q. Gong, J. E. Hylan, A. Jones, I. Linares, M. Postman, L. Pueyo, A. Roberge, L. Sacks, S. Tompkins, and G. West, “The large UV/optical/infrared surveyor (LUVOIR): decadal mission concept design update,” Proc. SPIE 10398, 1039809 (2017). [CrossRef]  

16. HabEx Study Team, “The habitable exoplanet observatory (HabEx) mission concept study final report,” 2019, https://www.jpl.nasa.gov/habex/pdf/HabEx-Final-Report-Public-Release.pdf.

17. The LUVOIR Team, “The large UV optical infrared surveyor (LUVOIR) final report,” 2019, https://asd.gsfc.nasa.gov/luvoir/resources/docs/LUVOIR_FinalReport_2019-08-26.pdf.

18. L. Pogorelyuk and N. J. Kasdin, “Dark hole maintenance and a posteriori intensity estimation in the presence of speckle drift in a high-contrast space coronagraph,” Astrophys. J. 873, 95–107 (2019). [CrossRef]  

19. B. Saif, D. Chaney, P. Greenfield, M. Bluth, K. Van Gorkom, K. Smith, J. Bluth, L. Feinberg, J. C. Wyant, M. North-Morris, and R. Keski-Kuha, “Measurement of picometer-scale mirror dynamics,” Appl. Opt. 56, 6457–6465 (2017). [CrossRef]  

20. J. J. Green, S. A. Basinger, D. Cohen, A. F. Niessner, D. C. Redding, S. B. Shaklan, and J. T. Trauger, “Demonstration of extreme wavefront sensing performance on the TPF high-contrast imaging testbed,” Proc. SPIE 5170, 38–48 (2003). [CrossRef]  

21. F. Zernike, “Diffraction theory of the knife-edge test and its improved form the phase contrast method,” Mon. Not. R. Astron. Soc. 94, 377–384 (1934). [CrossRef]  

22. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955). [CrossRef]  

23. R. H. Dicke, “Phase-contrast detection of telescope seeing errors and their correction,” Astrophys. J. 198, 605–615 (1975). [CrossRef]  

24. E. E. Bloemhof and J. K. Wallace, “Phase-contrast wavefront sensing for adaptive optics,” Proc. SPIE 5553, 159–169 (2004). [CrossRef]  

25. J. K. Wallace, S. Rao, R. M. Jensen-Clem, and G. Serabyn, “Phase-shifting Zernike interferometer wavefront sensor,” Proc. SPIE 8126, 81260F (2011). [CrossRef]  

26. J. K. Wallace, S. Crawford, F. Loya, and J. Moore, “A phase-shifting Zernike wavefront sensor for the Palomar P3K adaptive optics system,” Proc. SPIE 8447, 84472K (2012). [CrossRef]  

27. M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013). [CrossRef]  

28. M. N’Diaye, A. Vigan, K. Dohlen, J. F. Sauvage, A. Caillat, A. Costille, J. H. V. Girard, J. L. Beuzit, T. Fusco, P. Blanchard, J. Le Merrer, D. Le Mignant, F. Madec, G. Moreaux, D. Mouillet, P. Puget, and G. Zins, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. II. Concept validation with ZELDA on VLT/SPHERE,” Astron. Astrophys. 592, A79 (2016). [CrossRef]  

29. A. Vigan, M. N’Diaye, K. Dohlen, J. F. Sauvage, J. Milli, G. Zins, C. Petit, Z. Wahhaj, F. Cantalloube, A. Caillat, A. Costille, J. Le Merrer, A. Carlotti, J. L. Beuzit, and D. Mouillet, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. III. On-sky validation in VLT/SPHERE,” Astron. Astrophys. 629, A11 (2019). [CrossRef]  

30. O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592–614 (2005). [CrossRef]  

31. F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” J. Astron. Telesc. Instrum. Syst. 2, 011021 (2016). [CrossRef]  

32. X. Wang, F. Shi, and J. Wallace, “Zernike wavefront sensor (ZWFS) development for low order wavefront sensing (LOWFS),” Proc. SPIE 9904, 990463 (2016). [CrossRef]  

