Coronagraphic performance with segmented apertures: effect of cophasing errors and stability requirements

Summary

The effect of cophasing errors in a segmented aperture on the ability to achieve high contrast imaging is studied in the context of coronagraphic imaging of exoplanets. For angular separations within the diffraction limit of a single segment, it is shown that simple analytical expressions exist to both quantify the effect of segment cophasing errors on image contrast and to quantify how well segment cophasing errors can be optimally estimated with wavefront sensing technique(s) using light from the central star. When combined, these relationships can be used to derive requirements on the stability between segments, both for future space missions and ground-based telescopes with segmented apertures. Examples are presented for the pupil geometries of the Thirty meter Telescope (TMT) and Giant Magellan Telescope (GMT), representative of respectively highly segmented and moderately segmented pupils.

1. Introduction

Segmented apertures are an attractive technical solution for building large optical telescopes on the ground or for space. By enabling large telescope diameter, they also offer attractive performance for high contrast imaging of exoplanets and disks, thanks to the combination of fine angular resolution and large collecting area. In a previous study, it was shown that full performance coronagraphy is conceptually possible on segmented apertures, regardless of the size, number and arrangement of segments: in the absence of wavefront errors, direct imaging at high contrast (exceeding 1e10) was shown to be possible within 1 λ/D of the central source with full throughput. High contrast imaging is however very demanding in wavefront quality and stability, and the suitability of segmented mirrors - which are subject to segment cophasing errors - for high contrast imaging is thus unclear. The goal of this paper is to provide tools to assess this suitability, by first establishing a relationship between cophasing errors and image contrast degradation (section 2), and using this relationship to establish requirements for the cophasing errors stability (section 3).

2. Constrast degradation due to cophasing errors

2.1. Analytical expression for contrast due to cophasing errors

A telescope of diameter D consisting of N segments is considered here. For simplicity, we assume a circular aperture and equally sized segments, so that the segment diameter is d = D/sqrt(N). We consider a random cophasing error, decorrelated between segments: each segment phase is a random number with uniform probability distribution from -e rad to +e rad, so that the RMS phase across the pupil is e/sqrt(3) ~ 0.577 e rad. The speckle halo created by cophasing errors can be most easily derived by first considering a cophasing error dφ for a single segment. For small cophasing errors, the dephased segment creates a coherent light halo with central surface brightness (relative to the peak surface brightness of the full aperture PSF) equal to 1/N2 multiplied by dφ2. The halo decreases with distance from the optical axis following the diffraction pattern Aseg shape of the segment. The halos originating from separate segments add incoherently in intensity, provided that the cophasing error is decorrelated between segments.
Contrast(r) = Aseg shape(r d / λ ) / N dφ2 (1)
where Aseg shape(r d / λ ) is the diffraction pattern of a single segment, normalized to be equal to 1 for r=0. For simplicity, Aseg shape may be approximated as being equal to 1 for r<λ/d and 0 otherwise :
Contrast(r<λ/d) = dφ2 / N (2)

The effect of cophasing errors on image contrast is thus independant of the coronagraph concept in the separation range between the coronagraph's inner working angle (IWA) and the diffraction limit of a single segment. This is the regime that is studied in this paper, allowing simple analytical expressions to be derived to quantify how coronagraph performance degrades as cophasing errors increase. This is also the separation regime where potentially habitable planets may be first directly imaged.

Equation (2) is not valid outside of the segment diffraction limit λ/d, and the coronagraph design may, outside this range, be able to mitigate cophasing errors. For example, segment edges may be apodized in intensity to reduce sensivity to discontinuities in phases that occur with cophasing errors; this scheme would not have an effect within λ/d, but would greatly improve contrast outside λ/d in the presence of cophasing errors.

