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Abstract
The direct imaging of extrasolar planets requires extreme adaptive optics (ExAO) in the near-infrared wavelengths. The ExAO needs a high-efficient wavefront sensor (WFS) to measure the phase aberration accurately with a small number of photons. In addition, the WFS in the ExAO is required to run at high sampling rates of 1 − 7 kHz. To meet these requirements, we developed the birefringent point-diffraction interferometer (b-PDI) presented in our earlier paper. We tested the b-PDI in the laboratory with a polychromatic light source with wavelengths of 800 ± 100 nm. The b-PDI showed a relatively high efficiency, comparable to that of a fixed pyramid WFS. The b-PDI has a low calculation cost and a small readout region, which are suitable for high-speed sampling at 6.5 kHz.
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1. Introduction
The direct imaging of extrasolar planets is technically challenging. The planets are much fainter than their host stars, with a flux ratio of $10^{-4}-10^{-8}$ in the near-infrared wavelengths ($\lambda _{NIR}$). Also, the planets are located very close to the host stars with an angular separation of smaller than one arcsecond. The high contrast and small angular separation can be overcome by an aberration correction of $<\lambda _{NIR}/20$ accuracy with extreme adaptive optics (ExAO) on ground-based telescopes. An essential element in the ExAO is a wavefront sensor (WFS). The WFS is required to measure the phase aberration with high accuracy of $<\lambda _{NIR}/20$, a high sampling rate of $1-7$ kHz [1,2], and a high resolution of $\sim 10^3$ measurement points (subapertures) in an 8-m telescope aperture. Thus, the WFS needs high efficiency of photon-usage to measure the phase aberration accurately with a small number of photons from a faint star.
A high-speed measurement at $1-7$ kHz with a high resolution also requires a low calculation cost and a high frame rate of detectors. As for a high frame rate, CMOS detectors are potential candidates because they do not need a charge transfer. In particular, high-speed scientific CMOS (sCMOS) detectors have a circuit that simultaneously reads out the pixel values in each horizontal line. Hence, the frame rate of sCMOS detectors is further enhanced by minimizing the vertical size of the readout region for wavefront sensing. These high-speed capabilities realize the optimal wavefront sampling time [3] to maximize the contrast under a large wind velocity.
In terms of efficiency, the Zernike WFS [4] is superior to the pyramid WFS [5,6] and the Shack-Hartmann WFS. Guyon defined the sensitivity factor $\beta _p$ [3] as an indicator of the efficiency; a small value of $\beta _p$ indicates high efficiency. The sensitivity factor of the Zernike WFS is unity, whereas that of the pyramid WFS and the Shach-Hartmann WFS is $>\sqrt {2}$ and $>2$, respectively. The fact means that the Zernike WFS is the most efficient among them.
The Zernike WFS is a variation of point-diffraction interferometers (PDIs). The PDI employs a point-diffraction plate installed at the focal plane of a telescope. The plate is transparent and has a pinhole smaller than the Airy disk. The pinhole extracts an unaberrated beam (reference beam) from the incident beam under test. The rest of the beam (test beam) retains the phase informaation of the incident beam. The two beams interfere, and an interferogram arises on a detector. The intensity distribution of the interferogram can be converted to the phase.
So far, various PDIs, including the Zernike WFS, have been demonstrated for astronomical purposes. The first example is ZELDA [7,8]. This is a practical application of the Zernike WFS concept and thus has a high efficiency. It also has a low calculation cost and a small readout region. However, its dynamic range is restricted to $<0.5\lambda$ in the peak-to-valley (P-V) value at the sensing wavelength $\lambda$. This is because the intensity of the single interferogram gives only a sine of the phase. With the narrow range, ZELDA is not used to measure atmospheric aberrations. The second example is the vector-Zernike sensor [9]. The presented equation yields a low calculation cost but limits the dynamic range to P-V $<0.5\lambda$. The third example is the pupil-modulated PDI [10,11]. Its dynamic range is P-V $>1\lambda$ for low order phase aberrations. However, its efficiency is low because it does not utilize the rest of the beam, which is cut off when the focal plane mask extracts the reference beam. Additionally, its calculation cost is relatively high because of the Fourier transform.
