Toward the end of 19th century, catadioptric dialytes were to enlist a new design, whose originality and high level of correction will secure for it a special place and respect up to this day. After years of work on finding solution to the main problem of dialyte objectives - excessive lateral color - German optician Ludwig Schupmann had it finalized in the form of his "medial" (cross between refractor and reflector) telescope. All it took were two seemingly simple steps. The first was to let the front lens focus, and place the second, catadioptric element (CE) in its diverging light. The second step - a stroke of genius - to place an accessory lens at the front lens' focus.

This little lens did the impossible: by forming the image of the front lens in the continuum of optical train, it has brought their chief rays where they needed to be in order to cancel lateral chromatism - to the center of this image. Simply by placing the CE at the location of this image, its exit pupil is back on its surface, and lateral color disappears. At the same time, it also brings back together wavelengths widely scattered by front lens' secondary spectrum. However, that itself doesn't correct secondary spectrum of the front lens, since rays for different wavelengths are still at an angle, and need to exit the last element not only close together, but also nearly parallel in order for secondary spectrum to be cancelled. Since the CE has even more important task of correcting spherical aberration of the front lens, power of the CE needed for one of the two tasks will not coincide with power needed for the other one, except for one particular CE separation, if both elements (the front lens and CE) are made of the the same glass type.

There are several ways to correct this imbalance. They include:

1 - using somewhat different glass type for the two glass elements; for instance, if the front lens is crown, the rear element will need to have somewhat higher index of refraction and lower nominal dispersion (i.e. leaning toward flint, but still only at a fraction of the differential between standard crown and flint)

2 - adjusting the separation and, possibly, radii of the CE, until the secondary spectrum is minimized, if that can be done while still keeping spherical aberration acceptably low

3 - adjust system parameters until the secondary spectrum is corrected, and cancel spherical aberration created in the process by slightly aspherizing concave surface of the CE

4 - combining two or all three of the above; perhaps, using some other adjustments, such as changing a power of the front lens

The chromatism also can be corrected by moving the small concave mirror away from the focal zone of the front lens; however, since all three elements need to be tilted in order to make the image accessible, this adjustment should be kept relatively small, in order to avoid creation of significant aberrations resulting from tilt (tilt aberrations - primarily astigmatism - on the other two elements nearly offset each other; creating tilt errors at the mirror by moving it out of the focal zone would make offsetting more complex, possibly less efficient).

Side-benefit of tilting the elements is that it allows placing the diagonal flat out of the light converging from the CE, for an unobstructed-aperture system.

In any instance, the starting configuration for the Schupmann medial telescope is with the CE focusing right back at the front lens' focus. In such arrangement, spherical aberration and chromatism of the front and rear element are roughly balanced, and relatively small adjustments, mentioned above, are sufficient to fine-tune a system and bring it to a high level of correction.

How does it actually work? After all mystique surrounding it, it may sound all too simple. Neither the small focal zone lens (or mirror, as it doesn't make difference what type of optical element re-images the front lens) nor the reflecting surface, for which its object (image created by the preceding surface) and image nearly coincide at its center of curvature, induce any significant aberrations, on- or off-axis, except for astigmatism induced by the latter. The aberration calculation practically reduces to the front lens (for object at infinity) and the refractive surface of CE (first refraction for object at the front lens' focus and image at the reflective surface's center of curvature, reversed for the second reflection).

In the described setup, the system is to a large degree self compensating. Since refractions at the CE induce trace of coma and significant spherical aberration, the latter needs to be offset by spherical aberration of opposite sign at the front lens. For corrected coma (astigmatism remains low enough not to command special attention), the front lens needs to have minimized coma, and what is left of spherical aberration is to be dealt with through optimization. The system focal length nearly equals that of the front lens.

The simple recipe for the Schupmann medial telescope is, therefore, nearly coma-free front lens of the focal length desired for the system, small concave mirror placed at the lens' focus, to re-image the front lens at a distance usually 1/3 to 1/2 of the front lens' focal length, and catadioptric element whose focal length is nearly 1/2 of that distance, so that it focuses diverging light right back to its focus of origin.

