## G V gJ m

This angle is zero at the poles and maximum near t = ±n/4. In practice, liquid mirror bearings are levelled using a precise bubble level that is sensitive to the vertical direction defined by the acceleration vector g — ac where ac is the centripetal acceleration. The deviation of g — ac from the true vertical is corresponding to the sin2t term of (7.52). In this case, the required adjusting angle is just the cos t term; this term defines the direction of vector m + to coalign with g — ac.

Under this condition of cancelling the Coriolis effect, Astm 3 is removed, and, in a cylindrical reference (Z, r, 9) tangent to the mirror vertex with 9 = 0 in the terrestrial meridian, the mirror shape is

ZOpt

where p = 1 at the mirror edge. Applying the quarter-wave rule to the Sphe 3 term provides the limit in which a dioptric corrector does not need to compensate the spherical aberration; this gives a mirror size 2rm = Q^/[email protected] At the wavelength ^He-Ne, this would correspond to a 10 meter mirror at f/1.76. The Coma 3 term is fully negligible even for a 100 meter mirror.

These results remove the last fundamental obstacle to achieve diffraction-limited performance with extremely large Lmts [31]. Thus, Elmts may be foreseen for future astronomical observations from the ground or at a lunar pole.

7.7.2 Field Distortions and Four-Lens Correctors for LMTs

For Lmts, which are transit telescopes, the time exposure of a moving image on the detector is achieved by time delay integration, also called drift scanning. Since the image moves at a constant velocity because of the sidereal rotation, the electronic drive of the detector is operated at a constant rate for each pixel column by readout of the column reaching the detector edge.

of all telescope systems, Lmts are an unprecedented case requiring control of the third order distortion aberration - Dist 3 - (which is usually removed a posteriori by image processing) and also of the distortion effect due to non-rectilinear sky projection of the field. The correction of these distortion effects was first solved by E.H. Richardson.

Considering celestial equatorial coordinates (a, 8) and an Lmt located at the terrestrial latitude, say 0 < £ < 60°, the meridian lines - a = constant - are field imaged as straight lines all converging in a common point whilst declination lines -8 = constant - are projected as circles all centered at this point. The optimum drift scan integration of the image by the detector requires that the curvilinear field frames (a, 8) shall be optically transformed into strictly cartesian frames (x,y).

If an Lmt is at the Earth's equator, £ = 0, then only Dist3 = 0 is needed.

If an Lmt is at 0 < £ < 60°, in addition to Dist 3 = 0, the corrector lenses shall provide an asymmetrical distortion which is opposite to the above sidereal distortion. The correction is achieved by (i) setting the sag sx = -sa due to the north-south differential sidereal rate in the field, and (ii) setting the sag sy = -s8 due to the star trail curvature. These conditions can be simultaneously satisfied. Denoting the semi-field angles (pxm, Vym in a and 8 directions respectively, these sags are, from Hickson [30], sx = -2Qrm q>x m Vy mtan £, sy = Q rm vLtan£, (7.54)

where sx, representing the de-rotation sag of the meridian straight lines, is maximum at the diagonal field edges for which if the integration surface of the detector is square Vym = ±Vxm.

Richardson [56] has shown that fortunately both Dist 3 and asymmetrical sidereal distortion can be removed: the best result is achieved by a quadruplet lens that includes one of the lenses wedged. The design starts from a three-lens Wynne corrector [71] which annuls Sphe 3, Coma 3, Astm3, and Petz3, and consists of a positive meniscus followed by negative and positive lenses, all of the same glass. The four-lens Richardson corrector [29] introduces an off-axis wedged lens as fourth element near the focus. This corrector is presently used for typical fields of 20 x 20 arcmin but may be used for larger fields. This is a key component of all Lmts.