We developed and demonstrated a new WFS of the birefringent PDI (b-PDI) proposed in our earlier paper [12] (hereafter referred to as Tsukui+20). The point-diffraction plate in this concept utilizes a uniaxial birefringent crystal. The b-PDI has high efficiency because it utilizes all the incident photons except for those diffracted at large angles by the pinhole. It has relatively low calculation cost and vertically small readout region on an sCMOS detector. Its dynamic range is $>0.5\lambda$ because the phase is calculated with the arctangent function atan2, which does not degenerate between $-\pi$ rad and $+\pi$ rad. With these properties, the b-PDI is suitable for the WFS in the second-stage ExAO [12] to correct atmospheric aberrations. In this paper, we briefly review the principle of the b-PDI in Section 2. We describe the detail of the manufactured b-PDI in Section 3. We demonstrate the efficiency and dynamic range of the b-PDI by the laboratory experiments in Section 4. We discuss the result and compare the performance of the b-PDI to that of other WFSs in Section 5.
2. Birefringent PDI
The measurement principle of the birefringent PDI (b-PDI) is phase-shifting interferometry. This technique requires at least three interferograms with different phases between the test and the reference beams. The b-PDI generates four interferograms simultaneously with the birefringent point-diffraction beamsplitter (BPBS) shown in Fig. 1. The BPBS operates in the wavelength band whose central wavelength $\lambda _C$ is 800 nm and bandwidth is 200 nm. Note that the BPBS can operate at other wavelengths by modifying the design.
Fig. 1. Schematic drawing of the BPBS. (a) Perspective view. The optical axis of the birefringent crystal (TiO$_2$) is parallel to the $y$ axis. (b) Sectional view. The pinhole on the front surface is filled with the non-birefringent material (Nb$_2$O$_5$). The term $R_p$ and $d$ are the pinhole’s radius and depth, respectively.
The BPBS consists of an uniaxial birefringent crystal (TiO$_2$) and a non-birefringent material (Nb$_2$O$_5$). The optical axis of TiO$_2$ is arranged parallel to the $y$ axis. In this configuration, the refractive index is $n_e$ for the $y$-axis polarization (extraordinary ray; $e$) and $n_o$ for the $x$-axis polarization (ordinary ray; $o$). The BPBS has a pinhole on the front surface. The radius and depth of the pinhole are $R_p$ and $d$, respectively. The pinhole is filled with Nb$_2$O$_5$, which is not birefringent. The refractive index of Nb$_2$O$_5$ is $n$, and its thickness is $d$. The front surface of the BPBS is flat.
The BPBS is placed concentrically with the Airy disk at the focal plane. The pinhole on the BPBS extracts the reference beam, and the remaining beam is the test beam. The BPBS tilts slightly with respect to the optical axis to form a transmission path and a reflection path. In the transmission path, the optical path difference (OPD) inside and outside the pinhole causes a phase shift between the reference and test beams. Due to the birefringence of the crystal, the phase shift varies depending on the polarization directions. When the wavelength $\lambda _C=800$ nm and the depth of the pinhole $d=1.32\lambda _C$, the phase shifts for the polarizations $o$ and $e$ are calculated as follows, respectively:
The interferograms with phase shifts $\theta _o$ and $\theta _e$ are defined as $I_o^T(x,y)$ and $I_e^T(x,y)$, respectively. The $(x,y)$ is the position of the subapertures. The two interferograms are split with a Savart plate and arise on a detector without overlapping with each other. In the reflection path, the phase shift between the reference and the test beam is zero because the front surface of the BPBS is flat. The interferograms with polarization $o$ and $e$ are defined as $I_o^R(x,y)$ and $I_e^R(x,y)$, respectively. These interferograms are split with a Savart plate and arise on a detector. The system utilizes all the photons except for those diffracted at large angles by the pinhole and fall outside the interferograms.
The interferograms are converted to the phase $\delta _{rec}(x,y)$ of the incident beam at each subapertre $(x,y)$ with the following equation:
In the practical use, $I_o^R(x,y)$ and $I_e^R(x,y)$ in Eqs. (3) and (4) are replaced with the following $I_{ave}(x,y)$:
The substitution of $I_{ave}$ improves the signal-to-noise (S/N) ratio when $I_o^R$ and $I_e^R$ contain random noise.3. Manufacturing
3.1 BPBS
We manufactured the BPBS. Based on Tsukui+20, the pinhole was designed with a depth of $d=1.32\lambda _C=1.06\ \mu$m and a radius of $R_p=15\ \mu$m. This $R_p$ is 0.5 times the Airy disk radius at an F-number of 31 and a central wavelength of $\lambda _C=800$ nm. We made another pinhole of radius 200 $\mu$m filled with Nb$_2$O$_5$ to measure the parameters $T$ and $R$. Hereafter, the pinholes of radii 15 $\mu$m and 200 $\mu$m are called Pinholes A and B, respectively. We manufactured 24 Pinholes A and 24 Pinholes B on the front surface of a 20-mm square, 0.5-mm thick TiO$_2$ substrate. The back of the substrate is AR-coated; the reflectance is less than 1$\%$. We processed eight substrates to select the best Pinhole A from 192 samples.
