Millisecond Exoplanet Imaging, I: Method and Simulation Results
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Research Article Journal of the Optical Society of America A 1
Millisecond Exoplanet Imaging, I: Method and
Simulation Results
A LEXANDER T. R ODACK1,* , R ICHARD A. F RAZIN2 , J ARED R. M ALES1 , AND O LIVIER G UYON1
1 Steward Observatory, University of Arizona, Tucson, AZ 85721
2 Dept. of Climate and Space Sciences and Engineering, University of Michigan, Ann Arbor, MI 48109
arXiv:2105.06589v2 [astro-ph.IM] 29 Jun 2021
* E-mail: atrodack@email.arizona.edu
Compiled July 1, 2021
One of the top priorities in observational astronomy is the direct imaging and characterization of extra-
solar planets (exoplanets) and planetary systems. Direct images of rocky exoplanets are of particular
interest in the search for life beyond the Earth, but they tend to be rather challenging targets since they
are orders-of-magnitude dimmer than their host stars and are separated by small angular distances that
are comparable to the classical λ/D diffraction limit, even for the coming generation of 30 m class tele-
scopes. Current and planned efforts for ground-based direct imaging of exoplanets combine high-order
adaptive optics (AO) with a stellar coronagraph observing at wavelengths ranging from the visible to the
mid-IR. The primary barrier to achieving high contrast with current direct imaging methods is the quasi-
static speckles, caused largely by non-common path aberrations (NCPA) in the coronagraph optical train.
Recent work has demonstrated that millisecond imaging, which effectively “freezes” the atmosphere’s
turbulent phase screens, should allow the wavefront sensor (WFS) telemetry to be used as a probe of the
optical system to measure the NCPA. Starting with a realistic model of a telescope with an AO system and
a stellar coronagraph, this article provides simulations of several closely related regression models that
take advantage of millisecond telemetry from the WFS and coronagraph’s science camera. The simplest
regression model, called the naïve estimator, does not treat the noise and other sources of information loss
in the WFS. Despite its flaws, in one of the simulations presented herein, the naïve estimator provided a
useful estimate of an NCPA of ∼ 0.5 radian RMS (≈ λ/13), with an accuracy of ∼ 0.06 radian RMS in one
minute of simulated sky time on a magnitude 8 star. The bias-corrected estimator generalizes the regres-
sion model to account for the noise and information loss in the WFS. A simulation of the bias-corrected
estimator with four minutes of sky time included an NCPA of ∼ 0.05 radian RMS (≈ λ/130) and an
extended exoplanet scene. The joint regression of the bias-corrected estimator simultaneously achieved
an NCPA estimate with an accuracy of ∼ 5 × 10−3 radian RMS and an estimate of the exoplanet scene
that was free of the self-subtraction artifacts typically associated with differential imaging. The 5 σ con-
trast achieved by image of the exoplanet scene was ∼ 1.7 × 10−4 at a distance of 3λ/D from the star and
∼ 2.1 × 10−5 at 10 λ/D. These contrast values are comparable to the very best on-sky results obtained
from multi-wavelength observations that employ both angular differential imaging (ADI) and spectral
differential imaging (SDI). This comparable performance is despite the fact our simulations are quasi-
monochromatic, which makes SDI impossible, nor do they have diurnal field rotation, which makes ADI
impossible. The error covariance matrix of the joint regression shows substantial correlations in the exo-
planet and NCPA estimation errors, indicating that exoplanet intensity and NCPA need to be estimated
self-consistently to achieve high contrast.
© 2021 Optical Society of America
OCIS codes: 010.1080 Adaptive Optics, 010.7350 Wavefront Sensing
http://dx.doi.org/10.1364/ao.XX.XXXXXXResearch Article Journal of the Optical Society of America A 2
1. INTRODUCTION in post-processing in a relatively straightforward manner. Using
lasers in the laboratory to calibrate the NCPA tends to leave large
The goal of directly imaging planets and planetary systems at residual errors for many reasons, including differences between
optical and infrared wavelengths is among the top science pri- the laser path and path traversed by the starlight. For example,
orities of the coming generation of so-called “extremely large on a telescope with a segmented primary mirror it would be
telescopes" (ELTs), which will have primary mirror diameters rather difficult for the laser system to capture the piston and
ranging from D = 25 to D = 39 m. Due to the small angular tip/tilt errors on each segment that the starlight would expe-
separations between the planets and their host stars, as well as rience. The challenge of mitigating the quasi-static speckles is
significant differences in brightness between them, this imag- made further complicated by the fact that they are modulated at
ing problem is undoubtedly one of great remaining challenges kHz rates by the atmospheric aberrations that the AO system is
in astronomical optics. The most desired planetary targets for unable to correct.
detection and characterization are planets capable of support-
ing life. Visible and near-IR spectroscopy of reflected starlight
can reveal signs of biological activity in the planet atmospheric 2. SUMMARY OF CURRENT TECHNIQUES
composition. Such measurements could be obtained with fu-
ture large ground-based telescopes thanks to their collecting A. Differential Imaging
area and angular resolution, provided that the planets can be di- Currently quasi-static speckles are removed via background
rectly imaged. [1] Directly imaging them will require ultimately subtraction techniques that employ various types of differential
achieving a contrast (i.e., planet-to-star intensity ratio) of 10−10 imaging, the most common forms of which are Angular differential
or less (roughly independent of wavelength in reflected light) for imaging (ADI) and spectral differential imaging (SDI). SDI takes ad-
angular separations that approach the λ/D classical diffraction vantage of the fact that the point-spread function (PSF) stretches
limit of an ELT.[2–4] with wavelength (λ), while the location of a planet does not [11].
On the ground, imaging planets currently requires the com- As this PSF stretching is proportional to the distance from the
bination of high-order adaptive optics (AO) with a stellar coron- star, a weakness of this method is detecting targets at separations
agraph [5]. As shown in Fig. 1, the very first images acquired by approaching the angular distance of λ/D, as well as targets that
early stellar coronagraphs with AO systems showed swarms of are extended in the radial direction, due to self-subtraction arti-
speckles, many of which were vastly brighter than any planets facts. Furthermore, SDI is also limited by chromatic optical path
one might hope to see [6, 7]. Achieving the high-contrast goals difference [12–14]. ADI takes advantage of the field rotation
required to detect the desired targets in the face of these speckles that occurs over the course of the night when the instrument
remains an elusive, high-priority task decades later. At the heart rotator is turned off (Cassegrain focus) or adjusted to maintain
of this speckle problem lie quasi-static aberrations (QSA) in the fixed pupil orientation in a Nasmyth focus instrument, so that
optical system hardware. The temporal variability of the QSA the planet appears to rotate relative to the PSF [15]. Thus, to
results from thermal fluctuations and other variable conditions, the extent that aberrations do not evolve as the image rotates
such as the changes in gravitational stress as the telescope point- around the pointing center, correction can be achieved. Some
ing varies. As a result of the ever-changing conditions, the QSA known and characterized weaknesses of ADI are similar to SDI,
on ground-based platforms change on a wide variety of time in that star-planet separations close to λ/D are difficult to detect
scales, ranging from minutes to months [8]. because the planet must travel an arc length of several resolu-
tion elements within the observing period [16]. Other effects
The most problematic
caused by the time dependence of the planetary rotation rate
of the QSA are the non-
around the star due to field rotation, as well as artifacts from self-
common path aberrations
subtraction, induce a host of biases that affect astrometry and
(NCPA), occurring in op-
photometry calculations [14, 17, 18]. ADI also will remove circu-
tical components that are
larly symmetric components of the image, making the quality of
beyond the dichroic fil-
the results dependent on the spatial arrangement of the planets
ter (or beam splitter) that
and dust surrounding the target star. Furthermore, one of the
separates the light paths
largest issues with ADI is the fact that the quasi-static speck-
that go to the AO system’s
les change with time, which can greatly affect the subtraction
wavefront sensor (WFS)
accuracy.
