An Analysis of Fundamental Waffle Mode in Early AEOS Adaptive Optics Images¹

The nature of the smallest spatial scale waffle mode can be deduced from inspecting a "unit cell" of the WFS array. We assume a WFS subaperture that has four DM actuators, one at each corner of a square (see Fig. 1). The relationship between the wave‐front slope across this subaperture and the heights of the wave front at the four corners of the square contains the key to the waffle‐mode error, since the DM actuators at these corners will need to be moved by exactly half these heights to flatten the wave front across the subaperture. Of course the actuators could be placed anywhere relative to the subaperture grid, although real AO systems typically try to maintain the DM actuator‐to‐pupil alignment we describe if the actuator spacing and geometry matches the subaperture geometry. We stress here that waffle‐mode error does not depend fundamentally on the actuator location and geometry; it only depends on the sensor geometry.

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We label the corners of the subaperture with mnemonics generated from the cardinal directions, assuming north points toward the top of the figure. The corner heights are therefore h = (h_NE,h_NW,h_SE,h_SW). We assume that the wave front is flat across the subaperture, so we can immediately deduce the way our slope measurement s constrains the wave‐front heights at the corners:

where d is the subaperture spacing as well as the interactuator spacing. These are the two equations of condition of the problem. This relation between the four unknowns h and the two measured quantities s can be re‐expressed as

We use our only two observable quantities, s_x and s_y, to relate the heights across the diagonals of the subaperture. This leaves in these equations two undetermined quantities relating our unknown heights to the two data points; the equations of condition are rank deficient by 2.

It is not possible to determine all four heights in h with just two observables. If we face the engineering problem of having to command four actuators based on the data contained in s, we will have to add two arbitrary equations to the defining equations of the problem to solve them. That is, we have to add two more equations to the system of equations in order to obtain a solution for h. One such equation is easy to come up with: the physics of the problem indicates that the mean value of the four heights can be any value we choose, as it is the piston of the wave front. We can therefore add the constraint equation

to the equations of condition. Here P, the mean height of the wave front, is arbitrary.

Our original defining equations are difference equations, so it is scarcely surprising that the offset of the heights is unknown. Wave‐front slope data alone cannot determine the piston of the wave front. Let us assume a zero piston in our case (in any real system, we use common sense to set the mean voltages sent to the DM actuators):

We still need one more equation without data—a constraint equation—to solve our problem. This is the equation that determines how much waffle we introduce into our very simple AO system. So far we have set the slopes across the unit cell diagonals using data, and the mean piston using common sense. We must now guess how the means of the two sets of diagonally related actuators differ from each other, in the absence of new information (data). If the wave front is relatively flat across the subaperture (e.g., d≪r₀, where r₀ is the Fried length; Fried 1966), we can assert that the means are equal. This statement, expressed as the equation

adds the equation required to solve the system of equations relating our two WFS slopes to our four actuator heights completely. It determines how much waffle mode present in the atmosphere we let slip through our AO system, on average. Equation (3) is an "equation of constraint," since it does not involve any data. We note that ellipticity of the individual WFS subapertures' spots can yield data on the waffle‐mode error, although no AO system we know of uses this information in practice.

A singular value decomposition (SVD) method is often used to solve the underdetermined set of equations between WFS data and actuator heights. The physics of what really happens when we perform a wave‐front reconstruction with SVD is not apparent; this method does not distinguish between constraints and equations of condition.

Here we extend the description in § 2.1 to the simplest case in which we have more data than unknowns. For this we use a square aperture. While this is obviously unrealistic, our analysis of this case can be applied to arbitrary aperture geometries and leads to an understanding of the cause of unsensed modes in wave‐front reconstructors that use slope measurements as input to their algorithms.