Figure 2 shows the manufacturing process. In the process, the pinholes are first manufactured on the front surface of the TiO$_2$ substrate by SF$_6$-based reactive ion etching (RIE) [14]. The etching mask consists of two layers: photoresist and Cr. Then, the mask is removed by wet etching with acetone and nitric-acid-based etchant. Then, Nb$_2$O$_5$ is deposited on the front surface by an RF sputtering method. Finally, the front surface is manually polished, leaving the Nb$_2$O$_5$ film only inside the pinholes.
After the process, the shape of the front surface was measured with a stylus profilometer (Bruker DektakXT). The thickness $t$ and refractive index $n$ of the Nb$_2$O$_5$ film were measured with a microscopic spectrophotometer (JASCO MSV-5200). Based on these measurements, We selected several Pinholes A with a shape close to the designed value.
The selected multiple Pinholes A were set into the optical system described in Section 3.2. A tilted flat wavefront was injected into the optical system, and the contrast of the interferograms was evaluated. We selected Pinhole A ($\#$03-40), which produced the best contrast.
Figure 3 and Table 1 show the shape and phase shifts of Pinhole A ($\#$03-40). The surface of the Nb$_2$O$_5$ film is concave by 25 nm relative to the TiO$_2$ surface. This concavity causes a phase shift of $0.1\pi$ in the reflection path, which is acceptable. The phase shifts in the transmission path are redefined by considering the concavity $h$ as follows:
The term $n,t$, and $h$ are measured values. The term $n_o$ and $n_e$ are literature values [15].Fig. 3. (a) Optical microscopic image of Pinhole A ($\#$03-40). (b) Optical microscopic image of Pinhole B ($\#$03-41), in which the black areas are outside the field of view. In (a) and (b), red circles show the size of the Airy disk with a radius of 30 $\mu$m (a diameter of 60$\mu$m). The pinholes are illuminated with a white LED light source (incident type). (c) Schematic drawing of the cross-section of the manufactured pinhole. Note that the figure is stretched in the direction of the pinhole depth.
Table 1. Comparison between the manufactured pinholes and the design. PH: pinhole.
Pinhole B ($\#$03-41) adjacent to Pinhole A ($\#$03-40) is also shown in Fig. 3 and Table 1. The surface of the Nb$_2$O$_5$ film is a loose concave surface (P-V $\sim 70$ nm). The thicknesses of the films in Pinholes A and B are greater than the coherent length of the light used for the measurement of $R$ and $T$ (wavelengths of $800\pm 100$ nm). Thus, the films do not cause interference inside them. Therefore, the transmission and reflection coefficients in Pinhole B ($\#$03-41) are uniform and equivalent to those in Pinhole A ($\#$03-40).
We show the optical performance of Pinhole A ($\#$03-40). The optical path difference (OPD) map of the light transmitted through Pinhole A ($\#$03-40) was captured with a transmitted dual-beam interference microscope (Mizojiri Optical Co., Ltd. TD-series). The wavelength $\lambda _m$ of the light used in the evaluation was 546 nm. Figure 4 shows the result. The flatness of the OPD in Pinhole A is 20 nm RMS ($\lambda _m/27\sim \lambda _C/40$).
Fig. 4. OPD maps and their cross-section graphs of Pinhole A ($\#$03-40). (a) Ordinary ray. (b) Extraordinary ray.
3.2 Optical system of the b-PDI
Aiming to mount the b-PDI on the ExAO instrument SEICA [2] of the Seimei telescope [16], we built an optical system for the b-PDI. The WFS in SEICA requires $\gtrsim 25$ subapertures in the diameter of the telescope aperture because $D/r_0=25$, where $D$ is the diameter of the telescope aperture ($=$ 3.8 m) and $r_0$ is the assumed Fried parameter at the observation site ($=$ 0.15 m). The WFS also requires as few subapertures as possible (i.e. as wide subapertures as possible) to collect adequate photons from stars at each subaperture. Following these requirements, the b-PDI optical system has 492 ($26\times 26$) subapertures. The scalability and limiting factor of the subaperture number are discussed in Section 5.5.
The system is a modification of the optical system shown in Tsukui+20, which requires two detectors. The proposal to use two detectors is unsuitable for the target measurement frequency of 6.5 kHz. This is due to the difficulty of synchronization at the order of 6 kHz with the detectors in hand. To ensure the four interferograms are captured simultaneously, we designed the optical system in which the four interferograms appear on an sCMOS detector (Hamamatsu ORCA-Flash4.0v2). Furthermore, to maximize the frame rate (up to > 6.5 kHz), we minimized the vertical size of the readout region by aligning the four interferograms horizontally on the sCMOS detector.