and the coronagraph (see
Fig. 2). Because they oc-
cur downstream of the B. Focal Plane Wavefront Sensing and NCPA Compensation
Fig. 1. Coronagraphic AO image dichroic (or beam splitter), One way to eliminate the quasi-static speckles arising from the
(logarithmic intensity) from the
Lyot Project with the AEOS tele- the NCPA are either ex- NCPA is to perform some type of focal-plane wavefront sensing
scope of the bright star HD137704. perienced by the WFS op- with data obtained from the science camera, and then use a
The circles indicates the corona- tical train, or the coron- deformable mirror (DM) or other type of spatial-light modulator
graphic occultation region. The agraph optical train, but to apply a compensation to the wavefront [20–23]. One way to
∼ 1200 s integration time averages
over the atmospheric turbulence.
not both. The NCPA man- do this is to apply two or more DM commands, called probes,
λ = 1.6 µm. From [6]. ifest themselves as quasi- and measure the intensity for each probe, thereby setting up a
static speckles in the coron- simple regression in which the complex-valued electric field at
agraph’s science camera (SC), and several authors have shown the locations of interest is estimated. Once this field has been
them to be a major limitation in high-contrast imaging [9, 10]. estimated, the next step is to calculate a new DM command
The quasi-static speckles are particularly problematic, because, according to some merit function that creates a dark region. One
unlike the speckles created by the atmospheric turbulence, they such method is called electric field conjugation (EFC), but there
do not average to a spatially smooth halo that can be subtracted is now substantial literature on closely related methods [20–25].Research Article Journal of the Optical Society of America A 3
C. Millisecond Exposures
Millisecond exposure times with the science camera are a new
frontier in high contrast imaging, and they are becoming obser-
vationally attractive due to a new generation of noiseless and
nearly noiseless IR and near-IR detector arrays capable of mil-
lisecond read-out times [30, 31]. Millisecond exposures freeze
the swarms of speckles that arise due to atmospheric turbulence,
whereas longer exposures average these speckles into a smooth
halo. A planet and the stellar speckle exhibit radically different
behavior at millisecond time-scales, as shown in Fig. 3 [32]. This
is because at the center of a planet’s speckle pattern, the AO
system maintains the planet’s intensity at a nearly constant level
(given by the Strehl ratio), while the starlight at the planet’s
location exhibits much more volatility [33, 34]. Several methods
have been proposed to exploit this difference in temporal behav-
ior based on millisecond science camera time-series data alone
Fig. 2. Schematic diagram of an astronomical telescope with a closed- [35, 36].
loop AO system and a coronagraph. Modified from [19].
These methods are expected to be vastly more effective on space- Simple physical optics
based platforms than they have proven to be on ground-based arguments show that the
platforms due to the Earth’s atmospheric turbulence. Indeed, speckles caused by the
typical attempts to create an extended dark region on ground- NCPA are modulated by
based platforms only result in reducing the background starlight the AO residual at the
by a factor of about two. In addition, ground-based applications kHz time scale,[37] and
of such methods are quite slow, sometimes combating a few Frazin showed that know-
speckles at a time, and each DM probe must be remain in place ing the values of the AO
long enough to obtain a turbulent average of the intensity [21, residuals allows joint es-
22, 26]. The fundamental problem with ground-based probing timation of the NCPA
methods and EFC (and other related methods) is that the focal and the exoplanet im-
Fig. 3. Simulation of turbulent age from the millisecond
plane electric field (which probing seeks to estimate and EFC modulation of intensities at a sin-
seeks to cancel) is modulated by the atmosphere on the time- gle pixel of the science camera
science camera images
scale of a millisecond. from a stellar coronagraph. The [32]. Fig. 4 shows simula-
A new approach to estimating the NCPA is to replace the black solid line shows the time- tions of noise-free science
occulter (in the first focal plane of the coronagraph) with a phase series of the temporal variation of camera images with the
the planetary intensity. The dotted same NCPA being probed
mask, thereby turning the coronagraph into a Zernike wave- red line shows the stellar inten-
front sensor [27]. The Zernike wavefront sensor, combined with sity at the planet’s location. Both with different AO resid-
knowledge of the wavefront statistics and a coronagraph model, the planetary and stellar intensity ual wavefronts demon-
then provide sufficient information to estimate the NCPA. Of are normalized to have a mean of strating this phenomenon.
unity in this figure. From [32]. In 2013, independent pub-
the various approaches to determining the NCPA, the COFFEE
algorithm of Refs. [28, 29], which is also referred to as “phase lications by Frazin and
diversity," is most similar to the method simulated here. In Codona & Kenworthy proposed to exploit the WFS telemetry
COFFEE, a series of DM commands (called “pokes") is applied in addition to the millisecond science camera images for focal-
and a long-exposure (of sufficient duration to average over the plane wavefront sensing [32, 38]. The fundamental shortcoming
turbulence) image is acquired. Then, the NCPA coefficients of the methods of Frazin and Codona & Kenworthy is that they
are estimated via regression. The most important differences were unable to account for wavefront measurement error (WME).
between method in this paper and COFFEE are: This is to say, the WFS measurements only allow an imperfect
estimate of the phase of the wavefront impinging on the WFS
• Our method method uses millisecond science camera entrance pupil, not the actual phase. Specifically, any WFS ex-
telemetry, while COFFEE uses long (i.e., averaged over hibits spatial and temporal bandwidth limitations (the latter are
the turbulence) exposures. less important due to the kHz frame-rate), nonlinearity in the
phase of the wavefront, and noise. Outside of the high-contrast
• Our method uses the AO residuals (available from the mil- imaging problem, several authors have performed multi-frame
lisecond WFS telemetry) as probes, not DM pokes to pro- deconvolution with millisecond focal plane and WFS telemetry
vide the information needed to determine the NCPA. to remove atmospheric image distortion. These works use the
• Our method jointly estimates the NCPA and the exoplanet WFS-based estimates of the point-spread function (PSF) as an
image, thereby treating their joint error statistics, while the initial guess [39, 40]. (For approaches that do not make use of
COFFEE algorithm does not estimate the exoplanet image. WFS telemetry see, e.g., Refs. [41, 42]). The regression results
shown in this article are the first to employ a statistically rigor-
Note that both methods rely on coronagraph models and require ous treatment of the wavefront measurement error (WME), and
knowledge of the 2nd order spatial statistics of the AO residual details of the regression model are deferred to Part II due to their
wavefronts. complexity.Research Article Journal of the Optical Society of America A 4
t = 0 t = 150 t = 300 t = 450
5 0.0 5 0.0 5 0.0 5 0.0
λ/D
λ/D
λ/D
λ/D
0 −0.5 0 −0.5 0 −0.5 0 −0.5
−5 −1.0 −5 −1.0 −5 −1.0 −5 −1.0
−5 0 5 −5 0 5 −5 0 5 −5 0 5
λD
/ λD
/ λD
/ λD
/
Fig. 4. Log10 scale, noise-free focal plane images for different realizations of the instantaneous AO residual phases, showing the modulation of the
signal. The range of the color scales are limited to a factor of 10 to make the modulation easy to see.
