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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 17 — Aug. 21, 2006
  • pp: 7499–7514
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Optimal Fourier control performance and speckle behavior in high-contrast imaging with adaptive optics

Lisa A. Poyneer and Bruce A. Macintosh  »View Author Affiliations


Optics Express, Vol. 14, Issue 17, pp. 7499-7514 (2006)
http://dx.doi.org/10.1364/OE.14.007499


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Abstract

High-contrast imaging with adaptive optics (AO) for planet detection requires a sophisticated AO control system to provide the best possible performance. We evaluate the performance improvements in terms of residual error and point-spread function intensity provided by optimal Fourier control using detailed end-to-end simulation. Intensity, however, is not the final measure of system performance. We explore image contrast through analysis and simulation results, showing that speckles caused by atmospheric errors behave very differently in a temporal fashion from speckles caused by wavefront sensor noise errors.

© 2006 Optical Society of America

1. Introduction

One of the most active areas in astronomy in recent years has been the study of extrasolar planets. Since 1995 more than 160 planets have been discovered orbiting stars other than the sun [1

1. J. Schneider, The Extrasolar Planets Encyclopaedia, Tech. rep. (CNRS-LUTH, Paris Observatory, 2006). http://vo.obspm.fr/exoplanetes/encyclo/catalog.php.

]. The vast majority have been detected through measurements of the Doppler shift of the parent star caused by the gravitational influence of the planet and are found within five astronomical units of their parent star—different from the location of giant planets in our solar system.

A powerful complement to Doppler techniques would be the direct imaging detection of photons from the extrasolar planets themselves. Imaging detection is primarily sensitive to planets in wide orbits where scattered starlight is less of a problem. Imaging techniques require only a short period of time rather than a full orbit, and through photometry or spectroscopy can measure properties of the planet such as radius, temperature, and composition. Unfortunately such detection is extremely challenging. Jupiter, seen from outside our solar system, is 109 times fainter than the sun; a Jupiter analog in another solar system would be swamped by the light from its parent star, scattered by diffraction, optical imperfections, and atmospheric turbulence. At infrared wavelengths, younger and more massive planets will be brighter but still extremely challenging to detect with current telescopes and adaptive optics (AO) systems.

Truly imaging a large sample of extrasolar planets will require an AO system designed specifically for the high-contrast imaging task. This AO system will have dense actuator spacing, a coronagraph to control diffraction, and will be designed not merely to maximize Strehl ratio but to maximize detectability of a planet near a bright star. The spatially filtered wavefront sensor [2

2. L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004). [CrossRef]

] will prevent aliasing in the wavefront sensor (WFS). This will produce a “dark hole region” of reduced point spread function (PSF) intensity in the AO controllable band. Even with bright stars, such systems will typically have very small subapertures (d ≈ 20 cm) and high update rates (1000 – 2000 Hz) and hence still be photon-starved, particularly since many of the stars with the brightest planets are often distant and faint. As a result, these systems must employ optimal controllers to achieve the best possible performance from their available photons. We have proposed a system using an optimal Fourier controller (OFC) [3

3. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A 22, 1515–1526 (2005). [CrossRef]

]. The first major aim of this study is to explore how the gain optimization process works and affects performance.

No AO system can remove all the scattered light in the PSF of a target star. The ability to detect a planet will be limited by the amount and character of this residual light—not merely the Poisson noise set by photon statistics, but the properties of residual speckles present in the PSF [4

4. M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003). [CrossRef]

]. The second major aim of this study is to explore the qualities and temporal behavior of the science images produced by the AO system as a function of the residual phase error.

2. System specifications

2.1. Wavefront control

The control system uses the OFC scheme developed by Poyneer and Véran [3

3. L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A 22, 1515–1526 (2005). [CrossRef]

]. OFC consists of two portions. At every time step, the computationally efficient Fourier transform reconstruction [5

5. L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002). [CrossRef]

] (FTR) method is used to obtain the residual phase estimate from the slope measurements. FTR applies fast discrete Fourier transforms (DFTs) to the slopes and obtains the phase estimate by filtering. This filter consists of an inversion of the WFS process, a compensation for the deformable mirror (DM) response, and a control gain for each mode. The reconstructor works on a spatial frequency grid, where the frequencies k and l range from -N/2 + 1 to N/2, for reconstruction done on a N ×N grid.

Fig. 1. Block diagram of the AO control system for a complex Fourier modal coefficient. All modal signals are complex-valued. The WFS, FTR, and DM units are boxed off.

