A Reflective Telescope with an Unusual Freeform Aspheric Surface

David ShaferDavid Shafer Optical DesignFairfield, CT. 06824#203-259-1431shaferlens@sbcglobal.net

The Schmidt telescope has excellent aberration correction, on a curved image surface, but is long and requires a large size glass element for the front corrector plate. In can be made unobscured (by the image sensor) by using a decentered entrance pupil that is all to one side of the optical axis, as shown here.

Aspheric plate at center of curvature of spherical mirror

Optical axis image

Spherical mirror

Aperture stop

f/2.5

Suppose we fold the light path at the image and collimate the rays with a second spherical mirror M3, producing a beam that is now only ½ as big as the entrance pupil. Suppose also that the fold mirror M2 at the image is convex and has power that reimages the entrance pupil to be at the center of curvature of M3. Then neither M1 nor M3 has coma or astigmatism, because of the Schmidt principle. It can then be shown that this 3 mirror afocal system is also corrected for Petzval.The aspheric front plate has been

cut in half and only the top part is shown, in this unobscured system.

Exit pupil and center of curvature of M3

M3

M1

M2

Aperture stop

Now we are going to do something odd. The front aspheric plate can correct for the combined spherical aberration of mirrors M1 and M3. But we are going to do this in two parts. Suppose that instead of a single aspheric on the front plate we put one on the front and one on the back surface. And then let’s make the front aspheric only correct for mirror M1 and the back side aspheric only correct for M3.

M1

M2M3

Front aspheric only corrects for M1, back aspheric only corrects for M3

Both aspherics have the same vertex, on the optical axis

Instead of a single aspheric correcting for M1 and M3 together we separate it into two independent parts

The next step is to add another spherical mirror M4, with its center of curvature at the collimated exit pupil P from mirrors M1, M2, and M3. And an off-axis segment of another aspheric plate is placed at the exit pupil P position to correct for the spherical aberration of M4. The result is a system correct for spherical aberration and with no coma or astigmatism due to the Schmidt principle = M1, M3, and M4 all have a pupil at their center of curvature. All the components have a common optical axis. The final image is in a very inconvenient location.

All three concave mirrors have their own separate aspheric surface at their centers of curvature – two on either side of the large front aspheric plate and one on the small aspheric plate at P.

M1

M2

M3M4

P

Aspheric plate at aperture stop

Pupil and an aspheric plate

image

The next step is to shift the optical axis of the aspheric plate at P downwards and to also shift the mirror M4 downwards by the same amount so that its center of curvature stays on the optical axis of that aspheric plate at P. Then the final image is in a good location and is much more accessible.

imageOptical axis of aspheric plate

Now the aspheric plate at P and M4 have a separate optical axis from the large aspheric plate and mirrors M1, M2, and M3

P

M4

The next step is to replace the aspheric plate at P with an aspheric flat mirror Mp and fold the light path. We also tilt this mirror Mp so that M4 and M3 do not bump into each other.

Mp

M3

M4Image

Pupil P

f/2.5

The next step is a small one. The region around the mirror M2 is pretty tight, especially at the bottom edge of M2, with rays coming closer there than they do near the top edge of M2. So we tilt M3 to provide more clearance there. That drops its center of curvature down some and so we shift the aspheric axis of its aspheric on the large front aspheric plate downwards to compensate.

M2M3Mp

Remember that the aspheric on the front of the large aspheric plate corrects for M1 while the aspheric on the back side corrects for M3

increased

M1

M4image

Mp

Now we are ready for the really sexy part of this design development. The front plate has two aspherics on opposite sides and they no longer have the same axis. Both are imaged by M1, M2, and M3 onto Mp. So we transfer all of that front plate asphericity onto Mp, where it is imaged. Then we remove the large front aspheric plate.

Two axis aspheric plate

Now we have no front aspheric plate and all of its asphericity has been transferred to its image on Mp.The three aspherics in the system before now are all added together on Mp and the result is Mp is a freeform surface. But its asphericity is just the sum of three separate ordinary asphericdeformations that are overlaid on top of each other with shifted axes.

The final result is an unobscured reflective design with four spherical mirrors and one freeform aspheric mirror, ½ the size of the entrance pupil.

Because of the aberration theory used here we know that correction will be good since M1, M3, and M4 all have a pupil at their center of curvature. The freeform mirror corrects their combined spherical aberration. The design can now be computer optimized to make improvements, like moving M2 further away from the intermediate image

This can now be the aperture stop. There is no longer any need for an actual stop out at the center of curvature of M1

For a 200 mm diameter entrance aperture to the unobscured f/2.5 telescope the freeform aspheric mirror is ½ as large, or 100 mm. If it is tested at .6328u, in reflection, this is the wavefront contours with power removed. It is mostly astigmatism. If astigmatism is removed then this is what is left.

In summary, an unusual type of freeform aspheric surface has been described as well as its use in a design, where the freeform surface deformation can be decomposed into being the sum of two or more rotationally symmetric surface deformations with optical axes that are shifted to respect to each other. This unusual kind of surface arose by using aberration theory to construct a well corrected design. Once in place, however, there is no reason to not vary all the usual freeform aspheric coefficients as well as the other mirror radii and separations and let the design evolve into an even better corrected design. The main point of this design is to show how aberration theory can be used to develop new design types.