THE USE OF SPATIAL FILTERING WITH APERTURE MASKING INTERFEROMETRY AND ADAPTIVE OPTICS

An aperture mask is positioned in the pupil plane behind the AO system and transforms the full aperture into a sparsely populated set of sub-apertures (Figure 1(a)). The resulting image of the target is a set of over-lapping fringes called the interferogram. The amplitude and phase of each fringe correspond to the measurement of one particular component of the complex visibility with spatial frequency $\textbf {\em b}/\lambda$, where $\textbf {\em b}$ is the baseline vector and λ is the observing wavelength. Multiplying the complex visibility of specific baseline triplets formed by three sub-apertures creates bispectra (Lohmann et al. 1983), the argument of which is the closure phase (Jennison 1958; Cornwell 1989). Closure phases are robust against pupil-plane phase errors, which are a source of direct imaging point-spread function (PSF) calibration errors and speckle noise, and provide the mechanism for obtaining more precise astrometry and photometry with the aperture masking technique.

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Typical observations (including those in this paper) are conducted by taking one or more sets of target images interspersed with sets of one or more calibrators (unresolved stars near in the sky and of similar magnitude and color.) After the basic processing of the raw images (see, for example, Lloyd et al. 2006; Martinache et al. 2007; Bernat et al. 2010), the phase and log-amplitude of each fringe are extracted from the images and used to construct bispectra and closure phases. Mean values are obtained by averaging the quantities over a single set, and error estimates are derived from the scatter. Multiple sets are combined by weighted average. Calibration is performed by subtracting the closure phase and log-amplitude of the reference stars. Amplitudes are generally not used in companion searches because calibration is subject to stable atmospheric seeing and often is measurable to only 10%–50% (Tuthill et al. 2006). A binary model is fit to the closure phase data to minimize χ²; a positive detection results in measurement of the binary parameters (separation, orientation, and wavelength-dependent contrast ratio). Errors in binary parameters are often taken from the curvature of the χ² space at minimum. Many examples of this implementation for the detection of stellar companions can be found in the literature (Kraus et al. 2008; Bernat et al. 2010; Kraus et al. 2011).

For targets in which the AO system provides stable and mostly coherent (σ²_rms ≲ 0.1 rad²) correction of the wavefront on sub-aperture scales, this observing mode typically measured closure phases with an error scatter of 1°–2°in the H band (1.6 μm) after a few minutes on a bright target, equivalent to a contrast of detection of about 100–200:1 at λ/D. This technique is calibration limited by a systematic closure phase component of one to several degrees which likely arises from quasi-static instrumental wavefront errors. Additional calibrators usually decrease, but do not fully eliminate, this component. To account for this, an additional error term is added in quadrature to the closure phase errors until the best-fitting model yields a χ² of unity.

A critical requirement of the aperture masking design is that each pair of sub-aperture creates a unique interferometric baseline. The lack of baseline redundancy ensures that any spatial frequency measured can be traced back to the interference of a unique pair of sub-apertures (Haniff et al. 1987; Roddier 1986). Closure phases constructed by non-redundant baselines will be less affected by pupil-plane phase which would otherwise distort measurements of the spatial frequency phase. When two or more baselines contribute to the same spatial frequency, the power adds partially incoherently depending on the phase difference of each contributing baselines (e.g., Figure 1(b)). A random component will be introduced into the resulting spatial frequency phase which cannot be removed by closure phases (yielding a so-called non-zero closure phase). This component, termed redundancy noise, is largest when the redundant baselines are incoherent and zero when the they are coherent. Readhead et al. (1988) provide an extensive treatment of redundancy noise for seeing-limited imaging.

The mask cannot be entirely non-redundant. The finite sub-aperture size means that baselines are redundant at least within a sub-aperture (Figure 1(c)). With sub-aperture scale correction provided by the AO system, this sub-aperture redundancy noise is largely removed (Tuthill et al. 2006), as compared to the uncorrected case, but still gives rise to closure phase measurement errors due to the remaining spatial incoherence within the sub-aperture. In completely analogous fashion, temporal variations to the baseline phase during a single exposure create temporal redundancy which also give rise to closure phase errors (again, see Readhead et al. 1988).

It may be illuminating to contrast the uncorrected and corrected cases. With uncorrected observing, redundancy restricts sub-aperture sizes to smaller than the characteristic size of atmospheric turbulence and exposure times shorter than the atmospheric coherence time. AO removes both constraints since good correction supplies a stable, mostly coherent wavefront across the full aperture.

Current NRM masks are designed with sub-apertures on the order of the AO actuator spacing; this need not be the case and will likely not be so in upcoming extreme-AO aperture masking experiments. Atmospheric and AO residuals decorrelate on timescales much shorter than the exposure length and thus likely only contribute to the stochastic variation of closure phases from one image to the next. Changes in seeing between target and calibrator observations change the magnitude of the stochastic variability of closure phases (the signal to noise of the measurement), but do not introduce calibration offsets (Roddier 1986), and can thus be minimized by additional exposures. This is precisely why closure phases provide a robust measurable unlike visibility amplitudes, which are poorly calibrated if seeing changes (Tuthill et al. 2006).

