By Dave Stevick

http://bhs.broo.k12.wv.us/homepage/alumni/dstevick/stevpaul.htm

As a design with inherent third-order optical perfection, this arrangement of mirrors has been rediscovered numerous times since M. Paul's 1935 paper. Despite its superficial resemblance to a TCT, all mirrors are coaxial. The unobstructed view comes from using the telescope off-axis.

Using a flat to fold the light path, Bob Novack of Pittsburgh, Pa compacted the design and eliminated the diagonal mirror. Bob's plans for this 6" f/10 are here. | ||

Viewed from the eyepiece end, one notices the two finders on Bob's telescope. The finder on top is unity power. Bob is justly happy and proud of his telescope's performance | ||

A.L. Woods and Dave Stevick with Al's pioneering 6" f/12 Stevick-Paul telescope. The light path painted on the side of the telescope quietly answers the most frequently asked question. |

Internal construction details - *drawing by A.L.Woods*

Unobstructed optical systems offer unparalleled image contrast. Reflectors, while perfectly achromatic, typically have large central obstructions in the optical path. Several means have been used to free the reflector from the image deterioration caused from this central obstruction.

1. An eccentric pupil. This can take the form of an eccentric stop placed at the mouth of the telescope, or a Newtonian can have the primary figured to an off-axis section of a paraboloid. Entire complex systems have been constructed in this manner, usually of all-spherical optics as any other design would be an optician's nightmare.

2. Tilt the primary to throw the image clear of incoming light. This approach is utilized in the Herschelian, the Yolo, the Schiefspiegler, and the Tri-Schiefspiegler telescopes. Tilting the primary introduces large aberrations into the image which must then be balanced out with compensating aberrations from subsequent elements.

3. Use an axially-symmetric telescope off-axis. This approach seldom works well as off-axis aberrations in telescopes are normally quite large to begin with. However, a little-known system by the French designer, Maurice Paul, frees a paraboloid of all third-order aberrations except for a gentle field curvature. This system, used off-axis, yields paraboloidal quality images throughout. The Stevick-Paul telescope would be just another tilted-component telescope were it not for this phenomenal perfection of its image.

The **Stevick-Paul telescope** (SPT) is one of just a few elegant optical
systems and I prefer to think of it as a discovery rather than a design. It has aspects of
the Mersenne telescope (two confocal paraboloids) and also of the Schmidt camera. All
aberrations to the third-order total zero except for the Petzval surface which is gently
curved. The present embodiment is in fact the third time this basic optical arrangement
has been discovered.

The system was first described in 1935 by Maurice Paul who presented it as a zero-power
corrector for large astronomical telescopes. The two spherical mirrors are of mating
curve. Paul showed the concave tertiary in contact with the paraboloidal primary. Because
of the large obstruction of this axial arrangement, Paul considered the system of
theoretical value only.

Dr. J.G.Baker independently rediscovered the Paul system in the 1960s. He moved the two
spherical mirrors farther from the primary which reduces their diameter and obstruction, a
condition allowed for by Paul's equations, but his chief contribution was to add an
approximate fifth-order correction by turning the edge of the convex secondary. Since then
these designs have been known as Baker-Paul three-mirror systems.

In 1991, as my computer program to raytrace tilted-component telescopes (TCTs) was nearing final polish, I began searching for an improved Tri-Schiefspiegler design. By varying radii and spacing I systematically explored a variety of designs. One particular spacing change put parallel light between secondary and tertiary, a condition I felt was all wrong. Just to satisfy myself that this was indeed undesirable I proceeded to optimize the mirror tilts.

The results were not at all remarkable but curiosity led me to explore whether the parallel light would permit spacing changes between the second and third mirrors without the need to re-tilt them. Such was the case, but then I noticed that increases in this space were reducing both coma and astigmatism at the same time. When the sec.-tert. space became equal to the radius of the tertiary, all aberrations disappeared. (I should mention that both the second and third mirrors had by chance been given the same r.o.c.)

Spot diagrams which previously had been large blurs collapsed to single dots, unbelievable! It was a week before I accepted the results as valid, and more agonizing months passed while I assembled a prototype. Satisfied finally that what I had found was workable, I mailed the design to the Amateur Telescope Makers Journal.

Within days Dr. Richard Buchroeder responded confirming my work and pointing out that this telescope was an off-axis form of the Paul system. With his guidance I located Paul's original work and the more recent work of Dr. Baker. Since my design doesn't employ the turned edge of the Baker-Paul systems, Dr. Baker kindly allowed the design to be called the Stevick-Paul telescope.

