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We describe a method to reduce residual speckles in an adaptive optics system which add to the halo of the point spread function (PSF). The halo is particularly problematic in astronomical applications involving the detection of faint companions. Areas of the pupil are selected where the residual wavefront aberrations are large and these are masked using a spatial light modulator. The method is also suitable for smaller telescopes without adaptive optics as a relatively simple method to increase the resolution of the telescope. We describe the principle of the technique and show simulation results.
©2009 Optical Society of America
1. Introduction
The turbulent atmosphere causes phase variations across a wavefront propagating from an astronomical object to a ground based telescope. It is well known that these distortions degrade the imaging performance of the telescope and the whole field of adaptive optics (AO) has been developed to ameliorate these distortions. However, no AO system is perfect and the partially corrected point spread function (PSF) from a typical AO system consists of a diffraction limited core sitting on top of a much broader halo. The short exposure halo is made up from speckles which are averaged in a long exposure to produce a large (in angular extent) low level plateau which can limit the achievable signal to noise ratio of the detection of faint objects around bright stars.
In this paper we propose a method to reduce the halo by using adaptive pupil masking. Areas of the wavefront whose phase error is larger than some threshold value are completely blocked. If we mask some of the aperture then we are clearly reducing the throughput of the system, but we are also blocking the areas of the wavefront which tend to produce the halo rather than the central core of the PSF. We propose using an amplitude modulating spatial light modulator (SLM) as the active element. We show that blocking areas of the pupil can lead to both a reduction in the halo intensity and an increase in the central intensity.
There are other methods for improving the performance of telescopes which are related to adaptive pupil masking. In a binary adaptive optics system [1] areas of the pupil which are more than half a wave out of phase (modulo 2π) have a correction of π added to them. The basic philosophy behind binary AO is that all the parts of the wavefront which have a phase error of less than π are adding approximately constructively at the telescope focus - and therefore should be left unchanged. Areas of the wavefront which have an error of greater than π are adding destructively at the telescope focus and therefore if a correction of π is added then there will be a dramatic improvement of imaging performance. The proposed method of adaptive pupil masking is similar, except here we completely remove areas of the pupil with large phase errors, plus the criterium for masking is not necessarily a phase error of π. These are important differences as it means that this method is more suitable for use as an addition after a conventional AO system and before coronagraphic techniques - rather than binary AO which was proposed as a simple approach to full AO.
The adaptive pupil mask is also a variation on the Lucky Imaging-type techniques. This was first proposed by Fried [2] and consists of recording many short exposure images without adaptive optics. A fraction of these images are selected, according to their quality, and are coadded to produce impressive results [3]. The probability, P, of observing a ‘lucky image’ (an image with phase variance less than 1 rad2 as defined by Fried) can be calculated using ([2]),
for D=r 0>3:5, where D is the diameter of the telescope pupil and r 0 is the Fried parameter. As the telescope size increases the probability of observing a lucky image decreases with a strong function of D=r 0 which makes the method ideal for small telescopes but for larger telescopes the probability becomes so high that the method is unusable. A low order AO system increases the probability and has been demonstrated on larger telescopes [4]. Here we present a development of the standard lucky imaging method to increase its efficiency on telescopes of all sizes. Instead of temporally filtering the wavefront we spatially filter. This is similar to the work by Morossi et al. [5] who spatially select the best subapertures on a large telescope and co-add them in order to improve the resolution in the visible with AO systems configured for the IR. However, we do not co-add instead we simply block the subapertures conserving the full resolution of the telescope.