Any Fried geometry WFS system on a simply connected aperture will possess two families of actuator heights. Extending the unit cell to a case in which there are more data points than unknown actuator heights demonstrates this. The N_s WFS subapertures produce 2N_s slope measurements, since each subaperture yields an x and a y slope. With a square aperture, the simplest such case is that of a grid of four actuators on a side, enclosing a square of subapertures three to a side (see Fig. 2). We must deduce 16 actuator heights using 18 pieces of slope data. On the face of it, the problem appears to be overdetermined, but is in fact still underdetermined. Writing out the equations that relate slopes to actuator heights explicitly, the set of equations governing one family of heights is

and the equations for the second family are

where

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These two sets of nine equations each are disjoint—no individual unknown actuator height appears in both sets. We see by inspection that we can choose any one of the first set of heights arbitrarily. Setting h₁ to a fixed number sets every actuator height in the first set. Similarly, we can choose h₂ to be any value we please. Each of the two families of actuator heights therefore has an arbitrary net piston of its own (i.e., the mean of all the actuator heights in the family). This is the common‐sense expression of the statement that the matrix describing the system is rank deficient by 2. These pistons can be individually set to be equal, thereby selecting a "waffle free" solution. This solution still has arbitrary total piston. Again, we can select a zero‐piston solution from all the waffle‐free solutions by requiring the pistons of each family to be zero:

The atmospheric phase disturbance at any given instant does not typically possess zero waffle when viewed through the WFS subapertures. That is, if one regards the square WFS lenslets (projected back to the entrance pupil) as a checkerboard, the mean phase over the black squares will not equal the mean phase over the white squares. However, as Gavel (2003) shows, for subapertures that are small compared to r₀, the net waffle will be small compared to unity (waffle‐mode error has units of radians, since it is a mean of a phase). By enforcing a zero‐waffle solution on our actuator heights, we introduce some error in our fit of the wave front. This error produces the familiar waffle pattern appearing at an angular radius of √2(λ/2d) in the image (see § 3).

Here we consider waffle‐mode error as a phase aberration ϕ_w(x,y) over the pupil and trace the effect of the waffle mode on the AO‐corrected monochromatic wave front analytically. We use the notation of Sivaramakrishnan et al. (2002), which we restate here.

The telescope entrance aperture and all phase effects in a monochromatic wave front impinging on an optical system can be described by a real aperture illumination function A(x,y) multiplied by a unit modulus function A_ϕ(x,y) = e^iϕ(x,y). The aperture plane coordinates (x,y) are in units of the wavelength of the monochromatic light under consideration (to simplify the Fourier transforms), while the corresponding image plane coordinates (ξ,η) are in radians. Here phase variations induced by the atmosphere or imperfect optics are described by a real wave‐front phase function φ that possesses a zero mean value over the aperture plane:

We chose the location of the image plane origin at the centroid of the image PSF, which corresponds to a zero mean tilt of the wave front over the aperture (Teague 1982; Sivaramakrishnan et al. 1995).

We assume that the transverse electric field in the image plane is given by the Fourier transform (FT) of the field in the aperture plane (Goodman 1995). We indicate transform pairs by a change of case, where the FT of A is a and the FT of φ is Φ.

The "amplitude spread function" (ASF) of an optical system with phase aberrations is the FT of A_ψ = AA_ϕ and is given by a_ψ = a*a_ϕ, where * denotes convolution. This complex‐valued quantity is proportional to the amplitude of the incoming wave in the image plane. The complex number's phase at any point is interpreted as a relative phase difference in the wave front at that point in the image plane. The corresponding PSF of this optical system is p_ψ = a_ψa^*_ψ, where the superscripted asterisk (^*) indicates complex conjugation (see, e.g., Bracewell 2000 for the fundamental Fourier theory and Fourier optics conventions).