3. NUMERICAL SIMULATION method introduced by Assémat et al [46] via the hcipy package
[47].
In order to proceed with the verification of the regression model
As our model does not include the telescope’s beam reducing
described in Part II, a suite of simulation tools was developed in
optics, the first surface the atmospheric wavefront encounters
order to run numerical experiments using a millisecond imaging,
is that of the DM. We simulate the DM surface with 2D cubic
adaptive optics coronagraph. An end-to-end simulator was
splines, with the knots placed in a 20 × 20 grid. In this way, the
constructed in order to create turbulent wavefronts, as well as
knots play the role of the actuators and the height of any point on
models for the coronagraph and the AO system optical trains.
the surface is given by the spline interpolator function. With the
The AO system includes a deformable mirror (DM) followed
circular pupil inscribed on the DM surface, the resulting control
by a 4-sided pyramid WFS (PyWFS), and the coronagraph is
radius is 10 λ/D. The phase imparted by the DM is added to
a standard Lyot model, followed by a science camera. These
the pre-DM phase, φ p (w− t ):
simulations only require pupil planes and focal planes, and the
propagation between the two planes is computed via fast Fourier φ p (wt ) = φ p (w− DM
t ) + φ p,t , (2)
transforms [43].
where φDM p,t is the contribution of the DM to the phase at pixel p
A. Wavefronts and AO
and time t. The P × 1 vector containing all P values of φ p (wt ),
Fig. 2 shows a schematic diagram of a telescope with an AO φ(wt ) , is called the AO residual wavefront, however, we will
system and a coronagraph. The turbulent wavefront at the tele- also call wt by the same name for convenience. The P equa-
scope entrance pupil is taken to be at the science wavelength, tions shown in Eq. (2) define the parameter vector wt , implicitly
λ (chromatic issues will need to be considered in subsequent assuming that they can be inverted, which is a trivial matter
studies; a choice made to simplify the discussion in this work). for the PhasePixel representation used here. More complicated
We represent the entrance pupil of the telescope as comprised of representations may require a fitting step to find wt .
P pixels, with the phase at the pth pixel at time t represented as Following the DM, a beam splitter separates the coronagraph
φ p (w−
t ), where the
− superscript indicates that the wavefront
and wavefront sensor optical trains. In our simulations, the
has not yet impinged on the DM and only carries the effects of beam splitter’s only effect is to divide the available photon flux
atmospheric turbulence (the post-DM wavefront will not have between the optical trains. Moving down the AO system, the
the superscript). The function φ p and the parameter vector wt WFS telemetry then makes it possible to get an estimate (or mea-
together specify the wavefront. There are many ways to do this. surement) of wt , denoted as ŵt , with the “hat" ŵt designating
For example, φ p (wt ) could represent a sum over Zernike modes that it is an estimate of the vector of quantities that specify the
(evaluated at the location of pixel p), in which case the vector true phases wt . The wavefront measurement (determined by
wt would be a set of coefficients of the Zernike modes at time analysis of the WFS signal) can be modeled by the equation
t. Here we make a simpler choice: the pth component of wt is
the phase at pixel p and the φ p function just returns the value ŵt = W (wt ) + nt , (3)
of the pth component of the input vector. This was called the
PhasePixel representation in Ref. [44], where some of its compu- where W is a nonlinear measurement operator (even if the esti-
tational advantages are explained. With this representation, the mator itself is linear, as it is in these simulations, the WFS signal
complex-valued electric field in the pre-DM pupil plane is is nonlinear in wavefront phase values), and nt is the noise vec-
tor. The nonlinearity of W , aliasing, and finite bandwidth effects
u− − −
p ( wt ) = exp[ jφ p ( wt )] , (1) tend to make ŵ a biased estimate of w.
The regression model, which jointly estimates the NCPA and
where t is the time index in the range 0 to T − 1, with T being exoplanet image, makes essentially no assumptions about the
the total number of millisecond exposures. WFS architecture, allowing tremendous flexibility. In principle,
The temporal evolution of the on-sky turbulent wavefronts any information lost in the WFS is accounted for in the Monte
must be specified in order to model the functionality of the AO Carlo calculations (see below). In the simulation experiments
system. We have taken the spatial statistics of the turbulence below, the PyWFS is chosen in large part because it is being used
to be given by the von Kármán spectrum, with an outer scale for many modern AO systems including MagAO-X [48] and
parameter L0 , inner scale parameter l0 , and Fried parameter r0 SCExAO [49]. Ref. [44] provides linear and nonlinear regression
[45]. To keep the model simple, and maintain realism in the models for a PyWFS, and for this work we apply the linear
temporal evolution, we implement the infinite phase screen regression model. The input to the PyWFS regression modelResearch Article Journal of the Optical Society of America A 5
is the AO residual wavefront at time-step t, represented as wt ,
while the output is the estimate of the same, represented as ŵt ,
as summarized in Eq. (3). The PyWFS model includes circular
modulation of the beam, simulated as a discrete set of tips and
tilts. This formulation allows for the effects of noise, spatial
bandwidth, and aliasing to be included in the measurement of
the wavefront. An example of an AO residual phase, φ(wt ),
its measurement error, φ(wt ) − φ(ŵt ), and the corresponding Fig. 6. Schematic diagram of a Lyot coronagraph. Modified from [50].
PyWFS intensity at t, are provided in Fig. 5.
tical system, and thus provides for the freedom of many choices,
including Lyot, PIAACMC [51], APP [52, 53], and vector vortex
ϕ(wt ) (ϕ(wt) − ϕ(ŵ t)) [54] coronagraphs, to suppress host starlight. Indeed, a corona-
2 2 graph is not a mathematical necessity for running the algorithm
- although one would expect coronagraphic optical systems to
40 1 40 1 outperform non-coronagraphic ones.