The adaptive pupil mask could be deployed on any telescope, for example a small telescope without AO or a larger telescope with AO. For a telescope with AO the threshold can be set so that the piston cutoff is smaller than the residual piston after the AO correction in order to reduce the phase variance further. Small or medium sized telescopes are often left uncorrected due to the cost in dollars and complexity of a full AO system. The adaptive pupil mask could be used to improve the performance of these telescopes for a fraction of the cost. The reduced phase variance will result in reduced halo, increased peak intensity and reduced full width at half maximum (FWHM) of the PSF which will be useful in many areas of astronomy. For example it could be beneficial for multi-object spectrographs as the reduced FWHM will minimise the cross talk between the spectrograph channels. The example used in this paper to quantify the possible improvements is the detection of faint companions. We show that the signal to noise ratio is significantly improved by using the adaptive pupil mask. It should be noted that although we use the example of faint companion detection it is unlikely that the technique will be useful for terrestrial exoplanet detection. This is because the detection of extrasolar planets require extremely low residual phase variance in order to have sensible exposure times and when we obtain this using extreme AO other factors then start to become significant. Quasi static speckles ([6], [7], [8] and many others) appear due to the flaws in the optical surfaces and setup. These speckles are not static enough to be subtracted and not variable enough to average out. The temporal and spatial statistics of these speckles will be changed by the adaptive pupil mask changing the shape of the telescope pupil throughout an observation meaning that angular differential imaging [9] as used by Marois et al. [10] in the first direct imaging of an extra solar planet will no longer work. Scintillation is also a fundamental problem for the direct detection of extrasolar planets [11], [12] as this will alter the pupil function which will change the PSF. This was not included in the simulations or theoretical work as the effect is only significant when the phase variance is near to zero which we did not approach in the simulations.
In the following sections we present results of a simulation of the technique and a theoretical analysis which explains the critical threshold values.
2. Simulation
Using the AO simulation framework developed at Durham University [13] we have implemented and executed a full AO simulation including the adaptive pupil mask. The simulation consists of a single turbulent layer with a Fried parameter, r0, of 0.15 m and a wavelength, λ, of 500 nm. The phase screen is randomly evolving and is blown across the pupil of the telescope at 5 m/s. The segmented deformable mirror is modeled on the Durham ELECTRA mirror [14] allowing three degrees of freedom for each segment (piston, tip and tilt) with either 8×8 or 16×16 subapertures. The phase is measured via a Shack-Hartmann wavefront sensor (WFS) and a successive over-relaxation reconstruction algorithm estimates the phase map and passes the data to the mirror and the adaptive pupil mask. The mask is placed after the mirror and WFS pickoff in the pupil plane. The adaptive pupil mask will have the same geometry as the WFS (either 8×8 or 16×16 in this case). Ideally the subapertures will be a similar size to r 0 to achieve the optimum performance although this is not necessary. The pupil mask blocks the subapertures which have a reconstructed piston greater than a threshold value and updates at the same rate as the deformable mirror (every 5 ms). The threshold value chosen will depend on the strength of the turbulence in the atmosphere and the requirements of the user and will be discussed in section 3. Figure 1 shows the data flow of the simulation and the location of the pupil mask within the optical train.
Fig. 1. Block diagram for the adaptive pupil mask system. The mask is positioned in the pupil plane of the telescope after the deformable mirror (DM) and after the wavefront sensor (WFS) pickoff.
The examples below were chosen to show the mask in two different regimes. (1) The small telescope scenario (1 m without AO, with 8×8 WFS) and (2) the large telescope scenario (8 m with 16×16 AO). By blocking subapertures with a large instantaneous phase excursion the wavefront will be flatter. The result of which is a reduced PSF halo and an increased peak intensity. The extent of the PSF improvement is dependent on the wavefront variance after the blocking and so the the lower the wavefront phase threshold we choose to block the greater the fraction of the pupil is removed and the flatter the wavefront becomes. However, blocking the pupil will also reduce the total intensity of the PSF and modify the diffraction pattern. The optimum threshold is a balance between these two effects and can be found from plots of the full width at half maximum (FWHM) of the PSF (Fig. 2) and the peak intensity of the PSF (Fig. 3) as a function of the threshold value.