For this analysis, we assume that the total phase ϕ(x,y) is given by ϕ_AO(x,y) + ϕ_w(x,y), where ϕ_AO(x,y) is the residual atmospheric phase after correction by the AO system. Thus, the unit modulus function A_ϕ(x,y) can be written as

In order to understand the effects of waffle‐mode error on the PSF, we now look at a waffle aberration without any atmospheric aberration contribution. Setting ϕ_AO = 0, the aperture illumination function reduces to

In § 2.2, we present waffle‐mode error as a checkerboard pattern in phase with a period of 2d, where d is the WFS subaperture size in the Fried geometry. In the one‐dimensional case, we write this as a top‐hat function hΠ(x/d) convolved with series of Dirac δ‐functions spaced at a period 2d. Here h is a function of the atmospheric phase ϕ_atm, defining the "strength" of the waffle function. The periodic function describing our checkerboard thus takes the form

where we use the shah function SH(x/2d) to denote a series of Dirac δ‐functions at the required spacing (Bracewell 2000). The waffle phase error ϕ_w can then be written

where D is the telescope aperture diameter in units of wavelength. The wave amplitude in the image plane due to waffle mode is the FT of ϕ_w:

The above expression is the field due to the telescope aperture convolved with a sinc function sampled at θ = n/2d, where n is an integer. However, the function sinc(d θ) is zero for each even value of n (e.g., n = 2, 4, 6, ...). Thus, this fundamental waffle‐mode error results in the replication of the perfect PSF at a lattice of points at the angular period equal to the spatial Nyquist frequency of the WFS subaperture density, the relative intensity of each point being modulated by sinc(d θ) [with half of those points falling on the zeros of sinc(d θ)].

2.3.1. The PSF Expansion

Perrin et al. (2003) developed a general form for an expansion of an aberrated PSF p in terms of the Fourier transform of the phase aberration φ over the aperture. They express the PSF as a sum of individual terms in an infinite convergent series, with the nth term of that series given by

where *ⁿ is used to denote an n‐fold convolution operator. We refer the reader to the discussion of the second‐order expansion of the partially corrected PSF in Sivaramakrishnan et al. (2002) and in Perrin et al. (2003), noting here that while the first‐order term

is due entirely to the antisymmetric component of A_ψ, it does not contribute significantly to the waffle pattern. This is because the function a(x,y) is very small at points beyond a few λ/D from the center of the on‐axis PSF, and the first‐order term is multiplied by a (see Bloemhof et al. 2001 and Sivaramakrishnan et al. 2002 for further discussion of pinned speckles). This is borne out by simulations with pure waffle‐mode phase aberrations showing the symmetric terms of the PSF expansion (e.g., the p₂ and p₄ terms of Perrin et al. 2003) dominate, even when the phase error itself is large and antisymmetric. We discuss this further in § 3.1.2.

For traditional Shack‐Hartmann AO systems and WFR, waffle‐mode phase error present in the atmosphere remains unsensed and uncorrected, and thus leaks through the system. This is also true of the spatial filter we use to represent our AO correction. However, because of our code structure, we found it easier to add the waffle‐mode phase error back into the AO‐corrected residual phase after correction rather than before. We found no difference between these two approaches.

We calculated the waffle‐mode error ϕ_w present in our incoming atmospheric phase aberrations ϕ_atm by first representing the AEOS WFS subapertures as a checkerboard pattern over our aperture, with white and black squares corresponding to the two families of DM actuators (see Fig. 3). We then calculated the mean phase of the incoming wave front ϕ_atm over the white squares (ϕ₁), as well as the mean over the black squares (ϕ₂). The zero‐mean waffle phase error present in the atmospheric phase aberration is then constructed by assigning a phase ϕ_w defined by (ϕ₁ + ϕ₂)/2 over the white squares and (ϕ₁ - ϕ₂)/2 over the black squares. We then added ϕ_w back into our AO‐corrected phase arrays (ϕ_AO), producing an AO‐corrected wave front with waffle‐mode phase error.

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Since ϕ_AO is nonzero in this case, the aperture illumination function takes the form

The PSF corresponding to this illumination function is just the square of the absolute value of the Fourier transform of a_ψ.

We show this process graphically in Figure 4 over a portion of the AEOS pupil. We calculate the amount of zero‐mean waffle phase error present in the incoming phase screen (left) over the checkerboard array of subapertures. Without waffle‐mode phase error, the AO‐corrected wave front would just be dominated by residuals with spatial frequencies beyond those correctable by our spatial filter. However, the addition of waffle‐mode phase error produces a noticeable checkerboard pattern in the final, AO‐corrected wave front (right).