33. D. S. Doelman, F. F. Auer, M. J. Escuti, and F. Snik, “Simultaneous phase and amplitude aberration sensing with a liquid-crystal vector-Zernike phase mask,” Opt. Lett. 44, 17–20 (2019). [CrossRef]  

References

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  1. N. G. Orji, M. Badaroglu, B. M. Barnes, C. Beitia, B. D. Bunday, U. Celano, R. J. Kline, M. Neisser, Y. Obeng, and A. Vladar, “Metrology for the next generation of semiconductor devices,” Nat. Electron. 1, 532–554 (2018).
    [Crossref]
  2. B. Wu and A. Kumar, “Extreme ultraviolet lithography: a review,” J. Vac. Sci. Technol. B 25, 1743–1761 (2007).
    [Crossref]
  3. G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316–328 (2002).
    [Crossref]
  4. Y. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Diffraction phase and fluorescence microscopy,” Opt. Express 14, 8263–8268 (2006).
    [Crossref]
  5. S. W. Hell, M. Dyba, and S. Jakobs, “Concepts for nanoscale resolution in fluorescence microscopy,” Curr. Opin. Neurobiol. 14, 599–609 (2004).
    [Crossref]
  6. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30, 468–470 (2005).
    [Crossref]
  7. A. Canales-Benavides, Y. Zhuo, A. M. Amitrano, M. Kim, R. I. Hernandez-Aranda, P. S. Carney, and M. Schnell, “Accessible quantitative phase imaging in confocal microscopy with sinusoidal-phase synthetic optical holography,” Appl. Opt. 58, A55–A64 (2019).
    [Crossref]
  8. A. Abramovici, W. Althouse, J. Camp, D. Durance, J. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
    [Crossref]
  9. T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the LISA gravitational reference sensor,” Class. Quantum Gravity 26, 085008 (2009).
    [Crossref]
  10. F. Zhao, “Development of high-precision laser heterodyne metrology gauges,” Proc. SPIE 5634, 247–259 (2005).
    [Crossref]
  11. P. G. Halverson, D. R. Johnson, A. C. Kuhnert, S. B. Shaklan, and R. Spero, “Multichannel averaging phasemeter for picometer precision laser metrology,” Proc. SPIE 3740, 646–649 (1999).
    [Crossref]
  12. S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 62651I (2006).
    [Crossref]
  13. G. Ruane, D. Mawet, B. Mennesson, J. B. Jewell, and S. B. Shaklan, “Vortex coronagraphs for the habitable exoplanet imaging mission concept: theoretical performance and telescope requirements,” J. Astron. Telesc. Instrum. Syst. 4, 015004 (2018).
    [Crossref]
  14. B. Mennesson, S. Gaudi, S. Seager, K. Cahoy, S. Domagal-Goldman, L. Feinberg, O. Guyon, J. Kasdin, C. Marois, D. Mawet, M. Tamura, D. Mouillet, T. Prusti, A. Quirrenbach, T. Robinson, L. Rogers, P. Scowen, R. Somerville, K. Stapelfeldt, D. Stern, M. Still, M. Turnbull, J. Booth, A. Kiessling, G. Kuan, and K. Warfield, “The habitable exoplanet (HabEx) imaging mission: preliminary science drivers and technical requirements,” Proc. SPIE 9904, 99040L (2016).
    [Crossref]
  15. M. R. Bolcar, S. Aloezos, V. T. Bly, C. Collins, J. Crooke, C. D. Dressing, L. Fantano, L. D. Feinberg, K. France, G. Gochar, Q. Gong, J. E. Hylan, A. Jones, I. Linares, M. Postman, L. Pueyo, A. Roberge, L. Sacks, S. Tompkins, and G. West, “The large UV/optical/infrared surveyor (LUVOIR): decadal mission concept design update,” Proc. SPIE 10398, 1039809 (2017).
    [Crossref]