To quantify the improvements for a specific case in both of the following examples the threshold is chosen to maximise the peak intensity. The first example, Fig. 4, shows a 3D plot of the PSF from a 1 m telescope without a deformable mirror but using a WFS with 8×8 subapertures and the threshold was chosen to be ±1.8 radians. The intensity of the PSF is increased by 40 % and the FWHM is reduced from 0.58″ to 0.16″ with the diffraction limited FWHM being just 0.13″. On average 42 % of the pupil was blocked.
Fig. 2. Simulation results showing how the FWHM of the PSF is modified by the adaptive pupil mask. All areas of the pupil which have a phase error greater than the threshold (either positive or negative) are blocked. The plots show the FWHM for (a) a 1 m class telescope and (b) an 8 m telescope with AO. The solid horizontal black line at the top of the plot shows the FWHM without the adaptive pupil mask and the dashed line at the bottom shows the FWHM for a perfect system with no aberrations. For a large threshold only a small fraction of the pupil is blocked and there is little change of the PSF. As the threshold is reduced more of the pupil is blocked, the resulting wavefront is flatter and the FWHM is reduced. If the threshold is too low we block a large fraction of the pupil broadening the diffraction pattern due to the low fill factor of the pupil.
Fig. 3. Simulation results showing how the peak intensity of the PSF is modified by the adaptive pupil mask. (a) shows the peak intensity for the 1 m telescope without AO and (b) for the 8 m telescope with AO. The solid black line indicates the peak intensity of the PSF with no blocking. For low threshold values large fractions of the pupil are blocked and so the total intensity is also reduced by a large amount. For high threshold values the effect is negligible, but there is an intermediate value where the peak intensity is increased. The intensity is normalised to the peak value without a pupil mask.
Fig. 4. Example PSF from a 1 m telescope without AO with r0=0.15 m. On the top left is the original PSF from the telescope with no AO and on the top right is the PSF with no AO but using the adaptive pupil mask with a threshold value of ±1.8 radians. The bottom plot shows the radial intensity profiles of the two PSFs. The black dashed line is the original intensity pattern and the red line is the modified radial profile. The modified intensity is 40 % higher and the FWHM 4 times smaller than the original PSF.
Figure 5 is a 3D plot of the PSF from an 8 m telescope with a 16×16 AO system. The peak intensity of the PSF is increased by a factor 0.23 and the FWHM is reduced from 0.022″ to 0.01800 with the diffraction limited FWHM being 0.016″. The threshold was ±1.4 radians and on average 19 % of the pupil was blocked.
The advantage of this technique can be quantified by considering the case of the detection of faint companions. As a consequence of the reduced fraction of energy in the PSF halo the background count from the parent star at the position of the companion will be lower. Combining this with the increased peak intensity results in an improved contrast ratio which equates to either a higher signal to noise ratio (SNR) or a reduced exposure time to obtain a target SNR. As the simulation were all run for a simulation time of 100 seconds the results presented here are in terms of the possible gains in SNR. Figure 6 shows the SNR as a function of the threshold assuming no sky background and a detector with 100 % quantum efficency. The threshold for maximum SNR is different to the optimum threshold for peak intensity as it is a balance between maximising the peak intensity and minimising the FWHM. The simulation results are shown in table 1. The magnitude difference in each case is chosen so that the SNR after 100 s is 5 and the angular separation of the companions is 2λ/D. For a 1 m telescope this corresponds to a magnitude difference of 7.7 and a difference of 11.7 with an 8 m telescope. The adaptive pupil mask substantially increases the SNR in both cases, doubling it to 10.5 for a 1 m telescope and increasing it to 7.1 for the 8 m telescope. A four quadrant phase mask (FQPM) coronagraph [15] can be used to further increase the SNR for faint companion detection by reducing thelight from the parent star (we stress again that our proposed technique may not be suitable for detecting terrestrial exoplanets - but a coronagraph can be useful in general for detecting faint companions). The FQPM coronagraph is sensitive to pupil geometry [16] and so the adaptive pupil mask will mean that the coronagraph can not operate as effectively as it could. But the reduced wavefront variance after the pupil mask will also mean that the FQPM coronagraph will be more efficient [17]. The simulation results show that the reduced phase variance outweighs the effects of the changing pupil geometry and that the coronagraph actually works better after the adaptive pupil mask and so the combination of the pupil mask and coronograph results in an SNR of 10.6, twice the original value.