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3.1.1. Waffle Strength a Function of r₀

As part of our analysis, we examined how ϕ_w varies with the Fried length r₀. We chose eight values of r₀, giving us D/r₀ between 5 and 40 for the AEOS geometry, and created 100 independent realizations of Kolmogorov‐spectrum phase screens for each. We then calculated ϕ_w over the AEOS aperture for each phase screen realization.

Figure 5 shows the expected trend of increasing variance of ϕ_w with increasing D/r₀. For cases in which D/r₀ is greater than the number of WFS subapertures, the mean measured ϕ_w can be as high as ϕ_w = ±0.1 rad. More typical values are between ±0.04 rad. As we show in § 3.1.2, waffle‐mode error starts to cause a significant reduction in Strehl ratio when phase variations due to waffle grow beyond 0.1 rad. This suggests that WFRs that do not account for changing sky conditions throughout the night or that do not penalize waffle modes may not be able to accommodate these variations and may allow waffle and waffle‐like modes to build up, effectively enhancing the atmospheric waffle present in the observations.

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3.1.2. Effects of Waffle Mode on Strehl Ratio

To analyze the intrinsic effects of waffle‐mode phase error on the observed PSF, we assumed perfect AO correction (i.e., ϕ_AO = 0), with an introduced waffle error of ϕ_w = 0.01 rad of phase difference. This corresponds to a typical value of ϕ_w present in our simulated Kolmogorov‐spectrum phase screens with the AEOS WFS subaperture geometry. The result is a nearly perfect PSF attenuated by the regular pattern of satellite PSFs characteristic of fundamental waffle‐mode error. Here we note that other waffle and waffle‐like modes will produce additional patterns of satellite PSFs in the image plane (M. Britton 2004, private communication). We note in passing that the strengths of these additional modes will depend on the details of the WFRs used.

An example of our "waffle only" simulations can be seen in Figure 6, where the waffle‐mode phase error has been increased to 0.5 rad to show the grid of satellite waffle PSFs. We note the primary, secondary, and tertiary waffle PSFs appearing at the intersection of grid lines at a distance nλ/2d (where n = 1, 3, 5, ...) from the optical axis. As discussed in § 2.3, the waffle PSFs that fall on the grid points where n = 2, 4, 6, ... hit the zeros of the sampled function sinc(d θ) and do not appear in the final image.

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We define a waffle amplitude factor to describe the enhancement of the waffle mode beyond the atmospheric ϕ_w calculated from our simulated phase screens. In reality, describes the amplification of the waffle mode due to the choice of wave‐front reconstructor, imperfect WFS flat fields, the effects of immobile DM actuators, and so on.

We produced a series of PSFs with total phase errors ranging from 0.01 to 1.0 rad (peak to valley) by varying between 1 and 100, keeping ϕ_AO = 0 and ϕ_w = 0.01. We then calculated the Strehl ratios of these PSFs by measuring the peak pixel value of the centroided PSF in the reconstructed aberrated wave fronts as compared to that from an image formed by a perfect wave front with the same image plane sampling.

We present our Strehl ratio calculations in Table 1. We see little degradation in Strehl ratio for the cases with total waffle‐mode phase errors less than ∼0.10 rad—Strehl ratios in each of these examples remained higher than 0.99. Even for cases with 0.30 rad of total waffle, the calculated Strehl ratio remained above 0.90. It is only when the waffle‐mode phase difference grows beyond this value that pure waffle‐mode error becomes a major factor in reducing the Strehl ratio. This suggests that in the absence of other phase errors, fundamental waffle‐mode error may not be an important factor in limiting the correction (and hence the observed Strehl ratio) in AO imaging. However, as more high‐order AO (and ExAO) systems become prevalent, the use of WFRs that penalize or limit the growth of waffle modes in the AO system may be key to maintaining stable PSFs during AO observations. Additionally, the development of alternative methods of wave‐front sensing, such as the spatially filtered wave‐front sensor (Poyneer & Macintosh 2004) or the pyramid sensor (Ragazzoni 1996; Vérinaud et al. 2005), may be required to reach the contrast levels necessary to achieve ExAO science.