  16. HabEx Study Team, “The habitable exoplanet observatory (HabEx) mission concept study final report,” 2019, https://www.jpl.nasa.gov/habex/pdf/HabEx-Final-Report-Public-Release.pdf .
  17. The LUVOIR Team, “The large UV optical infrared surveyor (LUVOIR) final report,” 2019, https://asd.gsfc.nasa.gov/luvoir/resources/docs/LUVOIR_FinalReport_2019-08-26.pdf .
  18. L. Pogorelyuk and N. J. Kasdin, “Dark hole maintenance and a posteriori intensity estimation in the presence of speckle drift in a high-contrast space coronagraph,” Astrophys. J. 873, 95–107 (2019).
    [Crossref]
  19. B. Saif, D. Chaney, P. Greenfield, M. Bluth, K. Van Gorkom, K. Smith, J. Bluth, L. Feinberg, J. C. Wyant, M. North-Morris, and R. Keski-Kuha, “Measurement of picometer-scale mirror dynamics,” Appl. Opt. 56, 6457–6465 (2017).
    [Crossref]
  20. J. J. Green, S. A. Basinger, D. Cohen, A. F. Niessner, D. C. Redding, S. B. Shaklan, and J. T. Trauger, “Demonstration of extreme wavefront sensing performance on the TPF high-contrast imaging testbed,” Proc. SPIE 5170, 38–48 (2003).
    [Crossref]
  21. F. Zernike, “Diffraction theory of the knife-edge test and its improved form the phase contrast method,” Mon. Not. R. Astron. Soc. 94, 377–384 (1934).
    [Crossref]
  22. F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
    [Crossref]
  23. R. H. Dicke, “Phase-contrast detection of telescope seeing errors and their correction,” Astrophys. J. 198, 605–615 (1975).
    [Crossref]
  24. E. E. Bloemhof and J. K. Wallace, “Phase-contrast wavefront sensing for adaptive optics,” Proc. SPIE 5553, 159–169 (2004).
    [Crossref]
  25. J. K. Wallace, S. Rao, R. M. Jensen-Clem, and G. Serabyn, “Phase-shifting Zernike interferometer wavefront sensor,” Proc. SPIE 8126, 81260F (2011).
    [Crossref]
  26. J. K. Wallace, S. Crawford, F. Loya, and J. Moore, “A phase-shifting Zernike wavefront sensor for the Palomar P3K adaptive optics system,” Proc. SPIE 8447, 84472K (2012).
    [Crossref]
  27. M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
    [Crossref]
  28. M. N’Diaye, A. Vigan, K. Dohlen, J. F. Sauvage, A. Caillat, A. Costille, J. H. V. Girard, J. L. Beuzit, T. Fusco, P. Blanchard, J. Le Merrer, D. Le Mignant, F. Madec, G. Moreaux, D. Mouillet, P. Puget, and G. Zins, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. II. Concept validation with ZELDA on VLT/SPHERE,” Astron. Astrophys. 592, A79 (2016).
    [Crossref]
  29. A. Vigan, M. N’Diaye, K. Dohlen, J. F. Sauvage, J. Milli, G. Zins, C. Petit, Z. Wahhaj, F. Cantalloube, A. Caillat, A. Costille, J. Le Merrer, A. Carlotti, J. L. Beuzit, and D. Mouillet, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. III. On-sky validation in VLT/SPHERE,” Astron. Astrophys. 629, A11 (2019).
    [Crossref]
  30. O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592–614 (2005).
    [Crossref]
  31. F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” J. Astron. Telesc. Instrum. Syst. 2, 011021 (2016).
    [Crossref]
  32. X. Wang, F. Shi, and J. Wallace, “Zernike wavefront sensor (ZWFS) development for low order wavefront sensing (LOWFS),” Proc. SPIE 9904, 990463 (2016).
    [Crossref]
  33. D. S. Doelman, F. F. Auer, M. J. Escuti, and F. Snik, “Simultaneous phase and amplitude aberration sensing with a liquid-crystal vector-Zernike phase mask,” Opt. Lett. 44, 17–20 (2019).
    [Crossref]