Quasi-static instrumental wavefront errors contribute to all spatial scales and vary on timescales of tens of minutes (e.g., Bloemhof et al. 2001; Hinkley et al. 2007), or with movement of the telescope (Marois et al. 2005). As long as these errors remain static over a single exposure, large-scale wavefront changes (those larger than a sub-aperture) are removed by closure phases. One of the great advantages of using non-redundant masking to mitigate the quasi-static imaging problem is that closure phases require only a single image (they "self-calibrate"; Cornwell 1989). Instrumental errors change between observations of the target and calibrator, but only those smaller scale than a sub-aperture contribute to aperture masking calibration errors.

We propose that spatial filtering the wavefront before its propagation through the aperture mask can be used to reduce the inhomogeneity of the wavefront phase within the sub-aperture and, by virtue of being more coherent, reduce sub-aperture redundancy noise. Figure 2 illustrates the effect of an optimized pinhole spatial filter (discussed in the next subsection) to smooth the phase of a simulated AO corrected wavefront (Strehl ratio ∼50% in K_s). The portion of the wavefront sampled by each sub-aperture is much more uniform after spatial filtering.

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We anticipate that spatial filtering can lead to several measurable improvements of the closure phases, including decreases in systematic calibration error and stochastic variation from one image to the next. Measured visibility amplitudes, by virtue of their dependence on sub-aperture inhomogeneity, can show less degradation and be more robust to changes in seeing.

One possible implementation of the pinhole spatial filter, reminiscent of the four planes of a coronagraph (see Ferrari 2007 for a review), is shown in Figure 3. The pinhole, positioned at the center of the field and in what is usually referred to as the coronagraphic plane, blocks light beyond its inner transmission region. This acts to spatially filter the wavefront before entering the non-redundant aperture mask, located in a re-imaged pupil plane. (The mask location is equivalent to the Lyot-plane of coronagraphy.)

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The transmission profile of the pinhole is a top-hat function, with diameter d_spf which will be expressed for convenience in units of λ/D, with D the diameter of the telescope. At the location of the aperture mask, this is equivalent to the wavefront convolved by a Jinc function⁶ of characteristic diameter λ/d_spf.

Efficient spatial filtering aims to make the wavefront uniform within one sub-aperture of the non-redundant mask, and so one wants the characteristic diameter to be matched to the diameter of a sub-aperture, suggesting

$$\begin{equation} d_{{\rm spf}} \approx \frac{D}{d_{{\rm sub}}} \hbox{ (in units of $\lambda /D$).} \end{equation} \tag{ 1 }$$

A larger pinhole decreases the impact of the spatial filtering; a smaller pinhole restricts the field of view and begins to impinge on the signal we wish to measure.

This technique differs from the form of spatial filtering implemented in optical and infrared long baseline interferometers, where the signal from each telescope is injected into a single-mode fiber or pinhole before beam combination and extraction of the interferometric signals (du Foresto et al. 1997). The use of single-mode fibers on long baseline interferometers has shown increases in phase estimations by a factor of two, or more in bad seeing, despite the loss of flux (Tatulli et al. 2010). The implementation of this approach with non-redundant aperture masking would require the injection of light from each sub-aperture into a single-mode fiber or pinhole before recording the interferogram. Given the simplification of the single pinhole implementation, it offers, a priori, an appealing alternative (Keen et al. 2001).

2.3.1. Aperture Masking through a Pinhole

Given that the aperture mask is located in a pupil plane, the PSF of masking interferometry is invariant to target position over a wide field of view (several arcseconds). With the pinhole filter in place (in an image plane), the PSF and system response vary with target position relative to the pinhole, a smaller effective field of view.

The Appendix shows that the use of the pinhole filter alters the measured complex visibility of a point source:

$$\begin{equation} V_{{\rm spf}}(\textbf {\em b}) = T(\textbf {\em b}) \int d{\bm \theta }\Pi [{\bm \theta }/d_{{\rm spf}}] \times e^{2\pi i {\bm \theta }\cdot \textbf {\em b}/D} \times {\rm PSF}[\theta -{\bm \alpha }], \end{equation} \tag{ 2 }$$

where ${\bm \alpha }$ is the location of the point source on the sky (measured in angular units of λ/D), and ${\bm \alpha }=0$ corresponds to perfect alignment of the source in the pinhole. The baseline being measured is $\textbf {\em b}$, corresponding to a spatial frequency $\textbf {\em u}=\textbf {\em b}/\lambda$. The complex visibility is usually expressed as a function of the spatial frequency; here we present it as a function of baseline and wavelength for clarity. The telescope transfer function, $T(\textbf {\em b})$, gives the spatial frequency response of the aperture or aperture mask in terms of the baseline. It is defined explicitly in the Appendix. The top-hat function, $\Pi [{|{\bm \theta }|}/{d_{{\rm spf}}}]$, is equal to one for $({|{\bm \theta }|}/{d_{{\rm spf}}}) < ({1}/{2})$ and zero otherwise.

