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◄ 10.2. Sub-aperture corrector examples   ▐    11.2. Schmidt camera: aberrations ►

 

                           11. CATADIOPTRIC TELESCOPES WITH FULL-APERTURE CORRECTORS

Unlike sub-aperture catadioptric telescopes, combinations with spherical primary mirror are the most frequent form in arrangements with full-aperture corrector. They are particularly attractive, both, due to the ease of mirror fabrication and the possibility of influencing coma of a spherical mirror by the stop position - something that sub-aperture corrector catadioptrics can't take advantage of. The corrector can be used with either a single-mirror (Newtonian or camera) or with two-mirror (usually Cassegrain) arrangements. While there is a number of possible forms of the full-aperture corrector, the three types most often used for amateur telescopes are Schmidt corrector, Maksutov meniscus corrector and Houghton two-element corrector. The most common configuration are catadioptric Newtonian - Schmidt-Newtonian (SN), Maksutov-Newtonian (MN) and (quite rare) Houghton-Newtonian - as well as catadioptric Cassegrain systems: Schmidt-Cassegrain (SCT), Maksutov-Cassegrain (MCT) and (never saw one) Houghton-Cassegrain telescope (HCT).

Performance level achievable, as well as ease of manufacture, differ somewhat from one to another.

However, the simplest arrangement using full-aperture corrector is a camera, with the only other optical element required being a single concave mirror. By far the most popular arrangement is the Schmidt camera. A more compact alternative using full-aperture Schmidt corrector is the Wright camera. It can't compare to the Schmidt camera in the field quality department, but does have some other advantages.

11.1. Full-aperture Schmidt corrector: Schmidt camera

Back in 1930, Estonian-born optician Bernhard Schmidt succeeded in designing and making a full-aperture corrector for the spherical mirror. It resulted in a highly corrected optical system, known as Schmidt camera. Somewhat earlier, in 1924, Finnish astronomer Y. Väisälä, described similar arrangement, so this type of camera is sometimes called Schmidt-Väisälä (usually when incorporating field-flattener). Its concept is based on the unique property of a spherical mirror with the aperture stop at the center of curvature to be free from off-axis aberrations. The only image aberration remaining is spherical, and it can be cancelled by the appropriately figured lens corrector, placed at the mirror center of curvature. The lens element - called Schmidt corrector - has a very shallow aspheric curve calculated to give to the incoming wavefront just needed amount of deformation to result in a spherical reflected wavefront (FIG. 83). The only significant aberration induced by the Schmidt corrector is its corrective spherical aberration, resulting in the system free of four primary aberrations: spherical, coma, astigmatism and distortion. The only remaining aberration is field curvature.


 FIGURE 83: Schmidt corrector at the center of curvature of spherical mirror. The only remaining point-image aberration for a  sphere with the stop at its center of curvature is spherical aberration. It results from the aberrated wavefront Wa formed by the mirror alone. Role of the Schmidt corrector is to transform the incident flat wavefront W into W', so that after reflection from the mirror it becomes spherical (Wc), directing rays to a single point. In effect, the lens compensates for the optical path difference created at the mirror surface. Knowing that sphere produces wavefront that advances away from spherical at a rate of (ρd)4/4R3 with respect to the reference sphere centered at paraxial focus, needed Schmidt surface depth to have the wavefront corrected for paraxial focus is determined by z=(ρd)4/4(n-1)R3, adding the compensatory optical path length (n-1)z=(ρd)4/4R3. Other reference spheres have different rates of deviation, thus also different depth equation needed for the correction, according to a general relation z=(ρ4-Λρ)d4/4(n-1)R3. On the above illustration, the corrector is thinnest at the 0.707 zone, widening toward the edge and center. As a result, the wavefront is delayed most at the edge and the center, with the relative advance greatest at the 0.707 zone. Since the wavefront is relatively unchanged only at the .707 zone, rays from that wavefront point will be, after reflection, directed to its previous location - the mid point of the defocus L - as well as all other rays from now spherical wavefront (for instance, central wave is delayed just as much to meet the .707 wave in phase at the point of its intersection with the axis, and for the marginal wave, the same amount of delay has created the possibility to follow a slightly diverging path to the mirror surface, to the point from which it will be re-emitted toward the same focus, by arriving at the reference sphere centered at the mid focus at the same moment in which the central wave arrives at the mirror center). To the left, four out of many possible shapes of the Schmidt corrector surface (back side only), greatly exaggerated. From left to right, the neutral zone (NZ) - a zero slope zone with zero refraction and zero relative wavefront retardation (marked red) - is at ρ=1, √0.75, √0.5 and 0 of the pupil radius d normalized to ρ=1. Relative depth of the corrector curve, ρ4- Λρ2, as well as its shape - varies with the parameter 0 ≤ Λ ≤ 2, which determines the corrected focus location within the longitudinal mirror defocus L normalized to 2 (zero being at the paraxial focus). The parameter Λ determines NZ position at the corrector plate radius in terms of ρ as √Λ/2. Image surface Img is a curved Petzval surface, concentric with the mirror surface, thus of Rc=R/2 curvature radius.