The layout of the optical system is shown in Fig. 5. In the following, the $x$ and $y$ axes are perpendicular to the local axis of the beam and are horizontal and vertical, respectively. The polarization directions of the ordinary ray $o$ and the extraordinary ray $e$ are parallel to the $x$ and $y$ axes, respectively, as shown in Fig. 1. The incident beam (6.5-mm diameter) is focused by L0 and L0’ to create an Airy disk on the BPBS Pinhole A. The converged beam is F/31, and the Airy disk radius is 30 $\mu$m at $\lambda _C=$ 800 nm. The BPBS transmits and reflects the beam. The transmitted beam is bent at M0, collimated at L1, and further bent at M1. L1 creates the intermediate pupil IP1. On the other hand, the reflected beam is collimated at L2 and bent at M2. L2 creates the intermediate pupil IP2. The pupils IP1 and IP2 are at the same distance from L3. The beams from IP1 and IP2 pass HWP and SP, which separate the polarization directions $o$ and $e$. The function of HWP is described in detail later. The separated beams are relayed onto the detector by L3 and L4, creating four interferograms (Fig. 6). Each of the interferograms has a diameter of 26 pix (= 169 $\mu$m). Under these conditions, the Nyquist frequency is 13 cycles/pupil (c/p). The phase aberrations at $>13$ c/p are cut off by a square mask, which is placed in front of the BPBS and limits its clear aperture to $600\times 600$ $\mu$m. The BPBS Pinhole A is located at the center of the clear aperture.
Fig. 5. Layout of the b-PDI optical system. The optical system includes a focusing lens L0, a correction lens L0’, a square mask, the BPBS, a flat mirror M0, collimator lenses L1 and L2, flat mirrors M1 and M2, a half-wavelength plate HWP, a Savart plate SP, relay lenses L3 and L4, and a detector (Hamamatsu ORCA-Flash4.0v2). All are arranged in the horizontal plane. The BPBS is placed on the focal plane of L0. The center of the BPBS Pinhole A is on the axis of L0. The normal of the BPBS is inclined by 10 degrees to the axis of L0 in the $x-z$ plane. The inclination axis of the BPBS is parallel to the optical axis of TiO$_2$, which is parallel to the $y$-axis. The system from L0 to L4 fits on about the size of an A4 sheet of paper.
Fig. 6. Examples of the interferograms captured with the manufactured b-PDI. (a) A tilted flat wavefront with P-V = 1.6 rad (200 nm). (b) A sinusoidal wavefront with 2 c/p and P-V = 1.2 rad (150 nm). (c) The figure “SEICA” generated with a deformable mirror whose stroke is $\sim 2$ rad (250 nm). The interferograms in each image are $I_o^T(x,y), I_e^T(x,y), I_o^R(x,y)$, and $I_e^R(x,y)$ from left to right.
In the optical path, HWP rotates the polarization direction. The polarization direction of the ordinary ray ($o$) is rotated from 0 deg (parallel to the $x$ axis) to $-45$ deg. On the other hand, The polarization direction of the extraordinary ray ($e$) is rotated from $+90$ deg (parallel to the $y$ axis) to $+45$ deg. After this rotation, SP splits $o$ and $e$ horizontally and lines up the interferograms in the horizontal direction of the detector. Note that the interferograms can be aligned in the horizontal direction without HWP if the optical axis of the BPBS is in the $+45$ deg direction. However, prioritizing the ease of BPBS handling, we set the optical axis of the BPBS in the $y$ axis direction.
In order to prevent misalignment between the four interferograms and the detector pixel array, each optical element must have an alignment accuracy of a few microns or better. This accuracy is achieved with the custom parts with leverage mechanisms and the commercial precision feed screws.
Since this optical system contains many elements, the throughput is only about 70$\%$. However, we can apply the original optical system with a small number of elements proposed in Tsukui+20 if multiple detectors are synchronized at 6.5 kHz in the future. In this case, the throughput is expected to be $\sim 90\%$, assuming a loss of 2$\%$ per surface.
4. Experiment
4.1 Setup
We built a test bench (Fig. 7) to evaluate the b-PDI. In the test bench, the beam from the fiber source is collimated by CL and passes through the iris (6.5-mm diameter). RL1 and RL2 project the image of the iris onto the deformable mirror (DM; BMC 492-DM) at a magnification ratio of 1:1. The DM introduces phase aberrations to the beam. Then, the retractable FM1 switches the beam direction. When FM1 is moved out of the optical path, the beam directly enters the b-PDI. When FM1 is inserted into the optical path, the beam is reflected and enters the Shack-Hartmann WFS (SHWFS). In the SHWFS, RL3 and RL4 relay the beam to the WFS150-5C sensor head (Thorlabs), which contains a microlens array with $29\times 29$ effective elements. The SHWFS is calibrated with FM2 ($\lambda$/20), which replace the DM. The microlens array, the detector of the b-PDI, and the DM are in the conjugate plane.
