K1 = –1 – 2(1 – η)/m3 K2 = –[(m + 1)/(m – 1)]2 – 2mf/[(f – f1)(m – 1)3]      = –1 – 2[m(2m – 1) + 1 – η]/(m – 1)3,

(3)

Having calculated these coefficients, mirror surfaces can then be described by equations of the form:

x2 + y2 = 2Rz – (1 + K)z2,

(4)

R2 = 2f1/(m – 1),

(5)

R2 = 2(f – f1 – h)/[(mf + h)/(f – f1) – 2].

(6)

The vertex radii of curvature, R₁ and R₂, are the same for classical Cassegain and corresponding RC, but the conic constatnts are different and for classical Cassegrain they are:

K1 = –1 K2 = –[(m + 1)/(m – 1)]2.

(7)

The following table, Table 1., and Fig. 1. present example sets of coefficients of classical and two RC designs in which the secondary diameters are very nearly equal. The first two (i.e. Classical and Solution A) are used in this report for performance comparison basing on simulations with the technique of ray-tracing. The Solution A was obtained with certain modifications of Eqs. (1) and (2) and Solution B strictly according to these equations but starting with d = 8.94472 m to get final RC secondary diameter of 9.0 m. The Solution B gives considerably sharper focus, but in our case both solutions (A and B) seem equally suitable, and in certain respects Solution A is better than Solution B. This itself suggests that there must be a considerable room for choice of the conic constants and good chances for finding other solutions of interest that are far removed from original Ritchey and Chretien ideas. We shall return to this important aspect later in this report.

Table 1: Classical Cassegrain and Ritchey-Chretien Radio Telescope Data

 Aperture diameter, D                                     90.000 m
 Focal length of primary mirror, f                        31.500 m
 Focal ratio of primary mirror, f/D                        0.350  
 Subreflector diameter, d                                  9.000 m
 Height of secondary focus above main dish vertex, h       1.000 m
 
                                            Classical      Ritchey-Chretien
                                                        Solution A  Solution B 
 Subreflector radius                         4.50000      4.49950    4.50000 m
 Secondary magnification, m                  9.24790      9.31154    9.30740
 Prime focus to subreflector vertex, f1       2.97622      2.95785    2.95904 m
 Secondary focus to hyperboloid vertex, f2   27.52378     27.54215   27.54096 m
 Secondary interfocal length                30.50000     30.50000   30.50000 m
 R1                                         63.00000     63.00000   63.00000 m
 K1                                         -1           -1.00255   -1.002394
 R2                                          6.67412      6.62745    6.63046 m
 K2                                         -1.54377     -1.57496   -1.575293

Fig. 1: Example of geometry and dimensions of Ritchey-Chretien (in red; Solution A) and classical Cassegrain (in blue) telescopes for RTH. This figure is to scale. Note the smallness of differences in mirror shapes: the depth of both RC mirrors is smaller than CC ones by about 5 mm only (subreflector depths are 1.4334 m and 1.4380 m, for RC and CC, respectively).

Ray Tracing in an RC Telescope

The program OptiCass, developed a few years ago has a procedure for tracing ray paths between a feed and aperture plane with reflections from a hiperboloidal subreflector and paraboloidal main mirror. The procedure returns initial and final direction cosines, the pathlength and coordinates of the reflection point on paraboloid and coordinates on the aperture plane. These coordinates on aperture are used to transform (DFT) amplitude distribution into the telescope power pattern.

Almost all algorithms of this scheme had to be worked out anew because of different shape of main mirror and the fact that in RC the rays parallel to main mirror axis are no more reflected towards the primary focus. Still the general scheme, as described there, remains valid for the new procedure (provided the word 'paraboloid' there is replaced with 'primary hyperboloid'). The weakness of this modified procedure is that the solutions for rays very close to the vertices tend to be unstable when the conic coefficient K₁ nears –1 (i.e. the main dish turnes into paraboloid). Therefore, this version of OptiCass cannot be considered a true generalization of its progenitor, and hence it was named OptiCassRC. Nevertheless, OptiCassRC can be used to simulate also a classical Cassegrains by setting main mirror conic constant very, very close to –1 (with e.g. K₁ = –1.000001 90-m quasi-paraboloid depth of our RTH is smaller than true paraboloid depth by only 2 μm or 2E–6 m). This quasi-paraboloidal mode was used as a final check of the correctness of the ray tracing algorithm implementation in the new version of OptiCass.

RC Radio Telescope Properties

Fig. 2: The height of feed above the secodary focal plane (towards the subreflector) for which the least aberration (shown in Fig. 4) and spillover losses are found while simultaneously adjusting the direction of feed power pattern (Fig. 3).

Fig. 3: Correction of the direction of feed (with respect to direction to the centre or vertex of subreflector) for which the least aberration and spillover losses (shown in Fig. 4) are found.

Fig. 4: Losses of antenna gain as a function of feed lateral offset optimized in placement in z-coordinate and direction as shown in Fig. 2 and Fig. 3.