2019 (4)

A. Canales-Benavides, Y. Zhuo, A. M. Amitrano, M. Kim, R. I. Hernandez-Aranda, P. S. Carney, and M. Schnell, “Accessible quantitative phase imaging in confocal microscopy with sinusoidal-phase synthetic optical holography,” Appl. Opt. 58, A55–A64 (2019).
[Crossref]

L. Pogorelyuk and N. J. Kasdin, “Dark hole maintenance and a posteriori intensity estimation in the presence of speckle drift in a high-contrast space coronagraph,” Astrophys. J. 873, 95–107 (2019).
[Crossref]

A. Vigan, M. N’Diaye, K. Dohlen, J. F. Sauvage, J. Milli, G. Zins, C. Petit, Z. Wahhaj, F. Cantalloube, A. Caillat, A. Costille, J. Le Merrer, A. Carlotti, J. L. Beuzit, and D. Mouillet, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. III. On-sky validation in VLT/SPHERE,” Astron. Astrophys. 629, A11 (2019).
[Crossref]

D. S. Doelman, F. F. Auer, M. J. Escuti, and F. Snik, “Simultaneous phase and amplitude aberration sensing with a liquid-crystal vector-Zernike phase mask,” Opt. Lett. 44, 17–20 (2019).
[Crossref]

2018 (2)

G. Ruane, D. Mawet, B. Mennesson, J. B. Jewell, and S. B. Shaklan, “Vortex coronagraphs for the habitable exoplanet imaging mission concept: theoretical performance and telescope requirements,” J. Astron. Telesc. Instrum. Syst. 4, 015004 (2018).
[Crossref]

N. G. Orji, M. Badaroglu, B. M. Barnes, C. Beitia, B. D. Bunday, U. Celano, R. J. Kline, M. Neisser, Y. Obeng, and A. Vladar, “Metrology for the next generation of semiconductor devices,” Nat. Electron. 1, 532–554 (2018).
[Crossref]

2017 (2)

B. Saif, D. Chaney, P. Greenfield, M. Bluth, K. Van Gorkom, K. Smith, J. Bluth, L. Feinberg, J. C. Wyant, M. North-Morris, and R. Keski-Kuha, “Measurement of picometer-scale mirror dynamics,” Appl. Opt. 56, 6457–6465 (2017).
[Crossref]

M. R. Bolcar, S. Aloezos, V. T. Bly, C. Collins, J. Crooke, C. D. Dressing, L. Fantano, L. D. Feinberg, K. France, G. Gochar, Q. Gong, J. E. Hylan, A. Jones, I. Linares, M. Postman, L. Pueyo, A. Roberge, L. Sacks, S. Tompkins, and G. West, “The large UV/optical/infrared surveyor (LUVOIR): decadal mission concept design update,” Proc. SPIE 10398, 1039809 (2017).
[Crossref]

2016 (4)

M. N’Diaye, A. Vigan, K. Dohlen, J. F. Sauvage, A. Caillat, A. Costille, J. H. V. Girard, J. L. Beuzit, T. Fusco, P. Blanchard, J. Le Merrer, D. Le Mignant, F. Madec, G. Moreaux, D. Mouillet, P. Puget, and G. Zins, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. II. Concept validation with ZELDA on VLT/SPHERE,” Astron. Astrophys. 592, A79 (2016).
[Crossref]

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” J. Astron. Telesc. Instrum. Syst. 2, 011021 (2016).
[Crossref]

X. Wang, F. Shi, and J. Wallace, “Zernike wavefront sensor (ZWFS) development for low order wavefront sensing (LOWFS),” Proc. SPIE 9904, 990463 (2016).
[Crossref]

B. Mennesson, S. Gaudi, S. Seager, K. Cahoy, S. Domagal-Goldman, L. Feinberg, O. Guyon, J. Kasdin, C. Marois, D. Mawet, M. Tamura, D. Mouillet, T. Prusti, A. Quirrenbach, T. Robinson, L. Rogers, P. Scowen, R. Somerville, K. Stapelfeldt, D. Stern, M. Still, M. Turnbull, J. Booth, A. Kiessling, G. Kuan, and K. Warfield, “The habitable exoplanet (HabEx) imaging mission: preliminary science drivers and technical requirements,” Proc. SPIE 9904, 99040L (2016).
[Crossref]

2013 (1)

M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
[Crossref]

2012 (1)

J. K. Wallace, S. Crawford, F. Loya, and J. Moore, “A phase-shifting Zernike wavefront sensor for the Palomar P3K adaptive optics system,” Proc. SPIE 8447, 84472K (2012).
[Crossref]