From three baselines vectors, $\textbf {\em b}_1$, $\textbf {\em b}_2$, and $-\textbf {\em b}_1-\textbf {\em b}_2$, a closure phase is the argument of the product of the complex visibilities:

$$\begin{equation} \phi _{{\rm cp}} = \arg \left[ V(\textbf {\em b}_1)V(\textbf {\em b}_2)V(-\textbf {\em b}_1-\textbf {\em b}_2) \right]. \end{equation} \tag{ 3 }$$

The observed visibility amplitudes and closure phase change by an amount that depends on the position of the target with respect to the pinhole (its alignment) and the PSF. (e.g., Figure 4). The transmission of the pinhole blocks high spatial frequency content of the wavefront, even in the absence of wavefront errors. In this way, removal of the higher spatial frequencies can alias into changes in lower, baseline frequencies, similar to the effect observed by Poyneer & Macintosh (2004). Those on sub-aperture scales compete with closure phase measurements.

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As the PSF varies, so too will the aliased phase errors. This introduces a stochastic component to the closure phase errors which appears to undercut the effectiveness of the pinhole. A portion of the phase errors will also remain fixed and be strictly a function of the asymmetrical truncation of the target and its mean PSF. The latter we term misalignment error to distinguish it from other calibration components. Generally speaking, both will increase as the target is positioned further from the pinhole center.

Qualitatively, we can illustrate that the asymmetrical truncation of the target shifts the visibility and closure phase. Consider Figure 4(c). The visibility phase is influenced by the position of the target on the sky; the asymmetrical truncation of the target shifts its center of light, and so too the visibility phase. While closure phases are normally invariant to the position of the target, the pinhole breaks this invariance. The overall isotropy of the PSF and circular pinhole shape suggests that the misalignment error is symmetric about the origin. That is, just as aligning the target at two diametrically opposed positions will induce equal and opposite shifts in the center of light, the overall symmetry suggests equal and opposite phase errors as well.

The aliased phase errors are small as long as the majority of the flux resides within the pinhole and only competes with the closure phase measurements if the power aliases back into sub-aperture scale wavefront deviations. For this reason, shorter wavelengths will be preferentially advantaged due to their small PSF, assuming that the Strehl ratio is the same at each wavelength. Typically, AO residuals will also be larger for shorter wavelengths (producing a more substantial halo) and this partially counteracts the shorter wavelength advantage. More generally, this suggests that to minimize aliasing, the pinhole could be optimized to filter only frequencies that are not corrected by the AO system; in that case the pinhole size would be matched the AO actuator size instead of the masking sub-aperture size.

Each target of a binary will be truncated differently. The misalignment error can be calculated directly by replacing the single-star PSF in Equation (2) or Figure 4 with the binary image. This will be approximately the same as the error of each component individually. In other words, the misalignment error of the primary added to the misalignment error of the secondary multiplied by the secondary–primary contrast ratio. Assuming that the misalignment error is point symmetric about the origin, aligning the binary center of light near the pinhole center appears to be a viable strategy for minimizing the misalignment error.

If the misalignment component is large enough as to become the limiting component in closure phase noise, there are three approaches which can be employed for removing the component from science data. Empirically, the component can be calibrated by observing a single star aligned at the same location as the primary under similar AO performance. In practice, this may be limited by the accuracy with which one can repeatedly (visually) align a target within the pinhole, and the component arising from the (unknown) companion remains. Computationally, if the location of the primary is known along with an estimate of the PSF, then the misalignment component can be included in the mathematical closure phase model that is fit to data. This approach requires a direct image of the primary within the pinhole to determine its location and estimate its PSF. Both may detract from the overall efficiency of observing. Finally, a third option is to take several sets of aperture masking data with the target (center of light) at various alignments (i.e., dithering) close to the pinhole center. Specifically, if diametrically opposed alignments have misalignment errors opposite in sign, then the component could be averaged out.

In this course of our investigation of the pinhole filter, we also characterize the use of a tapered window function on masking images to reduce the impact of various noise sources and small-scale wavefront errors on closure phases. For example, we use a super-Gaussian function, exp (− kx⁴), with our experiments for its flat-top and quickly tapering Fourier transform. For concreteness, we will use this same function as an illustrative example here (Figure 5). The window function is described by the size of its half-width at half-maximum (HWHM) measured in pixels on the detector or in units of λ/D.

