1. Introduction

In just over a decade, more than 200 “extrasolar” planets have been discovered orbiting stars other than the sun, chiefly through rather indirect techniques such as radial velocity measurements that detect the recoil of the stellar primary [1]. Efforts are underway to provide direct imaging of planetary and other stellar companions, though this is a challenging undertaking: at visible wavelengths, an earthlike planet may have roughly 10^-10 the signal of its parent star. A classical technique for suppressing light from the primary is coronagraphy [2], and many innovative approaches are under development. Whether on the ground or in space, coronagraphs may be fed by adaptive optics (AO) [3] to control the wavefront aberrations induced by atmospheric turbulence or by flexure in beamtrain optics, respectively.

Even the best AO system will leave some fraction of remnant light in a halo around the stellar primary that will confuse the search for faint companions. A number of techniques for suppressing speckles and the image noise their fluctuations cause have been investigated. Here, some fundamental spatial symmetry properties are exploited.

2. Symmetry properties of speckles at high correction

2.1 Phase and amplitude speckle symmetries

At high adaptive correction, the speckle halo organizes itself into terms of distinct spatial symmetry, either even or odd, as can be seen by algebraic expansion of the Fourier-optical expression for the image of the bright central star [4–8]. A useful approximation here is

where 0≤A≤1 is the aperture (pupil) function, α<<1 is a small error in transmission amplitude (equivalent to a fractional intensity fluctuation 2α), and (ϕ<<1 is the (small) residual phase. This treatment assumes that all phase and amplitude errors are in the pupil plane. An overbar denotes two-dimensional Fourier transformation, an asterisk denotes complex conjugation, and |Ā|² is the point-spread function (PSF), much brighter than all the other terms. Several speckle terms containing more factors of the small quantities α or ϕ have been omitted from Eq. (1) because they are generally fainter at high correction. Since A, ϕ, α are real, their Fourier transforms are Hermitian, causing each term to have definite spatial symmetry. For the regimes to be considered in this paper, the brightest phase speckles come from the term linear in $\overline{\varphi }$ , 2Re{i A̅^* Aϕ̅}, which is spatially antisymmetric, and the quadratic term |Aϕ̅|², which is spatially symmetric. (The 4th term in Eq. (1), -Re{Ā ^* Aϕ ²̅}, is primarily a PSF correction plus true speckles that are fainter than those from the quadratic term [8].) Brightest of the amplitude speckles are the low-order terms |α̅|² and 2Re{Ā̅^* α̅}, each of which is spatially symmetric. The last two amplitude-speckle terms in Eq. (1) are spatially antisymmetric, but are bilinear in the small quantities α, ϕ and so are fainter than the linear phase speckles also retained here. (Later, in §3, simple image processing will remove the symmetric speckle species and leave the antisymmetric behind; the new speckle contrast floor will then be limited by linear phase speckles, the brightest spatially antisymmetric species.)

2.2 Speckle intensity estimates

Linear-term speckles can have negative intensity, but are “pinned” to diffraction rings of the PSF because they contain a multiplicative factor of the PSF amplitude, Ā. So they reduce diffraction-ring brightness locally, keeping total intensity non-negative. Although the total linear-term power in the halo vanishes, speckles fluctuate locally and cause speckle noise or contribute to the speckle-limited imaging contrast. Owing to the amplification by Ā, linear-term speckles will tend to be brightest in the inner halo; being of lowest order in Aϕ̅, they will always dominate in the limit of high adaptive correction (|ϕ̅| small). The PSF amplitude Ā for a round unobscured aperture is the familiar Airy amplitude. The “speckle amplitude” ϕ̅ and “intensity speckle amplitude” $\overline{\alpha }$ may be computed as a function of Strehl ratio (S) and deformable mirror actuator density (D/a, for pupil diameter D and actuator spacing a), which specifies the highest corrected spatial frequency and hence the halo diameter λ/a [9]:

D/a_α is the analogous spatial frequency characterizing intensity variations on the pupil. Typical intensities of individual speckles may be estimated to within a factor of ~2 by combining Eq. (1) and Eq. (2). Speckle noise is then directly proportional to speckle intensity [10]; for space-borne observatories, the speckle intensity level is directly of interest as the speckle-limited imaging contrast. Full numerical simulations of highly-corrected images at various (S, D/a) were made to determine typical degrees of net spatial symmetry in the phase speckle noise variance fraction; an example is shown in Fig. 1.

