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Considering the various spectrograph designs based on the Schmidt concept, the reflective design with axisymmetric aspherized gratings mounted in normal diffraction is of the highest interest because of the simple and compact design also presenting a high throughput and several other important advantages listed in Sect. 4.4.8.

4.4.4 Bi-Axial Symmetric Gratings (fi0 = 0)

For cases where the diffraction angle overpass the range, say, = ±5°, bi-axial symmetry gratings provide more efficient aberration corrections. This allows recovering residual images similar to the case of axisymmetric gratings and a resolving power such as given by (4.24).

^ The resolving power of all-reflective spectrographs working with bi-axisymmetric gratings out from normal diffraction onto the best-fit spherical focal surface and in the symmetry plane perpendicular to the grating lines is:

i dNc = 0.00043 sin2(a + ft)/Q3 + 0.011 ^/Q3, \ s = cos2^m, k = 3/2 i.e. M = 3/64Q2.

Such as with non-centered reflective Schmidts telescopes (Sect. 4.3.2), the grating surface is approximated by generating homothetical ellipses. Thus, the optical surface of the grating is

with s A

Biaxial symmetry gratings also apply to the design of all-reflective Schmidt-Littrow mountings (ft = —a) but do not seem to have been developed up to now. Generally, Littrow mountings concern high-resolution spectroscopy requiring long focal lengths with large incidence angles and slow f-ratios. This would lead to moderate asphericity gratings.

For spectrograph camera f-ratios up to f/2, in order to obtain a more accessible focus if necessary, a folding mirror may be added between the grating and the camera mirror. This holed mirror may fold either both incident and diffracted beams or only the latter. For cameras faster than f/2, the detector is fitted at the inner focal surface then requiring large size gratings (Sect. 4.4.11).

4.4.5 Flat Fielding of All-Reflective Aspherized Grating Spectrographs

Given the state-of-the-art of present detectors, it is quite impossible to have disposal of a sensitive surface other than a flat shape. Therefore, the cryostat window of the detector has to be replaced by a flattener lens. This operation requires, a small increase of the grating asphericity to compensate for Sphe 3 mode from the field flattener and a small repositioning of the camera optics towards the grating for Coma 3 compensation.

The flattener lens is of positive power. From Petzval's condition in Sect. 1.10.1 (see also Chretien [17], Born and Wolf [7]), setting the refractive index N;+1 = —Ni for a reflection at a surface numbered i of curvature c;, and also Nj+1 for the refractive index of the last medium, a null curvature is achieved if

Thus, with respect to the mirror curvature 1/R and curvatures c1, c2 of the singlet lens, and after substitutions, a flat field is obtained if where N is the lens refractive index of the central wavelength.

Although it is possible, in convergent beams, to design a singlet lens free from axial chromatism by a convenient balance of its c1; c2 curvatures, thickness, and back focal distance to the detector (Wynne [96]), ray-tracing optimizations show that the best field corrections are obtained with an almost convexo-plane singlet lens, i.e. c2 ~ 0, having the lowest possible thickness. For wide spectral range spectrographs, one of the most often used materials is fused silica, because of its low dispersion and high ultimate strength for a safe use as a cryostat window. Assuming a convexo-plane lens whose first surface of curvature c1 is passed by the light, a first approximation gives

A very slight change of c1; c2 somewhat improves the performance by giving c2 a small curvature that minimizes Astm 3 of the lens whilst its power c1 - c2 is unchanged.

The elastic deformation of the lens due to the cryostat vacuum does not significantly change in its thickness distribution and no elasto-optical effect modifies the image quality in the field. In the case of a lens flattener used as a cryostat window and thus submitted to the atmospheric pressure q ~ 105 Pa, the thickness of the lens must be carefully determined. Denoting a the half diameter of the lens, its axial and edge thicknesses must satisfy the following bending and shearing conditions where the stress a must be taken two or three times smaller than the ultimate stress ault (cf. Glass rupture and loading time in Sect. 5.2.5). For fused silica, ault = 700 x 105 Pa and Poisson's ratio v = 0.16.

4.4.6 Examples of All-Reflective Aspherized Grating Spectrographs

The hereafter presented spectrographs are not always strictly all-reflective systems since some of them use a singlet lens for field flattening and as a detector cryostat window. Nevertheless, the terminology "all-reflective" is almost correct because a singlet lens located just before the focal surface does not introduce a significant amount of chromatism but only slightly modifies the correction of the spherical aberration of the system.

• Laboratory spectrograph with k = 0 geometry gratings: Although a k = 3/2 geometry is preferred for the best aberration correction, the construction of the first aspherized grating was carried out with a geometry k = 0 by a tulip-form submaster (Lemaitre and Flamand [30]) for a laboratory spectrograph that, thus, used a plane-aspheric grating. A stainless steel submaster was polished flat when not stressed and a standard plane grating from Jobin-Yvon was replicated on it surface. In a second step, the sub-master was elastically aspherized by a central force in reaction with its free edge and then replicated on a Schott Zerodur rigid substrate. This replica is the final aspherized grating (see Chap. 5, Sect. 5.4.4).