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Multiplying the images by a window function that retains the intensity of the interferogram core but tapers at its edges greatly reduces the contamination of detector read noise and dark current, or scattered light from nearby targets not of scientific interest. Per-pixel Gaussian-distributed read noise, when Fourier transformed, results in a per-baseline Gaussian-distributed uncertainty. The per-baseline noise has a magnitude that is directly related to the total transmission of the window function. Therefore, a tighter window function removes more read noise.

However, complex visibilities are extracted from the Fourier transform of the image, so the application of a window function is identical to the convolution of the complex visibilities with the Fourier transform of the window function. This mixes the complex visibilities of the baselines and, ultimately, restricts the minimum window function size.

The aperture mask is designed to be non-redundant for baselines extending between the centers of sub-apertures; the finite size of the sub-apertures give rise to islands of transmitted power in the power spectrum, which we call splodges (see Figure 5, bottom left). The splodges of a completely non-redundant mask do not overlap, and hence the separation between neighboring splodge peaks is about 2 d_sub. (In practice, some overlap at the splodge edges may be exchanged for better coverage or larger sub-apertures.) Furthermore, neighboring splodges may arise from completely separate pairs of sub-apertures (i.e., separate baselines), and one would not assume any coherence between the baseline phases. For this reason, window functions narrower than about λ/2d_sub, or those without quickly tapered Fourier transforms, will mix the incoherent baselines of neighboring splodges and rapidly increase the error of closure phases, similar to redundancy noise.

The optimally sized window function will balance the reduction of read noise with the increase of redundancy noise.

Post-processing with a window function also reduces the impact of small-scale wavefront errors. This is most readily recognized by noting that the outer rings of the PSF encapsulate the high spatial frequency content of the wavefront; removing this flux also removes the high spatial frequency content of the wavefront.

This method of applying a post-processing window function to spatially filter differs in two major ways from implementing an on-telescope pinhole. Applying filtering in post-processing allows easy tailoring of the shape of the window function (including suppression of the tails of its Fourier transform), a task considerably harder with an apodized pinhole. The pinhole, however, filters the wavefront phase noise at each instant; the window function only filters some form of the time-accumulated phase noise.

Studies by Ziad et al. (2004) and Linfield et al. (2001) conducted with the Palomar Testbed Interferometer (PTI) and Hale 200'' Telescope confirm that the atmospheric turbulence power spectrum is approximately Kolmogorov with an outer scale most often within the range 10–50 m. The median-seeing Fried parameter (r₀) has been measured to be 9.0 cm at 550 nm, dropping to 3.8 cm during bad, but regularly observed seeing (Dekany et al. 2007).

Advection (wind) speeds drive the evolution of atmospheric structure within the sub-aperture on intervals between 0.1 and 1.0 s. Using measurements of the Palomar atmospheric temporal structure function over four nights, Linfield et al. (2001) derive wind speeds typically less than 4 m s⁻¹. For shorter timescales, the temporal structure function decays exponentially. The characteristic decay time, which scales with λ^6/5, has been measured in two surveys to vary between 15–80 ms (Ziad et al. 2004) and 60–150 ms (Linfield et al. 2001) in K_s.

Our simulations assume an outer scale of 50 m and Fried parameter value 9.0 cm, which correspond to 48 cm in K_s and 32 cm in H. We use wind speeds of 5 m s⁻¹ and temporal decay time of 60 ms.

We generate time-evolved Kolmogorov phase screens following the procedures of Lane et al. (1992) and Glindemann et al. (1993). These phase screens are characterized by three parameters: the Fried parameter of an instantaneous phase screen, the wind velocity which blows the static phase screen across the aperture, and an additional parameter driving the decorrelation of high spatial frequencies between time steps. From this last parameter emerges the exponentially decreasing temporal structure function for short timescales.

A single 431 ms exposure image is constructed by adding 24 sub-images, each of which is an instantaneous snapshot separated in time by 18 ms. We chose this time step to sufficiently sample the atmospheric evolution over relevant timescales: the wind sub-aperture crossing time is 84 ms; the coherence decay time is 60 ms.

The generated images are 512 × 512 pixels, designed to preserve the pixel scale of the PHARO detector. In frequency space, the 5.08 m aperture spans slightly less than 256 × 256 pixels in the K_s band. Phase screens of this size are generated; this corresponds to an inner scale of approximately 2 cm and sub-apertures approximately 450 pixels in area. We repeated several simulations with four times as many pixels and arrived at similar results; from these results we conclude that our simulation is well sampled.

The simulation generates one sub-image for each phase screen using an incoming monochromatic wavefront perturbed by the screen. After passing through the telescope aperture, the wavefront is corrected by AO, then propagates through the spatial filter, the aperture mask, and, finally, forms an image on the detector. We ignore read noise and photon noise. This simulation is similar to that used by Sivaramakrishnan et al. (2001).