2.3 Attenuation of linear-term speckles by coronagraph

For faint companion searches, adaptive correction may be coupled to coronagraphs, substantially modifying some speckle intensities. Specifically, speckles from the terms linear in ϕ̅ or in α̅ are suppressed, while quadratic-term speckles are little affected. The linear terms 2Re{ i A̅^* Aϕ̅} and 2Re{A̅^* α̅} contain a factor of the PSF amplitude, Ā, which the coronagraph naturally suppresses in suppressing the on-axis PSF, |Ā|². The action of the coronagraph at any given position in the focal plane may be described as reducing the PSF from |Ā|² to a new contrast level C=|Ā|²/R, with the suppression parameter R>1. Then the Airy amplitude Ā and each linear speckle term will be reduced [11, 12] by a factor √R. Considering only phase speckles for simplicity, the competition between the odd linear and even quadratic terms determines the net spatial symmetry of the speckle halo. At the highest correction, an adaptively-fed coronagraph is always dominated by antisymmetric linear-term speckles. But for high-performance coronagraphs at moderately high correction, suppression of the linear term will reverse the symmetry at typical operating parameters from antisymmetric to symmetric, with quadratic speckles then dominating.

Quantitatively, the degree of symmetry for phase speckles can be evaluated at high correction by considering only linear and quadratic terms, the former attenuated by the factor √R. Then if over many independent realizations of the phase screen an adaptive correction gives antisymmetric speckle noise variance fraction f, which is proportional to the antisymmetric fraction of the squared speckle intensity [10], the coronagraph converts this to a symmetric noise variance fraction f’ given by

This effect is illustrated in Fig. 1 for R=10 and 100, coronagraphic suppression values representative of extreme ground-based adaptive optics systems now under development.

 

Fig. 1. Spatially antisymmetric speckle noise variance fraction with no coronagraph (solid curve, “simple imaging”) as a function of Strehl ratio, for fixed D/a=16 (higher future D/a will give higher antisymmetry). Values are averages of many monochromatic short-exposures for small (0.25 λ/D) pixels in the innermost two rings of the PSF. The other two curves show the noise variance, though always antisymmetric at highest correction, becomes symmetric when linear-term speckles are suppressed by a coronagraph with PSF rejection R. (The coronagraph curves consider only linear and quadratic speckle terms, strictly valid only at high correction. For illustration, R is taken to be independent of correction.)

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3. Potential gains from symmetry-based speckle noise reduction

To apply these concepts, consider a space-based telescope equipped with a coronagraph but no deformable mirror or other high-order adaptive optics. Phase speckles are caused by slowly-varying telescope aberrations; speckle-limited contrast is simply the typical speckle intensity as a fraction of the peak of the primary star. High-quality conventional optics are assumed (λ/100 rms wavefront, or S~0.996) with spatial frequency content D/a~10. Coronagraph performance is assumed high, suppressing the PSF to contrast ~10^-7 on the 4^th Airy ring [13] (equivalent to R~7800), giving a predominantly symmetric speckle halo.

Equations (1) and (2) then give the intensities of all relevant phase and amplitude speckles. Near the 4^th Airy ring, the pre-coronagraph PSF amplitude is Ā~2.8×10^-2. The speckle amplitude is Aϕ̅~10^-2 anywhere in the halo. Assuming pupil-plane intensity variations of 2.6% [14] characterized by spatial frequencies ~10 cycles/aperture, the intensity speckle amplitude is α̅~2×10^-3 anywhere in the halo. So quadratic-term phase speckles |Aϕ̅|² dominate, with intensity (contrast) ~1.2×10^-4 , while linear-term phase speckles 2Re{ i Ā ^* Aϕ̅} have intensity ~5×10^-7 near the 4^th Airy ring. (Note that the PSF amplitude has been reduced by √R to capture the effect of the coronagraph, as discussed in §2.3. Also, each linear-term intensity has been reduced an order of magnitude from the small-pixel, monochromatic values of Eq. (2) to account for partial cancellation over an octave of spectral bandwidth on λ/D pixels.) Quadratic amplitude speckles |α̅|² will have intensity ~5×10^-6 anywhere in the halo, while linear amplitude speckles 2Re{Ā ^* α̅} will have intensity ~10^-7 near the 4^th Airy ring.

The initial speckle contrast floor is therefore C_n~1.2×10^-4, set by the bright quadratic phase speckles. Antisymmetrizing images through simple image processing will reject all the speckle species listed in the previous paragraph except linear phase speckles, which will then limit the contrast to C_n'~5×10^-7, an improvement by a factor of about 240. Source signal is halved when one symmetry component is rejected, so a companion intensity C_s=2C_n'~10^-6 would just match the final speckle contrast level C_n'. Hence, search sensitivity has been enhanced by about two orders of magnitude by using simple image processing, but no sophisticated optics, to reject the bright symmetric component of speckle noise.