3.1.3. Symmetric and Antisymmetric Waffle PSF Terms

In § 2.3.1, we presented the first two terms of the PSF expansion of Sivaramakrishnan et al. (2002) and noted that the symmetric terms of this expansion would dominate the PSF as waffle‐mode phase errors increased. To examine this, we made use of the simulated PSFs described in § 3.1.2 and differenced these PSFs with a perfect PSF (p₀), generated with no phase errors. The result for three of these cases (ϕ_w = 0.01, 0.03, and 0.10) can be seen in Figure 7 for aberrated PSFs with calculated Strehl ratios of 99.99%, 99.91%, and 99.0%, respectively.

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We can see that power in the satellite waffle PSFs is accounted for by the p_2,Strehl term

as illustrated by the on‐axis "hole." There is some evidence of the antisymmetric p₁ term in the diagonal patterns in these difference images, but their effect on the waffle PSF is small at these Strehl ratios. This antisymmetry is more obvious as Strehl ratios decrease to between 97.0% and 90.0%, but the p₁ term never dominates the intensity distribution.

The AEOS WFS is a Shack‐Hartmann system with an array of 84 μm lenslets, each producing a Hartmann spot that is imaged onto a 4 × 4 group of pixels on the WFS CCD. The deformable mirror is a 31.5 cm diameter Xinetics DM supported by 941 lead magnesium niobate (PMN) actuators spaced 9 mm apart on a square grid, with 35 actuators spanning the diameter of the DM (Roberts & Neyman 2002).

Because of initial concerns about pupil wander during AO imaging, early AEOS wave‐front reconstructors did not use the entire pupil for wave‐front reconstruction. Subapertures near the edge of the pupil were assumed not to be fully illuminated during each observation and were considered unreliable for active wave‐front reconstruction. To account for this, we modeled 33 active actuators (N_act = 33) across the pupil diameter instead of the full 35 actuators across the DM, giving us an undersized DM with a 28.8 cm clear area and providing us with a projected actuator spacing of an order of 0.11 m actuator⁻¹ at the primary mirror.

To mimic finite DM actuator throw, we assumed an actuator stroke limit of ±2.00 μm. While our simulations model 855 active actuators on the primary (33² actuators × π/4), the actual AEOS WFRs actively control 810 actuators, with actuators projected near the edges of the primary and secondary mirrors slaved to neighboring actuators. We did not model slaved actuators, nor did we model the effects of actuator influence functions, nonlinear actuator motions, or actuator hysteresis. We also do not account for system response time or calibration errors.

3.2.1. Seeing Measurements at Haleakala

The AEOS adaptive optics imaging data we used for our comparison comprises four sets of 25 closed‐loop I‐band observations of the K3 II star HR 7525 (γ Aql, V = 2.72) taken at two separate exposure times (48 and 73 ms) with the Visible Imager (VisIm) camera using a 2 mag neutral‐density filter. This star also served as the WFS target during these observations. These data were acquired on 2002 October 8 and span a total of 5.43 minutes from the first to the last exposure in the sequence. The images were processed (dark, bias, and flat‐field corrected) using the task ccdproc in IRAF.⁶

Measurements of r₀ taken with the University of Hawaii's Day/Night Sky Monitor (DNSM) during the night of our observations showed a mean measured r₀ = 11.44 ± 3.54 cm at λ = 0.500 μm from 320 data points over 9.22 hr. The median measured r₀ for this night was 11.97 cm. The mean zenith‐corrected r₀ for this night was r₀ = 13.17 ± 3.88 cm, with a median value of 13.73 cm. The nine measurements acquired just before, during, and after our observations showed a mean zenith‐corrected r₀ of 11.63 cm, with minimum and a maximum measured values of 8.84 and 13.60 cm, respectively.