We model the AO system as an instantaneous high-pass filter of the form A(k) = 1/(1 + (k_c/k)²), with a cutoff imposed by the Nyquist frequency of the actuator spacing, k_c = N_act/2D (about 1.4 cycles m⁻¹ at Palomar). This filter is applied to the Fourier transform of the phase screens and accurately reproduces the wavefront residuals observed under optimal operation at Palomar (Dekany et al. 2007). This overestimates the typical AO performance on faint targets, particularly the performance of tip-tilt suppression. To account for this, we apply the following modified AO filter function, which degrades the low spatial frequency AO response:

$$\begin{eqnarray} A(k) &=& \min \left[ \frac{1}{ 1 + (k_c/k)^{2}}, 0.05 \right], \nonumber\\ k_c &=& N_{{\rm act}}/2D. \end{eqnarray} \tag{ 4 }$$

To include the polychromatic effects, each wide band transmission filter is divided into a number of subintervals (generally four). A monochromatic image is generated for each subinterval (taking into account wavelength-dependent effects), and these images are added to form a single polychromatic sub-image. The sub-images are co-added to create a single 431 ms exposure.

This produces direct imaging Strehl ratios of 40%–50% in the H band and 60%–70% in K_s.

The PHARO instrument (Hayward et al. 2001) is especially suited to a pinhole implementation. PHARO has been designed with coronagraphic capabilities, giving access to both a focal plane and a pupil plane before the final focal plane (Figure 3).

The pinhole is placed in the focal plane before the aperture mask. Based on our simulations from the next subsection, a pinhole of angular diameter 0.779 arcsec was installed at the Lyot Stop in PHARO in 2008 June. This pinhole was chosen for optimal use at 1.6 μm (H band), corresponding to an angular size of 12 λ/D. The pinhole is of size 8.7 λ/D at 2.2 μm (K_s band).

With sub-aperture sizes of 0.42 m, the estimate from Section 2.3 suggests the optimal pinhole size to be D/d_sub ≈ 12λ/D. Using the simulation of the Palomar aperture masking experiment described in Section 3.2, we can determine the optimal pinhole size under typical observing conditions.

For various pinhole sizes, we simulated 100 H-band images of a 10:1 binary with companion separation 150 mas under typical Palomar observing conditions, without read noise, and with the primary aligned at the center of the pinhole. Images of a single star were simulated to be used as a calibrator. We analyzed these images both with and without a window function, and calculated the root-mean-squared (rms) residuals of the simulated closure phases fit to a model binary. Figure 6 shows these results for each pinhole size (with window function, solid line; without window function, dashed line). For reference, the rms obtained with no pinhole in place is included as a horizontal line.

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The spatial filter performance can be broken into following three classes:

• 1.  
  A small pinhole (≲10 λ/D) impinges on the field of view, rejecting enough of the light coming from the off-axis companion so that the closure phases will not match the model. (See also Section 2.3.1.)
• 2.  
  A very large pinhole (≳20 λ/D) provides very little filtering and the data are statistically similar to the unfiltered case.
• 3.  
  The operational pinhole range (10–20 λ/D) reduces the image to image variation of closure phases and produces data that better fit its model. The range 11–14 λ/D is most effective, reducing the fit residuals by roughly 25%.

These results agree with the estimate at the top of this subsection.

Finally, we note that because the window function itself provides some spatial filtering, the inclusion of the pinhole provides slightly less improvement if compared to the case in which no window function is used.

In Section 2.3.1, we showed that the asymmetrical truncation of the target by the pinhole alters the measured closure phases, even in the absence of wavefront errors, an effect we called misalignment error.

Equation (2) can be directly integrated to determine the misalignment error when a target is observed through the pinhole. Alternatively, we chose to simulate the effect to more accurately reflect the details of our analysis pipeline. For clarity, we present the rms closure phase deviation between the pinhole and non-pinhole values with the Palomar nine-hole mask (averaged over the set of 84 closure phases) in Figure 7.

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By inspection, we see that in the H band and at high levels of correction (Strehl ≳40%), the pinhole introduces less than 03 rms closure phase error at nearly any alignment within the 440 mas diameter pinhole. At a diameter of 12λ/D, less than 2% of the total flux resides outside the pinhole and very little aliases back into the baseline frequencies. The misalignment deviation is also insensitive to the orientation of the misalignment; the aperture mask is three-fold symmetric in physical space, but provides a uniform coverage of spatial frequencies. At moderate Strehl ratios, the misalignment deviation increases, but observationally, this increase is balanced by a larger benefit of spatial filtering.

The alignment requirements are tighter in K_s. Because the pinhole is designed for the H band, its size is only 8.9λ/D in K_s. Good positioning is even more important given that AO residuals and closure phase errors are usually smaller. Alignment can introduce up to 10 of closure phase errors if misaligned by 200–300 mas, or roughly 2–3λ/D.