The thickness of the tulip-form submaster was close to T40 ^ [p8/(3+v) -1/p2]1/3 [cf. Eq. (3.30)], a form that generates a pure p4-flexure and corresponds to M = 0, A2 = 0 and A4 = -1/4 in Table 4.1. This entails k = r^/r^ = 0, thus a null-powered zone located at the submaster vertex. In fact this form was slightly modified for also correcting the fifth-order spherical aberration. The thickness distribution generated the Sphe 3 and Sphe 5 modes but did not provide any curvature flexure Cv1 although a small amount arose for this latter mode, without any drawback, caused by the finite thickness of the central part of the tulip form. For

Fig. 4.14 Theoretical residuals of a 3,000-5,000 A spectrum with a plane-aspheric grating k = 0 and ft = 0 onto a curved field 44mm in length or 15° field. Camera beam f/2xf/1.8 at Aq. Although a much better aberration correction would occur with k = 3/2, this design provides a good correction in the direction of dispersion with residuals smaller than 25x5 ^m. The residual field balance leads to Sphe 3 at center and Astm 5 at edge

Fig. 4.14 Theoretical residuals of a 3,000-5,000 A spectrum with a plane-aspheric grating k = 0 and ft = 0 onto a curved field 44mm in length or 15° field. Camera beam f/2xf/1.8 at Aq. Although a much better aberration correction would occur with k = 3/2, this design provides a good correction in the direction of dispersion with residuals smaller than 25x5 ^m. The residual field balance leads to Sphe 3 at center and Astm 5 at edge p = 1, the corresponding simplysupported submaster edge was r = 85 mm. The ray-trace best fit of the design under-correction factor was found by setting s = cos ^m = 0.991 [instead of setting s = cos2 ^m for k = 3/2 in (4.24)] and the theoretical shape including the fifth-order submaster active deformation was that of (5.90) in Sect. 5.4.4.

This first aspherized grating, 1,200i.mm-1, 102x102mm, was installed in an experiment spectrograph designed as in Fig. 4.13d for optical evaluation. A slit and collimator mirror provided a parallel beam 2rm = 90 mm in diameter illuminated the grating mounted in a normal diffraction arrangement, A = 0° at Ao = 4,000 A corresponding to the incident angle a = 5o = 28.7°. The 90x 100 mm elliptical stop at the grating leads to camera beam focal ratios of f/2xf/1.8 for Ao and in the two principal directions. Fe- and Cd-spectra were obtained on curved films over the ultraviolet range 3,000-5,000 A with a 42 A/mm dispersion and a camera semi-field ^m = A = ±7.5°. The results from optical testing were found in full agreement with the theoretical image blurs (Fig. 4.14).

• Astronomical spectrographs with k=3/2 geometry gratings: The first aspherized grating built by the active optics replication process for astronomical observations was for the UV prime focus (Uvpf) spectrograph at Cfht in Hawaii and used an axisymmetric aspheric grating working in the order at diffraction angle Ao = 0 for A = 4,000 A.

The Uvpf spectrograph made the first run of the Cfht instrumentation in early 1980. It was originally observing with photographic ultraviolet enhanced films (Lemaitre [33]) with a nebular capability of 4 arcmin on the sky. The previous laboratory spectrograph (Fig. 4.14) served as a model for the development of aspherized gratings. The f/3.8 ratio of prime focus was reduced by the camera mirror to f/1.26 in the nebular direction. The dispersion was 55 A.mm-11 at 4,000 A. This spectrograph was updated for use with a CCD (Lemaitre and Vigroux [35], Boulade et al. [8]). This required introducing a convexo-plane field flattener lens as the cryostat window (Fig. 4.15).

The aspherization of the axisymmetric grating was carried out from replication of a vase-form active submaster stressed by air pressure (see Sect. 5.4.3).

Fig. 4.15 Optical design of the UVPF spectrograph of CFHT. The 1,200£mm~1 axisymmet-ric aspherized grating provided a spectral range XX [3,000-5,200A]. The normal diffraction at 4,000 A provides an f/1.10 camera beam in the dispersion direction. Originally, all the spectral range was covered by curved UV films 36 mm in length. Updating was performed in 1985 with a thinned CCD covering 890 A in one frame. Discrete slidings of the CCD and field flattener onto a cylinder allowed covering all the spectral range with three positions (Lemaitre and Vigroux [35])

Fig. 4.15 Optical design of the UVPF spectrograph of CFHT. The 1,200£mm~1 axisymmet-ric aspherized grating provided a spectral range XX [3,000-5,200A]. The normal diffraction at 4,000 A provides an f/1.10 camera beam in the dispersion direction. Originally, all the spectral range was covered by curved UV films 36 mm in length. Updating was performed in 1985 with a thinned CCD covering 890 A in one frame. Discrete slidings of the CCD and field flattener onto a cylinder allowed covering all the spectral range with three positions (Lemaitre and Vigroux [35])

The submaster was designed to generate the grating geometry k = r^/r^ = 3/2 which provides the best balance of the field aberration residuals. Its r0 = 71 mm built-in radius allowed the grating replication for use of an elliptical aperture area. This area is determined by the diameter of the collimator cross-section beam as 2rm = 2r0/%/k = 116 mm which corresponds to the grating width. In the other direction and for a normal diffraction angle, the collimator incidence angle a = 28.7° corresponds to a 132 mm grating length. Hence, the useful aperture of the grating -aspherized on a rigid Zerodur holed substrate - was an ellipse 116x132 mm in size.

The UVPF spectrograph design parameters for the CCD version are displayed in Table 4.3, and residual images in Fig. 4.16.

All-reflective spectrographs are also efficient for infrared studies and, of course, for broad-band spectral coverage. Having a number of optical surfaces reduced to a minimum, these designs are particularly appropriate for medium and low-resolution spectroscopy, i.e. faint object studies. A set of interchanged gratings provides various dispersions in the same instrument. Such pseudo-plane aspheri-cal gratings having rotational symmetry and in the form k = 3/2, have been built

Element |
Radius of curvature |
Axial separation |
Figure |

Primary mirror CFHT |
27,066. |
13,533. |
parabola |

Slit |

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