Fig. 1.37 Solid-folded (Left) and semi-solid (Right) Schmidt spectrograph cameras

If a concave back-surface mirror has a shorter radius on the front than on the back, it is called a Mangin mirror. Given a wavelength, the spherical aberration caused when light passes through the front is balanced by that caused by the reflection at the spherical back. Mangin mirrors are used in some spectrographs as the camera mirror.

If the glass is made thick, a very fast f-ratio can be designed such as solid or semi-solid Schmidt spectrograph cameras (cf. for instance Astronomical Optics by Schroeder [141]). For detector access, a solid Schmidt is designed in a folded form whilst a semi-solid Schmidt is designed in the Cassegrain form (Fig. 1.37). Compared to a classical Schmidt in the air where the focal ratio f/D is defined by the curvature of the mirror, the same curved back-surface mirror provides a focal ratio f/n'D which thus is faster by the refractive index n'. Therefore, since the efl is n' times shorter, in a direct imaging mode these systems provides a gain in detection of a factor n'2.

1.12.11 Field Derotator

If a telescope compensates for the Earth's rotation by altitude and azimuth motions, i.e. an alt-az mount telescope such as usually preferred for apertures larger than 5 m, the rotation of the field of view must be compensated by a field derotator. This is achieved by three plane mirrors all rotating around the field axis (Fig. 1.38). The processor may be substituted by an axial rotation of the focal instrument.

Fig. 1.38 Three-mirror field derotator

1.12.12 Pupil Derotator

If a coronograph is installed in a fixed position on the platform of a Nasmyth focus, the processor is a pupil derotator. Its rotation rate is opposite to that of a field derotator, whilst this latter device is not used. Thus, the residual aberrations of the telescope and the diffracted light from the spiders supporting the secondary mirror do not rotate at the detector during the observation.

1.12.13 Telescope Field Corrector

If Sphe 3, Coma 3, Astm3, and Petz3 and some higher-order aberrations are simultaneously corrected at a telescope focus, the processor is called a field corrector. Then the telescope benefits from an extended field of view. The first correctors were required for the prime focus. One may distinguish between corrector types for paraboloid-hyperboloid (PH) or for Ritchey-Chretien (RC) telescopes.

• Prime focus of PH telescopes: Only considering the correction of Coma 3 without introducing Sphe 3, F.E. Ross [135, 136] designed the first correctors of paraboloid mirrors for the 60-inch at Mount Wilson in 1933 and later the 200-inch at Palomar. His correctors included a thin low-power meniscus, with a strongly concave first surface, placed at some distance from a thin air spaced afocal doublet, itself at some distance from the focus. A Ross corrector provides a limited field extent since Astm 3 and Petz 3 still remain with also some chromatic differences for Sphe 3 and Coma3. Subsequently, A.B. Meinel [106] investigated the field correction by several aspherical plates and showed that Coma 3, Astm 3 and part of corresponding 5th-order terms can be corrected. Cancelling the four 3rd-order aberrations and some higher-order terms, C.G. Wynne [172] introduced in 1967 a four lens corrector with all lenses of the same glass and where the two first lenses are thin air separated whilst the next ones are two spaced lenses. Considering the two first lenses as equivalent to a positive lens, the power distribution positive-negative-positive provides the best arrangement for the correction of a large field. Faulde and Wilson [55] then showed that an equivalent performance is obtained with three spaced lenses whose central negative lens is with an aspherical surface. Epps et al. [50] introduced, for the Keck Telescope, a second aspherical surface in a three spaced lens corrector which included an option with atmospheric dispersion compensator. The widest field corrector presently built for a paraboloid mirror is 1° x 1 ° field at the Canada France Hawaii Telescope (CFHT).

• Prime focus of RC telescopes: For the prime focus of an RC telescope where the primary is an hyperboloid, the field correction is easier since Sphe 3 is available from the mirror to compensate for that of additional lenses. First, Gascoigne [64, 65] showed that both Sphe 3 and Coma 3 can be corrected by using an aspherical plate located at an appropriate distance from the focus. A Gascoigne corrector provides a limited field extent since Astm 3 and Petz 3 still remain with also some chromatic differences for Sphe 3 and Coma 3. Next, cancelling the four 3rd-order aberrations and some of the next order, Wynne [173] designed several correctors for large telescopes, showing that three spaced lenses in the same glass, with a power distribution positive-negative-positive, provides a wide field by only using spherical surfaces. Richardson [130] showed that given a field of view and f-ratio, there exists an optimum corrector size that matches the image quality requirement.