Thin lens imaging object at infinity generates zero coma if its shape factor, defined as q=(R2+R1)/(R2-R1), where R1, R2 are the front and rear surface radius of curvature, respectively, relates to the glass index of refraction n as q=(2n+1)(n-1)/(n+1). Substituting for the front lens' index of refraction n1 gives its needed shape factor q1, in general a biconvex lens with a strong front and weak rear surface radius of curvature. The value of q1 determines the R2/R1 ratio for the front lens as R2=(1+q1)R1/(q1-1), which after substitution in Eq. 1.2 gives R1=2(n1-1)ƒF/(1+q1), ƒF being the front lens' focal length. Spherical aberration of this lens is given by Eq. 50, which  further simplifies with n1 and q1 known.

Small concave mirror (as mentioned, it can also be a positive lens, but mirror produces more compact, practical configuration) placed in the focal plane of the front lens only needs to fulfill one requirement, to re-image the front lens - at which all three, the aperture stop, entrance pupil of the system and exit pupil for the mirror coincide - at an arbitrary distance L. Since its object is the lens at a separation equal to -ƒF, needed mirror focal length is, from Eq. 1.4, given by ƒM=LƒF/(L+ƒF). Note that all three values here are numerically positive, due to Eq. 1.4 being not coherent with the sign convention; in the standard Cartesian coordinate system all three distances, being measured from right to left, are numerically negative. With this in mind, needed radius of curvature of the concave mirror is RM=-2ƒM.

With lateral chromatism reduced to negligible at the exit pupil formed by the mirror, with near-zero coma and also negligible astigmatism induced by the front lens, the role of the optical element placed at mirror's exit pupil is to correct lens' spherical aberration and remaining secondary spectrum, while inducing no other appreciable aberrations of its own. Turned out, a job tailored for a negative meniscus with its inner concave surface made reflective: the catadioptric element. There is no strict requirement that the CE reflects light exactly back to the front lens' focus, but since nothing is gained if it doesn't, and convenience of the symmetry is lost, the concept of the two foci nearly coinciding is usually most practical. For the two foci to nearly coincide, the rear (reflecting) surface needs to reflect light right back where it apparently came from, and that is toward virtual image produced by the refracting surface in front of it. In other words, center of curvature of the reflecting surface needs to coincide with the virtual image. This also cancels spherical aberration and coma at the reflecting surface, leaving in only its low astigmatism.

Thus, we have a fourth surface - the front, refracting surface of the CE - placed at an arbitrary distance L from the mirror in the focus zone of the front lens, that needs to offset spherical aberration generated by the front lens, while forming a virtual image that coincides with the center of curvature of the reflecting lens surface following it. For the first task, its aberration coefficient for lower-order spherical for the front CE surface,

needs to have such value that 2sBdC4=-sFdF4, where n2 is the refractive index of the rear element, I3 the separation from the front CE surface to the image it forms (equal to the radius of curvature of the rear, reflecting surface R4 minus rear element's thickness), R3 is the third surface radius of curvature, dF, dC are the effective aperture radii for the front lens and catadioptric element, respectively, and sF the 3rd order aberration coefficient for spherical aberration at the front lens (Eq. 50). The coefficient is doubled because the aberration contribution from the second refraction at this surface is nearly identical when the light is reflected back at the same angle, thus doubling the total of aberration.

Note that the mirror to CE separation L is, according to the sign convention, numerically negative, while both CE radii and the R3 surface image separation I3 are all numerically positive.

Unfortunately, there is no simple way to express needed 3rd surface radius R3 in terms of other parameters. Since both, 3rd surface radius and its image separation are needed to calculate the aberration coefficient, process of iteration would require to calculate the appropriate value I3=n2LR3/[(n2-1)L-R3] for every value of R3, until the values for which the two coefficients are nearly identical are found. The sum residual of the two coefficients, Δs=sBdC4+sFdF4, determines the system P-V wavefront error of spherical aberration at best focus as W=Δs/4.

The final step in designing Schupmann is making the image accessible by tilting all three elements. Small concave mirror typically needs 4 to 5 degree tilt, catadioptric element about half as much, and front lens a fraction of it. Tilting the mirror doesn't induce appreciable aberrations, but tilting the CE does. This is why the front lens needs to be tilted as well, in the plane perpendicular to CE's tilt, in order to offset astigmatism induced by tilting the CE. If done correctly, tilting doesn't appreciably affect the mid-field aberration level. It does, however, increase off-axis astigmatism, and produces asymmetric, tilted image field typical for tilted systems. It also presents additional difficulty in achieving and maintaining proper system alignment.