2.2. Numerical verification on a highly segmented apertures

The TMT pupil is used here as an example of a highly segmented pupil. The TMT coronagraph design #1 given in this link is adopted, and offers, in the absence of cophasing errors or other wavefront aberrations, perfect rejection for an on-axis point source. The coronagraph is based on the PIAACMC technique, and offers full throughput and small sub-λ/D inner working angle (IWA). Figure 2.1 shows the result of the simulation with a cophasing error e=0.1 rad (corresponding to a 0.0577 rad RMS phase error).

If the TMT pupil were unobstructed and fully paved, it would consist of 491 segments. The expected contrast level in this example, according to equation (1), is therefore 0.05772/491 = 6.78e-6. As shown in figure 2.2, this prediction is in good agreement with the numerically computed PSF radially averaged contrast.

Fig 2.1: Effect of cophasing error on coronagraph performance. The entrance pupil amplitude (top left) and phase (top center) chosen in this example correspond to the TMT pupil with segment phases following the uniform probability distribution from -0.1 rad to +0.1 rad. The coronagraph output pupil (top right) shows that most of the light is diffracted outside the pupil, but a small amount of light remains in the pupil due to cophasing errors. After applying the Lyot mask (bottom left), the contribution of each segment to the coronagraphic leak is seen to be propotional to the square of the segment phase error. The final on-axis PSF (bottom center) shows that the speckle halo follows the envelop defined by the diffraction pattern of a single segment. 










    
 
Fig 2.2: Contrast as a function of angular separation for the example shown in figure 2.1. The contrast level predicted by equation (2) for the inner region of the PSF is shown as a horizontal line.














 2.3. Small number of segments 







    
 
Fig 2.3: Effect of cophasing error on coronagraph performance for the GMT pupil. See figure 2.1 caption for details.













    
 
Fig 2.4: Contrast as a function of angular separation for the example shown in figure 2.3. The contrast level predicted by equation (2) for the inner region of the PSF is shown as a horizontal line. 24 realizations of cophasing errors are shown.

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 3. Wavefront control 



 3.1. Wavefront sensing accuracy at the photon noise limit 


Measuring the phase of a single segment relative to the full aperture is fundamentally limited by the number of photon available for the measurement. With an optimally sensitive wavefront sensor, the error in the phase estimate, in radian, is equal to the inverse square root of the number of photon available over the segment. For a segmented aperture - as well as for a single segment -, the standard deviation in [radian] of the estimated phase is therefore:



σφ(t) =  1/sqrt(Nph seg) = 2 /  ( d sqrt(π 9.7e10 2.512-m t δλ) )  = 1/2760 (1/d) sqrt(1/(t δλ)) 2.512(mV-10)/2  [rad]

(3)



where d is the segment diameter [meter], t is the integration time for the measurement [second], δλ is the effective spectral bandwidth [μm], and mV is the star visible magnitude. For example, with a mV=10 star, a 10Hz sampling rate, a 0.1 μm effective spectral bandwidth and a d=1m segment size, cophasing errors can be measured with a 2nm standard deviation per measurement at λ=550nm.


 3.2. Stability requirement to meet contrast performance




Once the contrast requirement is set, equations (2) and (3) can be used to derive a stability requirement by first establishing the cophasing error requirement with equation (2), and then associating a corresponding timescale from equation (3). For example, if the contrast goal requires cophasing error to be no more than 1nm, and if measurement of a 1nm cophasing error requires 5 sec, then the requirement is that the segment phases must not drift by more than 1nm over a 5 sec period. In reality, active control of segment phases cannot be performed at the sampling frequency, and a factor 10 is adopted here to account for this: the stability timescale adopted here is chosen to be 10 times the value given by equation (3). The choice for the value of this factor is somewhat arbitrary, as its exact value is a function of the control loop details and the temporal power spectral density of the segment phase errors. In favorable cases where the segment phase error is a simple function of time (such as a drift at a constant rate, or a single frequency oscillation), the control loop can predict the input disturbance and the factor can become equal of less than 1. For simplicity, only the unpredictible part of cophasing errors is considered here, and it is assumed that any predictible component (such as a drift at a constat rate) can be efficiently removed.