It should be noted that a prior, independent, unpublished discovery of this system was made by German amateur Michael Brunn while exploring 4-mirror systems (Tetra-Schiefs - U.S. patent no. 5,142,417). Michael unfailingly uses binocular attachments while observing so ignored the system for lack of sufficient back focal length. Binocular attachments require 8 to 10 inches of in-travel for focusing.

Stevick-Paul telescopes are off-axis versions of Paul 3-mirror systems with an added flat diagonal mirror; figure 1 illustrates the configuration. It begins with a paraboloidal primary mirror of the same f-ratio as that desired in the final instrument. The secondary and tertiary mirrors are spherical and of mating curve. The radius of this curve is .4 the radius of curvature of the primary mirror.

The convex secondary mirror is placed just to the side of the light entering the telescope, leaving a little room for a light baffle. It is positioned afocally, so as to send parallel light on to the tertiary.

The concave tertiary mirror is positioned exactly twice as far to the side of the entering beam as was the convex secondary, and its own radius of curvature distant from the secondary. Because the tertiary mirror receives parallel light from the secondary, it forms an image at its focus, just as it would with starlight.

The focal plane lies within the system of mirrors, but is accessible to the eye with the inclusion of a flat diagonal. This fourth mirror does not obstruct light and corrects the reverted image that plagues telescopes having an odd number of reflections. It is angled to throw the image out of the telescope at 90 degrees to the sec.-tert. beam.

Although diffraction limited performance is possible down to about f/5.6, focal plane tilt, which increases with decreasing f-ratio, constrains the design to more moderate f-ratios. I have chosen to restrict my designs to f/10 or longer. This way image anamorphism is kept under 2 percent and focal plane tilt need not exceed 10 degrees.

The prototype telescope is a 10 inch f/6.2 design. This size was chosen because I already had the paraboloidal primary mirror and I wanted to test the system quickly; only two spherical mirrors had to be made. Eyepieces won't tolerate being tilted in the fat f/6 light cone, but it delivers an excellent image if the eyepiece is square. However, this leaves the top and bottom of the field out of focus and is why I recommend more moderate f-ratios be considered. With shallower cones of light the eyepiece can then be tilted to match the focal plane.

I employ a value for the radius of curvature of the spherical mirrors that is .4 that of the primary r.o.c.. This `radius ratio' has been found to be optimum, but telescopes are possible with different radius ratios.

Telescopes with radius ratios less than .4 are more compact, at least down to .33 where increasing pri.-sec. spacing lengthens the instrument again. They have a smaller primary tilt and smaller diameters for the spherical mirrors. However, they suffer a more steeply tilted focal plane.

Telescopes with radius ratios greater than .4 are longer. They have a larger primary tilt and larger diameters for the spherical mirrors. They do, however, enjoy a less steeply tilted focal plane.

By a stroke of good fortune, the eyepiece falls quite close to the balance point in an instrument designed with the preferred ratio of .4. It is in a convenient position for viewing and its arc of travel is short.

The tertiary mirror must be larger than the secondary by twice the clear field. Required minimum diameters of the spherical mirrors are determined with these equations:

[Eqn 1] | Sec. diam. = A*D+(1-A)*d | where: | A = radius ratio |

[Eqn 2] | Tert. diam. = A*D+(3-A)*d | D = primary diameter | |

d = diam. of clear field |

Mirror tilts have a fixed relationship among themselves. The primary tilt determines all the others. In a Stevick-Paul, the paraboloidal primary mirror must have a tilt a bit larger than that required for freedom from obscuration, to allow for an effective light baffle. The equations are as follows:

[Eqn 3] | Pri. tilt: | T1 = C/N | where: | C is selected from the table |

N = pri. f-ratio |

Mirror diam. | C |

4.25" | 37.08 |

6" | 36.24 |

8" | 35.67 |

10" | 35.29 |

12.5" | 34.97 |

16" | 34.67 |

[Eqn 4] | Sec. tilt: | T2 = k*T1 | where: | k = (1+A)/(2*A) |

A = radius ratio | ||||

[Eqn 5] | Tert.tilt: | T3 = T2-T1 |

[Eqn 6] | Focal plane tilt is approximated as 4*T3 |

The last three equations are valid for all possible values of the radius ratio. The numbers for the primary tilt are empirically derived to assure freedom from stray light and are valid for a ratio of .4 only.

The spaces between the surfaces can be found simply as:

[Eqn 7] | pri.-sec. | e1 = F1+F2 | where: | F1 = primary f.l. |

F2 = secondary f.l. | ||||

[Eqn 8] | sec.-tert. | e2 = R3 | R3 = tertiary r.o.c. | |

[Eqn 9] | tert.-f.p. | e3 = F3 | F3 = tertiary f.l. | |

note: | F2 is negative |

The focal plane lies within the system of mirrors and a flat diagonal is needed to throw it clear. The diagonal mirror is positioned in the cone of light converging to the focus, figure 2. If it is placed to catch all the light for a 1 inch diameter field, and at the same time, kept from infringing on any of the field rays between the secondary and tertiary mirrors (upper position), not enough residual cone will exist to get the focal plane free. The only solution to this dilemma seems to be an increase in the tilts of all the mirrors. This would lengthen the available cone of light but would increase the focal plane tilt dramatically.