ϕ[t]=x[t]grx+y[t]gry,
(1)

where these coefficients are given by

grx=1gd(gwx*gwx2+gwy2),
(2)

and

gry=1gd(gwy*gwx2+gwy2).
(3)

When the FTR coefficients are determined from measured WFS gains gwx and gwy and DM gain gd , the filter is termed custom. This is in contrast to theoretical filters based on analytic models of the elements, which we have used previously. These theoretical filters perform well but do not produce unity gain through the system. In our simulations we have generated such a custom filter and have used it.

C(z)=g1cz1,
(4)

2.2. Simulation

A detailed end-to-end AO system simulation with Fourier optics components is used to test new algorithms and evaluate system performance. This simulation incorporates a frozen-flow atmosphere and a full implementation of the AO system elements and wavefront control scheme. It is based on the design for a high-contrast imaging AO system for a D = 8 meter class telescope. The subaperture diameter is d = D/44, and the Fourier reconstruction uses a N = 48 grid for reconstruction and filtering. The block diagram of Fig. 1 describes the simulation’s behavior.

The base atmosphere is created given the Cerro Pachon atmospheric models [8

8. A. Tokovinin, “Modeling turbulence profile for GLAO,” Tech. rep. (Gemini Observatory, 2004).

] and condensed to two layers. Each large phase screen is created using a spectral factor method [9

9. E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE 1237 (1994) pp. 372–383. [CrossRef]

]. The screens are created to be large enough that the entire exposure time will fit in the width of the screen, as opposed to simply wrapping around the edges. The phase aberration in the pupil at any instant in time is available using subpixel shifting of the phase screens and ray tracing. We are not concerned with scintillation for this study, so ray tracing, not optical propagation, is sufficient for producing the phase error in the pupil plane.

The AO simulation runs at an internal system rate of 2 kHz. At each time step, the phase aberration in the pupil is phase-conjugated by the current surface of the DM and tip-tilt mirror. The residual phase is then sent to the science camera unit, which generates PSFs with a gray-scale Blackman function apodization on the amplitude to suppress diffraction. This is a computationally efficient way to model ideal coronagraphy.

The residual phase is sent to the WFS unit. The WFS contains a spatial filter, which is a square field stop of angular size λwfs/d in the focal plane before the lenslet array, where λwfs is the WFS wavelength and d is the subaperture size in the pupil. The spatial filter removes spatial frequency components in the phase that are beyond the ability of the system to measure or correct. As a result, it removes the aliasing error term and enables a dark region of improved correction inside the controllable band of the AO system.

After the spatial filter, the lenslet array generates spots on 2×2 pixel quadcells, with pixel size 1.8”. The WFS operates in the linear range given the pixel size. These pixel values are correctly generated to be integrated intensities over the area. Photon and read noise are added given system radiometry. The total number of received photons in a subaperture is based on a combination of standard photometric calibration to Vega, and an assumed transmission of the telescope and AO system and CCD efficiency of 0.468. The CCD has an assumed 8 electrons read noise per pixel. For a given star magnitude I, the signal-to-noise ratio (SNR) of the WFS measurement can be calculated using standard methods [10

10. S. B. Howell, Handbook of CCD Astronomy (Cambridge University Press, 2000).

]. In our specific case, the WFS SNR is exactly

SNRFI=0.146202(1090.4I)F(0.146202(1090.4I)F+256)12,
(5)

where F is the frame rate of the AO system.

3. Performance of the gain optimizer

The gain optimization process minimizes the residual error power, which is the variance of ε[t]. Assuming that the signal m[t] and the noises nx [t], ny [t] are wide-sense stationary (WSS) random processes, we can easily analyze how the control system modifies them using temporal PSDs and transfer functions. We will further assume for clarity of notation that the random processes are also zero-mean.

Pε(ω)=Pm(ω)1+c22ccosω)1+c2+g22c(1+g)cosω)+2gcos2ω).
(6)

The variance is found by integrating the PSD as

σε2=12πππPε(ω).
(7)

Though the exact formulation depends on the input PSD Pm (ω), the residual error variance will generally decrease as the gain is increased.

The noise has a different path through the system. We reasonably assume that the two noise signals are temporally white, of equal power, and uncorrelated, such that

Pnx(ω)=Pny(ω)=σn22.
(8)

The residual error ε[t] due to WFS noise has temporal PSD

Pε(ω)=σn22g2(grxgd2+grygd2)1+c2+g22c(1+g)cos(ω)+2gcos(2ω).
(9)

The integral for the error variance can be solved in closed form as

σε2=σn22g2(grxgd2+grygd2)g2(1+g)(1g)(1+g)2c2(1g).
(10)

As the control system gain increases, the error variance due to noise will monotonically increase. For very high gains (g > 0.85 in this case), the error variance will become very large due to the large overshoot in the transfer function.