This variation is apparent in our data, as measurements of the FWHM of the PSF in each image showed a median of 3.17 ± 0.26 pixels for 92 images, with a minimum FWHM of 2.77 pixels and a maximum FWHM of 4.49 pixels. Over its nominal optical bandpass (from 0.70 to 1.050 μm), the VisIm has a pixel scale of 00211 ± 0004 pixel⁻¹, giving us a median AO FWHM of 00669.

Unless otherwise noted, we have adopted the value r₀ = 11.60 cm to define our nominal seeing conditions.

We scale the value of r₀ to the I and H bands, following the relationship

where we assume the zenith angle ζ = 0 (Hardy 1998). In this way, we calculate r_I = 22.92 cm and r_H = 48.20 cm, and use these values in our simulations.

3.2.2. Estimating Waffle in AEOS I‐Band Data

To estimate the amount of waffle present in the AEOS I‐band data, we first required a knowledge of the Strehl ratios of the images themselves. We measured Strehl ratios by first simulating the diffraction pattern of the AEOS telescope, as computed from the analytical expression for the diffraction pattern of a centrally obscured circular aperture. We extracted photometry of the PSF and diffraction pattern and registered the PSF and the diffraction pattern to subpixel resolution by cross‐correlation. The PSF was then shifted to have the same subpixel center as the diffraction pattern, and the Strehl ratio was computed from the ratio of peak pixel intensities in both images (see Roberts et al. 2004 for a discussion of Strehl computation algorithms). Strehl ratios measured in this manner ranged from 0.17 to 0.28.

We measured the intensity of the peak pixel in each of the four satellite "waffle PSFs" observed in the images of HR 7525 and compared these measurements to the peak pixel of the on‐axis PSF. We found that the peak intensity of each waffle PSF was of an order of 0.25% of the peak intensity of the on‐axis PSF. The total flux measured in the waffle PSFs was ∼1.6% of the flux of the on‐axis PSF.

We then compared this result with the relative intensities of the waffle spots produced through simulation. Our simulations made use of five independent realizations of Kolmogorov‐spectrum phase screens, each with r₀ = 22.92 cm. Our AO correction model used the 0.9 power‐law filter in spatial frequency space, as described above in § 3. The residual phase that remained after AO correction (i.e., ϕ_AO≠0) was then used to simulate a PSF for each of the five incoming phase screens. These PSFs where then co‐added to produce our final image.

We simulated a set of monochromatic PSFs using this set of five phase screens, calculating ϕ_w from the phase screens themselves. We chose waffle amplitudes ranging from zero to atmospheric to the extreme (e.g., 100) and ran simulations using the same five input phase screens for each value of . We then examined the resulting images and assumed that the peak intensity of the simulated waffle PSFs corresponded to the peak wavelength of the I‐band filter transmission function used in the VisIm observations. We then chose the value of that produced differences in the peak waffle spot and central PSF intensities that best matched the observed values from the data.

Our simulated images are generated at a pixel scale of λ/4D, or 00125 pixel⁻¹, a nearly factor of 2 finer sampling than the true VisIm sampling at this wavelength (00211 ± 0004 pixel⁻¹). We rebinned the simulated data to match the VisIm pixel scaling using the IDL task frebin (Landsman 1993). We added Gaussian detector read noise (12 e⁻ rms) and dark current (22 e⁻ pixel⁻¹ s⁻¹) into our rebinned PSFs and modeled VisIm detector charge diffusion and optical effects by convolving our simulated images with a Gaussian profile of σ = 1.15 pixels, using the IRAF task gauss. This has the effect of limiting the maximum observable Strehl ratio in simulated VisIm data to S ∼ 0.70, which matches expectations for the VisIm and its optics as used in 2002 October.

We found that a waffle‐mode amplitude of 16 at λ = 0.88 μm best matched the intensity of waffle spots in these VisIm I‐band images.