2011 (1)

J. K. Wallace, S. Rao, R. M. Jensen-Clem, and G. Serabyn, “Phase-shifting Zernike interferometer wavefront sensor,” Proc. SPIE 8126, 81260F (2011).
[Crossref]

2009 (1)

T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the LISA gravitational reference sensor,” Class. Quantum Gravity 26, 085008 (2009).
[Crossref]

2007 (1)

B. Wu and A. Kumar, “Extreme ultraviolet lithography: a review,” J. Vac. Sci. Technol. B 25, 1743–1761 (2007).
[Crossref]

2006 (2)

Y. Park, G. Popescu, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Diffraction phase and fluorescence microscopy,” Opt. Express 14, 8263–8268 (2006).
[Crossref]

S. B. Shaklan, J. J. Green, and D. M. Palacios, “The terrestrial planet finder coronagraph optical surface requirements,” Proc. SPIE 6265, 62651I (2006).
[Crossref]

2005 (3)

P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30, 468–470 (2005).
[Crossref]

F. Zhao, “Development of high-precision laser heterodyne metrology gauges,” Proc. SPIE 5634, 247–259 (2005).
[Crossref]

O. Guyon, “Limits of adaptive optics for high-contrast imaging,” Astrophys. J. 629, 592–614 (2005).
[Crossref]

2004 (2)

E. E. Bloemhof and J. K. Wallace, “Phase-contrast wavefront sensing for adaptive optics,” Proc. SPIE 5553, 159–169 (2004).
[Crossref]

S. W. Hell, M. Dyba, and S. Jakobs, “Concepts for nanoscale resolution in fluorescence microscopy,” Curr. Opin. Neurobiol. 14, 599–609 (2004).
[Crossref]

2003 (1)

J. J. Green, S. A. Basinger, D. Cohen, A. F. Niessner, D. C. Redding, S. B. Shaklan, and J. T. Trauger, “Demonstration of extreme wavefront sensing performance on the TPF high-contrast imaging testbed,” Proc. SPIE 5170, 38–48 (2003).
[Crossref]

2002 (1)

G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316–328 (2002).
[Crossref]

1999 (1)

P. G. Halverson, D. R. Johnson, A. C. Kuhnert, S. B. Shaklan, and R. Spero, “Multichannel averaging phasemeter for picometer precision laser metrology,” Proc. SPIE 3740, 646–649 (1999).
[Crossref]

1996 (1)

A. Abramovici, W. Althouse, J. Camp, D. Durance, J. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[Crossref]

1975 (1)

R. H. Dicke, “Phase-contrast detection of telescope seeing errors and their correction,” Astrophys. J. 198, 605–615 (1975).
[Crossref]

1955 (1)

F. Zernike, “How I discovered phase contrast,” Science 121, 345–349 (1955).
[Crossref]

1934 (1)

F. Zernike, “Diffraction theory of the knife-edge test and its improved form the phase contrast method,” Mon. Not. R. Astron. Soc. 94, 377–384 (1934).
[Crossref]

Abramovici, A.

A. Abramovici, W. Althouse, J. Camp, D. Durance, J. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
[Crossref]

Aloezos, S.

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N. G. Orji, M. Badaroglu, B. M. Barnes, C. Beitia, B. D. Bunday, U. Celano, R. J. Kline, M. Neisser, Y. Obeng, and A. Vladar, “Metrology for the next generation of semiconductor devices,” Nat. Electron. 1, 532–554 (2018).
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B. Mennesson, S. Gaudi, S. Seager, K. Cahoy, S. Domagal-Goldman, L. Feinberg, O. Guyon, J. Kasdin, C. Marois, D. Mawet, M. Tamura, D. Mouillet, T. Prusti, A. Quirrenbach, T. Robinson, L. Rogers, P. Scowen, R. Somerville, K. Stapelfeldt, D. Stern, M. Still, M. Turnbull, J. Booth, A. Kiessling, G. Kuan, and K. Warfield, “The habitable exoplanet (HabEx) imaging mission: preliminary science drivers and technical requirements,” Proc. SPIE 9904, 99040L (2016).
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A. Abramovici, W. Althouse, J. Camp, D. Durance, J. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
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Marois, C.