• Prime focus of spherical primary telescopes: Large segmented telescopes like the Hobby-Eberly Telescope (Het) require correcting the huge amount of Sphe 3 and the next aberrations. For a typical field of view of 4-5 arcmin, two holed pairs of concave mirrors provide the correction (Ramsey et al. [122], Sebring et al. [143]). These aspherical mirrors have their concavity in front together, and work as image transports by successively re-imaging the field at a second and third focus.

• Cassegrain focus of RC telescopes: The Cassegrain focal surface of an RC telescope can be made flat by equalizing the curvature of the mirrors (cf. the Petzval sum in Sect. 1.10.1). In this special form, the aplanatic combination is called a flat field Ritchey-Chretien. This was realized by I.S. Bowen with the 2.5 m Irenee du Pont de Nemours Telescope (Table 1.1). The next correction is Astm 3 which was obtained by Bowen and Vaughan [21] with a single aspherical plate located at some distance before the focus. This requires a slight re-design of the mirrors which thus departs from a strict RC form to provide an extended field Ritchey-Chretien. Because of the remaining chromatic difference of Astm3 and other residual aberrations, Wynne [176] further replaced the aspherical plate by two spaced aspheric lenses of different glass. Examples of such extended field telescopes are the ultraviolet space telescope Galex 1.2° field, and the Sloan Digital Sky Survey Telescope (Sdss) 2° x 1.5° field (Table 1.1).

A detailed account on telescope field correctors is given by Wilson [170].

1.12.14 Atmospheric Dispersion Compensator

If the processor is two prismatic units which can increase or decrease the amount of dispersion to match that of the atmosphere, this system is called an atmospheric dispersion compensator (Adc). Two concepts are used for ADCs: (1) two prism pairs which are counter rotated, and (2) two thin single prisms with a variable axial separation.

In a study of refraction, Claudius Ptolemy (^ 140 AD in Alexandria) mentioned the effect of atmospheric refraction; at that time, the refraction was assumed as proportional to the incidence. Tycho Brahe made some attempts to measure the atmospheric refraction, but his correction of the parallax (3 arcmin) for the determination of the planet position was totally false which postponed its refraction correction. The first model of the atmospheric refraction goes back to J-D. Cassini (1662), who assumed that the atmosphere was a medium with constant index na abruptly ending at a fixed elevation ha before the "aither." Although Cassini's model only requires use of Snell's law by appropriate choice of the two constants (na, ha), we may notice that

Table 1.8 Refractive index of the air versus wavelength A for p0=1013.25x 102 Pa, 70=273.15 K, and pw=0 from Allen and Cox [3]

A [nm] 300 400 500 600 700 800 1,000 2,000 (n-1) 106 307.6 298.3 294.3 292.2 290.9 290.1 289.2 288.0

For a water vapor pressure pw= 550Pa, subtract 0.2 to (n-1) 106; for other values apply a linear correction. For other temperatures and pressures multiply (n-1) by p70/p0T.

his resulting refraction angles - difference of two arsin functions - are remarkably accurate for zenith distances up to 45°.

Considering a more realistic model, let us denote n the refractive index of the air and z the zenith distance, the atmospheric diffraction angle is accurately represented for z < 80° by y = [n(A, p, T,pw) - 1] tanz, (1.94)

where A, p, T , and pw are the wavelength, air pressure and temperature, and water vapor pressure at the observation place. The values of the refractive index in Table 1.8 are extracted from Allen's Astrophysical Quantities [31].

The atmospheric dispersion is the variation of the refraction in a given wavelength range (A1, A2). This transforms a star image into a spectrum, and the purpose of an ADC is to correct this degradation. From (1.94), the angular spectrum caused by the atmospheric dispersion may be expressed as

From the ultraviolet atmospheric cut-off to the infrared limit of sensitivity of a CCD detector, corresponding to the spectral range 300-1,000 nm, one obtains from Table 1.8 the angular dispersions

Ay{z=45°} = 3.79 arcsec, Ay{z=60°} = 6.57 arcsec.