Fig. 5. Example PSF from an 8 m telescope equipped with a 16×16 AO system with r0=0.15 m. On the top left is the original PSF from the telescope and on the top right is the PSF with AO and the adaptive pupil mask and a threshold value of ±1.4 radians. The bottom plot shows the radial intensity profiles of the two PSFs. The black dashed line is the original intensity pattern and the red line is the modified radial profile. The modified intensity is 23 % higher and the FWHM is reduced from 0.022 ” to 0.018 ”. As the fraction of the light in the core has been increased the halo component is seen to be reduced.
Fig. 6. SNR obtained as a function of threshold for observing a faint companion at 2λ/D and δm=7.7 with a D=1 m telescope (a) and δm=11.7 with a D=8 m telescope (b). The solid black line shows the SNR for the un-masked system.
Table 1. Simulation results for a combination of different telescope diameters and instruments to show the achievable SNR with each system. The target separation was chosen to be 2λ/D and the magnitude difference (δ Mag) of the binary system was selected so that the signal to noise ration (SNR) of the system without the mask (control) was 5. The adaptive pupil mask (APM) can then increase the SNR dramatically. For the 8 m telescope the SNR with a four quadrant phase mask coronagraph (FQPM) and the combination of the APM and FQPM is shown.
3. Threshold selection
It is important to calculate the optimum phase threshold for the system. This could be done in real time on sky by a trial and improvement strategy however it would be useful to be able to calculate the optimum threshold from measurements of the immediate seeing. In order to do this an analysis of the performance of the mask as a function of wavefront variance is required. The point spread function (PSF) assuming on-axis observations can be calculated by,
where MTF atmos is the atmospheric modulation transfer function, MTF tel is the telescope modulation transfer function and I′T=IT is the ratio of the modified total intensity, I′T, to the original total intensity, IT, and is the fraction of the pupil which is not blocked by the mask.
The atmospheric modulation transfer function, MTF atmos, can be calculated from the phase structure function using the following [18],
where the MTF is now shown as a function of position in the pupil, r, and Dϕ(r) is the phase structure function and is given by [19],
where ϕ(r) is the phase at radial position r and ϕ(r+r′) is the phase at a different position in the pupil separated by a distance r′. r is related to the focal length, f, and the spatial frequency, κ, by r=λfκ.
Fried [19] calculated the phase structure function for uncorrected wavefront to be,
and for large r,
where L 0 is the outer scale of the turbulence and σ2 is the phase variance of the wavefront. At separations greater than the outer scale the structure function converges to a constant. For partially corrected wavefronts the structure function is no longer given by equations 5 and 6. An AO system will reduce the phase structure function for low spatial frequencies, the effect of which can be modelled by a high pass filter, H(κd=2),[20] as shown in Fig. 7(a),
where d is the diameter of the subapertures and Jn is a Bessel function of the first kind of order n. The partially corrected phase structure function is given by Greenwood [20] as,
where Fϕ is the phase power spectrum and assuming a Kolmogorov power spectrum (i.e. an infinite outer scale, zero inner scale and negligible amplitude effects) it will be of the form [20]
Equation 8 becomes,
where x=r/d and u=κd in order to bring the d/r 0 term to the front and so that the integral does not depend on the atmospheric parameters. The phase variance of a tip/tilt corrected wave-front is σ2=0,134(d/r 0)5/3 and so it can be seen that the coefficient of the structure function is determined by the wavefront variance and Eq. (10) can be written as,
Increasing r 0 will reduce the wavefront variance and also lower the saturation level of the structure function. Figure 7(b) shows the partially corrected structure function for d/r 0=1 and it is seen that this converges to a value of 0.268 which is consistent with 2σ2. From this we can confirm that Eq. (10) converges to 2σ2 and for a partially corrected wavefront Eq. (6) becomes
Fig. 7. Theoretical plots to show the effect of an AO system on the wavefront structure function. (a) is the AO high pass filter function as defined by Greenwood [20]. Low spatial frequencies are removed by the AO system and high spatial frequencies propagate. (b) shows the uncorrected structure function (black line) and partially corrected structure function (red line). The partially corrected structure function saturates when r>d as large spatial scale deformations (low spatial frequencies) have been removed by the AO system as seen in (a).