3.1. AO simulation parameters

To assess performance improvement, the end-to-end simulation is used with a reasonable atmospheric model. The simplified Cerro Pachon model, condensed into two layers, is used with better than median conditions. Layer 1 has r 0 of 24.6 cm and moves in frozen flow at speed 2.3 m/s along and angle -124 degrees from the x-axis. Layer 2 is weaker, with r 0 of 34.1 cm, but with a higher wind speed of 16.5 m/s at an angle of -95 degrees. This gives a total r 0 of 18.7 cm. The dominant wind direction is clearly visible in the AO-corrected PSFs.

Performance of the system is tracked with telemetry of residual error power and short- and long-exposure apodized PSFs. This was done for operation at 1 kHz and 2 kHz, and for a natural guide star of magnitude I from 4 to 9.

Fig. 2. Optimal gain filters for a range of NGS magnitudes and two frame rates. The filters are centered so that the lowest frequency is in the middle. The optimal gain varies with modal spatial frequency, spanning in most cases a range of 0.4 gain units.

3.2. Results: Necessary range of gains

In all except the lowest-SNR cases, the optimal gains span a range of around 0.4 gain units. The optimal gains do not in general go above 0.8, because the overshoot in the transfer function becomes too large. This means that the optimal gains take up essentially half of the available range. This large range emphasizes the need to do gain optimization. If a uniform gain is used across all spatial frequencies, many, if not most, frequencies will have substantially the wrong gain.

Another interesting thing shown in Fig. 2 is the strong structural similarity in gains as the NGS magnitude changes. Changing the magnitude simply changes the level of WFS noise power. As long as the SNR is not too low (e.g., I = 9 here) the optimal gains are simply scaled as I changes.

3.3. Results: Improvement in PSF intensity by region

The optimal gain PSFs have equal or lower intensity at all regions in the controllable band out to λ/(2d), except the core, than the uniform-gain PSFs. The in-band error is the error in spatial frequencies out to frequency 1/(2d), set by Nyquist. We will examine this for both signal and noise error reduction through two animated figures.

Fig. 3. [Animated Figure: 1.3 MB] Gain optimization that reduces temporal errors in the PSF, I = 5, 2 kHz case.
Fig. 4. [Animated Figure: 1.9 MB] Gain optimization that reduces noise errors in the PSF, I = 8, 2 kHz case.

The second animation, Fig. 4, shows the 1 sec PSF for the I = 8, 2 kHz case, where WFS noise is the dominant error source. The PSFs are shown on the same color scale as in Fig. 3. With uniform 0.3 gains, the PSF’s controllable region is dominated by a smooth, symmetric halo of WFS noise. It is smooth because 2000 realizations of WFS (1 s at 2 kHz) have been averaged out. This default gain of 0.3 is too high. The gain optimization lowers all gains into the range 0.05 to 0.25. The PSF after optimization has a visible butterfly pattern from the temporal error. There is still some noise halo, but at a greatly reduced level. The median in-band rms error before gain optimization is 69.6 nm rms. After gain optimization it is 58.4, a reduction of 37.9 nm. There is a slight reduction in the butterfly area, but the biggest gain is in the orthogonal noise-dominated region along the x-axis. In this region the PSF intensity is reduced by up to 80%; the PSF intensity is just one-fifth of what it had been. Because the optimization minimizes the residual error power, it maximizes the Strehl. In the signal-dominated case, the already high Strehl improves from 96.1% to 96.3%. In the noise-dominated case, the Strehl ratio improves from 91.0% to 92.8%. These small increases in Strehl disguise the significant reduction in intensity in the dark hole, emphasizing that for high-contrast imaging, the Strehl ratio is too simple a metric.

3.4. Benefits of gain optimization

4. Image contrast and speckles

In this specific planet-imaging application, we are concerned with the ability to detect a planet (if it exists) in a ~ 1 hour exposure. This depends strongly on the light level in the controllable region of the PSF, but also on the smoothness of that halo in a region. As shown above, the gain optimizer results in PSFs with reduced intensity. But the speckles produced by atmospheric and noise errors have very different structures on short time scales. In order to properly evaluate system performance for planet detection, we need a theoretical framework adequate for determining the ability to detect a planet and for verifying the simulation results as reasonable.

To illustrate the performance evaluation problem, consider a short and a long exposure PSF from the simulation, as shown in Fig. 5. The first PSF is a 10 ms exposure; the other is a 5 s exposure. These two PSFs have nearly the same level of scattered light in the controllable region, though the short exposure is slightly lower due to random fluctuations. Despite the similar levels of intensity, the structure of the PSFs is very different. In the short exposure PSF there are many bright speckles and many dark regions. In the long exposure the overall structure is much smoother. In both images a simulated planet has been inserted at a peak intensity 2 × 10-5 times less than the PSF peak value. It is located in the upper right region at 0.6” separation. The smoothness of the long exposure makes the target planet much more clearly detectable.