The presence of a companion adds an additional misalignment error, although this error will be attenuated by the secondary–primary contrast ratio. Therefore, particularly for high-contrast binaries, the misalignment errors provided here give good rule-of-thumb indications of how well a target must be positioned within the pinhole. Given that aperture masking observations at Palomar typically yield 1°–2° of closure phase, targets must be positioned close to the pinhole center during K_s-band observations.

Finally, we note that the percentage of blocked flux is a weak function of the target alignment, and a strong function of the AO performance and size of the direct imaging halo (Figure 7, right panel). In the observations of Section 5 that compare pinhole and non-pinhole observations, the percent loss of flux is a measurable quantity. From the results of this plot, it is clear that large flux drops can be attributed to momentary drops in AO performance or very large shifts in alignment. Astrometric jitter on the scale of hundreds of mas is never seen. Practically, if a series of images contains one or a few with large flux drops, and the target was initially well aligned, then identifying large flux drops is a useful proxy for excluding frames taken with poor AO performance.

For our experiment, we use a super-Gaussian function, exp (− kx⁴), for its flat-top and quickly tapering Fourier transform. We describe the window function by the size of its HWHM, w_pix, measured in pixels on the detector.

The optimal window function size finds the balance between eliminating read noise and increasing redundancy noise. We present a measure of the effectiveness of various window function sizes in the presence of various levels of read noise in Figure 8.

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We simulated 100 images (256 × 256 in size) of a 10:1 binary with companion separation 150 mas and of a calibrator in the same fashion as described in Section 4.2 but without a pinhole in place. To each set of images we added Gaussian read noise with a per-pixel noise level ranging from 0.2% to 5.0% of the peak intensity. (This corresponds to targets of 7th to 10th magnitude when taking 6 s exposures on the PHARO detector.) The images were processed with window functions of various sizes and we calculated the rms residuals of fits to a model binary. In Figure 8, we plot the ratio of the measured rms with a window function to the same set of images without the window function. For all levels of read noise, an optimally sized window function improves the model fits. We caution that, because the effect of read noise on closure phases scales with image size and the choice of default image size is arbitrary, the results can only be evaluated qualitatively.

Several features are apparent. First, the optimal window function is approximately λ/d_sub, or ∼12λ/D for the Palomar aperture mask, with tighter windows for the high-noise cases; one expects the optimal window size to be a function of the aperture mask. This makes qualitative sense: the interferogram image is a set of interference fringes under an Airy function envelope of characteristic size λ/d_sub. One would expect the optimal window function to crop out those pixels with signal to noise less than one. Beyond λ/d_sub, the intensity of the interferogram drops below a few percent of its peak value, comparable to the read noise.

Second, too small a window function quickly adds redundancy noise into the measurements. This turning point is near 0.5 λ/d_sub.

Third, even in the absence of read noise, a window function decreases closure phase noise (solid curve). This demonstrates the capacity of the window function to spatial filter the wavefront phase noise, even though the improvement is only about 3%–4%.

To emphasize the utility of the window function as a spatial filter of wavefront errors, we simulated another set of images in which the wavefront is static over the exposure. With only spatial variation of the wavefront, the window function reduces closure phase errors by nearly 20% (Figure 9).

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By comparing these results to those obtained with an optimized pinhole (25%), it is clear that the pinhole and window functions only provide comparable spatial filtering when the wavefront phase errors is static during an exposure. When accounting for the more realistic, dynamic wavefront, the pinhole provides superior spatial filtering.

Misaligning the peaks of the window function and interferogram introduces a tiny amount of closure phase error. Even an unrealistic misplacement of 10λ/D (about 26 pixels on PHARO at 1.6 μm) with a tight window of size 0.7 λ/d_sub produces an error below 006. This will likely be relevant for observations taken by space telescopes.

Images of the pinhole were taken at several periods throughout the run from which its location on the detector can be measured. The location of the pinhole remained accurate to less than two pixels for the entire run, with the exception of two instances in which the Slit wheel appeared to lodge the pinhole at an alternative location. This was easily repaired by putting the Slit wheel into its "HOME" position.

Explicit (direct) images of the target within the pinhole were taken only rarely. However, the locations of the interferogram centers provided an accurate location of the binary targets on the detector. Drift of the target was typically less than one-half pixel during a set of 20 images (about 2 minutes). Comparison of the target and pinhole locations indicate that our alignment was accurate to within four pixels of the measured pinhole center. We conclude that the largest potential source of misalignment error arises from the initial pointing accuracy within the pinhole, and not jitter of the target or pinhole during data taking.

The effect of the pinhole was seen to alter several observables. Most directly, when observing reference stars, the pinhole induced a drop of target flux by roughly 35% in H and 15% in K_s. The pinhole is located after the AO system in the focal plane, hence these values reflect the percentage of the direct imaging PSF which falls outside the pinhole. These percentages are consistent with the pinhole size and the level of AO performance (Strehl ratios of approximately 10% in H and 30% in K_s). Binary targets show a similar drop in flux despite their larger size, which we can take as an indication that their center of light is well aligned.