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Mawet, D.

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G. Ruane, D. Mawet, B. Mennesson, J. B. Jewell, and S. B. Shaklan, “Vortex coronagraphs for the habitable exoplanet imaging mission concept: theoretical performance and telescope requirements,” J. Astron. Telesc. Instrum. Syst. 4, 015004 (2018).
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F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” J. Astron. Telesc. Instrum. Syst. 2, 011021 (2016).
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J. K. Wallace, S. Crawford, F. Loya, and J. Moore, “A phase-shifting Zernike wavefront sensor for the Palomar P3K adaptive optics system,” Proc. SPIE 8447, 84472K (2012).
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A. Vigan, M. N’Diaye, K. Dohlen, J. F. Sauvage, J. Milli, G. Zins, C. Petit, Z. Wahhaj, F. Cantalloube, A. Caillat, A. Costille, J. Le Merrer, A. Carlotti, J. L. Beuzit, and D. Mouillet, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. III. On-sky validation in VLT/SPHERE,” Astron. Astrophys. 629, A11 (2019).
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B. Mennesson, S. Gaudi, S. Seager, K. Cahoy, S. Domagal-Goldman, L. Feinberg, O. Guyon, J. Kasdin, C. Marois, D. Mawet, M. Tamura, D. Mouillet, T. Prusti, A. Quirrenbach, T. Robinson, L. Rogers, P. Scowen, R. Somerville, K. Stapelfeldt, D. Stern, M. Still, M. Turnbull, J. Booth, A. Kiessling, G. Kuan, and K. Warfield, “The habitable exoplanet (HabEx) imaging mission: preliminary science drivers and technical requirements,” Proc. SPIE 9904, 99040L (2016).
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M. N’Diaye, A. Vigan, K. Dohlen, J. F. Sauvage, A. Caillat, A. Costille, J. H. V. Girard, J. L. Beuzit, T. Fusco, P. Blanchard, J. Le Merrer, D. Le Mignant, F. Madec, G. Moreaux, D. Mouillet, P. Puget, and G. Zins, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. II. Concept validation with ZELDA on VLT/SPHERE,” Astron. Astrophys. 592, A79 (2016).
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M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
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N. G. Orji, M. Badaroglu, B. M. Barnes, C. Beitia, B. D. Bunday, U. Celano, R. J. Kline, M. Neisser, Y. Obeng, and A. Vladar, “Metrology for the next generation of semiconductor devices,” Nat. Electron. 1, 532–554 (2018).
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G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316–328 (2002).
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Obeng, Y.

N. G. Orji, M. Badaroglu, B. M. Barnes, C. Beitia, B. D. Bunday, U. Celano, R. J. Kline, M. Neisser, Y. Obeng, and A. Vladar, “Metrology for the next generation of semiconductor devices,” Nat. Electron. 1, 532–554 (2018).
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N. G. Orji, M. Badaroglu, B. M. Barnes, C. Beitia, B. D. Bunday, U. Celano, R. J. Kline, M. Neisser, Y. Obeng, and A. Vladar, “Metrology for the next generation of semiconductor devices,” Nat. Electron. 1, 532–554 (2018).
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Patterson, K.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” J. Astron. Telesc. Instrum. Syst. 2, 011021 (2016).
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Paul, B.

M. N’Diaye, K. Dohlen, T. Fusco, and B. Paul, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor,” Astron. Astrophys. 555, A94 (2013).
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Peters, A.

T. Schuldt, M. Gohlke, D. Weise, U. Johann, A. Peters, and C. Braxmaier, “Picometer and nanoradian optical heterodyne interferometry for translation and tilt metrology of the LISA gravitational reference sensor,” Class. Quantum Gravity 26, 085008 (2009).
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A. Vigan, M. N’Diaye, K. Dohlen, J. F. Sauvage, J. Milli, G. Zins, C. Petit, Z. Wahhaj, F. Cantalloube, A. Caillat, A. Costille, J. Le Merrer, A. Carlotti, J. L. Beuzit, and D. Mouillet, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. III. On-sky validation in VLT/SPHERE,” Astron. Astrophys. 629, A11 (2019).
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G. E. Sommargren, D. W. Phillion, M. A. Johnson, N. Q. Nguyen, A. Barty, F. J. Snell, D. R. Dillon, and L. S. Bradsher, “100-picometer interferometry for EUVL,” Proc. SPIE 4688, 316–328 (2002).
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Poberezhskiy, I.