The image statistic that describes this detectability is contrast, a term for which there are several definitions in use. For a reasonable planet detection rule, we need an estimate of the variance of the background in which that planet might appear. In our case, this background is the controllable region of the PSF, excluding the PSF core. We will not use a general intensity-based metric, such as those that are appropriate for the space-based imaging cases where static speckles dominate the PSF [11

11. J. J. Green and S. B. Shaklan, “Optimizing coronagraph designs to minimize their contrast sensitivity to low-order optical aberrations,” in Techniques and Instrumentation for Detection of Exoplanets, D. R. Coulter, ed., Proc. SPIE 5170 (2003) pp. 25–37. [CrossRef]

].

The first way to estimate the background variation is to take a single long-exposure PSF (generated by the simulation) and to average spatially over the speckles in a sensible fashion. This spatial-averaging method has the advantage that it requires only a single PSF; it is particularly relevant for a detection strategy that involves the analysis of a single science image. However, it does assume that spatial variations in a single PSF will be comparable with temporal variations at a specific location; in other words, it assumes that the process is ergodic.

Fig. 5. Short and long exposure PSFs from the end-to-end simulation. Both have the same standard deviation of intensity in the controllable region (dark hole) but a planet is more easily detected in the long exposure image.

Another way to estimate image contrast is to do temporal averaging. For this, a large set of sample images of the desired exposure time is used to evaluate the sample variance for a specific location of interest. This set is drawn over random atmospheres and will give an estimate of the temporal variations at a specific location. Temporal averaging is most appropriate when multiple images will be used for detection decisions. However, this technique requires much more computation and as such may be prohibitively expensive.

A third technique would be to use an analytic or semi-analytic approach, which would obviate the need for long end-to-end simulation runs. Spatial-frequency domain modeling is a type of analytic approach. Versions of this include the linear systems spatial-frequency method of Ellerbroek [12

12. B. L. Ellerbroek, “Linear systems modeling of adaptive optics in the spatial-frequency domain,” J. Opt. Soc. Am. A 22, 310–322 (2005). [CrossRef]

] and the spatial power spectrum approach of Jolissaint et al. [13

13. L. Jolissaint, J.-P. Véran, and R. Conan, “Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach,” J. Opt. Soc. Am. A 23, 382–394 (2006). [CrossRef]

]. Another analytic approach looks directly at the statistical behavior of the PSF intensity. Examples include the theoretical work of Aime and Soummer [14

14. C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. 612, L85–L88 (2004). [CrossRef]

] and the experimental work of Fitzgerald and Graham [15

15. M. P. Fitzgerald and J. R. Graham, “Speckle statistics in adaptively corrected images,” Astrophys. J. 637, 541–547 (2006). [CrossRef]

]. Neither of these approaches has entirely what we want; the spatial-frequency domain techniques are not designed to analyze the PSF intensity temporal behavior, and the statistical PSF methods do not incorporate detailed knowledge of an AO system.

Aspects of three methods (spatial-averaging, temporal-averaging, and the analytic approach) are used in concert to analyze the temporal behavior of the PSF intensity. The residual error output of the AO control system consists of atmospheric errors and WFS noise errors. These two types behave very differently as a function of time, and this difference should be taken into account when evaluating final system performance.

4.1. AO simulation parameters

To study the temporal behavior of the PSF intensity, the simulation is run in a special short-exposure mode. Like the long-exposure simulations previously described, the simulation generates the instantaneous PSF at every time step (2 kHz). For speckle studies, these are accumulated into 10 ms exposures which are saved, producing a 100 Hz sampling of the PSF evolution. For this study 12 different realizations of the above-median atmosphere were run, each for 5.12 seconds. This was done for the I = 6, 2 kHz case and also for the same case but with no WFS noise, so that the atmospheric behavior could be examined directly. One 5.12 second exposure with no atmosphere and only WFS noise was also run.

4.2. Image behavior as exposure time increases

We begin with a basic statistical model for how the PSF intensity behaves in short- and long-exposure images. The PSF intensity at any location is modeled as a random process with a specific mean and autocovariance function. This is a discrete-time model because the simulation is discrete-time, but the concepts here can easily be converted to a continuous-time model.

The PSF intensity at a specific point in time is represented by a random process i[t]. The exact mean and autocovariance of this time series will be discussed later, but it is worth noting now that this random process is not zero-mean. A long exposure image will be the average of many of these consecutive values to preserve normalization of intensity. This long-exposure value for T consecutive frames will be

pN[t]=1Tl=0T1i[tl].
(11)

Since this is a linear filter, if i[t] is a WSS random process, the same techniques as used in Section 3 can be applied to determine the PSD and variance of pN [t]. Given a temporal PSD for the instantaneous PSF intensity Pi (ω), the temporal PSD for a long-exposure is

Pp,T(ω)=Pi(ω)1T2[sin(ωT2)sin(ω2)]2.
(12)

Our quantity of interest, the variance of the long-exposure intensity, is then

σp,T2={12π1T2ππPi(ω)[sin(ωT2)sin(ω2)]2}mi2,
(13)

where mi is the mean of i[t].