The solution is to nestle the diagonal mirror up against the central beam of light passing between the secondary and tertiary mirrors (middle position). Therefore, the diagonal intrudes into the light at the lower edge of the field, but is out of the way at the center and the upper part of the field. The diagonal mirror is then slid along the central beam toward the tertiary mirror (lowest position) until it catches only enough of the rays forming the top edge of the field to produce a one-half magnitude loss due to vignetting. These two compromises, acceptable intrusion at the lower part of the field, and acceptable vignetting of the edge of the field, have minimized tilt angles while still providing sufficient free cone to the focuser.

In this design, free cone is more important than residual cone. The length of cone which can be fielded by the diagonal mirror is the residual cone. I define free cone as that part of the residual cone which exists on the observers side of the parallel sec.-tert. beam. I look for at least 4 inches. A low profile focuser would be required if less free cone were available.

A light baffle surrounds the converging cone of light from the primary mirror, fitting into the notches made where the several light paths cross. Trying to keep the baffle out of the road of all field rays would result in the need for higher tilt angles. Following the philosophy of permitting a small loss in illumination at the edge of the field, the baffle is sized to just pass the central beam intact. This means vignetting technically begins immediately away from the center of the field, but because the baffle is so far from the focus, it blocks less than 10 percent of the light at the field edge. Vignetting arising from the placement of the diagonal mirror is far greater than that from the baffle.

With the baffle thus sized, the required primary tilt is not much larger than the theoretical minimum. By raytracing the light that just makes it through the baffle, I look to see that it approaches no closer than 1 inch to the field center. This means a field 2 inches in diameter is free of stray light. Even 2 inch O.D. eyepieces would thus be protected from washout. The table of primary tilts reflects this design philosophy.

Tolerances for spherical aberration are the same in a Newtonian and a Stevick-Paul telescope since the primary f-ratio is preserved in the final system; therefore, any long focus primary mirror that could be left spherical in a Newtonian may also be left spherical in this telescope.

The best way I know to show what is involved in making one of these telescopes is to lead you through the design process. I am currently building an 8" f/12.

The paraboloidal primary mirror should have a focal length of 96 inches, but the grinding and polishing of the primary is already done and I have missed my target focal length by two inches. It is only 94 inches so we are now talking f/11.75. With a primary r.o.c. of 188 inches, the auxiliary mirrors need a curve of 75.2 inches radius. This is the value I will shoot for while grinding.

Starting with equation 3, and selecting the proper value of C from the table, we divide by 11.75, and assign a primary tilt of 3.04 degrees (slightly rounded up). At a radius ratio of .4, k has a value of 1.75. Using equation 4, we find the secondary tilt will be 5.32 degrees. Equation 5 yields a tertiary tilt of 2.28 degrees. The focal plane tilt (Eqn 6) works out to 9.12 degrees, very close to the exact value of 9.07 degrees derived from raytracing. We shall find later that focal plane tilt may be reduced somewhat if we are willing to accept the introduction of a small amount of astigmatism into the field.

Next, we may easily calculate the spaces between surfaces with equations 7, 8, and 9. We mustn't forget to assign a negative focal length to the convex secondary mirror.

The design so far looks like this:

Mirror | Radius | Figure | Tilt | Dist. to next surface |

1 | 188 | Para | 3.04 | 56.4 |

2 | -75.2 | Sph | 5.32 | 75.2 |

3 | 75.2 | Sph | 2.28 | 37.6 |

Our first questions at this point are, "What are the mirror diameters, especially the diagonal, and where does the diagonal mirror go?". Equations 1 and 2, for mirror diameters, are for the clear field and the diagonal mirror was not included there. The design field has a one-half magnitude loss at the edge and cannot be used in these equations.

The diameter of the clear field which results in the prescribed loss at the edge of the design field can be found in the accompanying table. I have chosen to design for a 1 inch diameter field. Smaller telescopes suffer if this value is adhered to rigorously. Therefore, I downsize the design field to 3/4 inch for a 6 inch telescope, and to 1/2 inch for a 4.25 incher. Without this concession, these sizes wouldn't have enough free cone except at higher tilts, or at the longer f-ratios.