7.7.3 LMT Concepts with MDMs for Off-Zenith Observations

A paraboloid mirror observing a 10 arcmin typical field of view, with a field center located at several degrees from its axis, would present high amplitude Coma3, Astm 3 and several higher-order aberrations. The design of an off-axis dioptric corrector would be extremely difficult to achromatize. In order to observe with a liquid mirror at a substantial off-zenith angle, a preliminary study by Wang [65], Moretto [47] investigated the case of off-axis corrections with a three-mirror system. The telescope consist of a parabolic primary, and two additional tilted mirrors - secondary and tertiary - having biaxial symmetries.

The starting design is derived from a Paul-Baker telescope [4, 54]. In the original Paul form [59, 70], the concept starts off with a primary-secondary afocal pair, that is concave-convex confocal paraboloids which provide an anastigmatic Mersenne beam compressor. Let us define the beam reduction ratio by k = R2/Ri . Paul added a tertiary whose center of curvature is placed at the vertex of the primary. In collimated beams the tertiary can be used spherical is similarly as a Schmidt spherical mirror. The spherical aberration of the tertiary can be compensated by the secondary. If R2 = R3, we can modify the parabolic secondary into a sphere since the Sphe 3 contributions are just opposite. The system is again free from Sphe3. It was first noted by Paul that the system is also free of Coma3 and Astm3. The focal surface curvature is 2/Ri since the Petzval sum is 2(1/Ri - 1/R2 + 1/R3). The Paul system can be generalized to give also a zero Petzval curvature with R2 = R3. It was first noted by Baker that the 3rd-order aberrations of this modified system remains at zero if the spherical secondary is replaced by an ellipsoid whose conic constant is ck2 = -1 + (R2/R3)3 = -1 + (1 - k)3 so that the tertiary still receives collimated beams and remains a sphere (ck3 = 0). The first large Paul-Baker telescope was built

Fig. 7.12 Optical design of a three-mirror telescope with doublet-lens corrector for observaions at 5° from zenith with 15 arcmin FOV. The liquid primary is 4 m in diameter. The secondary and tertiary mirrors are 1.15 m in diameter; their design with vase-form MDM allows in-situ aspher-izations. The rms size of residual images is smaller than 0.5" over the FOV [45]

Fig. 7.12 Optical design of a three-mirror telescope with doublet-lens corrector for observaions at 5° from zenith with 15 arcmin FOV. The liquid primary is 4 m in diameter. The secondary and tertiary mirrors are 1.15 m in diameter; their design with vase-form MDM allows in-situ aspher-izations. The rms size of residual images is smaller than 0.5" over the FOV [45]

at the instigation of Angel et al. [3] as a transit automatized instrument for CCD sky surveys. This telescope [44] has a 1.8 meter aperture, f/2.2 and is flat fielded over 1°. With its beam reduction ratio of k ~ 1/3, one has R2 = Ri/3, R3 = Ri/2, cK1 = -1, ck2 = -0.704 and ck3 = 0.

The study by Moretto [45, 46] of a telescope concept for off-zenith observations with a liquid mirror is derived from Paul-Baker flat-fielded anastigmats. The concave secondary and tertiary are vase-form MDMs. The x,y field compensations for the distortion and sky projection has been carried out by including a doublet-lens corrector (spherical lenses) to allow the CCD drift scan imaging. This design has a

4 meter primary and a beam reduction of 3.5 at the secondary for observations up to

The first MDM was designed with the parameters of Fig. 7.2 for the development of such secondary and tertiary mirrors. This 12-arm MDM was built and various obtained Clebsch-Seidel modes were controlled by interferometry. The resulting co-addition of Tilt 1 mode and four flexural modes is shown by the interferogram of Fig. 7.13.

The stress-strain linearity - Hooke's law - of the Fe87Cr13 alloy for the 12-arm MDM has been verified by experiment in generating a maximal stress a = 50daN.mm-1 into the outer ring. This test was performed for the Astm3 mode z22 = A22 r2 cos 20. If the rigidity of the inner disk is neglected, then the stress-strain relation derived from the thicker ring is

leading to a ptv deformation sag of z22 = ±1.7mm at clear aperture radius a = 80 mm. This large deformation was found to lie within the elasticity domain, then showing the large amplitude capability of active methods.