Fig. 7. (a) Layout of the test bench. The size is 1.5 $\times$ 0.8 m. The test bench consists of a fiber source, a collimator lens CL, an iris (6.5-mm diameter), relay lenses RL1 and RL2, a deformable mirror (DM; BMC 492-DM), the b-PDI, a retractable flat mirror FM1 ($\lambda$/20), a flat mirror FM2 ($\lambda$/20), and a Shack-Hartmann wavefront sensor (SHWFS). The SHWFS consists of relay lenses RL3 and RL4, and a Thorlabs WFS150-5C sensor head with $29\times 29$ subapertures. The DM has 492 ($24\times 24$) actuators. The b-PDI has $26\times 26$ subapertures. (b) Photograph of the test bench.
We use two different fiber sources. For the SHWFS, a bright diode laser with a wavelength of 800 nm is applied to reduce noise. The measurement error of the SHWFS with this source is $5\times 10^{-2}$ rad (7 nm) RMS, which is determined by measuring a spherical wavefront with a known radius of curvature. This error is primarily systematic. For the b-PDI, on the other hand, a tungsten-halogen lamp is applied to simulate the weak stellar flux with a low coherency. The band-pass filters in the lamp unit limit the wavelengths to 800$\pm$100 nm.
When we inspect the point spread functions (PSFs) at the focal plane, we replace the SHWFS with a focal plane imager. The imager consists of an F/31 imaging lens and an sCMOS camera (Hamamatsu ORCA-Flash4.0v3). The configuration yields a PSF sampling of FWHM $=3.8$ pix.
4.2 Measurement of the transmission/reflection coefficients
The phase calculation (Eqs. (3–5)) requires the parameters expressed as the ratios of the transmission and reflection coefficients. These paremeters must be measured before the calculation. Specifically, the parameters in the Table 2 are to be measured.
Table 2. Required parameters.
In the measurement, a collimated beam with wavelengths of 800$\pm$100 nm was injected into the b-PDI. The BPBS was displaced laterally and placed so that Pinhole B or the bare TiO$_2$ crystal surface overlapped the Airy disk. The resulting pupil images were captured with the detector and photometrically measured. The parameters were calcurated by the ratio of the photometric values, as shown in the third row of Table 2. The $IB$ and $IC$ are the photometric values of the pupil images produced by Pinhole B and the bare TiO$_2$ crystal surface, respectively. The subscripts $o$ and $e$ denote the polarization direction, and the superscripts $T$ and $R$ denote the transmission and reflection path, respectively. The fourth row of Table 2 summarizes the results.
4.3 Systematic error
We evaluated the systematic error of the b-PDI by measuring known wavefronts. The systematic error was calculated from the difference between the known phase $\delta _{0}(x,y)$ and the phase $\delta _{rec, av}(x,y)$ measured by the b-PDI under a high S/N condition.
The wavefront under test was a sinusoidal wavefront with a spatial frequency of 8 c/p. The sinusoidal wavefront was generated with the DM and measured with the SHWFS to determine the phase. The tip/tilt components were removed from each of the 10 measured frames. The 10 frames were then converted to $26\times 26$ pixels and averaged to be $\delta _0(x,y)$. The measured amplitude was $\simeq 0.24$ rad ($\simeq 30$ nm), which means P-V $\simeq 0.48$ rad ($\simeq 60$ nm). As mentioned earlier, the SHWFS has the systematic error of about $5\times 10^{-2}$ rad (7 nm) RMS.
We measured the wavefront with the b-PDI under the following conditions. By adjusting the integration time of the detector, the interferograms with a high S/N ratio were captured. The average number of photons incident on one subaperture was $2.3\times 10^4$ in the number of photo-electrons; the details of the photometry are described below. Ten frames of interferograms were converted to the phase $\delta _{rec}(x,y)$ with Eqs. (3–5). The tip/tilt components were removed from each frame. The 10 frames of the phase were then averaged to be $\delta _{rec, av}(x,y)$. We evaluated the RMS value of the difference between $\delta _{rec, av}(x,y)$ and $\delta _{0}(x,y)$. Here, 21 subapertures were affected by a dead actuator on the DM and thus excluded.