For these simulations
0 0 we choose a Lyot coron-
20 20
−1 −1 agraph (See Fig. 6) with
Instantaneous SC Image a binary focal plane mask
10 −2.0
0 −2 0 −2 (FPM) occulter that is cir-
0 25 0 25 5 −2.5 cular in shape with a ra-
dius of 1.5 λ/D, and a
PyWFŜIntensity
λ/D
0 −3.0
Lyot Stop with a diame-
−5 −3.5 ter 90% of the pupil di-
−10 ameter. The complex-
1.0 −4.0
100 −10 0 10 valued electric field at
λ/D the lth science camera de-
50 0.5 tector pixel, vl , can be
represented in terms of
Fig. 7. Log scale 1 ms exposure
science camera image in contrast the complex-valued elec-
0 tric field in the coron-
0 100 units of an 8 magnitude source at
λ = 1 micron with 10% spectral agraph entrance pupil,
bandwidth. u p (wt , a), via the linear
Fig. 5. Example of wavefront measurement with the simulated Py- coronagraph operator, a
WFS. top left: True AO residual (radian). top right: Error in measured L × P matrix, D = { Dl p } (L is the number of science camera pix-
AO residual (radian). bottom: Intensity at PyWFS detector, in normal- els, P is the number of pixels in the discretization of the entrance
ized units.
pupil):
P −1
Progressing to the coronagraph from the beam splitter, the
wavefront that arrives at the entrance pupil of the coronagraph
vl (wt , a) = ∑ Dl p u p ( w t , a ) , (5)
p =0
is the same as the wavefront at the WFS, but with the addition
of the NCPA phase. For simplicity, the cumulative sum of any where u p (wt , a) is given by Eq. (4). Note that Eq. (5) is the form
NCPA is treated as a single phase error in the coronagraph of a matrix-vector multiplication and is ideally suited to being
entrance pupil. This can be thought of as the total phase error carried out on a graphics processing unit (GPU). D comes from
induced by the individual optical elements in the coronagraph, a computational model of the coronagraph optical train. For a
aberrations upstream of the beam splitter that are not corrected Lyot coronagraph, such a model is provided by the operator:
by the AO system (e.g., island modes), as well as aberrations in
the WFS optical train that are flattened by the AO system (for
D = {F {S × F [M × F ]}} , (6)
which the inverse of would appear in the coronagraph optical
train). The electric field in the coronagraph entrance pupil is where F is the 2D discrete Fourier Transform operator, S is a
thus given by mask representing the Lyot Stop, and M is a focal plane mask.
Finally, the noise-free (i.e. true) representation of the intensity
u p (wt , a) = exp[ jφ p (wt ) + jθ p ( a)] , (4)
impinging on the lth pixel of the science camera is given by
where a is the column vector (with Na components) containing
the coefficients that specify the NCPA, and θ p is the function Il (wt , a) = vl (wt , a)v∗l (wt , a) , (7)
that specifies the phase of the NCPA in the pth pixel in the
pupil. In these simulations, in which aberrations are represented where ∗ denotes the complex conjugate.
by Zernike polynomials, θ p ( a) represents the weighted sum of The regression approach allows the intensity Il (wt , a) in
Zernike polynomials evaluated at pixel p. Eq. (7) to be nonlinear in wt , but it does require linearization in
a (which can be refined iteratively). The linearization about the
B. Coronagraph point a0 is represented as:
The regression algorithm simulated below makes no assump-
tions about the type of coronagraph architecture that is in the op- Il (wt , a) ≈ dl (wt ) + Al (wt ) a (8)Research Article Journal of the Optical Society of America A 6
where dl (wt ) = Il (wt , a0 ) and Al (wt ) is a 1 × Na row vector Note that A(wt ) is different in Eqs. (12) and (13); in the former
representing the gradient A(wt ) is L × Na , whereas in the latter it is L × ( Na + Np ). It is
the latter form that allows the joint estimation of the NCPA and
∂Il (wt , a) exoplanet brightness coefficients. The details of constructing the
Al (wt ) = . (9)
∂a a0 system matrix can be found in the Part II article.
With this notation, Al (wt ) a is a scalar product between the
C. Regression Approach
row vector Al (wt ) and the column vector a. The L values of
dl (wt ), one for each detector pixel, are concatenated into the In this article, simulated kHz measurements from the science
column vector d(wt ) and, similarly, the L row vectors Al (wt ) camera and WFS are used as inputs into a regression model
are stacked to form the L × Na matrix A(wt ). A(wt ) is called the based on Eq. 13 to jointly estimate the NCPA and the extended
system matrix, and d is called the zero point vector. The simplest image of circumstellar material (including planets). We define
situation is when the linearization point a0 is all zeros, in which estimation to mean the solving Eq. 13 for x̂, an estimate of the
case d corresponds to the science camera intensity without any true vector x, which consists of the Na NCPA basis function coef-
NCPA present. ficients discussed above, as well as the Np exoplanet brightness
This intensity is normalized such that a point source prop- coefficients.
agated to the final focal plane without a coronagraph has a This article provides simulations of the estimation methods
maximum value of unity, centered at the origin of the science defined in Part II:
camera. This gives the effective units of the science camera • Ideal estimation
measurements to be in “contrast" as defined as the ratio of coro-
nagraphic intensity from our simulated source divided by the • Naïve estimation
non-coronagraphic intensity of an on-axis point source. Scaling • Bias-corrected estimation
in this fashion allows for direct calculation of contrast values.
Finally, we simulate the science camera’s measurement of the In ideal estimation, the true AO residuals (wt ) are used in the
intensity by adding photon counting (shot) noise and detector regression equations, resulting in an unbiased estimator (this is
readout noise. An example of such a measurement can be found the method of Frazin [32]). In the real world, ideal estimation is
in Fig. 7. The number of photons incident on the detector is not possible because only a measurement of the wavefront, (ŵt ),
determined by choice of target star magnitude and science wave- can be known. The ideal estimate will be used as a benchmark.
length, optical system beam splitter ratio, and chosen values In naïve estimation, the measurements of the AO residuals are
for system throughput, which are specified in Table (1). The simply plugged into the regression equations wherever they call
intensity, Il (wt , a), in units of contrast, is converted to units for the true values of the AO residuals. The fact that estimates
of photons. σRN is then chosen to represent the readout noise of the AO residuals are not the same as the true AO residuals
in photon counts per pixel, per read out (every 1 millisecond). is ignored in naïve estimation. It is important to understand
The standard deviation of the intensity, in photon units, in the that even if the wavefront measurements are unbiased, meaning
science camera measurement, is thus given as that the expected value of the wavefront measurement error
is zero, the naïve estimator is still biased for reasons that are
explained in Part II. In our simulations, it turns out that naïve
q
2 ,
σl,ph = Il,ph + σRN (10)
estimation works rather well for estimating large (0.5+ radian
allowing for the calculation of the noise term in the lth pixel via RMS, or ≈ λ/13) NCPA, but fails to produce precise results
sampling from a zero-mean, σl ph Normal distribution. This noise when estimating small (< 0.1 radian RMS, or ≈ λ/62) NCPA or
term, nl , separate from the noise term in the WFS measurement the exoplanet image, as will be seen below.
in Eq. (3) because they come from different detectors, is then Unlike the naïve estimate, the bias-corrected estimate treats the
converted back to units of contrast and added to the result of Eq. wavefront measurement error (WME) under the assumption that
(7), giving the final, measured intensity in the lth pixel, again in the spatial statistics of the AO residual wavefronts are known.