It can be shown from these results that, even for a telescope located at the high elevation of 4,000 m, the image degradation must be corrected for an upgraded imaging.

• Counter rotated prism pairs: A first form of ADC is with two N-shaped prisms which form a counter rotated prism compensator (Fig. 1.39). Each prism pair does not deviate the beams for the wavelength corresponding to the mean refractive index of the central spectral range. Such designs were developed by Wynne [174], Epps et al. [50] as an option included to a wide-field corrector, Wynne and Worswick [175], D-q. Su [151], Bingham [15], Wang and Su [165] and others.

• Varying separation thin prism pair: A second form of ADC is with two thin prisms whose axial separation is varied (Fig. 1.40). This concept proposed by Beckers was designed by Avila and Rupprecht [7] who called it a linear atmospheric dispersion compensator (LADC). Compared to the previous concept by counter

Fig. 1.39 Left: Schematic of counter rotations of each prism pair in an Adc. The maximum compensation is when the prism pairs co-add their dispersion, at a=90° (B: blue, R: red). Right: The four possible forms of an Adc shown at maximum zenithal compensation zmax (most dispersive glass shown in gray)



Fig. 1.40 Schematic of an Adc with two thin prisms and varying separation - Ladc/Vlt (after Avila, Rupprecht and Beckers [7])

rotation, the design is much longer but shows the advantage of avoiding tilting of the pupil axis and, with only one glass (silica), benefits from higher throughputs from ultraviolet to infrared.

In an alt-az telescope, the Adc is usually located in series with the field derotator.

1.12.15 Adaptive Optics

• Seeing-limited and diffraction-limited imaging: Ground-based astronomical observations suffer from wavefront degradations caused by the Earth's atmosphere. Thus, a large telescope with passive optics can only provide a seeing-limited imaging whilst its aperture diameter should theoretically allow the much higher angular resolution of diffraction-limited imaging. For extremely short exposures, a seeing-limited star image is made of a collection of speckle elements whose individual size is that of an Airy pattern. Most observation sites located at near-sea-level elevations show a 1-2 arcsec seeing. For exceptionally good sites at 4,200 m elevation (Mauna Kea) the seeing is 0.5-1 arcsec.

If the processor is a telescope mirror whose shape, orientation, and position can be modified by actuators, controlled either by an open- or a closed-loop system working at a low time frequency, say, f < 0.1 Hz, then the system is called an active optics system.

If now the local shape and tip-tilt positions of a mirror are modified by a closed-loop control at high time frequency, say, f > 50 Hz, which compensates for the image degradation caused by turbulence in the Earth's atmosphere, then the system is called an adaptive optics system.

• Atmospheric seeing corrections with deformable mirrors: Use of a telescope with adaptive optics corrections allow obtaining a sharp image of a star, i.e. close to the Airy pattern. This is usually called high angular resolution imaging. For this purpose the deformable mirror (DM) of an adaptive optics system must be located at or near a reimaged pupil of the telescope input pupil (Fig. 1.41).

In some recent telescopes, a first DM is the Cassegrain mirror which thus is the telescope input pupil. Present studies of extremely large telescopes (ELTs) include a large pupil DM in their main mirror train.

Realtime wavefront sensing of stars - dichroically separated or at the telescope field edge - and fast algorithms for the wavefront reconstruction allow driving the DM actuators. The closed-loop control also implicitly compensates for low aberration residuals and vibrations of the telescope optics, say, for a fraction of a wavelength. If there are not enough bright stars in the field of view, the light of a laser guide star (LGS) allows wavefront analysis by use, for instance, of the Na doublet lines retro-reflected by the atmosphere.

Because the turbulence effects are mainly due to several dominant atmospheric layers, a single layer wavefront correction only provides an isoplanatic field of ~ 30 arcsec in the near infrared. Using several DMs coupled with several wavefront guide stars allow increasing the isoplanatic field up to 2-3 arcmin. Such a technique is called multi-conjugate adaptive optics (MCAO).

Realtime control algorithm

Wavefront Sensor

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