Pri. size | Design field diam. | Clear field diam. |

4.25" | 1/2" | .28" |

6" | 3/4" | .45" |

8" | 1" | .62" |

10" | 1" | .50" |

12.5" | 1" | .36" |

16" | 1" | .18" |

These values of clear field should be considered minimums. Larger telescopes (above 8 inches), or ones with long f-ratios may well be capable of a larger clear field (up to as much as the design field), and still yield enough free cone for the focuser.

Right now, let's see what the minimum required mirror diameters are in the 8 inch f/11.75 telescope. Using a clear field diameter of .62 from the table in equations 1 and 2, the answers are:

sec. diam. = 3.572" tert. diam. = 4.812"

Because an oversized secondary mirror may be offset sideways (it is spherical, after all), and because there is plenty of space for an oversized tertiary, I will be using standard 4.25" and 5" blanks.

We now need to know the required size of the diagonal mirror, its proper placement, and the available free cone. The easiest way to determine these is from a full-scale drawing. Such a drawing is necessary anyway for actual construction of the telescope.

All Stevick-Paul telescopes follow simple rules of construction. Refer to figure 3. We already know the distances between the surfaces. We need only calculate a1. This is the first axis distance. It is determined from the primary tilt as:

[Eqn 10] a1 = sin(2*T1)*e1 where: e1 = pri.-sec. space

The second axis distance will be twice the first.

[Eqn 11] a2 = 2*a1

The third axis distance will be one-half the first.

[Eqn 12] a3 = a1/2

It couldn't be simpler.

Using equation 10 for the first axis distance, we find that its value for the 8"
we are designing is 5.9737... inches. This is so close to 6 inches that we shall call it 6
inches even. It is always permissible to round up this figure slightly for constructional
ease. Our axis distances are then:

a1 = 6", a2 = 12", a3 = 3".

We have enough information now to locate the centers of the surfaces on a large sheet of paper. Use the long edge of the paper as a guide; light should enter the telescope parallel to this edge. Mark the center of the primary mirror first, a bit higher up than its semi-aperture from the bottom edge. Draw a line the length of the paper through this point, and parallel to its lower edge. Then, using the known spacings and axis distances, locate the other three centers. Draw the path of the principal ray with a long straight edge, connecting the centers of the four surfaces (You may have already drawn the one between the secondary and tertiary surfaces in order to locate the focus.)

To accurately lay in the light rays of interest, we need to angle the mirror surfaces just as they will be in the final instrument. Bisect the three angles, then with a square or protractor, draw the three mirror surfaces at right angles to these constructs. The focal plane, which for now lies inside the system of mirrors, has no angle to bisect but is quite easy to draw in with a carpenter's square, as it is always oriented perpendicular to the incoming light.

Using a ruler, mark the full diameter of the primary mirror, then place tick marks to locate the diameter of light at the edges of the clear field on the last three surfaces. For the secondary and tertiary mirrors, this is the required minimum diameters for these mirrors. For the focal plane this will be the clear field itself.

Lastly, place another set of tick marks on the secondary and tertiary mirrors that represent the diameter of the central beam of light only. It has a value of A*D and is the same for both of them. For the 8" telescope the value is 3.2 inches.

[Eqn 13] | central beam diam. = A*D | where: | D = pri. diam. |

A = radius ratio |

Taking again our straight edge, we shall lay in some of the light paths. In order to avoid confusion, not all possible light rays will be drawn. Start by drawing in the full beam of light entering the telescope. These two rays will be parallel to the edge of the paper, and of the same diameter as the primary mirror.

The next lines to be drawn will connect the edges of the primary mirror to the corresponding tick marks on the secondary surface marking the diameter of the central beam. At this point we have produced the first V notch marking the location of the top edge of the light baffle.

Starting where we just left off, draw parallel lines from the secondary mirror to the same set of tick marks on the tertiary surface. We can now see the second V notch, inverted from the first, that marks the location of the bottom edge of the light baffle.

Changing now, we pick up the wider set of tick marks on the tertiary mirror and connect them to the marks previously placed in the focal plane indicating the edge of the clear field. Of the two rays we just drew, the upper one intersects the lowest (closest to the primary) of the parallel rays between the secondary and tertiary mirrors right where the top edge of the diagonal mirror is to be placed.

To send light across the parallel beam at right angles to it, the diagonal will have a tilt somewhat less than 45 degrees.

[Eqn 14] diagonal tilt T4 = (45-T3)

Again, in the 8" f/11.75, this mirror will be tilted 42.72 degrees. This optical angle is measured against the principal ray passing through the center of the diagonal mirror.

