A&A 404, 379-387 (2003)
Subaru Telescope, National Astronomical Observatory of Japan, 650 North A'ohoku Place, Hilo, HI 96720, USA
Received 13 January 2003 / Accepted 20 March 2003
In this paper, an alternative to classical pupil apodization techniques (use of an amplitude pupil mask) is proposed. It is shown that an achromatic apodized pupil suitable for imaging of extrasolar planets can be obtained by reflection of an unapodized flat wavefront on two mirrors. By carefully choosing the shape of these two mirrors, it is possible to obtain a contrast better than 109 at a distance smaller than from the optical axis. Because this technique preserves both the angular resolution and light gathering capabilities of the unapodized pupil, it allows efficient detection of terrestrial extrasolar planets with a 1.5 m telescope in the visible.
Key words: techniques: high angular resolution - stars: planetary systems - telescopes
Unfortunately, the large contrast (about 109 in visible, and 106 in thermal infrared) between ETPs and their parent star, along with the small separation (0.1'' for a Earth-Sun system at 10 pc) makes direct detection very challenging. Ground-based telescopes of reasonable size (less than 100 m diameter) cannot image ETPs in the visible because of the strong wavefront errors due to atmospheric turbulence, and ground-based thermal infrared observations lack the required sensitivity.
Two solutions are seriously considered to image ETPs from space:
One such coronagraphic technique, "classical'' pupil apodization (CPA), aims at modifying the pupil transmission function by placing an amplitude mask in the pupil plane to reduce the amplitude of the PSF wings. This adaptation of the pupil to the problem of high-contrast imaging can allow the PSF wings to be reduced to 10-10 of the central surface brightness at a distance of a few . Many pupil apodization masks have been proposed for this purpose (Jacquinot & Roizen-Dossier 1964; Nisenson & Papaliolios 2001; Aime et al. 2001; Spergel 2001; Kuchner & Traub 2002), and optimization techniques to generate these masks have been developed (Gonsalves & Nisenson 2002). Unfortunately, this technique reduces the flux sensitivity (a large fraction of the light is absorbed by the apodization mask) and the angular resolution (the transmission at the edges of the apodized pupil is lower than at the center) of the telescope. Moreover, in many CPA designs, only part of the field of view is suitable for planet detection.
In this paper, an alternative to the CPA technique is proposed: phase-induced amplitude apodization (PIAA). Instead of producing the apodized pupil by using an amplitude mask in the pupil plane, the pupil is modified by reflection on 2 mirrors (which can be the primary and secondary mirrors of the telescope) whose phase aberrations are chosen to produce an exit pupil which greatly reduces the intensity of the PSF wings.
The PIAA technique is first briefly presented in Sect. 2. In Sect. 3, the properties of the images obtained by this technique are studied, and several solutions are presented to use a PIAA telescope as a wide field coronagraphic imager. In Sect. 4, the performance of the PIAA technique for ETPs detection is studied, and the technical feasibility of the technique is discussed in Sect. 5.
|Figure 1: Schematic representation of the Classical Pupil Apodization (CPA) technique (top) and the Phase Induced Amplitude Apodization (PIAA) technique (bottom).|
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In the PIAA optical design represented in Fig. 1, the role of the primary mirror is mainly to modify the light distribution across the pupil, while the secondary mirror is used mostly to correct the phase aberrations introduced by the primary mirror. These two separate functions are in reality somewhat shared between the two optical elements. As shown in Fig. 1, the PIAA technique can be used to remove the central obstruction from the pupil.
The PIAA technique is similar to the beam shaping techniques developed to modify laser beam profiles (Shealy 2002). A similar apodization can also be produced by a secondary mirror designed to correct the spherical aberration of a primary mirror (Goncharov & Puryayev 2002).