Simulations and calculations using a sample of direct images show that the percentage of blocked flux is a weak function of the target alignment, and so large flux variations do not indicate a high level of astrometric jitter. Instead, this percentage is a strong function of the AO performance (and size of direct imaging halo). Low transmission (below 40%) well identified individual images blurred by temporary drops in AO correction. High flux transmission well predicted good fits to models (e.g., Figure 10).

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The use of the pinhole generally reduced the variance of closure phases and increased the amplitudes for the observed single stars. These results are compared to a set of simulated observations in Figure 11.

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The closure phase standard deviation is reduced by 10% and 19% in the H band (1.6 μm) and the K_s band (2.2 μm), respectively. A larger reduction is expected at the longer wavelength, as the pinhole is relatively smaller and provides more aggressive spatial filtering. Although not displayed, the rms residuals of the data fit to their model values also reduced by 10%–20% indicating better calibrated data. The pinhole filtering increases the visibility of log-amplitude by 14% and 18% on the longest baselines at the H band and the K_s band, respectively. Each of these confirm that the spatial filter is reducing closure phase noise from sub-aperture redundancy.

In both cases the simulation predicts a larger reduction in variance, although the results across wavelengths are consistent. The simulation does not include the time-variation of AO residuals during a single exposure, particularly slow tip-tilt correction of the PALAO system (Bloemhof et al. 2001). This can produce several interferograms in a data set with large closure phase errors which cannot be improved by spatial filtering.

We observed four previously characterized binaries with well-defined orbits with and without the pinhole in several bandpasses: GJ 164 (Martinache et al. 2009), G 78-28 (Pravdo et al. 2006), GJ 623 (Martinache et al. 2007), and GJ 802B (Ireland et al. 2008). The characterized orbits were used to predict the location of each companion on the observing date using a Monte Carlo simulation to account for the uncertainty of the orbital parameters. Each target was observed to obtain 20 images (6 second exposures) in several bandpasses in 2009 September. Three of the four known binaries were successfully resolved (Table 1). The high-contrast companion to GJ 802B could not be resolved at the correct separation and contrast ratio. The quality of each set can be quantified by the rms of the difference between the data closure phases and the best-fitting closure phase model of a binary. The astrometry of the best-fitting binary model can also be compared to the astronomy predicted by the orbital parameters in the literature.

GJ 164 was imaged for three sets near dawn with the pinhole and one set without. All but the first sets of data suffered poor AO correction, were visibly less sharp, and had flux levels drop by nearly 75%. The binary could be well resolved in all three sets with the pinhole, but the latter sets yield much worse fits and larger systematic errors and are not used in the analysis. The fit parameters to GJ 164 share a degeneracy with a spurious set of parameters (ΔH = 0.030, at nearly the same location), and limit the quality of parameter errors which can be derived from the fit. Instead, the magnitude contrast was held fixed to the values determined by Martinache et al. (2007). The observations of reference stars to calibrate GJ 623 varied in AO correction and quality between pinhole and non-pinhole measurements, and so this target is presented without calibration in order to compare the two performances.

The median closure phase scatter (stochastic errors) decreased for each observation by roughly 20%–25%, which indicates that the pinhole operated effectively to minimize closure phase errors from AO residuals. However, the data fits to binary models showed no decrease in rms residuals, whereas the fits to single-star data decreased by 10%–20%. Of the eight total measurements, four showed an increase in rms, although only two K_s-band observations increased by more than 10%. The observations that increased in rms are those in which either the companion was aligned far from the pinhole center or the companion is comparatively bright. We prefer to discuss this measurement in terms of the amount of unattributed systematic error that needs to be added in quadrature to the stochastic errors to yield the rms values, i.e., (rms)² = 〈σ²_ϕ + σ²_sys〉. This is listed in the next to last column of Table 1. A systematic component of 10 specific to binary targets would account for the discrepancy of performance.

Misalignment of the target within the pinhole contributes to the increased residuals. Because each binary system could be resolved by direct imaging, the targets were aligned with their center of light as close to the pinhole center as could be accurately judged by eye. This necessarily placed each star off-center of the pinhole. The astrometric alignment of each component can be inferred from the center of the interferograms and is listed in Table 2. As discussed earlier, the alignment of the components is determined by the original placement of the observer and moves comparatively little during a set of images.

We simulated each binary with and within the pinhole displaced by the amounts in Table 2. The atmospheric seeing was tuned to fit Strehl ratio estimates from direct images taken throughout the night: 40% in K_s and Brγ bands, 15% in H and CH4_s bands, and 10% in the J band. The astrometric jitter of the simulated images was consistent to those observed (and less than 20 mas per 20 images for most targets). Each simulation produced enough images until all errors reached a steady state; the estimates of the misalignment error are included in the last column of Table 1. Additional contributions from misaligned calibrators are not included.