A protractor can be used to draw in the diagonal but the easiest way is to score around the residual cone with a hobby knife letting the bottom cut extend a bit farther than we think is needed. Fold the flap of paper thus released at right angles across the parallel beam, beginning the fold at the intersection point. Crease the paper once it is properly oriented. We can now measure the free cone available on the eyepiece side of the parallel beam. Now unfold the paper and scribe the diagonal along the crease. The major axis of the diagonal mirror can now be measured. Divide by 1.4 to get the required minor axis diameter.

Some amateurs may opt to send the light out the side panel of the telescope instead of across the parallel beam. This is perfectly permissable. The focuser will then be in close proximity to the altitude axis resulting in a nearly stationary eyepiece. In this configuration the diagonal mirror will be on its side with one side up against the parallel beam. For this, fold the cone of paper straight back on itself instead of across the parallel beam, crease, then measure the required minor axis directly.

We can now finish the specifications of the 8" f/11.75 telescope. Measuring the diagonal mirror from the paper and dividing by 1.4 shows it must have a minor axis of 1.6 inches. We will specify the next larger standard size of 1-5/8 inches. The top of this slightly oversized mirror will be kept against the parallel beam while the rest sticks down. We can also measure the separation of the diagonal and tertiary mirrors along the principal ray. It is 28.75 inches, leaving 8.85 inches of residual cone. Almost 5 inches of free cone is available.

Here then is the finished design.

8 inch | f/11.75 | ||||

Mirror | Diam. | Radius | Figure | Tilt | Dist. to next surface |

pri. | 8 | 188 | para | 3.04 | 56.4 |

sec. | 4.25 | -75.2 | sph | 5.32 | 75.2 |

tert. | 5 | 75.2 | sph | 2.28 | 28.75 |

diag. | 1.63 | flat | flat | 42.72 | 8.85 |

f.p. | 1 | 9.1 |

A Stevick-Paul telescope requires a light baffle to function effectively. Without one, two fields of stars would be seen superimposed on each other. The higher power field is the true one. The one of lower power results from light striking the tertiary directly. A baffle restricts the false field from coming into view.

This baffle may be of one piece, or two piece construction. It should be made of a thin stiff material. If of one piece, the baffle needs an elliptical cutout. This opening can have a much wider minor axis than that necessary to cradle the light. Only the major axis needs to be tight onto the central beam. The vignetting from the baffle is thus restricted to the tangential plane. The sides of the field will be unhampered.

To see just how big the cutout needs to be, draw a line connecting the V notches catercorner across the converging cone of light. Measure its length. That's the required major axis. Now place the ruler square across the principal ray at its intersection with this slanted line. Measure the width of the cone at this location. Make the minor axis of the cutout considerably larger than this value. Bend the baffle ends at the edges of the opening so that it can be mounted without the excess protruding into the optical path.

If the baffle is to be in two pieces, each working edge should be cut to a curve rather than straight across. This way diffraction effects will be minimized when the edges of the baffle intrudes slightly into the field. A two-piecer has the advantage of not needing to be bent at the ends for mounting purposes.

All necessary information to build one of these telescopes has already been presented. However, a few simple improvements to this completely workable design are possible.

The diagonal mirror we drew and measured produces a one-half magnitude loss around the entire edge of the design field. (Usually 1 inch in diameter.) It is obvious when we look at the full-scale diagram that there is room for an oversized mirror. We can take advantage of this to limit the half magnitude of vignetting to just the top of the field by using the next larger size available diagonal. The important thing to remember is to keep the upper edge against the parallel beam and let the rest stick down.

A second improvement is possible in those telescopes with more than enough free cone. The table of clear field diameters allows us a maximum of free cone, but with some compromise as relates to vignetting. Smaller telescopes, under 8", may be extremely tight here and require a low profile focuser. Larger, or longer, instruments may have free cone to spare. When this is the case, we may profitably re-design for a larger clear field, thereby lessening the vignetting.

The last modification we can make will reduce the tilt of the focal plane. This reduction comes with a string attached; we must allow some astigmatism to creep into the field. Image quality at the center is unaffected.

Changes in the sec.-tert. spacing have no effect on the the central image. This rather long space completely eliminates all linear coma and astigmatism. By purposely reducing this distance, we allow the introduction of astigmatism, and concurrently, a small amount of coma into the field. It is the introduced astigmatism that reduces the focal plane tilt. The aberrations resulting from this move will not exceed the Rayleigh tolerance even at the edge of the field for apertures up to 12.5 inches.

Just how far we can go in shortening the instrument can only be found from the full-scale layout, but is approximately 14 per cent. The tertiary and diagonal mirrors must move together as a unit. They are slid along the parallel beam toward the secondary. No angles are changed. The limit is reached when the diagonal mirror threatens to infringe on the light between the primary and secondary mirrors. Reduction of focal plane tilt is typically 7 per cent for a 14 per cent shortening.