Fig. 5: Height of first sidelobe and comatic distortion as a function of lateral offset at optimal placement and orientation of feed (see Fig. 2 and Fig. 3).

The results presented in Figs. 4 and 5 seem to prove that the RC design performes indeed significantly better than Cassegrain in respect of aberration losses as well as of magnitudes of coma and first sidelobe.

RC Solution A             Classical Cassegrain	RC Solution B

Fig. 6: (Left figure, RC Solution A and Classical Cassegrain) Distortions of power pattern of Ritchey-Chretien (left panels) and classical Cassegrain (middle panels). The distortions (shown in red) represent difference between power pattern at given offsets and offset-free power pattern. Note the different scales of each panel as indicated next to the vertical bars (readable here). The two uppermost panels (in blue) contain the patterns themeselves at largest analysed feed offset (4.5 m), the lower one showing more details of bottom part of the same pattern (main beam being here cut-off at 9% of the maximum power). Click on this image to see it in a higher resolution. The three figures on the right, RC Solution B, contain similar results for RC but using Solution B of Table 1. Here only distortions at feed offsets of 4.5 and 0 m are shown. Note that at 0 m offset (all bottom panels) this solution has an order of magnitude smaller residuals than the Solution A, but for nonzero offsets their patterns are very much the same. The circular inset at top right of the three upper panels represents all the rays traced, with their phases (colour) and amplitudes (brightness), as seen on aperture plane after traversing the path from feed (rays reflected from the secondary but missing the primary mirror contribute to spillover loss and are marked red).

Sensitivity of 90-m Antenna at 15 GHz to Misalignments

                                 RC  Class.
Subreflector z_offset -0.002 m  1.43  1.39
Subreflector z_offset +0.002 m  1.46  1.39
Subreflector x/y_offset ±0.01 m 2.52  2.43
Subreflector tilt ±0.2°         2.50  2.69
Feed z_offset -0.2 m            3.99  3.90
Feed z_offset -0.1 m            1.02  0.99
Feed z_offset +0.1 m            1.04  1.01
Feed z_offset +0.2 m            4.11  4.00
Feed tilt ±1°                   1.59* 3.06*
Feed tilt ±2°                   6.23* 9.22*

                                 RC  Class.
Subreflector z_offset -0.002 m  0.41 -1.36 
Subreflector z_offset +0.002 m  1.64  1.44
Subreflector x_offset -0.005 m -2.27 -3.21
Subreflector x_offset +0.005 m  2.96  3.76
Subreflector y offset ±0.005 m  0.48  0.45
Subreflector x-z_tilt -0.02°    1.24  1.40
Subreflector x-z_tilt +0.02°   -1.18 -1.38
Subreflector y-z_tilt ±0.2°    -0.01  0.64
Feed z_offset -0.1 m            0.66  0.72
Feed z_offset +0.1 m            0.69  0.62
Feed x-z_tilt offset -1°        1.57* 2.27*
Feed x-z_tilt offset +1°        1.50* 2.22*
Feed y-z_tilt ±1°               1.34* 2.20*

Ritchey-Chretien Would be More Effective at Higher Radio Frequencies

From the previous sections it can be concluded that RC idea does not give as significant improvements as could be hoped from experience at optical wavelengths. The reader might take this as an indicatation of a flaw in our implementation of the RC idea itself. However, our algorithm, as implemented in OptiCassRC, produces very nice improvement over classical Cassegrain at very high frequencies, say 100 GHz or higher. Here, in Fig. 7a., is an example of performance of the two designs at λ = 0.3 mm (1000 GHz).

Fig. 7a: Left: Off axis power pattern of classical Cassegrain telescope and, Right:, the same for Richey-Chretien at (unrealistically short) wavelength of 0.3 mm. Both the aberration losses and distortions are smaller in RC telescope by an order of magnitude.

Further ray tracing simulations at range of frequencies exhibited quite interesting behaviour of RC design which seems to be really superior to classical solutions. Over a wide span of frequencies it does have smaller aberration losses (those due to phase errors), distortions and sidelobes than Cassegrain.

Fig. 7b: Off axis characteristics of the 90-metre radio telescope computed for classical Cassegrain (CC) and Richey-Chretien (RC) designs as functions of observing frequency. Note the excellent performance of RC above 100 GHz.

Here are more systematic results obtained for three telescopes, the other two having the same optical geometries as RTH90 but scaled up and down by the factor of 10.

Fig. 7c

Fig. 7d

Fig. 7e

Thus, it seems, as we go towards longer wavelengths the RC system gradually loses its effectiveness.

Modified Ritchey-Chretien Design

K2 = (K1 + 1)(D/d)4[R2/(2f)]3 – [(m + 1)/(m – 1)]2.

(8)

Fig. 8. and Fig. 9. give more details on performance of so modified RC telescope. Compare also the HPBW of 1.01 arcminutes here with 1.09 to 1.14 arcminutes of RC and classical Cassegrain in Fig. 6.