units of contrast, as This statistical knowledge is used to create a set of Monte Carlo
wavefronts that form the basis of the technique. To the extent
yl (wt , a) = Il (wt , a) + nl . (11) that the Monte Carlo wavefronts are representative of the ran-
To apply the regression methods, we use linearization given in dom process that generates the true AO residual wavefronts, the
Eq. (8), and we stack the L SC measurements given by Eq. (11) technique converges to an unbiased estimate with a large sam-
into a column vector yt : ple size. In other words, when the knowledge of the statistics
is exact, the bias-corrected estimator is unbiased (and nearly as
yt = A(wt ) a + d(wt ) + nt , (12) good as the ideal estimate). If the knowledge of the statistics
is “good" but not exact, the bias-corrected estimator, as is the
where nt is vector that represents noise in the SC measurements
case for the simulations shown below, retains a small bias. The
at time t. When we consider T exposures and allow t to be an
details of all three estimators are given in Part II.
index from 0 to T − 1, Eq. 12 allows one to estimate a, the NCPA
The simulations presented herein are divided into Phase A
coefficients. Further, as shown in Part II, the system matrix
and Phase B. The Phase A simulations are designed to showcase
can be extended to include p, the Np × 1 column vector coeffi-
the potential for correcting large NCPA via the naïve estimate
cients corresponding to the image of the exoplanetary system,
with only one minute of observed data. Thus, in Phase A, the
which we will call the exoplanet image for convenience. When we
regression model only includes the NCPA, as per Eq. (12), and
include the exoplanet image in Eq. (12), it takes the form:
requires no knowledge of the statistics of the wavefronts. The
Phase B simulations are based on Eq. (13), and are designed
a
yt = A(wt ) x + d(wt ) + nt , where x = . (13) to demonstrate the full potential for simultaneously estimating
p small NCPA and the exoplanet image.Research Article Journal of the Optical Society of America A 7
D. Simulation Parameters cal experiments.
In the experiments described below, the sensing and science The NCPA for the simulated experiments is chosen to be the
wavelengths are both chosen to be Y band, centered around linear combination of 6 radial orders of Zernike polynomials
1.036 µm. The Y band was essentially an arbitrary choice cor- (Noll indices 4-36, so, Na = 32), following Eq. (4). When needed
responding to a possible MagAO-X instrument observing case; for the joint estimate, a 13 × 8 object source grid correspond-
the methods discussed would work at shorter or longer wave- ing to the Np = 104 sky-angles for which the scene (exoplanet
lengths. Although modern ExAO systems typically use separate brightness coefficients) are to be estimated, is used. This source
wavelength bands for the WFS and coronagraph optical trains, grid is a collection of point sources created such that their images
we chose to use the same band to match the current expectation propagated through the coronagraph are spaced by 1λ/D, and
for a first on-sky implementation of this method. This mitigates have an inner edge at 3λ/D at the closest, and an outer edge
chromatic effects that may arise when these functions are per- at 10λ/D. The choice of this scene is to demonstrate how the
formed at different wavelengths. A 6.5 meter, circularly symmet- techniques presented can recover both point sources (exoplanets)
ric aperture with no central obscuration nor spiders is chosen and extended sources (such as circumstellar disks), located at a
to define the telescope pupil. The light is taken to be quasi- few times the classical optics resolution limit of λ/D. The bright-
monochromatic. The photon fluxes, which are needed for noise est point is (1 × 10−4 ) in the contrast units, while the faintest
statistics, calculated from an apparent magnitude 6 or 8 sources point is (1.8 × 10−6 ), with an average brightness of 3.9 × 10−5 .
with a 10% spectral bandwidth and a 50% overall throughput.
This results in a total of 28,700 photons/ms for the magnitude 4. PHASE A
8 source and 182,200 photons/ms for the magnitude 6 source. The objective of the first experiment performed is to estimate and
70% of the photon flux was apportioned to the WFS, while 30% compensate for the NCPA. In this experiment, we use Eq. (12) to
was apportioned to the coronagraph. This non-standard way estimate the coefficient vector a. Compensating for the NCPA
of splitting the photons was chosen so that changing from a using only one minute of observation time provides a good
magnitude 6 to magnitude 8 target in the simulations would not starting position for the more demanding combined estimation
incur a penalty in the resulting Strehl ratio from the AO system. of the residual NCPA and the exoplanet image in "Phase B,"
We did not run experiments with other ratios. described in Section 5.
The AO system uses a leaky integrator control system with
a loop gain of 0.35, with a two frame correction delay, and pro- A. Estimating the NCPA
vides a closed-loop Strehl ratio of ∼ 0.73 in Y band. The tur-
bulence parameters are specified in Table 1. The control radius NCPA Coefficient Estimates
of the AO system (defined by the maximum correctable spatial
0.4
frequency by the DM) is 10λ/D in both the x and y directions,
0.2
leading to a square corrected region, and a residual bright halo
Estimated Value
outside. A 1 kHz frequency is chosen for both the wavefront 0.0
and the science camera measurements, and they are assumed −0.2
to collect simultaneous telemetry, meaning that yt , the vector
−0.4 T uth
of science camera intensities at all L pixels measured at time- Naive: Magnitude 6
−0.6 Naive: Magnitude 8
step t, results from the wavefront wt , with measurement ŵt .
These choices are representative of modern ground-based AO 5 10 15 20 25 30 35
Noll Index
systems equipped with high frame rate science cameras. This
AO performance, and the resulting raw contrast provided by the
Fig. 8. The estimated aberration Coefficients from the simulated
Lyot Coronagraph without any NCPA present, is consistent with experiments, following five relinearization iterations. Included are the
the current theoretical performance of such a simple integrator true, starting NCPA coefficients, and their naïve estimates using both a
control system given by Males and Guyon [55]. magnitude 6 (RMS error of 2.14 × 10−2 radian) and magnitude 8 source
The turbulent wavefront phase, φ( w− (RMS error of 5.95 × 10−2 radian).
t ), is generated follow-
ing Section 3.A and corresponds to a 50 × 50 pixel grid inscribed Phase A simulates a 1 minute observation, consisting of
with a circular aperture with a radius of 25 pixels. The total num- 60, 000 synchronized millisecond exposures in the science cam-
ber of pixels in this inscribed circle is P = 1976. The turbulence era and WFS, the purpose of which is to provide a coarse esti-
model used in the simulation is a phase only atmosphere, with mate of large NCPA, which have an RMS phase of 0.52 radian.
