◄ 10.2.3.3. Maksutov-Cassegrain   ▐    10.2.3.5. MCT off-axis aberrations ►
 

10.2.6.4. Maksutov-Cassegrain aberrations: axial

Aberrations of a Maksutov-Cassegrain system are a sum of the aberrations of the meniscus corrector and the two mirrors. In that sense, they are no different than system aberrations of a Schmidt-Cassegrain, given with Eq. 113-119, with the only difference coming from the aberrations induced by a corrector. System aberration coefficients for two mirror systems with meniscus corrector are found from the sum of the coefficients for the three system elements. The main difference, as already mentioned, is that higher-order spherical aberration (HSA) is significantly greater with the meniscus-type corrector, making the third-order expressions for spherical aberration more approximate even with relatively slow mirrors. Designing a two-mirror Maksutov-Cassegrain is considerably more calculation extensive also due to the relatively significant corrector power.

Lower-order spherical aberration (LSA) of a two-mirror system with full aperture Maksutov-type corrector is a sum of spherical aberration contributions of the corrector and two mirrors. Thus, the system aberration coefficient can be written as s=sL+sP+sS, with sL, sP and sS being the aberration coefficients for the meniscus, primary and secondary mirror, respectively. The system wavefront error is given as W=sD4/64 as the best focus P-V wavefront error (D being the aperture diameter).

Lower-order spherical aberration aberration coefficient for the meniscus is, according to Eq. 121.1, approximated by

sL~-(n+2)/8nƒLR12,

with the meniscus focal length ƒL=-n2R1R2/(n-1)2t, n being the glass refractive index, R1 and R2 the front and back surface radius and t the lens center thickness. Since the lens focal length ƒL is numerically negative, the meniscus aberration coefficient is numerically positive, expressing over-correction.

For the primary, the lower-order aberration coefficient is given by

sP=(1+K1)a'4/4R3,

with K1 being the primary conic and σ the corrector-to-primary separation in units of the primary's radius of curvature RP. The fourth power factor a'=1+(σRP/ƒL)+[(n-1)t/nR1], with σ being the primary-to-corrector separation in units of the primary's radius of curvature - shows dependence of the primary aberration contribution on the corrector location vs. primary. The farther, and/or the thicker corrector, and/or the stronger corrector's radii, the larger primary diameter needed to grasp diverging cone of light, and the larger its aberration contribution (FIG. 188). This factor is neglected in calculations for the single-mirror system (Eq. 122-125), because it partly offsets with the factors neglected in Eq. 121.1, thus making the approximation more accurate. It will be included in the following consideration.

It should be noted that the purpose of going through the Maksutov two-mirror system aberration coefficients is primarily to show how it generates the aberrations, not to have them precisely calculated; as already mentioned, this can't be done by calculating the third-order aberrations alone.

For spherical primary K1=0 and the aberration coefficient is approximated by sP~a'4/4R3. It is numerically negative (under-corrected), opposite in sign to the lens aberration.

For the secondary, the aberration coefficient expression is identical to that for the SCT secondary (Eq. 113.1). However, due to the relatively significant negative power of Maksutov corrector, all three parameters, height of the marginal ray at the secondary k (in units of the aperture radius), secondary radius of curvature in units of the (effective) primary's r.o.c. ρ and secondary magnification m, have changed with respect to those for the two mirrors alone. The increased height of the marginal ray at the primary, coupled with the slight reduction in its effective relative aperture, results in the increased height of the marginal ray at the secondary as well, and larger k. Due to the primary focal length being effectively reduced by a factor of ~[1-(σR/ƒL)], ρ is also larger, as well as the effective secondary magnification m. Denoting the changed parameters as k', r' and m', the system aberration coefficient for spherical aberration can be approximated as:

with the first, second and third bracket quantity being the aberration contribution of the meniscus, primary and secondary mirror, respectively, and R1, RP and RS the radii of curvature of the first meniscus surface, primary and secondary, respectively. The relation is for spherical surfaces; conics can be added, as given for the SCT primary and secondary. Contributions of the meniscus and the secondary are of the same sign (positive), and opposite to that of the primary. For given primary mirror, the system needs to be configured so that its contribution is cancelled by those of the meniscus and secondary. With R1/R2~1, the meniscus' focal length is approximated by ƒL~(nR1)2/(n-1)2t, and needed first meniscus radius - after substituting ρ'RP for RS - can be approximated as:

The extraction is not straightforward, as all three of the secondary's parameters are affected by the corrector's power, an unknown beforehand. Probably the simplest initial configuration can be arrived at by starting out with the primary and meniscus alone. Taking again R1/R2~1 and ƒL~(nR1)2/(n-1)2t, the meniscus lens focal length ƒL is approximated in units of the primary radius of curvature as

ƒL~[n(n+2)RP3/2t]1/2/(n-1)a'2.

For n~1.5 and an average value for a'2 of ~1.1, it gives the meniscus focal length ƒL~3(RP3/t)1/2, which in turn gives the appropriate front lens surface radius as R12~3(n-1)2(tRP3)1/2/n2 or, for n~1.5, R12~(tRP3)1/2/3. Rear corrector radius for best chromatic correction is, as before, R2=R1-[1-(1/n2)]t/0.97, t being the corrector center thickness.