Fig. 8: Left: Power pattern of modified Richey-Chretien telescope at best position near focus (31 cm below it), Right: The same for the feed placed 4.5 m off axis. (The quantity 'v2v' is the vertex to vertex distance between the two mirrors.)

Fig. 9: Off axis pattern distortions in modified RC of Fig. 8.

Results presented in this report so far, for speed of computing, were obtained with a small number of rays traced (about 1300) and with low resolution (80 pixels in u/x direction) of final patterns. Here are similar results computed with these parameters being nearly maximum possible in OptiCass presently (over 11000 rays and 200 pixels). In addition, both versions of the program, OptiCass and OptiCassRC, were somewhat improved with respect to initial ray distribution on the aperture. Earlier the rays covered the area up to half their spacing away from dish edge; now they go up to half the spacing beyond the edge (the rays close to the edge are waighted, besides the usual illumination function, by the approximate fraction of circular area around them that remains on the dish). Therefore, these data should be considered more realistic.

The new results are given in the form of Figs. 10. to 15. They all were computed for wavelngth of 2 cm. Each of the four curves correspond to a set of conic parameters detailed in Table 3.

Table 3: Conic Constants of Four Variants of Mirror Curvature of 90-metre RTH

                                K1          K2     R2 [m]  v2v [m]
Classical Cassegrain      -1.00000001 -1.5437704 6.67412 28.52378
Ritchey-Chretien          -1.0023936  -1.5752934 6.63046 28.54096
Modified Ritchey-Chretien -1.0065000  -1.6152331 6.63046 28.54096
Modified Ritchey-Chretien -1.0065000  -1.6118407 6.53000 28.54096

Fig. 10: The height of feed above the secodary (final) focal plane (towards the subreflector) for which the least aberration and spillover losses (shown in Fig. 12) are found while simultaneously adjusting the direction of feed power pattern (Fig. 11).

Fig. 11: Correction of the direction of feed (with respect to direction to the centre or vertex of subreflector) for which the least aberration and spillover losses (shown in Fig. 12) are found.

Fig. 12: Losses of antenna gain as a function of lateral feed offset optimized for z-coordinate and direction as shown in Fig. 10 and Fig. 11.

Fig. 13: Height of comatic distortion as a function of lateral offset at optimal placement and orientation of feed (see Fig. 10 and Fig. 11).

Fig. 14: Height of first sidelobe as a function of lateral offset at optimal placement and orientation of feed (see Fig. 10 and Fig. 11). The sidelobe is measured along the direction of feed offset on both sides of the main lobe and the higher value is taken. In the case of the two modified RCs the sidelobe on the side further off telescope symmetry axis, as it grows with incerasing offset, it gradually merges in the telescope main beam to disappear altogether, so that there is a downward jump and the values shown at greater offsets represent level of the second sidelobe.

Fig. 15: Half-power beamwidth as a function of lateral offset at optimal placement and orientation of feed (see Fig. 10 and Fig. 11).

What About Anastigmatic Aplanat?

K1 + 1 = 4 (m2 + β) (1 + β)/[m (m – β)]2

(9)

Ray tracing simulations indicate that the gain losses for such a design and for feed offsets smaller than about 3 m are slightly smaller than with heretofore analysed solutions, and similar or somewhat worse above this offset (although still better than in the case of CC). However, see note in the last paragraph about insufficiency of these simulations.

According to theory the extent of angular coma (ASC or sagittal coma, to be precise) and angular astigmatism (AAS) in a two mirror telescope whose mirrors fulfill the condition of absence of spherical aberration, are as follows (see Tab. 6.7 of Schroeder):

ASC = [1 + (K1 + 1)m2(m – β)/(2(1 + β))]Θ/(4 m f/D)2

(10)

AAS = [(m2 + β)/(m(1 + β)) – (K1 + 1)m(m – β)2/(4(1 + β)2)]Θ2/(2 m f/D)

(11)

These functions for RTH geometry are plotted in Fig. 16. for a range of the angle and the K₁ constant.

Fig. 16: Coma (left) and astigmatism (right) as a function of the angular offset (the Θ range is 0 to 2 deg and it corresponds to about 0 to 5 m of feed lateral offset) and of the primary conic constant, K1, for 90-metre radio telescope. Coma zeroes a little below K1 = –1.002 and astigmatism zeroes at about K1 = –0.95. Note that where astigmatism is zero coma is about twice as big as astigmatism is at coma's zero.

Since, as evidenced by this figure, theoretical coma increases faster towards K₁ for which astigmatism gets to zero than astigmatism decreases, one should not expect improvement of RTH telescope properties of anastigmat aplanat type. However, the above mentioned ray tracing simulations suggest that there is certain improvement. Unfortunately, these simulations cannot be taken as conclusive since geometry of such aplanat is too different from classical Cassegrain (the main mirror is not completely illumined by subreflector). Although we believe the results from simulations are not far from truth, this type of design would need further study to be seriously considered as a candidate for RTH.