Gaussian statistics can be used to describe independent atmospherically induced fluctuations in the phase of a wavefront. An AO system will remove low order modes but as the atmospheric turbulence is fractal, within an inertial range between the inner scale and outer scale, the residual phase deformations after AO correction can also be described by Gaussian statistics. So whether or not the adaptive pupil mask is used in conjunction with an AO system the piston distribution (P(ϕ)) will be a Gaussian with variance, σ2 ϕ. The adaptive mask will block the subapertures with the largest phase, truncating the Gaussian at +/- T, where T is the threshold piston value. The variance will therefore be reduced to,
where ϕ̄ is the mean piston of the wavefront. We can now plot the residual variance after the mask as a function of the initial variance entering the adaptive mask and the threshold chosen (Fig. 8(a)). By knowing the input wavefront variance and choosing a threshold value we can select the resultant variance we require. However, it is important to take account of the changing diffraction limited PSF and in scenarios where the observer is photon starved the intensity reduction may also be important (Fig. 8(b)).
Fig. 8. The performance of the pupil mask is defined by the balance between reducing the residual wavefront variance and minimising the fraction of the pupil being blocked. The plot on the left shows the relationship between initial variance and residual variance for a number of thresholds. The lower the threshold the greater the reduction in variance. The plot on the right shows the relationship between the initial variance and the fraction of the pupil which is blocked for a given threshold. A low threshold will result in a decreased wavefront variance but it will also require blocking a large fraction of the pupil, reducing the total intensity in the image and changing the diffraction limited PSF.
The masked phase structure function, D ϕ ;APM (r/d), will have the same form as Dϕ (r/d) but will be scaled so that it does not deteriorate as rapidly and will now converge to 2σ2 T. By substituting Eq. (13) into Eq. (12) the phase structure function for partially corrected wavefronts becomes,
and for uncorrected wave-fronts,
Numerical analysis of the phase structure functions indicate that they all converge towards a constant value of Dϕ (r>d)=2σ2 where d is the subaperture size in the case of partially corrected wavefronts or the diameter of the telescope (assuming d<L 0) for uncorrected wavefronts. From Eq. (3) it follows that the atmospheric transfer function also converges to a constant value,
Figure 9(a) shows the MTF atmos for a number of values of d/r 0. The curves can be decomposed into a Gaussian with a dc bias. The atmospheric component of the PSF will be a central peak defined by the dc offset plus a Gaussian halo with width inversely proportional to the width of the MTF atmos Gaussian component. As all the curves correspond to the same total intensity the fraction of energy within the core is given by the value of the dc offset, in this case the convergent value of the MTF atmos, and when the phase variance is low (<1.6 radians2) the Maréchal approximation tells us that this constant is equal to the Strehl ratio. The adaptive mask will raise the convergent value (Fig. 9(b)) increasing the fraction of energy in the diffraction limited core. However, the effect this will have on the final PSF also depends on the telescope modulation transfer function as the Fourier transform of the MTF tel will be the diffraction limited PSF. MTF tel is given by the autocorrelation of the pupil function and as the adaptive pupil mask changes the shape of the telescope pupil MTF tel will also be modified. The greater the fraction of the pupil that is blocked the narrower the MTF becomes (Fig. 10) due the lower fill factor in the pupil. The effect of which is to reduce the peak intensity and broaden the diffraction limited PSF.