If the intensity is temporally white, Pi (ω) is constant and this integral is easily determined to be 1/T, by Parseval’s theorem. This gives us the averaging out with exposure time that we would expect. If Pi (ω) is not white, we need to consider how the integral behaves as the exposure time increases. When T = 2, the sinc-squared function is very broad. As T increases, the lowpass filter rapidly narrows and suppresses the high temporal frequencies. Once Pi (ω) is relatively uniform over the width of the lowpass filter, the integral behaves exactly as in the white noise case: the variance drops off as 1/T. The level of the variance is set by the power at the lowest temporal frequencies of Pi (ω). We can identify the point at which the intensity variance drops off to one-half of the instantaneous exposure level. This is our analytic basis for the speckle lifetime, a concept that was introduced by Racine et al. [16

16. R. Racine, G. A. H. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999). [CrossRef]

].

This will be done using the temporal-averaging contrast method. Using the 100 Hz sampling of the PSF described above, the intensity of the PSF at a specific location is tracked through the long exposure. Figure 6(a) shows the PSF intensity for the location 0.45” above the PSF core. The Atmosphere line is for the no-WFS noise simulation, and the WFS Noise line is for the no-atmosphere simulation. The intensity caused by the atmospheric residual error has large and much slower fluctuations than the WFS noise. We characterize the temporal behavior of the PSF intensity by using estimated temporal PSDs for each pixel in the image. These are calculated with the averaged periodogram method[18

18. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time Signal Processing (Prentice Hall, 1999).

] over the entire 5.12 second interval. This produces a sampled temporal PSD with a frequency spacing of 0.195 Hz and a maximum frequency of 50 Hz. Figure 6(b) shows the estimated temporal PSD for the same location. Based on the two separate simulations, the temporal behavior for the intensity due to only atmosphere and only WFS noise are shown. The two temporal PSDs have substantially different structures.

Fig. 6. (a) (left) PSF intensity at a location 0.45” above the PSF core, sampled at 100 Hz. Atmosphere is for simulation with no WFS noise. WFS noise line for simulation with no atmospheric error. (b) (right) the estimated temporal PSDs for PSF intensity at the same location as shown in (a).

Using these temporal PSD estimates, the pixel variance as a function of exposure time can be computed using Eq. (13). The results of this are in Fig. 7(a). The atmospheric component has initially flat variance, since in a short exposure an atmospheric-induced speckle stays relatively constant. As the phase screen blows by the aperture, the phase content changes. Averaging begins to occur and for exposures past 0.25 seconds, the pixel variance follows the 1/T behavior. The WFS noise-induced intensity follows the 1/T behavior for the range of exposure times.

Fig. 7. (a) (left) estimated intensity variances as a function of exposure time, calculated with Eq. 13 and the data shown in Fig. 6b. The WFS noise has the 1/T drop-off, but the atmospheric variance is flat before dropping off after 0.25 seconds. (b) (right) Intensity variances drawn from a spatially-averaged contrast metric at 0.45” distance for cumulative exposures of increasing length. Atmosphere data points are over 12 difference atmospheres, with error bars. These results, determined with a completely different method, confirm the results shown (a).

Given the outputs of the simulation, the speckles due to atmospheric errors behave very differently than the speckles due to WFS noise. To get a reasonable estimate of the contrast given residual atmospheric errors, a sufficiently long exposure needs to be examined so that the low temporal-frequency terms have begun to average out.

4.3. Intensity as a function of residual modes

The PSF expansion is a mathematical way to represent the PSF, generated from a complex field of arbitrary amplitude and phase, in terms of the Fourier transforms of that amplitude and phase. It allows for improved understanding of the high-contrast imaging scenario as well as enabling mathematical analysis in a tractable form. Given the complex field a(x,y)exp[(x,y)], where both a and ϕ are functions of two spatial variables in the pupil plane, the PSF in the image plane is

PXY=AXY+jAXY*ΦXY0.5AXY*ΦXY*ΦXY+2.
(14)

First, since the amplitude is real and symmetric, complex conjugation on A(X,Y) is not necessary. Second, the spatial variables will be dropped unless necessary for clarity. This makes the PSF

PA2+2Im{A[A*Φ*]}+A*Φ2Re{A[A*Φ**Φ*]}
+Im{[A*Φ][A*Φ**Φ*]}+0.25A*Φ*Φ2
(15)