This gives first approximation of the meniscus radii needed to correct primary's spherical aberration. Adding the secondary offsets more of the aberration of the primary, requiring somewhat weaker corrector for the system. The degree of weakening can be approximated, most simply, by neglecting the corrector for a moment, and figuring out the aberration coefficient for secondary mirror that would be produced by a system similar to the intended MCT in regard to the relative aperture, obstruction ratio etc. (the level of aberration will increase on both mirrors after the introduction of meniscus lens, but the proportion will not change significantly). The resulting ratio of the secondary to primary aberration contribution, given by s'~sS/sP (Eq. 154) is the ratio of reduction in the corrector's contribution. Since the aberration contribution of the corrector is (approximately) inversely proportional to the forth power of R1, the above value for the front lens radius needs to be increased by a factor of [1/(1+s')]1/4.

The above procedure would result in the first outline of a system, most likely not yet at the optimization level. The main reason is that the meniscus radii approximation would nearly cancel lower-order spherical, which leaves higher-order spherical aberration - significant with faster mirrors - unbalanced. Thus, radii adjustments are needed to produce usable system. Since it is normally the corrector's properties that need to be adjusted in the first place, the calculation is complicated due to the changes at the corrector causing changes in all other parameters down the optical train: a', k, ρ and m, hence in the aberration contribution of the two mirrors as well. In order to maintain needed geometric properties of the system, effects induced by the adjustments of the corrector's power need to be compensated for with the appropriate changes in mirror separation. Obviously, the two-mirror Maksutov system is more complicated than single-mirror system, and warrants even more the use of ray tracing software for system optimization and final verification.

Still, MCT system approximation that usually need only minor optimization can be determined using empirical relation given for Maksutov camera. Needed front corrector radius for the primary mirror alone is determined from Eq. 126 and Eq. 128 (slightly modified Eq. 128, as R1~[1-2τ-0.01F)R1" should be better suited for faster MCT primaries). This value is then corrected for the aberration contribution of the secondary mirror, by multiplying it with [1/(1+s')]1/4 - noting that s' is according to Eq. 154 numerically negative - and the rear radius value is then obtained from Eq. 128.1. It can be all summarized in an initial approximation for the corrector surface radii given as:

R1~R(1-2τ-0.01F)[τ(n+2)(n-1)2/2n3(1+s')]1/4, and R2=R1-{[1-(1/n2)]t/0.97}

As shown in the section on off-axis aberrations, coma and astigmatism are inherently low in a typical MCT system (k~0.25, m~4) with separated spherical secondary.

In the arrangement with an aluminized spot on the back of corrector as the secondary (Gregory-style), R2=RS and R1=R2-[1-(1/n2)]/0.97. Thus, changes in the radius affect similarly both, corrector's and contribution of the secondary's: they either raise or fall in a similar proportion. The system geometric properties are also maintained by manipulating the corrector/secondary-primary spacing.
 

An interesting aspect of the commercial Maksutov-Cassegrain is the question of its star test. There is a notion that its optics has special properties, making it sort of exception in that its intra and extra focal pattern are not supposed to be identical, even when it is near perfectly corrected. Or, put somewhat differently, that it doesn't need to have near-perfect star test for near-perfect performance.

The answer to this special status is in its higher order spherical aberration. Due to its steeply curved optical surfaces, especially those of the meniscus corrector, Maksutov-Cassegrain systems generate 6th-order spherical aberration that can't be cancelled (w/o aspheric surface terms), only minimized by balancing it with the 4th-order aberration. While roughly as much noticeable in the star test as the lower-order spherical aberration for given P-V wavefront error (FIG. 189), the balanced form is considerably less detrimental to image quality.

In an MCT, the higher-order aberration originating mostly at the corrector, needs to be minimized by balancing it with a similar amount of lower-order aberration. Ideally, they are near-optimally balanced one against another, in which case the error reduction factor is about 0.2 (1/5 of the HSA aberration RMS error alone, balanced with nearly identical amount of the opposite in sign LSA). Thus, if an MCT system is, say, 0.95 Strehl, it has ~1/28 waves RMS of combined higher-and lower-order aberrations. In a near perfectly balanced system, each form alone would be over 1/6 wave RMS. Of course, it is possible that the two are less than perfectly balanced, which makes quantifying the star test more difficult, unless the higher-order component is clearly dominant, when it tests similarly as LSA.

The difference between balanced higher-order (BHSA) and lower-order (primary) spherical aberration (LSA), at a similar error level, is that the former affects contrast at lower MTF frequencies somewhat more, and those closer to the mid-range somewhat less than LSA. This is due to BHSA's more steeply curved wavefront edges, causing energy transfer farther away from the Airy disc.

This mysterious property of the seemingly compromised star test combined with an excellent performance level, is also characteristic of other systems with steeply curved optical surfaces, like apo refractors. Schmidt-Cassegrain telescopes also can have the higher-order component present, if the higher-order term is not accurately put on the corrector.

◄ 10.2.3.3. Maksutov-Cassegrain   ▐    10.2.3.5. MCT off-axis aberrations ►

Home  |  Comments