There is a direct parallel between the Schmidt curve and parabolizing. The most efficient mirror parabolizing method is working the center and the edges of a sphere the most, gradually reducing glass removal to a minimum at the 0.707 zone. This surface modification causes relative advance of the wavefront that culminates at the 0.707 zone, and diminishes to zero at the edge and the center, resulting in a corrected, spherical shape of the wavefront. This same wavefront modification is accomplished by placing the neutral zone at 0.707 radius of the Schmidt corrector (in fact, the curve of change of a spherical surface in parabolizing is of the same type as the curve polished into a Schmidt corrector, only shallower). Consequently, for a given mirror, the theoretical maximum thickness of glass needed to be removed from mirror center and the edge, when parabolizing, is smaller by a factor of (n-1)/2 from the (maximum) Schmidt corrector depth at the 0.707 zone. This holds true for any corrector/parabola pair with identical final focus location.

The Schmidt corrector curve radial profile (depth variation) is determined by:

                                                                   z=(r4-Λr2)d4/4(n-1)R3                                  (101)

with Λ being the arbitrary parameter determining neutral zone position, r the height in the pupil normalized to 1, d the pupil (aperture) radius, n the glass index of refraction and R the mirror radius of curvature. Alternatively, the curve depth can be expressed as:

      z=[(rd)2/2Rc]+ [b(rd)4/8(n'-n)] + b'(rd)6/16(n'-n) =
                                           = [(rd)2/2Rc]+ A1(rd)4 + A2(rd)6                           (101.1)

with Rc being the corrector radius of curvature, b and b' the third- and fifth-order aspheric coefficient (for the ray aberration; fourth- and sixth-order for the wavefront), and n' the index of refraction of the exit media (media to the right of the Schmidt surface, normally air, with n'=1).

As these relations show, depth of the corrector curve is smallest with the neutral zone placed at 0.707 radius (0.866 radius neutral zone placement requires corrector deeper by a factor of 2.25). This neutral zone position - as it will be explained in more details ahead - also results in minimized spherochromatism.

The corrector radius of curvature is given by:

                                                               Rc=4(n'-n)/Λbd2 =1/2ΛA1d2                             (102)

with the third-order aspheric coefficient b=2/R3, and n'=1 for the aspheric surface on the back of corrector. Optimized for the small effect of corrector's radius of curvature,

                                                                     b=2(1-Λ/16F2)/R3                                      (103)

with F being the mirror F-number (f.l./D). The fifth-order aspheric coefficient b'=6/R5. The two aspheric coefficients  A1 and  A2 determine the Schmidt corrector shape, according to Eq. 101.1. The aspheric coefficients are obtained from the aberration coefficients for third- and fifth-order spherical aberration of the system set to zero,

                                                              s3=-b/8 + (1-Λ/16F2)/4R3 = 0                              (104)  and

                                                                    s5=-b'/16 + 3/8R5 = 0                                     (104.1)

with the left side of the coefficient (b factor) being the corrector aberration contribution, and the right side that of the mirror (the slightly lower third-order mirror coefficient results from its effective aperture slightly reduced due to the ray divergence created by the corrector). The third- and fifth-order system P-V wavefront error at the paraxial focus are W3=s3d4 and W5=s5d6, respectively.

The significance of the fifth-order term is in correction of higher (fifth-order ray, sixth order wavefront) aberrations. Those include axial spherical, as well as oblique (lateral) spherical, and wings, the higher-order astigmatism as it was named by Schwarzschild. They both increase with the square of off-axis height in the image space, and set the limit to field quality. The latter has the P-V error larger by a factor of 4n, n being the glass refractive; since it varies with cosθ, θ being the pupil angle, the off-axis aberration in the Schmidt camera peak along the tangential plane (the one determined by the chief ray and optical axis, for which θ=0). 

For a 200mm f/2 Schmidt camera, the amount of higher-order axial spherical aberration that can't be corrected with the third-order surface term alone is ~1/4 wave P-V.

Follows more detailed account of aberrations of the Schmidt camera.


 ◄ 10.2. Sub-aperture corrector examples   ▐    11.2. Schmidt camera: aberrations ►

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