If reflective optics are used (mirrors), the apodized pupil formed by the PIAA technique is perfectly achromatic. This achromaticity is simply due to the achromaticity of the geometric laws of reflection on a mirror. Because the PIAA optics maintain a zero optical pathlength difference across the pupil (see Sect. 2.2.1), the PIAA technique does not introduce phase aberrations, regardless of the wavelength considered. Therefore, the PSF obtained by a PIAA telescope, except for a wavelength proportional scaling factor, is identical at all wavelengths.
The first step of the geometric construction is to establish a correspondence between the radius in the entrance pupil and the radius in the exit pupil: for a light ray entering the entrance pupil at a radius r1, what is the radius r2 at which this light ray is exiting the exit pupil? This problem is solved by measuring the total flux of the entrance pupil enclosed inside the radius r1, and choosing r2 such that this flux equals the total flux of the exit pupil enclosed inside r2:
|Figure 2: A method for geometric construction of the PIAA primary and secondary mirror shapes for circular-symmetric pupils.|
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This geometric construction is equivalent to solving the following differential equation:
|A(t) = r2(t)-r1(t),||(5)|
|B(t) = M2(r2(t))-M1(r1(t)),||(6)|
Just as in the case of "classical'' telescopes, the mirror shapes computed by the technique presented above can be modified to be used in an off-axis telescope by simply adding a constant slope across the mirror. In the case of a classical telescope, this operation transforms a parabola into an off-axis parabola ( ), and it can be applied on both the primary and secondary mirrors. It is easy to check that this operation only "steers'' the whole beam, but does not introduce phase aberrations or modifies the light amplitude distribution in planes perpendicular to the central light ray.
|Figure 3: Radial profile f2(r) of the PIAA exit pupil (top) and radial profile of the corresponding PSF (bottom) for the example considered. The radial profile of the unapodized pupil of equal total flux is shown as a dashed line, and its diameter d is used to measure the distances in the PSF (unit is ).|
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|Figure 4: Optical designs for the PIAA example studied: telescope with central obstruction (upper left) and off-axis telescope (upper right). Both images show a cut of the mirror shapes and the density of light rays between the two reflections. A few light rays are represented entering the telescopes and the corresponding light rays exiting the telescope are shown. The shape of the optics is shown for the telescope with central obstruction: deviation of the primary mirror to a parabolic fit (center) and shape of the secondary mirror (bottom).|
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Figure 4 shows two possible optical configurations: on-axis telescope or off-axis telescope. In both cases the distance between the 2 mirrors is less than the diameter of the primary, and the primary is therefore close to a fast parabola (F/D<1), as shown in Fig. 4, center: for a 2 m diameter on-axis primary mirror, the difference between the primary mirror and a perfect parabola is less than . This deviation from a perfect parabola is mostly seen in the outer edge of the primary, where the surface if "bent'' to reflect a small amount of the light into the outer part of the secondary (the "wings'' of the f2 function). The shape of the secondary mirror (Fig. 4, bottom) is also peculiar, and its central part is approximated by a cone (the slope of the mirror does not tend to 0 at small distance from the optical axis) in the case of a telescope with central obstruction.
|Figure 5: Degradation of the PSF quality with distance to the optical axis for the example studied in Sect. 2.3. The brightness scale is linear and identical for all images.|
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The PIAA technique redistributes the complex amplitude of the light in the entrance pupil to form the PIAA exit pupil. The incoming wavefront of an on-axis point source is flat and its phase is constant across the pupil, and because the PIAA does not introduce phase aberration, this property is still true in the PIAA exit pupil. However, the geometric redistribution of light between the entrance pupil and the exit pupil of the PIAA does not preserve the constant phase slope of a wavefront from an off-axis source. The PSF is therefore not translation-invariant as shown in Fig. 5: the PSF becomes less concentrated for off-axis sources.
This effect does not affect the performance of a PIAA imager for detection of ETPs within a few times because the loss of PSF quality is moderate to small in this central region, and the image of the star (defined by the PSF of an on-axis source) remains very contrasted (contrast better than 109 at in the example considered). However, wide field imaging performance of the PIAA imager is seriously affected by this effect. In the particular example studied, the point source detection limit for planets at large angular separation would be affected, especially in the presence of a strong background (exozodiacal light for example).