Our calculation of the misalignment error requires knowledge of the absolute positions of the pinhole and target components, and an accurate measure of the PSF. Given the small jitter of targets, our calculation is limited by imperfect knowledge of the PSF. Even still, the quality of correction can fluctuate in timescales of minutes, as can be observed by viewing successive direct images or other observables such as pinhole transmission (Figure 10). Given these assumptions, misalignment contributes systematic components of 04–08 to the H- and K_s-band binary observations. We also note that simulations demonstrate large ≳ 10 alignment errors for slightly lower Strehl ratios (≲10% in H and ≲20% in K_s). These challenges instead warrant the development of an observing strategy to calibrate alignment errors empirically.

The binary parameters measured with and without the pinhole, and those predicted by the system orbits, are in good agreement. In each case, the spatial filter data fit to slightly higher contrasts ratios (≈5%). This may indicate that the companion flux was skewed closer to the pinhole edge and its flux truncated by a few percent more, which is consistent with the measured locations of the objects within the pinhole (Table 2). No bias was found in the relative astrometry.

Generally speaking, the lower closure phase errors brought on by the pinhole filter ought to lead to proportionally lower errors in binary parameters and higher contrast of detection. The pinhole demonstrably decreases closure phase measurement error by 20%–25% and indicates that, if the systematic component can be minimized, the pinhole can provide an increase of contrast by 20%–25% using the same length of observation. However, the misalignment error limits the increase in precision. In practice, we scale the closure phase measurement errors until the χ² of the best fit is unity to account for any known systematic errors. Therefore, to maintain the increased precision in this analysis, the decrease in closure phase measurement error must be matched by a similar decrease in rms fit residual (or systematic component). Since the fit residuals did not improve, the precision in the binary parameters did not improve in all cases.

In light of this investigation, the pinhole can prove to be effective in two scenarios. When seeing or correction is poor, and closure phase observations do not reach the calibration limit, a pinhole filter will clearly provide higher signal-to-noise closure phases and more efficient data taking.

At the other extreme, the pinhole may be valuable for improving very high Strehl ratio (≳80%) observations with extreme-AO systems. The high spatial frequencies filtered by the pinhole contribute to a larger proportion of the overall wavefront errors in this scenario. The pinhole could be optimally tuned to the AO actuator size, much larger, and alignment requirements would be substantially relaxed. Most importantly, the aliasing of high spatial frequency content into sub-aperture scale phase errors is a higher order effect of the wavefront (Poyneer & Macintosh 2004). At very high levels of correction, the aliased power is a smaller component of closure phase errors.

We can arrive at a rough estimate of a pinhole performance on extreme-AO systems by simulating a higher-order AO system, such as the 3368-actuator Palomar 3000 system (P3K; Dekany et al. 2007). As mentioned earlier, our simulation does not account for the sluggish tip-tilt correction of the current Palomar system and overpredicted the pinhole performance as a result. The P3K system will have upgraded response by using the current 241 actuator system to perform low-order correction, and thus provide more stable correction. Using the specifications of this system, we simulated a 10:1 binary separated by 150 mas in the H band under typical conditions (see Section 4.2). A pinhole of size 12λ/D decreased stochastic closure phase errors to 20% nominal values, i.e., a five-fold increase in signal to noise. When compared to our simulations of the current AO system (a two-fold increase), the pinhole can impact extreme-AO observations favorably.

For very high Strehl ratio observations, misalignment error will be lower and less sensitive to changes in correction (019 in this example). Empirically calibrating misalignment by observing a reference star at the same position as the primary is only a partial solution (as it does not take into account the companion), likely limited by the pointing accuracy of the observer, and decreases the overall efficiency of the observations. Instead, observations should be taken by dithering the center of light around the pinhole center to minimize this systematic component. Using diametrically opposed target placements successfully reduced the misalignment component to 011 in simulation, a reduction of nearly half.

Finally, one must also consider the optimal pinhole size to survey across the near-infrared bands. For example, if the effective range of pinhole sizes is narrow enough (e.g., 11–14 λ/D for this experiment) a single pinhole may not accommodate multiple wide band infrared filters. In those cases, multiple pinholes tuned to specific observation bands will be most effective.

Next generation AO systems with aperture masks, such as Palomar 3000 (Dekany et al. 2007) and Gemini Planet Imager (Macintosh et al. 2008), provide actuator densities high enough to correct wavefront errors on sub-aperture scales. Design and optimization of these aperture masking experiments that balance sub-aperture size and number against AO correction and the limitations of systematics require a more detailed understanding of how sub-aperture wavefront errors impact closure phases, and where resources should be focused. Extreme-AO systems are anticipated to provide highly stable correction down to scales of 8 cm and thus may motivate smaller and additional sub-apertures, providing more spatial frequency coverage and resolution. This balance will depend on the spatial and temporal power spectra of the corrected wavefront residuals. These data should be accumulated and their impact on closure phase errors should be examined.