An example of this monochromatic simulation with full AO correction, both without and with waffle, can be seen in Figure 8. We used this as the starting point to extend our monochromatic simulations to full I‐band simulations.

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3.2.3. Simulating I‐Band Images

We simulated a set of I‐band images by first generating 36 monochromatic images across the bandpass, from λ = 0.700 to 1.050 μm (the effective cutoff wavelength of the VisIm detector), in increments of 0.01 μm. We used the same Kolmogorov‐spectrum phase screens for each image at a given λ_i, scaling the phase difference of each phase screen relative to the bandpass central wavelength; e.g., ϕ_atm(˙λ_i/λ_ref), where λ_ref is the peak wavelength of the filter transmission function in the bandpass. We maintained the same 0.9 power‐law AO filter in spatial frequency space as before, with N_act = 33 actuators across the primary and a waffle‐mode amplitude of 16. We did not consider the effects of slaved or dead actuators as part of our simulations, but did include a ±2.0 μm DM stroke limitation in our analysis.

We rebinned the pixel scale of each monochromatic PSF by the ratio of the simulated wavelength to the reference wavelength to account for the wavelength‐dependent resolution changes inherent in Fourier methods. We then multiplied each rebinned λ_i PSF by the transmission expected for a standard Johnson/Bessell I‐band filter transmission function obtained from SYNPHOT in STSDAS,⁷ and combined the set of monochromatic PSFs for a given phase screen into a single broadband PSF using the IRAF task imcombine.

We generated five broadband PSFs in this manner and co‐added these PSFs into a single "short exposure PSF." Assuming a speckle correlation time of an order of a few tens of milliseconds, these five PSFs, when averaged, would approximate a ∼100 ms PSF. We note that longer exposures can be simulated in this way by adding statistically independent realizations of PSFs together, but would lack any correlated or longer term effects (such as quasi‐static speckles) without the addition of static sources of phase errors.

With the methods described above, our simulated Kolmogorov wave fronts produce PSFs with Strehl ratios between 0.544 and 0.635, with purely atmospheric phase disturbances. When waffle‐mode phase error with = 16 is introduced, the Strehl ratios decrease to between 0.406 and 0.556 for the same phase screens. When convolved with the Gaussian filter, the Strehl ratios of our simulated PSFs were further reduced by 70%, to between 0.284 and 0.389. However, the resulting final images produced waffle spots with the same morphology and intensity distribution as in VisIm images (Fig. 9). The total flux in the visible waffle spots, ∼1.8% of the flux of the central PSF, is in good agreement with the ∼1.6% value measured in the VisIm data. The peak intensities of each waffle spot, ∼0.25% of the peak of the central PSF, is also in agreement with the VisIm data.

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While the simulated Strehl ratios remain higher than expected for the r₀ values observed in the VisIm images from this night, we note that our simulations only account for two sources of phase error: residual atmospheric variations and the fundamental waffle mode. We do not account for pointing variations or pupil wander in the telescope beam, nor do we employ a realistic model of the DM. Abreu et al. (2000) noted that pinned and slaved actuators may provide as much as a 10% hit to the observed Strehl, while actuator influence functions and actuator dynamics will also limit the quality of the wave‐front reconstruction. Any observed noncommon path aberrations between the AEOS WFS/DM and the VisIm will also degrade the observed Strehl ratios, as will the effects of the VisIm detector pixel modulation transfer function (MTF).

We stress here that these VisIm images, acquired early in the AEOS AO system lifetime, did not benefit from the waffle‐suppressing WFRs that are now in use at AEOS (G. Smith 2003, private communication).

Images obtained more recently in I, J, H, and K bands have shown waffle‐mode error in AEOS imaging to be at or near nominal (atmospheric) levels (Perrin et al. 2004).