ϕxyt=εcos[t]cos(2πD[kx+ly])+εsin[t]sin(2πD[kx+ly]),
(16)

where ε cos[t] is just the real part of ε[t] and ε sin[t] is the imaginary part of ε[t]. The Fourier transform of this phase is

ΦXYt=12{ε[t]δ(XλkD,YλlD)*ε*[t]δ(X+λkD,Y+λlD)}.
(17)

ik,t[t]A2klAklεsin[t]+ε[t]24.
(18)

This is of course a simplification. We have ignored other terms, such as the higher-order scattering of light from outside the controllable region that we have previously identified [2

2. L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004). [CrossRef]

] as having the potential to contribute non-negligible intensity. However, for the purpose of illuminating significant temporal behavior of the PSF intensity, this approximation is adequate.

Given this formulation of the pixel intensity as a function of the random residual phase error modes, the statistical behavior of the intensity can be analyzed. Since the image-formation process is not a linear process, a transfer function cannot be used. Direct calculations, however, will obtain the mean, autocorrelation, and autocovariance functions and hence the PSD of the pixels. This will be done assuming that the random processes are zero-mean and Gaussian, which allows evaluation of the higher moments. At this point the notation is clearer when the statistics are given in terms of the real and imaginary parts of ε, namely ε cos and ε sin. The variance of the error is the sum of two parts

σε2=σε,cos2+σε,sin2.
(19)

Furthermore, the correlation between the cosine and sine parts is given by E[cs]. This correlation is nonzero in the case of the atmosphere-induced residual error. The mean intensity is just diffraction plus the power level of the residual error

E[i]=A2kl+σε24.
(20)

The variance of the instantaneous exposure is

σi2=18{σε,cos4+σε,sin4+2(E[cs])2}+A2klσε,sin2.
(21)

These expressions are useful in evaluating how the error terms due to the atmosphere and the WFS noise (which are uncorrelated sources) combine in the PSF. Consider two uncorrelated errors, ε 1 from signal and ε 2 from noise. The expected intensity is simply the sum of the individual terms

E[i]=A2kl+14(σε12+σε22).
(22)

This is what we would expect, as uncorrelated noise sources should add in the expected PSF intensity. Analyzed individually, the PSF intensity variance from each term is simply Eq. 21 rewritten:

σi12=18{σε1,cos4+σε1,sin4+2(E[c1s1])2}+A2klσε1,sin2,
(23)
σi22=18{σε2,cos4+σε2,sin4+2(E[c2s2])2}+A2klσε2,sin2.
(24)

However, when those two error random processes are added together as ε 1 +ε 2, the resulting intensity variance is not simply the sum σi12 +σi22, but rather

σi2=σi12+σi22+{14(σε1,sin2+σε2,cos2+σε1,sin2σε2,sin2)+12E[c1s1]E[c2s2]}.
(25)

The significance of this extra term depends on the relative power levels in residual signal and noise error random processes. When the power levels are close, this extra variance term becomes significant. If the signal and noise residuals have equal power, the PSF intensity variance can be twice the simple sum of the individual variances. This extra-variance phenomenon is visible in the simulation outputs in the temporal PSDs when power levels are similar.

When considering two error sources in the AO system (i.e., atmospheric signal and WFS noise), the expected PSF intensity is well-approximated by the diffraction pattern plus the power level of that specific spatial frequency. However, the variance of the PSF intensity (and hence the contrast of the PSF and the ability to detect planets) is not necessarily the sum of the individual variances. This only occurs in specialized situations, but since any extra variance will degrade contrast, this term should be analyzed. The end-to-end simulation run with both inputs automatically takes into consideration this extra-variance, but analytic methods, which sum over independent error sources, may not.

5. Gain optimizer and contrast

The above formulation for PSF intensity variance as a function of the atmospheric and WFS noise phase error random processes and the AO control system provides a framework for evaluating the performance of OFC in regard to contrast. Such an analytic treatment is beyond the scope of this study, but we will provide a direct comparison of the simulation results for long-exposure images. The test case is I = 6, 2 kHz operation. This case has a mix of gains being raised (up to 0.5 in the butterfly) and lowered (down to 0.15 orthogonal to the butterfly). A 2 s exposure PSF was accumulated from the 100 Hz short exposure series to generate an estimate of the PSF intensity. The same 100 Hz short exposure series was used as described above to generate an estimate of the image contrast in the 2 s exposure, using the temporal-averaging contrast method.