Figure 8 shows the measured results and the difference. For the sinusoidal wavefront (8 c/p, P-V $\simeq 0.47$ rad), the RMS value of the difference was $8.4\times 10^{-2}$ rad (11 nm). This RMS value is the root-sum-square of the systematic errors of the SHWFS and the b-PDI. Considering that the SHWFS has a systematic error of $\sim 5\times 10^{-2}$ rad (7 nm) RMS, the systematic error of the b-PDI is $\sim 7\times 10^{-2}$ rad (9 nm) RMS. The systematic error could be reduced by a calibration with a flat wavefront.
Fig. 8. Comparison between $\delta _{rec,av}(x,y)$ and $\delta _{0}(x,y)$. The map $\delta _{rec, av}(x,y)$ is the average of the 10 frames of the phase, which are measured with the b-PDI under a high S/N condition. The map $\delta _{0}(x,y)$ is the average of the 10 frames of the phase, which are measured with the SHWFS under a high S/N condition.
The average photon number was estimated by the following photometry. By shifting the PSF core onto the bare TiO$_2$ crystal surface $\sim 150\ \mu$m away from Pinhole A, we obtained the pupil images at the reflection path with less interference effect. The pupil images were captured at the same integration time as the above interferograms. The photometry of these images yielded the average number of photons (in the number of photo-electrons) in the reflection path. The average number of photons in the transmission path was estimated using the parameters $s_o$ and $s_e$ (Table 2). The sum of these numbers is the average number of photons in all the paths. Dividing this sum by the area ratio of the subaperture to the incident pupil, we got the above value ($2.3\times 10^4$ electrons). This value equals the average number of incident photons per subaperture when the throughput of the b-PDI optics is 100$\%$.
4.4 Statistical error
We evaluated the statistical error of the b-PDI by measuring sinusoidal wavefronts. Noise propagation theory predicts that the statistical error $\sigma _{stat}$ [rad] of the b-PDI is expressed by the following equation:
In the case of the zonal wavefront measurement in the b-PDI, $N_e$ is the average number of photons (in the number of photo-electrons) incident on one subaperture. The terms $C_p$ and $C_r$ are the coefficients of error propagation from photon noise and readout noise, respectively. The term $N_r$ is the readout noise; $N_r=1.6$ electrons in this experiment.As the prediction, we ran simulations using the numerical model [12], which reflected the actual Pinhole A ($\#$03-40) geometry shown in Table 1. The values of $C_p$ and $C_r$ were predicted from the simulations with photon noise or readout noise at $N_e>300$. Table 3 shows the result with sinusoidal wavefronts. The spatial frequencies of the wavefronts were 2, 4, and 8 c/p. The amplitudes of the wavefronts were 0.16 rad (20 nm), which means P-V $=0.32$ rad (40 nm).
Table 3. Predicted $C_p$ and $C_r$ from the simulations.
In the experiment, the sinusoidal wavefronts are shaped with the DM. We measured the wavefronts with the b-PDI under the following conditions. By adjusting the integration time of the detector, $N_e$ was set in a range of $24-2.4\times 10^4$. The $N_e$ was estimated in the same way as in Section 4.3 and scaled by the integration time. At each integration time, 100 frames of interferograms were captured. However, 10 frames were captured at $N_e=2.4\times 10^4$ electrons. The phase $\delta _{rec}(x,y)$ was calculated at each subaperture $(x,y)$ with Eqs. (3–5). This calculation used two types of numerical masks: fixed and low-count masks. The fixed masks excluded the eight subapertures that hit the beam’s edges and the 12 subapertures that overlap the dead actuator on the DM. The low-count masks excluded subapertures with
This operation excludes physically meaningless count values. The tip/tilt components were subtracted from the calculated phase $\delta _{rec}(x,y)$.The experimental values of the statistical error $\sigma _{exp}$ were calculated from $\delta _{rec}(x,y)$. First, $\delta _{rec}(x,y)$ at $N_e=2.4\times 10^4$ electrons was averaged over 10 frames and defined as the reference phase $\delta _{Ref}(x,y)$. Next, the difference between $\delta _{Ref}(x,y)$ and $\delta _{rec}(x,y)$ at $N_e<2.4\times 10^4$ electrons was taken, and its RMS value was calculated. This RMS value is $\sigma _{exp}$. Note that $\sigma _{exp}$ contains only the statistical errors because the difference canceled the systematic errors. The median, maximum and minimum values of $\sigma _{exp}$ were selected from the 100 frames at each integration time.