Fig. 9. The atmospheric modulation transfer function after AO correction depends on the wavefront variance, defined by the d/r0 ratio. The left plot shows the atmospheric modulation transfer function for a range of d/r0 values. A lower ratio means the AO system is capable of better correction and so will converge at a higher level. Equation (16) states that the MTF atmos converges to exp(-σ2) which using the Maréchal approximation indicates the fraction of energy within the diffraction limited core. The plot on the right shows how the MTF atmos is modified by the pupil mask for a single value of d/r 0=4. As the threshold is reduced the system rejects more subapertures and so the residual wavefront variance is reduced, increasing the fraction of energy in the PSF core.
The actual PSF is the Fourier transform of the product of the two modulation transfer functions normalised by the fraction of the total intensity which is blocked (Eq. (2)). We can now calculate the expected PSF from the input parameters (immediate seeing, telescope pupil function and number of subapertures) and calculate the optimum threshold for maximum peak intensity. Given extra parameters for a binary system (magnitude difference and separation) the optimum threshold for faint companion detection can also be calculated.
4. Conclusions
We have presented a novel technique for improving the quality of a PSF in terms of increasing peak intensity and reducing the halo. Light from areas of a telescope pupil which are out of phase will not add to the core but instead create a diffuse halo. By blocking the appropriate subapertures we obtain a much flatter wavefront and by controlling the extent of the blocking we can maximise the peak intensity and minimise the PSF halo. If we block too much the diffraction limited PSF becomes broader and the peak intensity will be compromised. If we do not block enough there will be subapertures with large piston remaining. The performance of the adaptive pupil mask is most dramatic in systems with a large fraction of energy in the halo but can also provide significant improvements for higher Strehl images.
Fig. 10. The adaptive pupil mask will have an effect on the telescope transfer function. The magnitude of this effect will depend on the fraction of the pupil that is blocked. The black line in the plot is the telescope MTF for a circular aperture, the blue line shows the MTF for a telescope with a central obscuration 1/4 the diameter of the primary and the red lines show the extent of modication due to the pupil mask with 20 %, 40 % and 60 % of the pupil blocked.
The optimum threshold is a function of initial phase variance and the ratio of subaperture size to telescope diameter. A theoretical explanation of the pupil mask has been developed in order to estimate the optimum phase threshold as a function of initial phase variance. The two examples shown in this paper are for two different scenarios but the technique will work in any instance where the wavefront is properly sampled.
In simulations the peak intensity for a 1 m class telescope can be increased by 40 % and the FWHM reduced by 76 % to near the diffraction limit. This was done by blocking any subaperture with a piston excursion greater than 1.8 radians. For an 8 m class telescope equipped with AO the adaptive pupil mask can increase the peak intensity by 23 % and the FWHM reduced from 0.022 ” to 0.018 ”. The reduced FWHM and increased peak intensity is beneficial for the direct imaging of faint companions as the contrast ratio will be reduced.
Simulations show that the SNR for a 100 s exposure observing a faint companion at an angular separation of 2λ/D from the primary star with a magnitude difference of 7.7 on a 1 m telescope is 5. The inclusion of the adaptive pupil mask double this to 10.5. A binary system of the same separation but magnitude difference of 11.7 on an 8 m telescope also has a SNR of 5. The addition of a four-quadrant phase mask coronagraph results in an increase to 6.6 which is comparable to the addition of the pupil mask. If the pupil mask is used before the coronagraph a SNR of 10.6 is achieved due to the reduced FWHM resulting in a more efficient coronagraph.
Digital Micromirror Device [21] technology is now reaching a very developed stage and could easily handle the update rates and chip sizes required of the adaptive pupil mask and could be used in the pupil plane of the telescope to reflect the appropriate sections of the wave-front out of the optical path. For a telescope with no AO capabilities the pupil mask could be implemented with a beam splitter, a Shack-Hartmann wavefront sensor and a DMD. A telescope already equipped with an AO system can share the wavefront sensor and so only requires positioning the DMD in the optical path after the deformable mirror.
Acknowledgments
We are grateful to the Science and Technology Facilities Committee (STFC) for financial support (JO).
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