As discussed in Section 3, the PSF intensity is lowered in the controlled region everywhere that the optimized gain is not the default gain of 0.3. However, for the contrast this is not the case. Everywhere the gain was increased by the optimizer, the resulting contrast is improved. Everywhere the gain was decreased, however, the resulting contrast is worse. In the region where the optimal gains were lowered, the atmospheric residual error power is now higher. Because the speckle decorrelation time is non-zero, the atmospheric speckles dominate the PSF contrast. This is illustrated in Fig. 8, with slices through the PSF contrast given by the intensity standard deviation. In the butterfly region the contrast has improved by a factor of 2, but in the opposite region contrast has been degraded by a factor of 1.5. The reason why contrast is worse in part of the PSF is because the gain optimization metric minimizes total residual error power, not contrast. In this region, though the WFS noise error term dominates, the atmospheric component still dominates the contrast. For high-contrast imaging, a contrast-optimizing controller is needed. We are currently investigating adaptive OFC to optimize image contrast, using the formulation of Section 4.3 to develop the gain optimization metric.

Fig. 8. Contrast of 2 s PSF, before and after gain optimization. (a) (left) Slice along y-axis through butterfly region. Gain optimization has improved contrast by a factor of 2. (b) (right). Slice along x-axis through region of little atmospheric error. Gain optimization has decreased contrast by a factor of 1.5.

6. Conclusion

PSF intensity, however, is not the final measure of system performance for planet detection. Contrast, defined here as the variation in intensity in the PSF for a given exposure time, is what we want. PSF speckles caused by atmospheric phase errors behave very differently in a temporal fashion from speckles caused by WFS noise phase errors. While the WFS noise errors lead to a 1/T drop-off in intensity variance as a function of exposure time for even short exposures, the atmospheric speckles have a much longer decorrelation time. These speckles can dominate the contrast, even when error power is low, as shown in Section 5. As a consequence, the error-minimizing metric of OFC does not optimize contrast.

Acknowledgments

The authors thank the reviewers for their useful comments, particularly regarding the clarity of the manuscript. We also thank Julia W. Evans for her helpful suggestions regarding the organization of this paper. This work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract W-7405-Eng-48. The document number is UCRL-JRNL-219806. This work has 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.

References and links

1.

J. Schneider, The Extrasolar Planets Encyclopaedia, Tech. rep. (CNRS-LUTH, Paris Observatory, 2006). http://vo.obspm.fr/exoplanetes/encyclo/catalog.php.

2.

L. A. Poyneer and B. Macintosh, “Spatially filtered wave-front sensor for high-order adaptive optics,” J. Opt. Soc. Am. A 21, 810–819 (2004). [CrossRef]

3.

L. A. Poyneer and J.-P. Véran, “Optimal modal Fourier transform wave-front control,” J. Opt. Soc. Am. A 22, 1515–1526 (2005). [CrossRef]

4.

M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, “The structure of high Strehl ratio point-spread functions,” Astrophys. J. 596, 702–712 (2003). [CrossRef]

5.

L. A. Poyneer, D. T. Gavel, and J. M. Brase, “Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform,” J. Opt. Soc. Am. A 19, 2100–2111 (2002). [CrossRef]

6.

E. Gendron and P. Léna, “Astronomical adaptive optics, 1: Modal control optimization,” Astron. Astrophys. 291, 337–347 (1994).

7.

P.-Y. Madec, “Control Techniques,” in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University Press, 1999) pp. 131–154.

8.

A. Tokovinin, “Modeling turbulence profile for GLAO,” Tech. rep. (Gemini Observatory, 2004).

9.

E. M. Johansson and D. T. Gavel, “Simulation of stellar speckle imaging,” in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE 1237 (1994) pp. 372–383. [CrossRef]

10.

S. B. Howell, Handbook of CCD Astronomy (Cambridge University Press, 2000).

11.

J. J. Green and S. B. Shaklan, “Optimizing coronagraph designs to minimize their contrast sensitivity to low-order optical aberrations,” in Techniques and Instrumentation for Detection of Exoplanets, D. R. Coulter, ed., Proc. SPIE 5170 (2003) pp. 25–37. [CrossRef]

12.

B. L. Ellerbroek, “Linear systems modeling of adaptive optics in the spatial-frequency domain,” J. Opt. Soc. Am. A 22, 310–322 (2005). [CrossRef]

13.

L. Jolissaint, J.-P. Véran, and R. Conan, “Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach,” J. Opt. Soc. Am. A 23, 382–394 (2006). [CrossRef]

14.

C. Aime and R. Soummer, “The usefulness and limits of coronagraphy in the presence of pinned speckles,” Astrophys. J. 612, L85–L88 (2004). [CrossRef]

15.

M. P. Fitzgerald and J. R. Graham, “Speckle statistics in adaptively corrected images,” Astrophys. J. 637, 541–547 (2006). [CrossRef]

16.