A hybrid CPA/PIAA optical configuration is also possible. For example, the central part of the f2 function can be obtained by classical apodization while the wings of the f2 function are obtained by the PIAA technique. This hybrid configuration would extend the field of view at the expense of a loss of light.
|Figure 6: Schematic representation of the PIAA coronagraph.|
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If the occulting mask is removed, the effect of the 4 mirrors cancel each other and the PIAA coronagraph exit pupil is identical to the telescope entrance pupil. The PSF is therefore translation-invariant and of good quality across a wide field of view. An off-axis source is not affected by the occulting mask because its image misses the small mask in the first focal plane, but the central source's light is blocked by the occulting mask, whose size is chosen to match the size of the on-axis PSF.
The optical quality requirements are much more strict for the two mirrors before the mask than after the mask: the PSF wings must be kept very faint at the first focal plane. Therefore, the addition of two extra optical elements to build a PIAA coronagraph does not pose serious difficulties, and the wavefront accuracy for these two mirrors can be as low as without significant loss of performance.
This technique is very similar to the coronagraphic technique used in the interferometer concept studied by Guyon & Roddier (2002), where pupil densification (Labeyrie 1996) is used to first adapt the interferometer's sparse pupil to a coronagraph, and pupil redilution is then used to restore the entrance's pupil shape. This later step is essential to restore the wide field of view that is lost in the pupil densification process.
|Figure 7: Constraints imposed on the function f2 and on the on-axis PSF. The grey area represents the permitted values for both f2(r) and the PSF radial profile and the thick curves represent examples of permitted functions. In the pupil radial profile, the dashed line represents the unapodized unobstructed pupil of diameter d, and of total flux equal to the total flux of the PIAA exit pupil.|
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Figure 7 shows how these constraints were applied with five parameters:
The example shown in Sect. 2.3 was obtained by this algorithm with a set of parameters which yields a PSF suitable for ETPs detection in the imager configuration ( , X = 0.8, LIM = 10, C = 10-9, R = 1.0). Although the field of view of this particular example is well suited for ETPs detection with a small telescope ( ), wide field imaging would require either a different choice of values for and X, or a coronagraphic configuration.
To estimate the performance of the PIAA and CPA techniques for direct planet imaging, the detection signal to noise ratio (SNR) is computed for a representative case, with the diameter of the telescope as the main variable. Four configurations were tested under the same conditions:
|y0(x) < |y| < y0(x)+yw(x)||(8)|
|Figure 8: CPA pupils adopted for the comparison between PIAA and CPA technique performance. The pupils are shown on the upper row and the corresponding PSFs are shown in the lower row (logarithmic scaling).|
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To compute the SNR of a companion detection, only photon noise is considered, and the noiseless PSF is supposed to be perfectly known. For a given image of the star/planet system, the best detection SNR is obtained by computing the optimally weighted sum of the PSF-subtracted image pixel values. The weight of each pixel in the sum is proportional to the square of the SNR on that pixel and the resulting detection SNR is
The four optical configurations are tested on a mv=5 star with a planet 109 times fainter at a separation of 0.1 . The central wavelength is and the spectral bandwidth is . No zodiacal or exozodiacal light was included in this simple simulation, and the detectors are assumed to be perfect (no readout noise).
|Figure 9: Telescope diameter needed to detect () a companion 0.1 from a mv=5 star with a flux ratio of 109. The central wavelength is and the bandwidth is .|
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The results of the performance analysis is shown in Fig. 9. The 3 CPA configurations require telescopes of diameters between 5 and 7 meters to detect () the ETP in one hour exposure time, while the PIAA configuration reaches the same performance with a 1.5 m diameter telescope. This result is explained by careful analysis of the Table 1.