 Cophasing requirement 



 
The cophasing requirement obtained from equation (2) is:


dφ = sqrt( N x Contrast)

(4)


This relationship is only valid within the central focal plane area defined by the diffraction spot of a single segment. At larger angular separations, cophasing errors can be significantly larger, and the coronagraph design can impact the relationship between cophasing errors and contrast (by apodization of individual segments for example). 





It is important to note that the  cophasing error required to meet a given contrast level is independant of telescope diameter. With a larger number of segments, the cophasing error can be larger, as the corresponding speckle halo is spread over a larger area in the focal plane. 






 Timescale, stability requirement 

The associated timescale, including the factor 10 described above, is, assuming that wavefront sensing is performed at λ=550nm:


t = 1.31e-6 x 2.512mV-10 / ( Contrast x D2 x δλ )

(5)


While a larger number of segments (smaller segment size d) corresponds to fewer photon per segment to perform the phase measurement, it also corresponds to a larger allowable cophasing error. The two effect perfectly cancel, yielding, a  timescale which is, for a fixed telescope diameter D, independant of the number of segments N 





Examples are given in table 1, assuming that the wavefront sensing is done at λ=550nm with an effective spectral bandwidth δλ=0.1μm.








 TABLE 1: Segment cophasing requirements

  Telescope diameter (D) & λ  Number of Segments (N) Contrast Target Cophasing requirement Stability timescale 





 Ground-based telescope 



 10 m, 1.6 μm 36 1e-6  mV=8  1.5 nm 21 ms


 30 m, 1.6 μm 10 1e-6  mV=8  0.8 nm 2.3 ms


 30 m, 1.6 μm 1000 1e-6  mV=8  8.1 nm 2.3 ms



 Space-based telescope 



 4 m, 0.55 μm 10 1e-10  mV=8  2.8 pm 22 mn


 8 m, 0.55 μm 10 1e-10  mV=8  2.8 pm 5.4 mn


 8 m, 0.55 μm 100 1e-10  mV=8  8.7 pm 5.4 mn



 




Ground-based telescope systems, thanks to their moderate contrast goal (fundamentally limited by atmospheric turbulence) and the longer wavelength typically adopted for high contrast imaging, can tolerate nm-level cophasing errors. Optical measurement of cophasing errors at this level is achieved in a few ms for the mV=8 target considered in Table 1. High contrast imaging systems in space would be much more demanding, with allowable cophasing errors of a few pm. The timescale for the measurement of these errors would also be significantly longer (several minutes), and in fact comparable to the exposure time required for the detection of an Earth-size planet in reflected light.







 Conclusions 



In segmented telescopes, the relationship between segment cophasing error and PSF contrast at small angular separation (within the diffraction limit of individual segments) is simple and independent of coronagraph design. Requirements on the allowable cophasing errors and, for an actively controlled system, the open loop stability necessary to meet this requirements, can then be derived. 



One key result of this paper is that, with the assumption that cophasing errors are uncorrelated between segments, the allowable cophasing error grows with the number of segments. Correlated cophasing errors (created for example by large scale flexures of the structure holding the segments) were not considered in this study, and could invalidate this conclusion. While we have only considered wavefront control systems using starlight for optical metrology, internal metrology using laser interferometry could be required if the temporal bandwidth required for active control of cophasing errors cannot be achieved with starlight.



The conclusions of this study also apply to segment tip-tilt errors, as the physical origin of unwanted diffracted light in the PSF is phase steps at the interfaces between segments. These phase steps can be produced either by cophasing or tip-tilt errors on the segments. At angular separations larger than the diffraction limit of a single segment, the coronagraph design may be tuned to mitigate sensitivity to segment cophasing errors, and no universal relationship between image contrast and cophasing error exists.