As just shown, symmetry-based image processing can deeply suppress speckles. But particularly for faint primary stars, integration times must be long enough to overcome shot (photon) noise on the pre-processing intensity of the brightest speckle species:

where SNR is the final signal-to-noise ratio desired, r^* the rate of detected photons from the primary, C_n the noise contrast set by initial speckle intensity, and C_s the contrast describing the signal to be detected. (The two subtracted frames both have shot-noise fluctuations.) In the visible, λ=400–800 nm, r^*~3×10⁸ photons/s are detected from a m=5 G star by a ~20m² telescope having overall efficiency 5% [15]. To exploit the speckle suppression described in the previous section, and realize signal contrast C_s~10^-6 with SNR=7 from an initial speckle contrast C_n~1.2×10^-4, would require integrating for just ~20 seconds.

4. Practical feasibility of symmetry-based speckle noise suppression

Operationally, an image is symmetrized/antisymmetrized by adding/subtracting a spatially inverted copy. This section examines the fundamental photon-noise-limited precision with which this can be done, and shows it is sufficient to realize the gains described in §3. First, in locating the image center, monochromatic diffraction-limited speckles are grossly characterized as Airy patterns of diameter FWHM~1.03/D or as Gaussian spots with HW1/e σ_sp~FWHM/(2√(ln2))~0.62λ/D. The minimum possible uncertainty in the single-axis centroid location of a Gaussian spot of size σ_sp when only photon noise is considered is [16]

where N_ph= C_nr^* t is the number of photons per speckle, C_n the initial contrast, and r^* the rate of detected photons from the primary star, as in §3. (Broadband radial speckle streaks are located transversely to roughly this accuracy.) Locating the center between symmetric speckle pairs improves the accuracy by √2, and combining measurements from all speckle pairs in the halo gives a further improvement by √(N_sp/2), where N_sp≈0.342(D/a)² is the number of speckles in a halo characterized by actuator density (D/a) [17]. The overall precision of locating the speckle halo’s center of symmetry is then

The new remnant speckle noise floor due to imperfect subtraction of image and inverted copy, given the misregistration Δ just calculated, is found from a linearization of two shifted Gaussians valid for small arguments: the spurious intensity is I₀Δ/σ_sp, where I₀=C_n is the intensity of each Gaussian speckle. The resulting post-processing contrast is

equal to ~10^-8 assuming t~100 s for the illustrative space-borne observatory (with initial speckle contrast floor C_n~1.2×10^-4) and the representative primary star (r^*~3×10⁸ photons/s) of §3. This value is small compared to the final noise contrast limited by linear-term speckles, C_n'~5×10^-7, so frame registration accuracy is adequate to realize this speckle contrast improvement. This is an important check on realistic speckle suppression: past studies have assumed perfect image antisymmetrization [18] around a numerically-defined image center.

5. Conclusions

More ambitious observatories might incorporate active optics to increase S and D/a, thus reducing the initial speckle-limited contrast C_n. A λ/160 correction (S=0.9985), D/a=100, and R=7800 on the 4^th Airy ring imply C_n~5×10^-7 and a final contrast C_s=2C_n'~6×10^-8, limited by the antisymmetric linear-term phase speckles that survive image processing (Fig. 2). Integration times to overcome speckle and PSF shot noise are still quite short, t~25 seconds.

 

Fig. 2. Improvement in speckle-limited search contrast with the image antisymmetrization described in §3, as a function of initial correction, for R=7800 on the 4^th Airy ring. Also plotted is the potential limit imposed by centroiding errors, as described by Eq. (7), assuming speckle lifetimes of t~1 second. Contrast values are based on the analytic expressions of Eqs. (1) and (2). At the highest correction considered here, giant extrasolar planets are observable.

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Symmetry-based speckle filtering might profitably be combined with speckle nulling. The antisymmetrization presented in this paper can suppress amplitude speckles over a broad spectral band more readily than speckle nulling can [15, 19]. Centroiding errors in this case remain tolerable because the permissible integration time in Eq. (7) may be quite long, limited by the timescale of spacecraft flexure. This hybrid approach might significantly improve limiting search contrasts, to a level perhaps approaching the 10^-10 contrast needed for terrestrial planet detection, while adding little mission complexity to that needed for nulling.

Suppressing speckles by exploiting their spatial symmetries at high adaptive correction is an inherently simple and reliable approach requiring image processing but no special hardware. Strong suppression is available at corrections contemplated for space-borne coronagraphic observatories, and the photon fluxes typical of stars that are candidates to have exoplanets are sufficient to permit that suppression to be realized. Use of the symmetry-based techniques presented here on an uncorrected but high-quality space-based telescope could significantly enhance companion detection with no special hardware beyond the coronagraph, making it a potentially interesting addition to a general-purpose observatory.