Table 1: Main properties of the PIAA configuration and 3 CPA configurations. IWD stands for Inner Working Distance, and is the smallest angular distance at which a 109 contrast is reached in the PSF. The Discovery space is the fraction of the field of view usable for faint companion detection.
The Inner Working Distance (IWD), which is the angular separation at which the contrast of the PSF reaches 109, is about 3 times larger for the CPAs than for the PIAA configuration. This means that, in the visible, the minimum telescope diameter required to detect ETPs in reasonable exposure times is about 4 to 5 meters for CPA configurations while it is only 1.5 meter for the PIAA configuration. In addition, the low transmission and small discovery space of the CPA configuration yield a comparatively lower efficiency compared to the PIAA configuration. The product of the transmission by the discovery space is a good approximation of this efficiency: for the CPA configurations, this efficiency is about 0.1 (responsible for an increase of the exposure time by a factor of 10).
The fact that ETPs can be detected with a 1.5 m telescope is not surprising when the number of incoming photons is considered. In the typical example studied above, the ETP photon flux is about , and about 1200 photons can be collected by a 1.5 m telescope in 1 hour.
The phase of the exit wavefront (the apodized wavefront in case of the PIAA and CPA techniques) needs to be flat to within approximately for efficient ETPs detection (Kuchner & Traub 2002). Larger wavefront errors can be tolerated at the expense of increased exposure times, if the wavefront is still stable enough to allow an accurate calibration of the PSF. Scattering by the optics also needs to be very low, especially in the mirrors that are the closest to the PIAA exit focal plane, and good spatial uniformity of the reflectivity of the mirrors is essential.
Since the polishing of full-size (1.5 meter diameter primary) PIAA optical elements is currently very challenging, an optical configuration with a "classical'' off-axis afocal telescope feeding small PIAA optics is probably preferable. In this optical configuration, the required optical quality of the telescope optics (primary and secondary mirrors) is identical to the required optical quality for CPA telescopes or more "classical'' coronagraphs aimed at detecting ETPs. However, the smaller size (by a factor of 3) of the telescope allowed by the use of the PIAA technique makes the polishing of these optical elements significantly easier. The even smaller size of the "apodizing'' mirrors, which could be only a few centimeters in diameter, makes it possible to efficiently use high accuracy polishing techniques, such as ion beam figuring. The polishing of the PIAA optics offers similar difficulties as the polishing of "wild aspherics'' optical element in imaging systems.
The PIAA technique can also be used to adapt the telescope's pupil to a coronagraph. For example, it can produce a pupil suitable for the phase mask coronagraph (Roddier & Roddier 1997), which can only reach full coronagraphic extinction with a suitably apodized pupil (Guyon & Roddier 2000; Aime et al. 2002; Guyon & Roddier 2002; Soummer et al. 2003). This pupil adaptation to the coronagraph can also be performed on the densified pupil of an interferometer. Another interesting application is to increase the coupling efficiency between a telescope and a single mode fiber in interferometers. By producing a suitable unobstructed apodized exit pupil, the PIAA technique can increase the coupling efficiency by 22% for an unobstructed telescope pupil, and by 93% for the CFHT pupil (0.44 central obstruction).
The PIAA technique could also be used on a smaller size telescope (about 0.5 m to 1 m diameter), or at a longer wavelength, to detect Jupiter-like planets. The technique could also replace Lyot coronagraphs on ground-based telescopes with adaptive optics for scientific projects requiring lower contrast ratios. The PIAA technique could be implemented with two small high-order deformable mirrors on "classical'' telescopes, and these two mirrors could simultaneously correct the wavefront aberrations of the telescope's optics and reduce the intensity of the PSF diffraction wings.
The author is thankful to Katherine Roth and Thomas Kane for suggestions to improve the manuscript, and to Lyu Abe and Koji Murakawa for discussions about this idea. The author is also thankful to the referee for many useful comments.