7.8 MDMs as Recording Compensators for Holographic Gratings

### 7.8.1 Holographic Gratings Correcting Aberrations

The recording process for obtaining holographic diffraction gratings requires the formation of an interference pattern which is frozen into the photosensitive layer of the grating substrate. Straight interference lines allow recording of plane gratings. The use of a Rowland mounting allows recording of concave gratings free from Astm3 by curved and variable spaced interference lines. For correcting several aberrations, at least one of the two recording wavefronts must be aspheric. Up to now, the formation of an aspherical wavefront requires the design and construction of special optical systems providing exactly the opposite shape of the wavefront to be corrected. Classically, such a compensating system is complex, expensive and furthermore, only usable for making a particular grating. Also, various types of aberrations cannot be simultaneously achieved with such optical systems, therefore leading to great difficulties for correcting high-order aberrations with holographic gratings. However, an exception is given in "Third generation Rowland holographic mounting" by Duban [16, 20, 21], which uses two spherical auxiliary holographic gratings to produce the two aberrated recording wavefronts. This allows aberration correction of the recorded spherical grating up to and including some fifth-order modes. The construction of the auxiliary gratings for the recording of a corrected grating increases the cost and the auxiliary gratings remain dedicated to this.

Recording methods for making aberration corrected holographic gratings are greatly simplified by using a plane MDM upon one of the two recording beams. MDM compensators provide easily the superposition of many aberration modes by active deformations. The available Clebsch-Seidel modes allow a higher degree of correction than the Rowland mounting. Aberration compensators based on plane MDMs provide a general method for recording corrected diffraction gratings without requiring the above sophisticated optical systems.

As an example of the method, we have considered the recording of the holographic gratings of the Hst Cosmic Origins Spectrograph (Cos) [25, 48]. Very substantial improvements in the image quality have been found by Duban [15, 17, 19] by use of a MDM as a recording compensator. The result is that with such holo-corrected gratings :

^ Many higher-order aberrations can be simultaneously corrected,

^ The residual monochromatic images of the spectra have much smaller areas.

7.8.2 Design Example for the COS Gratings of HST-Recording Parameters

The gratings of the Cosmic Origins Spectrograph must correct the original spherical aberration of the Hst. Before correction, the maximum amount of this aberration is up to 200-times larger than that of the spectrograph alone. No holographic correction can reduce such an amount over the whole spectral range. Therefore, it is not possible to keep the grating substrates purely spherical; we have introduced fourth and sixth degree deformations on the grating substrates, i.e. z40 and z60 terms.

Since the Cos incident beam is located 5.40 arcmin off the Hst optical axis, we also have been led to correct the Hst field aberrations, mainly Astm3 since the Hst produces an astigmatism length of 1.20 mm. This correction is made holo-graphically, by using a MDM to record the gratings onto the Optimized Rowland Mounting [21]. The mounting cancels Astm 3 at two points P1 and P2 of the spectrum. In a very general result which is also valid for the Cos gratings, this mounting is the only one really suitable for obtaining the Astm 3 compensation.

For three Cos gratings, Table 7.3 shows the spectral data of the spectrograph. Table 7.4 gives the grating parameters, where N is the groove density in l. mm-1, R the radius of curvature of the grating substrates in mm, A0 the laser recording wavelength, i the incidence angle at the Hst, a and ¡5 the recording angles in degree. Table 7.5 gives the deformation coefficients of the grating substrates in mm-n+1. Substrates of gratings #1 and #2 are identical. Table 7.6 gives the deformation coefficients in mm-n+1 and the incidence angle iMDM upon the MDM.

The optical design of each corrected grating minimizes the blur images for five wavelengths. For grating #1, the wavelengths are those listed in Table 7.3 and correspond - from left to right - to Amin,P1, the middle of the spectrum Amed, P2, and Amax, plus two other intermediate wavelengths around the central one. The correction of astigmatism at points P1 and P2 is evident (Fig. 7.14).

Grating |
¿min |

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