3.2.4. The Magnitude of Waffle‐Mode Phase Error

Following Sandler et al. (1994), we note that for Strehls ≳0.30, the final observed Strehl ratio of a system can be considered as a multiplicative combination of Strehl ratios, due to successive uncorrelated sources of error. Thus,

where σ²_ϕ1 is due to one class or family of aberrations (i.e., high‐order aberrations), while σ²_ϕ2 is due to another class of aberrations uncorrelated with the first, etc.

Neglecting the optical effects of the VisIm, we can calculate the mean square wave‐front error for atmospheric phase disturbances, σ²_ϕAO, from the mean Strehl ratio of our simulated images:

If we then assume σ_ϕAO = 2πz/λ, where λ = 0.880 μm, we find z (the residual rms wave front of the AO‐corrected atmospheric phase screen) to be 101.9 nm. If we then consider the addition of waffle‐mode phase error with = 16, we find

With S_ϕAO = 0.589 as above, S_ϕw = 0.829, and thus, σ²_ϕw = 0.188. Following the method above, we calculate the residual rms wave front due to waffle‐mode phase error to be z_w = 60.7 nm.

While our simulations were able to reproduce the general morphology of the early AEOS AO system PSF, as well as the shape and energy seen in the primary waffle PSFs, we realize our work does not fully reproduce the AEOS optical system, WFS geometry, or WFRs. In fact, the presence of immobile DM actuators in the AEOS pupil imprints a phase and an amplitude error across the entire pupil. Abreu et al. (2000) has estimated the Strehl hit to AEOS AO system performance due to these immobile actuators at ∼10%. Initial work has suggested these failed actuators can mimic localized waffle modes described by Gavel (2003).

Given a DM surface map taken with an interferometer, it is relatively straightforward to calculate the waffle present at the DM surface. Comparing this to the strength of the waffle seen in real images will indicate whether WFRs tailored to minimizing the effects of failed actuators will help to significantly improve the image quality. A reconstructor that drives the wave front to the best‐fitting plane through the failed actuators' heights on the DM is one obvious strategy. However, such a strategy would adversely affect the usable stroke of the DM and thus would be beneficial only when the seeing is good.

Circular aperture boundaries and partial subaperture obscuration by edges and secondary support spiders will affect the equations relating WFS slopes to actuator commands. Other waffle modes can be introduced by secondary support structure obscurations. In the extreme case, a completely obscured line of subapertures partitioning the pupil into disconnected sections can result in free‐floating piston between the sections in the reconstructed wave front. While this form of error might be less obvious, it can adversely affect Strehl ratios and the dynamic range of companion searches and debris disk studies because of its proximity to the core of the reconstructed PSF. These quadrants must somehow be explicitly stitched together by the reconstructor algebra using the mathematical equivalent of extra equations of constraint. Partially illuminated subaperture data should be ignored or used differently than WFS centroid data from fully illuminated apertures in order to reduce their contribution to undesirable behavior of the reconstructor.

The authors are grateful to D. T. Gavel, G. Smith, and P. E. Hodge for helpful discussions, and to the staff of the Maui Space Surveillance System for their assistance in taking these data. We are also grateful to L. W. Bradford, T. A. ten Brummelaar, J. R. Kuhn, N. H. Turner, and K. Whitman for the use of the Kermit image used in this paper. The authors wish to thank the anonymous referee for suggestions that have improved this manuscript. The authors also wish to thank the Space Telescope Science Institute's Research Programs Office, its Director's Discretionary Research Fund, and its generous Visitors' Program. R. S. and M. D. P. are supported by NASA Michelson Postdoctoral and Graduate Fellowships, respectively, under contract to the Jet Propulsion Laboratory (JPL), funded by NASA. The JPL is managed for NASA by the California Institute of Technology. The research presented here was supported by the STScI Director's Discretionary Research Fund, NSF grants AST‐0088316 and AST‐0215793, and AFRL/DE through contract F29601‐00‐D‐0204. Our work has also been supported by the National Science Foundation Science and Technology Center for Adaptive Optics, managed by the University of California at Santa Cruz, under cooperative agreement AST‐9876783. This research made use of the SIMBAD database, operated at CDS, Strasbourg, France.