We compared the experimental results $\sigma _{exp}$ with its predicted value $\sigma _{stat}$ (Eq. (13)). Figure 9 shows the results. The experimental $\sigma _{exp}$ and its predicted value $\sigma _{stat}$ are consistent in the region $N_e\gtrsim 100$. At $N_e=60$, the experimental $\sigma _{exp}$ exceeded the predicted $\sigma _{stat}$. This implies nonlinear error propagation, with effective $C_p$ and $C_r$ being larger than the values in Table 3, respectively. These elevated values were also observed in the simulations. For example, for the 4 c/p sinusoidal wavefront, the simulated effective $C_p$ and $C_r$ are 1.4 and 4.1 at $N_e=60$, respectively. These simulated values are consistent with the experimental value $\sigma _{exp}$ at $N_e=60$. On the other hand, the experimental value $\sigma _{exp}$ was lower than the predicted value $\sigma _{stat}$ at $N_e=24$. This is because the low-count masks eliminated the subapertures with large noise and small count values below 1 electron, suppressing the overall error (see Fig. 9(d)). In this region, the effective $C_p$ and $C_r$ are smaller than the values in Table 3, respectively. The low-count masks removed $\sim 30\%$ of the subapertures at $N_e=24$ and $\sim 5\%$ at $N_e=60$.
Fig. 9. (a)(b)(c) Comparison of the experimental value $\sigma _{exp}$ of the statistical error with its predicted value $\sigma _{stat}$. The horizontal axis is the average number of photons per subaperture $N_e$ (in the number of photo-electrons). The graphs show results with the sinusoidal wavefronts (2, 4, and 8 c/p), as noted in each panel. The gray dashed line represents the predicted $\sigma _{stat}$, the black dots are the median of $\sigma _{exp}$, and the error bars are their maximum and minimum values. The $N_r$ represents the readout noise; $N_r=1.6$ electrons in this experiment. (d) Experimental values $\sigma _{exp}$ of the statistical error with different thresholds for the low-count masks. For the black dots, the low-count masks are applied at $<2$ ADU ($\simeq$ 1 electron). For the gray dots, the low-count masks are applied at $<1$ ADU ($\simeq$ 0.5 electron)
4.5 Dynamic range
We evaluated the dynamic range of the b-PDI, using the sinusoidal wavefronts generated with the DM. The spacial frequency was 1, 2, and 4 c/p.
First, we measured the wavefronts with the SHWFS. Ten frames were measured. The tip/tilt components were removed from each frame. The 10 frames were then converted to $26\times 26$ pixels. The measured phase of the SHWFS at frame number $j$ was defined as $\delta _{SH,j}(x,y)$.
We then measured the wavefronts with the b-PDI. By adjusting the integration time, the number of photons was set to $N_e\gtrsim 1.2\times 10^4$ electrons. Ten frames of interferograms were captured. The phase was calculated at each subaperture with Eqs. (3–5). The measured phase of the b-PDI at frame number $j$ was defined as $\delta _{rec,j}(x,y)$.
We then calculated the magnification factor $M_{j}$ of the b-PDI relative to the SHWFS at frame number $j$. The optimal value as $M_j$ minimizes the following evaluation function $D(m)$:
In other words, $M_j$ satisfies the following equation: Therefore, $M_j$ is defined asFinally, we captured the PSF with the focal plane imager to calculate the Strehl ratio. The calculation was based on Method Six described by Roberts et al. [17].
Figure 10 shows the result. Since the arctangent function atan2 does not degenerate in the range of $1\lambda _C$ in P-V, $M_j>0$ was expected in the same range. However, we observed $M_j=0$ at P-V $<1\lambda _C$, and the measured phase was almost flat. The $M_j=0$ occurred when the Strehl ratio was $\sim 0$ at any spatial frequency. This is because the reference beam disappears, and the contrast of the interferogram takes $\sim 0$ at the Strehl ratio $\sim 0$. At this point, interferometry can not be performed. This effect sets a practical upper limit of the dynamic range. Thus, the dynamic range is P-V $<0.6\lambda _C-1.0\lambda _C$ for the sinusoidal wavefronts with $4-1$ c/p. In the dynamic range, the wavefront phase is calculated without being affected by the degeneracy of the equation.
Fig. 10. (a) Magnification factor $M_j$ of the b-PDI, plotted against the P-V value of the phase measured with the SHWFS. (b) Magnification factor $M_j$ plotted against the Strehl ratio of the PSF. In each panel, the black dots represent the median of $M_j$, and the error bars are their maximum and minimum values. The line types identify the spatial frequency of the wavefronts: solid line for 1 c/p, dashed line for 2 c/p, and dashed-and-dotted line for 4 c/p.