In some ways the Stevick-Paul telescope defies optical logic. Unlike other unobstructed telescopes, this instrument shouldn't be scaled down from a larger design; stray light could enter the field. It scales up with impunity however.

Don't worry about making it too large; aperture limits are unreal. At f/10 for instance, a 160" aperture telescope is easily possible, albeit not practical. At f/15, 800" is the limit. At f/20, one-twelfth of a mile.

Established principles say it is better to err on the long side with radii. However, in a Stevick-Paul, it is better to err on the short side while grinding the auxiliary spherical mirrors. This results in better baffling and a bit more free cone. On the other hand, gross error on the long side may cause stray light to enter the field, requiring an increase in the tilts.

Mismatches in radii of the two spherical mirrors, such as may creep in during polishing, are of no consequence. More important is a departure from the target radius which will require a slight adjustment of the spacings. Shift the secondary mirror slightly to preserve parallelism of light after its exact focal length is known [Eqn 7]. Likewise, space the tertiary mirror to its value of radius of curvature [Eqn 8]. In addition, if you've erred more than 1 percent on the long side with the spherical mirrors, assign a larger primary tilt, increasing it by a percentage equal to the increase in r.o.c.. This will keep stray light out of the field.

Should you find that your contemplated telescope has insufficient free cone to suit your needs, there are three ways to improve it. Increase the aperture, the f-ratio, or the tilts (axis distances). The last method should be reserved for 4.25" telescopes which should then be designed at f/12 or above to keep focal plane tilt to acceptable levels.

Here's a summary of the steps already covered.

1. Choose an aperture and f-ratio. (approx. f/10 minimum is recommended) Make the paraboloidal primary mirror. You may already have a long focus mirror you wish to use.

2. Based on the r.o.c. of the finished primary mirror, make the mating spherical mirrors. Shoot for a radius of .4*R1. Choose blanks to fit your criterion of clear field. Next larger standard size is O.K.

3. When the r.o.c. of the finished spherical mirrors is known exactly, calculate the spacings between surfaces and the axis distances. Make a full-scale drawing.

4. Incorporate any changes you wish to make, then measure the diagonal mirror. Don't forget to divide by 1.4 to find the required minor axis diameter. Larger sizes are O.K. to use. Be certain you have enough free cone for the focuser.

Making the paraboloidal primary mirror is straightforward and I won't deal with that here. The two spherical mirrors should be ground together to maintain matching radii. The finished tertiary may then serve as the test plate for the secondary.

Most telescopes will need different diameter blanks for the secondary and tertiary mirrors. The working of unequal size disks is covered in the literature and may offer no real problems. Tools are necessary for the pitch laps and should be curved roughly to match the mirrors. These can be made of wood, plaster, cement, glass, or whatever the ATM prefers. I used glass tools and ground them to the same radius of curvature as the mirrors. During the fining process, I periodically ground each mirror against its own tool. By this occasional working of the same size disks together, I was certain of maintaining a spherical curve.

The two mirrors should come through fine grinding with the same radius of curvature. If these curves maintain themselves through polishing, then interference testing of the convex secondary is possible by using the tertiary as a test plate. I ended up with 30 fringes of concavity. Frustrated, I returned to grinding the secondary mirror on its tool in the hopes of shortening the radius of curvature enough to compensate. This left me with about the same number of fringes of convexity.

Defeated in my efforts to test the secondary mirror by interference, I searched for an alternative. The test I eventually devised for the convex surface may be of interest to you if your mirrors suffer a like fate (see footnote).

We will consider now some of the practicalities of construction. With four reflections, some or all of the mirrors should have enhanced coatings. The prototype doesn't and the light loss is more noticeable than I expected.

Housing of your finished mirrors will call for ingenuity as no best method has yet been determined. Because the prototype was stubby, I opted for a simple box. Stevick-Paul telescopes most closely resemble classical Schiefspieglers, so a long tube with the primary mirror slung underneath may suit. You may also, if you wish, pattern your construction after that used for Tri-Schiefspiegler telescopes. Bearing in mind that none of these have exactly the shape of a Stevick-Paul, valuable source material is still to be had in Telescope Making #4, #16, #28, and Sky & Telescope for December 1969 and February 1975. Here again, this design is new and you are on your own.

Mirror cells that permit adjustment in two orthogonal planes are desirable; see the article by Bob Cox in T.M.#6. We know that the sec.-tert. spacing is quite immune to even large changes in its dimension, but this is not true of the pri.-sec. space. This spacing must be quite close to specifications to preserve the parallelism of the sec.-tert. beam. Therefore, either the primary or the secondary cell must be adjustable longitudinally. I chose to have the primary mirror mounted on spring loaded bolts so that it could be moved back or forth by tightening or loosening all the nuts.