F. Shi, K. Balasubramanian, R. Hein, R. Lam, D. Moore, J. Moore, K. Patterson, I. Poberezhskiy, J. Shields, E. Sidick, H. Tang, T. Truong, J. K. Wallace, X. Wang, and D. Wilson, “Low-order wavefront sensing and control for WFIRST-AFTA coronagraph,” J. Astron. Telesc. Instrum. Syst. 2, 011021 (2016).
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L. Pogorelyuk and N. J. Kasdin, “Dark hole maintenance and a posteriori intensity estimation in the presence of speckle drift in a high-contrast space coronagraph,” Astrophys. J. 873, 95–107 (2019).
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Popescu, G.

Postman, M.

M. R. Bolcar, S. Aloezos, V. T. Bly, C. Collins, J. Crooke, C. D. Dressing, L. Fantano, L. D. Feinberg, K. France, G. Gochar, Q. Gong, J. E. Hylan, A. Jones, I. Linares, M. Postman, L. Pueyo, A. Roberge, L. Sacks, S. Tompkins, and G. West, “The large UV/optical/infrared surveyor (LUVOIR): decadal mission concept design update,” Proc. SPIE 10398, 1039809 (2017).
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Prusti, T.

B. Mennesson, S. Gaudi, S. Seager, K. Cahoy, S. Domagal-Goldman, L. Feinberg, O. Guyon, J. Kasdin, C. Marois, D. Mawet, M. Tamura, D. Mouillet, T. Prusti, A. Quirrenbach, T. Robinson, L. Rogers, P. Scowen, R. Somerville, K. Stapelfeldt, D. Stern, M. Still, M. Turnbull, J. Booth, A. Kiessling, G. Kuan, and K. Warfield, “The habitable exoplanet (HabEx) imaging mission: preliminary science drivers and technical requirements,” Proc. SPIE 9904, 99040L (2016).
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Pueyo, L.

M. R. Bolcar, S. Aloezos, V. T. Bly, C. Collins, J. Crooke, C. D. Dressing, L. Fantano, L. D. Feinberg, K. France, G. Gochar, Q. Gong, J. E. Hylan, A. Jones, I. Linares, M. Postman, L. Pueyo, A. Roberge, L. Sacks, S. Tompkins, and G. West, “The large UV/optical/infrared surveyor (LUVOIR): decadal mission concept design update,” Proc. SPIE 10398, 1039809 (2017).
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M. N’Diaye, A. Vigan, K. Dohlen, J. F. Sauvage, A. Caillat, A. Costille, J. H. V. Girard, J. L. Beuzit, T. Fusco, P. Blanchard, J. Le Merrer, D. Le Mignant, F. Madec, G. Moreaux, D. Mouillet, P. Puget, and G. Zins, “Calibration of quasi-static aberrations in exoplanet direct-imaging instruments with a Zernike phase-mask sensor. II. Concept validation with ZELDA on VLT/SPHERE,” Astron. Astrophys. 592, A79 (2016).
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Quirrenbach, A.

B. Mennesson, S. Gaudi, S. Seager, K. Cahoy, S. Domagal-Goldman, L. Feinberg, O. Guyon, J. Kasdin, C. Marois, D. Mawet, M. Tamura, D. Mouillet, T. Prusti, A. Quirrenbach, T. Robinson, L. Rogers, P. Scowen, R. Somerville, K. Stapelfeldt, D. Stern, M. Still, M. Turnbull, J. Booth, A. Kiessling, G. Kuan, and K. Warfield, “The habitable exoplanet (HabEx) imaging mission: preliminary science drivers and technical requirements,” Proc. SPIE 9904, 99040L (2016).
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Raab, F.