L0 = 25 m and r0 = 0.3 m. The phase φ(wt ) is constructed as The Phase A estimate uses the naïve regression model, which
the sum of 6 layers (as scintillation effects are ignored to simplify has unwanted bias that increases increases as the wavefront mea-
the discussion in this work), each with their own wind direction surement error gets larger. The initial linearization point [see
vector, corresponding to a 10 m/s speed. The choice of 6 layers Eq. (9)] was taken to the zero vector, ie., a0 = 0. The Phase A
collapsed together in the entrance pupil helps ensure that each experiment was repeated for both a magnitude 6 and magnitude
phase screen is unique. 8 target source. The AO system being simulated is not photon
The science camera (in the final focal plane of the corona- starved at magnitude 8, so although fewer photons are being
graph) consists of a 67 × 67 pixel square array, with 0.39 pixels observed compared to the magnitude 6 source, the Strehl ratio
per λ/D sampling. A value of 0.3 ph/pix/read of readout noise only decreases from 0.76 to 0.73 when going from the magnitude
is assumed. This number of pixels in the detector frame, as well 6 source to the magnitude 8 source. The magnitude 8 source has
as the readout noise value, running at a 1kHz frame rate, is a larger wavefront measurement error, which leads to larger bias
within specifications for modern, commercially available EM- in the naïve estimate.
CCD cameras running at an acceptable gain setting. Table (1) Fig. 8 shows the results of the naïve estimate of the aberra-
summarizes the general parameters that are used in the numeri- tion coefficients composing the NCPA, following a 5 iterationResearch Article Journal of the Optical Society of America A 8
treatment of the nonlinearity (see Section 4.B), for both the mag-
Simulation Parameter Symbol Value or Type nitude 6 and 8 sources. The root mean squared (RMS) error of
Pixels in Entrance Pupil P 1976 the magnitude 6 source estimates of the aberration coefficients
is 0.0214 radian, whereas for the magnitude 8 source the RMS
True AO Residual wt vector of length P error is 0.0595 radian, approximately a factor of 3 worse.
WFS Measurement ŵt vector of length P
B. Treating the Nonlinearity
Pupil Diameter D 6.5 m
Coronagraph Operator D Lyot Treating the Nonlinearity
0.4
Fourier Transform Operator F
0.2
M
Coefficient Value
FPM binary;
radius: 1.5λ/D 0.0
Lyot Stop S binary; −0.2
xa0
xa1
diameter: 0.9D xa2
−0.4 xa3
WFS Model W Modulated PyWFS Truth
−0.6
5 10 15 20 25 30 35
PyWFS Modulation Radius 3 λ/D Noll Index
Science Camera SC EMCCD Fig. 9. Successive relinearization points, x a,n , for the Phase A exper-
Pixels in SC L 4489 iment, starting from all zeros, and then using the previous iteration’s
estimate.
SC/WFS Cadence 1kHz, synchronized
In order to effectively treat the nonlinearity in the NCPA, the
SC Pixel Array Shape 67 × 67 pixels
regression algorithm is run iteratively, successively relinearizing
SC Pixels per λ/D 0.39 about the updated estimate values. Note that each iteration can
be computed using the same set of observed data. The number
SC Readout Noise σIph 0.3 photon counts/px
of iterations that are needed can be determined by monitoring
Wavelength λ Y band (1µm) when the change in the updated linearization point from one
iteration to the next becomes insignificant. This process is illus-
Beam Splitter Ratio 70/30 trated in Fig. 9, starting about the zero vector, and continuing
Source Apparent Magnitude 8 (and 6) using the previous iteration’s estimate as the new linearization
point. The only cost of adding relinearization steps is compu-
Source Spectral Bandwidth 10%̇ tation time since new observations are not needed. In these
Photons per ms in SC 8, 718 (55, 008) simulations, three iterations effectively estimated the aberration
coefficients, but we applied five for good measure.
Photons per ms in WFS 20, 342 (128, 351)
Fried Parameter r0 0.3 m C. Compensating for the NCPA
Once we have estimated the NCPA, we can use a non-common
Inner Scale l0 0.01 m
path DM (such as the one in MagAO-X [56]) to compensate for
Outer Scale L0 25 m them. A DM command that performs the compensation can
be calculated by a least-squares fit of the NCPA optical path
Turbulence Model von Kármán; 6 layers; difference (OPD) map to the DM command modes. The result-
no scintillation; ing phase maps in the coronagraph entrance pupil after DM
frozen flow compensation for our simulations are shown in Fig. 10 in the
Strehl at λ = 1µm 0.73 top right (magnitude 6 source) and bottom (magnitude 8 source)
panels. As we expect from the the RMS error of the estimated
AO Correction Delay 2 Frames aberration coefficients, compensating the NCPA via the magni-
NCPA Zernike Modes 4−36 (6 radial orders) tude 6 source estimate does a better job, achieving a reduction in
RMS phase due to the NCPA from 0.52 radian to 0.0495 radian,
NCPA RMS 0.52 radian a 10.5× improvement. Using the magnitude 8 source estimate
True Object Grid 13 × 8 points achieves a reduction in RMS phase from 0.52 radian to 0.11 ra-
dian, a 4.74× improvement. In order to proceed in Phase B, we
Grid Locations x: [3λ/D, 10λ/D ] will adopt the better compensated NCPA while the magnitude
y: [−6λ/D, +6λ/D ] 8 source in order to demonstrate being able to estimate a small
Spacing of 1λ/D NCPA jointly with the exoplanet image. This choice could rep-
Relinearization Point x a,n iteration n lineariza- resent an observation sequence where the NCPA is calibrated
tion point on a bright star, and then the telescope would slew to a fainter
science target. If this is done, any change to the NCPA due to the
slewing would likely remain small, especially if the calibration
Table 1. Summary of the defining simulation parameters for the
performed numerical experiments. star is close to the target on sky. If additional NCPA is incurred
through this slewing, it can be estimated while observing theResearch Article Journal of the Optical Society of America A 9
Starting NCPA Compensated Mag 6
3
Phase (radian) 2.0 NCPA Phase (radian) 2.0
Magnitude of RMS Error Compenasted RMS
3
1.5 1.5
Source (radian) Phase (radian)
2 2
1.0 1.0 6 0.0214 0.0495
1 1
0.5 0.5
8 0.0595 0.11
Meter
Meter
0 0.0 0 0.0
−0.5 −0.5
−1 −1
−1.0 −1.0 Table 2. Summary of the resulting root mean square (RMS) error
−2 −2 using the naïve estimate based on 1 minute of observations to solve for
−1.5 −1.5
−3 −3 −2.0
the NCPA, and the residual RMS phase left after using the estimates to
−2.0 −3 −2 −1 0 1 2 3
−3 −2 −1 0 1 2 3 compensate the NCPA. Note the starting RMS phase was 0.52 radian.
Meter Meter
Compensated Mag 8
NCPA Phase (radian)
3 2.0 Section 4. Phase B gathers 240, 000 millisecond synchronized ex-
2
1.5 posures in both the WFS and science camera telemetry streams.
1.0 The same process of treating the nonlinearity described above is
1
0.5 performed here, starting by linearizing the regression equations
Meter
0 0.0 with the Na NCPA coefficients and Np exoplanet coefficients set
−0.5 to zero (see Eq. (13)), and successively updating with the previ-
−1
−1.0 ous iteration bias-corrected estimate, for a total of five iterations.