Collimation of the finished instrument will perhaps be easier than with the TCT types. This is because there are no moves that preferentially affect the astigmatism or the coma. When one is right, both are right. You may sight through the telescope from either end and adjust the mirrors one at a time to place the circle of light where it belongs on each element. Do this in the daylight, or if at night, set a flashlight on the eyepiece holder.

When the eye is centered, the beam between the secondary and tertiary mirrors will just graze the diagonal. If the diagonal mirror is oversize, its active area will be somewhat above center. If the secondary mirror is oversize, don't forget to allow for its offset. Light in the primary and tertiary mirrors will be centered.

Having angled the mirrors to get a rough collimation, we must now get the correct pri.-sec. space. It can be set exactly with the following procedure.

Remove the secondary mirror and its mounting plate. Use the tertiary and diagonal mirrors as a Herschelian, to focus on starlight coming through the hole where the secondary used to be. Replace the secondary mirror and check the focus. Shift the primary mirror as needed to put the focus in the same position.

Astigmatism is the most easily recognized aberration. It results in elliptical disks inside and outside of focus. At first you will likely see focal lines. These should be either vertical or horizontal. If a star's image has a tail sticking out at 45 degrees, or some other angle, this means that one or more mirrors is tilted left or right of the center line. Correct this with the sagittal screws before going farther. Once you have the image imperfectons limited to the precise vertical and horizontal directions, use the meridional screws on the primary cell.

The primary tilt may be too high or low when the image has been corrected. You will know this is the case if the telescope doesn't look straight out of the opening. To cure this, tilt one of the other mirrors a tiny bit with its meridional screws, then remove the astigmatism just introduced by using the primary screws only. After a few trial moves, everything should collimate to specs.

The light baffle(s) should not intrude on either the top or bottom of the central light cone when the telescope is properly collimated. If it does, shift the baffle up or down appropriately.

If you have made a truly long f-ratio telescope, you may be able to ignore the focal plane tilt. Shorter f-ratios will require that you tilt the eyepiece to match the focal plane. The above referenced articles describe ways to do this.

Let me emphasize that you don't need a computer to design a Stevick-Paul. The tables for primary tilt angles and clear field are all you need. Work only with spacings and axis distances. The full-scale drawing must be the last word on the design anyway, as it is the pattern against which the finished telescope is assembled. If you are serious about building one of these telescopes and have questions, please write.

My profound thanks to Don Carron who has meticulously read the many versions of this manuscript and offered invaluable editorial advice.

David Stevick

2224 Charles Street

Wellsburg, WV 26070

[1] R. A. Buchroeder, "A New Three-Mirror Off-Axis Amateur Telescope", Sky & Telescope, vol. 38, no. 6, pp. 418-423, December 1969

[2] R. A. Buchroeder, "Technical Report Number 68", Optical Sciences Center, University of Arizona, May 1971

[3] R. E. Cox, "Telescopes with Unobstructed Light Paths", Sky & Telescope, vol. 43, no. 2, pp. 117-120, February 1972.

[4] A. Kutter, "A New Three-Mirror Unobstructed Reflector", Sky & Telescope, vol. 49, no. 1, pp. 46-49, January 1975

[5] M. Paul, "Systèmes correcteurs pour réflecteurs astronomiques", Revue d'optique, vol. 14, no. 5, pp. 169-202, May 1935.

[6] J. G. Baker, "On Improving the Effectiveness of Large Telescopes", IEEE Transactions, vol. AES-5, no. 2, pp. 261-272, March 1969.

[7] M. Brunn, "Unobstructed All-Reflecting Telescopes of the Schiefspiegler Type", U.S. Patent 5,142,417, Aug. 25, 1992

[8] R. A. Schmidt, "Constructing a 10" Kutter Schiefspiegler", Telescope Making #4, pp. 8-13, Summer 1979.

[9] R. E. Cox, "Better Mirror Cells", Telescope Making #6, pp. 32-39, Winter 1979/80.

[10] R. E. Cox "The Kutter Tri-Schiefspiegler", Telescope Making #16, pg. 10, Summer 1982.

[11] A. L. Woods, "How to Build a Tube for the Tri-Schiefspiegler", Telescope Making #16, pp. 11-17, Summer 1982.

[12] A. L. Woods, "Collimating the Kutter Tri-Schiefspiegler", Telescope Making #16, pp. 18-21, Summer 1982.

[13] R. J. Wessling, "Building a 12.5" Buchroeder Tri-Schiefspiegler", Telescope Making #28, pp. 32-43, Fall 1986.

[14] S. W. Johnston, "Construction of a Second 12.5" Tri-Schiefspiegler", Telescope Making #28, pp. 44-51, Fall 1986.