A. Abramovici, W. Althouse, J. Camp, D. Durance, J. Giaime, A. Gillespie, S. Kawamura, A. Kuhnert, T. Lyons, F. Raab, R. L. Savage, D. Shoemaker, L. Sievers, R. Spero, R. Vogt, R. Weiss, S. Whitcomb, and M. Zucker, “Improved sensitivity in a gravitational wave interferometer and implications for LIGO,” Phys. Lett. A 218, 157–163 (1996).
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Appl. Opt. (2)

Astron. Astrophys. (3)

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Supplementary Material (1)

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» Supplement 1       Supporting Technical Information

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Figures (6)

Fig. 1.
Fig. 1. Schematic of the ZWFS testbed; see text for details of operation. The interaction of the ZWFS dimple and point spread function (PSF) is emphasized in the detailed view. Equipment details: IFO, 4D Inc. 4020 HSPS Interferometer; DM, Boston Micromachines Multi-3.5 Deformable Mirror; ZWFS/D, Etched Fused Silica, Silios Inc.; ZWFS/C, Andor iXon Ultra 897 EMCCD; L1–L4, lenses; Newport Inc. PAC0XX (various); FM, fold mirror: Thorlabs PF2011-F01; BS, beam splitter: Thorlabs BSW27.
Fig. 2.
Fig. 2. Schematic of data acquisition process. An individual measurement in either the “ON” or “OFF” command (CMD) comprises an average of ${M}$ individual frames. The differential wavefront estimate is the difference of these two measurements averaged over ${ N}$ cycles.
Fig. 3.
Fig. 3. Measurement validation of ZWFS measurements using a commercial Twyman–Green interferometer (IFO). Wavefront error maps are shown for the ZWFS and IFO at three discrete points and highlight the good agreement between the two techniques (colormap units in nm). Deviations occur above wavefronts errors of approximately 20 nm RMS due to intensity inversions in the ZWFS signal (highlighted with dashed circles).
Fig. 4.
Fig. 4. (a) Mean (solid curve) and standard deviation (shaded area) of detected wavefront over 70 independent trials. The insert highlights the sub-picometer repeatability of the system. The dashed line represents the mean determined from a numerical model of the experiment, in the absence of atmospheric effects. (b) Wavefront error maps representing one trial of a ZWFS measurement for varying levels of averaging.
Fig. 5.
Fig. 5. Standard deviation of RMS wavefront difference over 70 independent trials. The blue line is experimental data, and the orange line is from numerical simulation. The measurement precision increases with the number of averages, ${N}$. Because the experiment was performed in air, over small averages, the precision is dominated by the time-varying atmosphere. Note that over large averages, the slope of the experimental variance more closely matches that of the numerical simulations.
Fig. 6.
Fig. 6. Numerical simulation demonstrating the dynamic range of the ZWFS for varying levels of (a) tilt and (b) focus. Top row, phase reconstruction estimate, with errors apparent for amplitudes greater than 160 nm PV. The corresponding motion of the PSF relative to the ZWFS dimple is also noted. Bottom row, differential phase estimate when a 240 pm PV checkerboard pattern is applied on top of the dominate tilt/focus terms. Errors in the reconstruction are highlighted by dashed circles. All colormap units in nm.

Tables (1)

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Table 1. Camera and Imaging Parameters of the ZWFS Testbed

Equations (9)

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ϕ A ϕ B = π 2 + arcsin [ I I A I B 2 I A I B ] ,
u ( x ) = A P ( x ) e i ϕ ( x ) ,
u B ( x ) = L [ u ( x ) ] e i π / 2 = i A b ( x ) e i β ( x ) ,
u A ( x ) = H [ u ( x ) ] = A P ( x ) e i ϕ ( x ) A b ( x ) e i β ( x ) ,
ϕ ( x ) β ( x ) = π 4 + arcsin [ I ( x ) I P ( x ) 2 I b ( x ) 2 2 I P ( x ) I b ( x ) ] ,
I n i = 1 M m = 1 M I m i ,
Δ ϕ n = ϕ ( I n ON ) ϕ ( I n OFF ) .
Δ ϕ N = 1 N n = 1 N Δ ϕ n .
σ t = σ s 2 + σ d 2 + σ r 2 .

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