−2
−1.5
Note that after compensating for the NCPA with the DM, the
−3 −2.0
−3 −2 −1 0
Meter
1 2 3 average level of stellar light in the science camera where the
exoplanet image is being estimated is ≈ 3 × 10−4 in contrast
units.
Fig. 10. top left: Starting NCPA phase (in radian), with RMS 0.52
radian, in the coronagraph entrance pupil. top right: The phase in A. The Naïve Estimate
the coronagraph entrance pupil with the magnitude 6 source naïve
estimate used for compensation. The residual phase has RMS 0.0495
radian. bottom: The phase in the coronagraph entrance pupil with True
the magnitude 8 source naïve estimate used for compensation. The Image Naive Es ima e
residual phase has RMS 0.11 radian. ×10−4 1.0 ×10−4
6 6
8
Y Grid Loca ion in λ/D
Y Grid Loca ion in λ/D
4 0.8 4
target star, as will be discussed in the following section. The 2 2 6
0.6
resulting time averaged images in the science camera in contrast 0 0 4
units for the cases of pre- and post compensation, can be seen in 0.4
−2 −2 2
Fig. 11. 0.2
−4 −4 0
−6 0.0 −6
3 5 7 9 3 5 7 9
Time Average SC Image Time Average SC Image X Grid Loca ion in λ/D X Grid Loca ion in λ/D
Pre-Compensation −1.0 Post-Compensation −1.0
10 −1.5 10 −1.5 Fig. 12. The estimated exoplanet imaging coefficients from the simu-
5 −2.0 5 −2.0 lated Phase B experiment, following five relinearization iterations. left:
the true object source grid. right: Naïve estimate of object source grid.
−2.5 −2.5
λ/D
λ/D
0 0
All units are in contrast.
−3.0 −3.0
−5 −5
−3.5 −3.5 As we have seen in the Phase A experiment, the naïve esti-
−10 −10 mate can perform quite well. A combination of factors including
−4.0 −4.0
−10 0 10 −10 0 10 but not limited to WFS measurements of reasonable quality and
λ/D λ/D
the large NCPA, provided conditions under which the naïve
estimate recovers a useful estimate of the NCPA. However, we
Fig. 11. Log scale Time averaged images in the science camera for the found that when using the naïve estimate to solve for the ex-
Phase A and Phase B experiments in contrast. left: the manifestation
oplanet image coefficients and smaller NCPA coefficients, the
of the 0.52 radian RMS (≈ λ/13) NCPA is present, and shows a sig-
nificant degradation to the coronagraphic image. right: the estimated bias from the WME renders the estimate unacceptable, as can be
NCPA has been applied to a DM to compensate, leaving behind the seen in Fig.12 from the Phase B experiment. In this figure, the es-
manifestation of a 0.0495 radian RMS residual NCPA. timated imaging coefficients have been reshaped and plotted as
a 13 × 8 image to allow for easy comparison to the object source
grid. It is clear this is not a useful estimate of the exoplanet
5. PHASE B: JOINT ESTIMATION image, as it results in an RMS error in brightness (contrast units)
of 1.63 × 10−4 , not to mention the fact that the morphology of
The second experiment performed is to jointly monitor the resid-
the image is almost entirely lost.
ual NCPA left over after Phase A, while also estimating any
present exoplanetary image. Recalling the results from Section 4,
B. The Bias-Corrected Estimate
the NCPA present in the coronagraph optical train has an RMS
phase error of 0.0495 radian, and the time average image in the B.1. Choosing Monte Carlo Wavefronts
science camera can be seen in the right frame of Fig. 11. Phase As described in Part II, the bias-corrected estimator uses a set
B is a 4 minute observation of a magnitude 8 star taking place of Monte Carlo wavefronts created using the knowledge of the
directly after the 1 minute Phase A observation described in spatial statistics of the AO residual wavefronts. To generate theResearch Article Journal of the Optical Society of America A 10
Monte Carlo wavefronts, we calculate the mean and covariance be able to leverage the NCPA estimates).
matrix of all of the T AO residual wavefront vectors, the set of
which is represented as {wt }. While it is not possible to know 1e−3 Imaging Estimates
the exact mean and covariance matrix of the AO residuals on a 1.0 xtrue
Ideal
real system (since only the measurements {ŵt } are available), 0.8 Nai e
Bias-Corrected
we ignore that complication for the purposes of this study. We 0.6
Estimated Value
then draw the Monte Carlo wavefronts from a P-dimensional
0.4
multivariate normal with sample mean and covariance of the
AO residual wavefronts. The Monte Carlo wavefronts are thus 0.2
not temporally correlated like the true AO residual wavefronts, 0.0
but there is no need for them to be, so long as their distribution
−0.2
has the same moments as the time-averaged moments of the 0 20 40 60 80 100
Grid Location
stochastic process governing the true AO residual wavefronts.
Although our Monte Carlo wavefronts have the same 2nd Imaging Estimates
1e−4
order statistics as the T = 240, 000 AO residual wavefronts in 1.2
xtrue
Ideal
the 4 minute on-sky data set, (apart from the error due to the Bias-Corrected
1.0
finite number of Monte Carlo samples, which is mitigated by
Estimated Value
using 480, 000 separate Monte Carlo samples), the accuracy of 0.8
the bias-corrected estimate is limited by the fact that the true 0.6
AO residual wavefronts did not obey multivariate normal statis- 0.4
tics. We know this because the univariate (i.e., single pixel) AO 0.2
residual values failed the Anderson-Darling test for univariate
0.0
normality. (Univariate normality of all of the individual vari- 0 20 40 60 80 100
Grid Location
ables is a necessary, but not sufficient, condition for multivariate
normality.) These statistical details are discussed in Part II.
1e−5 Difference of Truth and Imaging E timate
1
B.2. Estimating the Exoplanet Image
0
−1
E timated Value
T ue
Image Bias Co ected Estimate |t ue - estimate| −2
×10 −4
1.0 ×10
−4
1.0
×10
−5
6 6 6 −3
5
Y G id Location in λ/D
Y G id Location in λ/D
Y G id Location in λ/D
4 0.8 4 0.8 4 4
−4
2 2 2 Ideal
0.6 Bia -Corrected
−5
0.6 3
0 0 0
0.4 0.4
0 20 40 60 80 100
2
−2 −2 −2 Grid Location
−4 0.2 −4 0.2
−4 1
−6 0.0 −6 0.0 −6
3 5 7 9 3 5 7 9 3 5 7 9 Fig. 14. top: The exoplanet brightness coefficients for each of the three
X G id Location in λ/D X G id Location in λ/D X G id Location in λ/D
estimators for the Phase B experiment, following five relinearization
iterations. middle: The same as the top, except the naïve estimates are
not displayed. The error bars provided by the ECM are too small to
Fig. 13. The estimated exoplanet imaging coefficients from the sim-
ulated Phase B experiment, following five relinearization iterations, be seen. The partial χ2 value for the ideal estimate is 0.76, and for
reshaped in to the 13 × 8 grid. Left: the true object source grid. Mid- the bias-corrected estimate it is 225. bottom: The difference of the es-
dle: Bias-corrected estimate of object source grid. Right: The absolute timated coefficients and the true values for both the ideal and bias-
value of the error in the estimate. All units are in contrast. corrected estimators
Performing the bias-corrected estimate shows a marked im- The vector of exoplanet image coefficients is plotted in Fig. 14.