5. Discussion
5.1 Statistical error and efficiency
The coefficient $C_p$ describes the contribution of photon noise to the statistical error. The experiments showed that the manufactured b-PDI has $C_p$ of $\gtrsim$ 1.0, which is $\gtrsim 1.4$ times larger than that of a Zernike WFS [7]. This means that the b-PDI is $\gtrsim 1.4$ times more sensitive to photon noise than a Zernike WFS. Thus, the sensitivity factor $\beta _p$ of the manufactured b-PDI is $\gtrsim 1.4$. In other words, the efficiency of the b-PDI is comparable to that of a fixed pyramid WFS.
The further enhancement of the b-PDI efficiency is a future work. For example, the following modifications are to be invesigated. (1) Further optimization of the pinhole depth $d$ under chromatic effects. (2) Modifying the reconstruction algorithm to avoid the strong error propagation at $N_e\sim 60$.
5.2 High-speed capability
The calculation cost of the b-PDI is relatively low. The cost is $\mathcal {O}(N_{SA})$, where $N_{SA}$ is the total number of the subapertures. This is as low as the cost of a Zernike WFS, and that of the fastest algorithm of a fixed pyramid WFS [18].
The b-PDI has a relatively small readout region in the vertical direction on a detector. The size in the direction is $N_{1d}$ pixels, where $N_{1d}$ is the number of subapertures filling the aperture diameter. A Zernike WFS and a fixed pyramid WFS require $N_{1d}$ and $>2N_{1d}$ pixels in the direction, respectively. The fewer pixels in the vertical direction of sCMOS detectors reduce the time for readout and yield a higher frame rate.
5.3 Dynamic range
The b-PDI dynamic range is $0.6\lambda _C-1.0\lambda _C$ for low-order sinusoidal wavefronts. The range is wider than those of a Zernike WFS and a fixed pyramid WFS ($<0.5\lambda _C$). The dynamic range was limited not only by the atan2 function (P-V $<\lambda _C$) but also by the Strehl ratio. This is because the reference beam disappears at a Strehl ratio of $\sim 0$, resulting in unsuccessful interferometry. This effect is inevitable and limits the dynamic range of any PDI.
Note that we do not willingly use phase unwrapping algorithms to reduce the calculation cost. However, if the cost is not an issue, there is room to use the algorithms to overcome the limitation of the atan2 function.
Table 4 summarizes the comparison.
Table 4. Comparison between the b-PDI and other WFSs. $C_p$: coefficient of error propagation from photon noise. $\beta _p$: sensitivity factor.
5.4 Low-count masks
The numerical low-count masks reduce the statistical errors when the b-PDI captures a low number of incident photons ($N_e\sim 24$ electrons). In the experiment, the subapertures with signals equivalent to $\lesssim 1$ electron were masked. The masked subapertures were $\sim 30\%$ of the total at $N_e=24$ electrons. In the future, detectors with smaller readout noise ($<1$ electron) will reduce the number of the masked subapertures. Simulations predict that the fraction of the subapertures masked when the readout noise is 0.3 and 0 electrons is 21$\%$ and 12$\%$, respectively. In the practical use in a closed-loop ExAO, the unmeasurable phase values in the masked subaperture can be compensated for by the phase values measured at the previous loop.
5.5 Scalability and limiting factor
ExAO in a larger telescope requires more subapertures and higher resolution of the interferograms because $D/r_0$ increases. The scalability of the b-PDI resolution is limited by the detector size; the diameter of each interferogram must be smaller than a quarter of the detector width.
6. Conclusions
The ExAO requires a wavefront sensor with a high sampling rate, high efficiency, and a wide dynamic range. We manufactured the b-PDI, which can run at 6.5 kHz with a relatively small readout region and a low calculation cost. We evaluated its efficiency and dynamic range by the laboratory experiments. The experimental results showed the following characteristics. The statistical error was explained by $\beta _p\gtrsim 1.4$. Thus, the b-PDI has relatively high efficiency, comparable to that of a fixed pyramid WFS but lower than that of a Zernike WFS. The dynamic range for sinusoidal wavefronts varied with spatial frequency and was $0.6\lambda _C-1.0\lambda _C$. The range is broader than that of a Zernike WFS and a fixed pyramid WFS ($<0.5\lambda _C$). The b-PDI’s dynamic range is limited by the arctangent-type equation and the Strehl ratio of the incident beam.
Funding
Japan Society for the Promotion of Science (JP16H03957, JP18H03719); Japan Science and Technology Agency (JPMJSP2110).
Acknowledgments
This study is supported by the development team of Seimei 3.8-m segmented telescope, Kyoto Prefectural Technology Center, Mizojiri Optical Co., Ltd., and 3D Model Advanced Processing Center of Osaka Electro-Communication University.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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