[15] J. M. Sasian, "Variations on the Schiefspiegler", Telescope Making #43, pp. 18-25, Winter 1990/91.

[16] J. Sasian, "The World of Unobstructed Reflecting Telescopes", ATM Journal #1, pp. 10-15, Fall 1992.

[17] M. Brunn, "Die Entwicklung des Schiefspieglers", Sterne und Weltraum, pp. 647-651, Aug.-Sept. 1993 and pp. 808-812, Nov. 1993

Comments welcome -- *dstevick@juno.com*

Convex surfaces may be tested by the knife edge from behind if we test through a known good concave surface of similar radius (S & T Jan. 1970). Since such a concave surface already exists on the tertiary mirror, all that is needed is to `fuse' the two spherical mirrors together with household glycerine whose refractive index is identical to Pyrex. The glycerine will fill in any scratches or pits remaining on the backsides.

A custom stand is required. Mine took one evening to throw together from 1/2" plywood and pine strips. The photographs should speak for themselves on this matter. Notice that I purposely angled the test stand slightly forward. I no longer feel this is necessary; supporting the rear disk behind its center of gravity should insure that the two stay in contact.

To test the convex backside, the Foucault tester must be closer to the assemblage than it would be to test the concave front. Under test, the shadows will have a beautiful indigo color while striae remain uniformly gray. Test for a null.

Practice will show you just how much glycerine to pour on the back of the secondary before putting it on the rack. Squeeze the sides of the disks to spread it out. Slide the convex off when done testing. The concave mirror isn't removed from the stand until the figuring is finished. Plain water will clean up the secondary for the next round of figuring.

Back to Fabricating the Optics

Ratio A=0.45, Radius of curvature of correcting spheres to radius of curvature of primary.

To my knowledge this is the first FSP telescope. It functions essentially the same as the standard Stevick-Paul* (which also uses 4 mirrors) except the design is more compact and with the slight increase in `A' the tilt of the focal plane is slightly decreased. An F15 design can use a spherical primary and would help further simplify construction.

The location and tilt angles of the mirrors cancel the on and off axis astigmatism and coma. Spherical aberration is removed by the primary, put back in by the secondary, and then finally removed by the ternary mirror. The distortion, image tilt and field curvature which remain along with standard higher order aberrations are relatively minor in nature.

By Bob Novack, 143 Wolfe Run Rd., Cranberry Twp. PA, 16066 Tel.(412) 828-9040 x220

*The Stevick-Paul off axis Reflecting Telescope, D. Stevick, ATMJ Issue 3, pp4-9 (1993)

I have had it (the FSP) to several star parties and have always received favorable comments on the clarity of the view...I had QSP put their 96% enhanced coating on all 4 of the mirrors. So far I have been very pleased with the coating's performance and durability.

The image tilt has never been noticed by a member of the public. The scope has many times been complimented as giving the clearest view on the hill.

...unless one uses an image tilting lens the fastest SP for purely visual use is about F10. The F10 to F15 range is still fast relative to most other TCT's, except maybe Yolos and there one has to do fancy glass work or have a warped sense of humor. A folded F12-F15 with 3 spheres may be the easiest off-axis scope that can be made with reasonable speeds i.e., avoiding very long radii.

The FSP is baffled between the entering beam and the secondary with a black velvet covered plate just inside the telescope entrance. Just a plain black baffle is placed in the triangular area between the secondary and the flat and runs parallel to the beam between the secondary and the flat. The scope was used to look at Jupiter which was not far from the moon and gave excellent results. The scope can be used on bright sunny days to look around the neighborhood and gives clear contrasty images. The flat being where it is to some extent serves as a baffle. I have chamfered off its hip and have painted the visible part black. The edges of all the other mirrors have also been blackened. Every little bit helps.

The flat was made after the primary was polished out and brought to a sphere. I had two 4-inch tools left over from making a refractor doublet so the sides were brought to the same measure with the spherometer. They were polished and brought to good smoothness using the Ritchey-Common set up using the spherical primary. I had previously made three 6-inch flats, by the "one on the other method", and by comparison of the fringes and solving the simultaneous equations, hopefully arrived at a 1/10 wave flat.

The two spherical mirrors were interferometrically compared, after the concave was nulled, until the convex sphere gave good straight fringes.

The hardest part was parabolizing the F10 mirror. A sphere (at F10) is almost good enough but not quite. The shadows on the Foucault test are very soft. I used as narrow a slit as possible to enhance the shadow contrast...Now that I know what I was getting into Al Woods was correct by going a little longer (F12) and leaving his mirror a sphere. However it was a good experience.

I can receive email here at work at: *<bob_novack.ametek@notes.compuserve.com
>*

Bob Novack