provement over the naïve estimate, recovering much of the detail The top frame shows the coefficients for each of the estimates
in the true signal, as can be seen in Fig. 13. The true image is in as well as the truth values, and the bias of the naïve estimate is
the left frame, the bias-corrected estimate is in the middle, and rather evident. In order to see the much smaller residual bias
the error in the estimate is in the right frame. The RMS error in the bias-corrected estimate, the middle frame in this figure
for the bias-corrected estimate of the exoplanet brightness coef- does not contain the naïve estimate values. The bottom frame
ficients, averaged over all of the points is 9.5 × 10−6 , a 17.25× in this figure shows the estimated values subtracted from the
improvement over the naïve estimate. Looking more closely at true values. As explained above, this residual bias in the bias-
the error in the estimated coefficients, there are three grid points corrected estimate is largely due to the fact that the true AO
at a distance near 3 λ/D, where the PSF is bright, with an RMS residuals do not obey multivariate normal statistics, while the
error of about 3.5 × 10−5 . The 13 grid points with a distance Monte Carlo wavefronts do.
near 10 λ/D, where the PSF is not a bright, have an RMS error of In principle (i.e., assuming that the Monte Carlo wavefronts
about 4.2 × 10−6 . Note these the RMS errors in the estimates are have the same statistics as the true AO residual wavefronts),
indicative of the 1 σ contrast achieved, since the estimate error the bias-corrected estimate should be nearly as accurate as the
has very little dependence on the exoplanet brightness (assum- ideal estimate, but here we see that the ideal estimate (which
ing that it is much less bright than the star). The ideal estimates is unbiased) is very close to the true values, with an RMS error
of the exoplanet coefficients are better by roughly a factor of 20. in contrast units of 8.83 × 10−7 . The plots in Fig. 13 and corre-
Note that we did not attempt to reduce the PSF brightness with sponding RMS error numbers show that the simple assumption
dark hole techniques (in principle, dark hole techniques should that the statistics of w are governed by a multivariate normal,Research Article Journal of the Optical Society of America A 11
PSF Subtracted T ue
Image Ext acted Signal |t ue - estimate|
SC Image −1 6
×10 −4
1.0
6
×10
−4
1.0
6
×10
−5
1.5
Y G id Location in λ/D
Y G id Location in λ/D
Y G id Location in λ/D
4 0.8 4 0.8 4
10
−2
2 2 2
0.6 0.6 1.0
0 0 0
0.4 0.4
−2 −2 −2
5
0.5
−3 −4 0.2 −4 0.2
−4
−6 0.0 −6 0.0 −6
3 5 7 9 3 5 7 9 3 5 7 9
X G id Location in λ/D X G id Location in λ/D X G id Location in λ/D
λ/D
0 −4
−5 Fig. 16. Extracting the exoplanet signal from the PSF subtracted focal
−5 plane. Left: The true object source grid. Middle: Extracted signal from
doing perfect PSF subtraction. Right: The absolute value of the error
−6 in the extracted signal. All units are in contrast.
−10
−7 by the off-axis PSF. While one could attempt deconvolution to
mitigate the blurring, instead, we used our knowledge from the
−10 −5 0 5 10 simulation that the exoplanet image consists of point sources
λ/D located on a grid. This allowed us to sidestep deconvolution
simply by extracting the values of the image in Fig. 15 at the grid
Fig. 15. Log scale focal plane image in contrast units after subtracting locations. The result of that extraction is seen in the middle frame
a perfect PSF from the science image. The perfect PSF is created by of Fig. 16, with the true source grid pictured for convenience in
using the same sequence of wavefronts as the science image. the left frame, and the corresponding error in the right frame.
The subsequent measurement of the RMS error for this extraction
with mean and covariance matrix calculated from the 4 minute is 7.63 × 10−6 . Referring to Table (3), this result can be directly
set of millisecond wavefronts, is a useful assumption (at least compared to the RMS error for the various estimate techniques
when compared to naïve estimation), but that it is not adequate presented, showing that the perfect PSF subtraction result quite
to remove all of the bias. is comparable to that of the bias-corrected estimate, with the
RMS error of the bias corrected estimate being about 25% larger.
B.3. Comparing to Perfect PSF Subtraction
The RMS error of the ideal estimate turned out to be 8.6× better
In this section, we compare the bias-corrected estimate with than the perfect PSF subtraction.
a more common method, PSF subtraction. To proceed with While we were able to use knowledge of point-source loca-
this discussion, we first define some terms that will be used. tions to sidestep the deconvolution issue after performing the
PSF stands for "point-spread function," which for our purposes PSF subtraction, this would not be possible on-sky, and it is im-
amounts to the time-average image in the science camera that portant to understand the our estimation methods automatically
would be observed without any planets. It carries the effects of include the blurring effects that off-axis PSF has on the planetary
diffraction, NCPA and the AO residual wavefronts. The science image.
image is the time-average of the science camera telemetry, and it
is shown in the right-hand frame of Fig. 11. We further define
the perfect PSF: Estimate Type NCPA Estimate Exoplanet Image
RMS error Estimate RMS
• it is created using the same sequence of wavefronts as the (radian) error (brightness)
science image
Naïve 1.13 × 10−2 1.63 × 10−4
• it has the exact same NCPA as the science image Bias-Corrected 4.39 × 10−3 9.45 × 10−6
• it has the same noise realizations as the science image Ideal 1.82 × 10−4 8.83 × 10−7
• it excludes all light coming from the circumstellar material Perfect PSF N/A 7.63 × 10−6
(“the exoplanet scene”). subtraction
Thus, the perfect PSF is the ultimate image to subtract from the
Table 3. Error metrics for a simulated 4 minute observation. Sum-
science image, as it perfectly removes starlight and noise. In mary of the resulting root mean square (RMS) error using the naïve,
particular, subtracting the perfect PSF does not suffer from self- bias-corrected, and ideal estimates to jointly solve for the NCPA and
subtraction artifacts associated with differential imaging (see exoplanet image. The results for using ideal PSF subtraction for esti-
Section 2.A), and which arise from having to measure a PSF to mating the exoplanet image are also included
subtract on-sky. Of course, the perfect PSF cannot be known out-
side of simulation studies, but it is a useful benchmark. Fig. 15
shows the resulting difference image of subtracting the perfect C. Jointly Monitoring the NCPA
PSF from the science image following the convention described The regression equations for the Phase B experiment were set
in Section 3.B. up to jointly estimate the exoplanet image and the NCPA. The
Despite the perfect PSF subtraction, the image in Fig. 15 is NCPA had an RMS of 0.0495 radian, which is the uncorrected
missing much of the detail present in true image seen in left portion from Phase A. The estimated coefficients from each esti-
frame of Fig. 13. This is due to the fact that the image is blurred mator are plotted in Fig. 17. Starting our analysis on the naïveYou can also read