R. Racine, G. A. H. Walker, D. Nadeau, R. Doyon, and C. Marois, “Speckle noise and the detection of faint companions,” Publ. Astron. Soc. Pac. 111, 587–594 (1999). [CrossRef]

17.

B. Macintosh, L. A. Poyneer, A. Sivaramakrishnan, and C. Marois, “Speckle lifetimes in high-contrast adaptive optics,” in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson and M. Llyod-Hart, eds., Proc. SPIE 5903 (2005) p. 59030J. [CrossRef]

18.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time Signal Processing (Prentice Hall, 1999).

OCIS Codes
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(350.1260) Other areas of optics : Astronomical optics

ToC Category:
Focus Issue: Adaptive Optics

History
Original Manuscript: March 16, 2006
Revised Manuscript: May 17, 2006
Manuscript Accepted: May 18, 2006
Published: August 21, 2006

Citation
Lisa A. Poyneer and Bruce A. Macintosh, "Optimal Fourier control performance and speckle behavior in high-contrast imaging with adaptive optics," Opt. Express 14, 7499-7514 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-17-7499


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References

  1. J. Schneider, The Extrasolar Planets Encyclopaedia, Tech. rep. (CNRS-LUTH, Paris Observatory, 2006), http://vo.obspm.fr/exoplanetes/encyclo/catalog.php.
  2. L. A. Poyneer and B. Macintosh, "Spatially filtered wave-front sensor for high-order adaptive optics," J. Opt. Soc. Am. A 21, 810-819 (2004). [CrossRef]
  3. L. A. Poyneer and J.-P. Véran, "Optimal modal Fourier transform wave-front control," J. Opt. Soc. Am. A 22, 1515-1526 (2005). [CrossRef]
  4. M. D. Perrin, A. Sivaramakrishnan, R. B. Makidon, B. R. Oppenheimer, and J. R. Graham, "The structure of high Strehl ratio point-spread functions," Astrophys. J. 596, 702-712 (2003). [CrossRef]
  5. L. A. Poyneer, D. T. Gavel, and J. M. Brase, "Fast wavefront reconstruction in large adaptive optics systems with use of the Fourier transform," J. Opt. Soc. Am. A 19, 2100-2111 (2002). [CrossRef]
  6. E. Gendron and P. Léna, "Astronomical adaptive optics, 1: Modal control optimization," Astron. Astrophys. 291, 337-347 (1994).
  7. P.-Y. Madec, "Control Techniques," in Adaptive Optics in Astronomy, F. Roddier, ed. (Cambridge University Press, 1999) pp. 131-154.
  8. A. Tokovinin, "Modeling turbulence profile for GLAO," Tech. rep. (Gemini Observatory, 2004).
  9. E. M. Johansson and D. T. Gavel, "Simulation of stellar speckle imaging," in Amplitude and Intensity Spatial Interferometry II, J. B. Breckinridge, ed., Proc. SPIE 1237 (1994) pp. 372-383. [CrossRef]
  10. S. B. Howell, Handbook of CCD Astronomy (Cambridge University Press, 2000).
  11. J. J. Green and S. B. Shaklan, "Optimizing coronagraph designs to minimize their contrast sensitivity to loworder optical aberrations," in Techniques and Instrumentation for Detection of Exoplanets, D. R. Coulter, ed., Proc. SPIE 5170 (2003) pp. 25-37. [CrossRef]
  12. B. L. Ellerbroek, "Linear systems modeling of adaptive optics in the spatial-frequency domain," J. Opt. Soc. Am. A 22, 310-322 (2005). [CrossRef]
  13. L. Jolissaint, J.-P. Véran, and R. Conan, "Analytical modeling of adaptive optics: foundations of the phase spatial power spectrum approach," J. Opt. Soc. Am. A 23, 382-394 (2006). [CrossRef]
  14. C. Aime and R. Soummer, "The usefulness and limits of coronagraphy in the presence of pinned speckles," Astrophys. J. 612, L85-L88 (2004). [CrossRef]
  15. M. P. Fitzgerald and J. R. Graham, "Speckle statistics in adaptively corrected images," Astrophys. J. 637, 541-547 (2006). [CrossRef]
  16. R. Racine, G. A. H. Walker, D. Nadeau, R. Doyon, and C. Marois, "Speckle noise and the detection of faint companions," Publ. Astron. Soc. Pac. 111, 587-594 (1999). [CrossRef]
  17. B. Macintosh, L. A. Poyneer, A. Sivaramakrishnan, and C. Marois, "Speckle lifetimes in high-contrast adaptive optics," in Astronomical Adaptive Optics Systems and Applications II, R. K. Tyson andM. Llyod-Hart, eds., Proc. SPIE 5903 (2005) p. 59030J. [CrossRef]
  18. A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-time Signal Processing (Prentice Hall, 1999).

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