PAGE 13

Copyright – P. J. Smith

But permission is given to distribute this material in unaltered form as long as it is not sold for profit.

Various Schemes for

Testing Cassegrain Secondaries.

Examples based on Cassegrain designs on Page 12.

by Peter John Smith.

A variety of schemes with emphasis on Near Null tests

using far smaller and less extreme auxiliary optical

elements than the classic Hindle Test.

NOTE. Many of the diagrams are exaggerated to show salient features.

In practice, light cones are often narrower than shown.

APOLOGY. Some of these tests are the similar to, or are derived from other published tests.

Rather than use some commonly accepted names for these tests, I have opted for a descriptive name instead.

This is in part because many of these tests may be traced further back to earlier workers.

Contents

Testing of surfaces

Concave Surfaces, Concave Aspheres, Convex surfaces

Null tests

Why a Null, Problems with a Null, What is a useful Null, Sensitivity, Testing of Convex Hyperboloids

Review of possible convex hyperboloid tests

Testplate Method, Size range of convex secondaries considered, Hindle test, Units and conventions, Restraints

Simple reverse Foucault test

Double pass refraction test by Autocollimation

Nulling with spherical mirror by reverse reflection

Analysing Null position and quality

Conclusions

Testing of surfaces

Testing concave surfaces is so taken for granted that a preliminary review is in order.

We are familiar with Foucault or related tests of concave surfaces where fortuitously the surface under test requires no auxiliary optical elements. This is because rays from a point source naturally spread out to fully illuminate all of the surface which then reconverges the rays back along their original path to a point focus.

For a concave sphere, an aberration free image occurs when source and image are superimposed at the Centre of Curvature.

This unique scheme is so simple that we often take its advantages for granted.

1. Since source and image are superimposed no precise measurements are needed to position and space them.

2. There are no auxiliary optics to introduce unknown errors.

3. Since the image is aberration free, this is a true null test. As long as there are a minimum of off axis errors, it is very trustworthy.

Concave Aspheric Surfaces

We may extend this test to other conics by one of two methods.

1. Deliberately introducing a calibrated amount of Spherical Aberration.

2. Modifying the optical setup so a true null results.

The first method is our usual scheme for parabolising. Since we cannot place the source at infinity, it is impossible to null test directly. But, subjected to the same test as a spherical mirror, spherical aberration results as shown.

If we deliberately ensure the 'null' contains the correct amount of Longitudinal Spherical Aberration, the mirror will be a the required paraboloid.

The second method is illustrated by the test for an ellipse where a perfect null results from separating source and image position.

Convex Surfaces

Convex surfaces always produce diverging rays from a point source. Thus it is impossible to use a Knife Edge directly

This problem may be overcome by introducing some auxiliary optical complexity. One scheme is a converging lens in the light path.

Unfortunately, this lens, even if perfectly formed, introduces some longitudinal spherical aberration (LA).

The combination is still quite useful, however, because it is possible to calibrate the amount of aberration introduced by the lens. This method was first used successfully to test convex hyperboloids by Gaviola. He used a crown element from a quality large refractor as the auxiliary lens. Figuring must result in calculated amount of LA.

While this method is a real option if the secondary is made of non homogenous glass, some of the methods described later, using optical glass, are more attractive.

As an interesting aside, this basic technique has been used to test small extreme spheres such as steel balls or glass hemispheres as found in oil immersion microscope objectives.

The complexity of the auxiliary lens system is needed to reduce aberrations. A good quality microscope objective can often achieve a similar result.

Why a Null?

Testing is a delight. No complex calculations are needed to interpret the results. The final test is unambiguous.

Small irregularities stand out well and are much easier to interpret.

Problems with a null test?

1. When deep curves are under test we need great care, cunning, and often auxiliary optics such as a beam splitter to retain an essentially perfect null. In practice, if Source and Knife Edge are slightly separated, the null is not perfect. Each case requires careful evaluation.

2. Any slight set up error may well cancel the advantages of the simple null test.

3. We may be lulled into a sense of false security by a poorly set up 'null'.

In a few optical situations we meet a fortuitous set of geometrical properties that produce perfect nulls. These include testing at the Centre of Curvature of a sphere, imaging from one focus of an ellipse to the other and the overlaying of a focus of a Hyperbola with the centre of curvature of an auxiliary Hindle sphere.

Often, in practice, more complex optical systems are needed to produce workable nulls and we may soon end up with an optical Rube Goldberg machine.

In this case Radii of curvature errors, Spacing errors, and Alignment errors may turn our cunningly contrived Null setup into a liability. Hubble should be a constant reminder to us all in this context where, I believe, an incorrectly spaced Offner nulling device was the culprit which voided the other notable attributes of its quality surfaces..

What is a useful 'Null' ?

The following examples may help us decide what is a useful null and point out some anomalies.

No.	Comment	PV	RMS	LA(mm)*

1	Perfect	0	0	0
2	Excellent	0.021	0.008	0.008
3	Satisfactory ?	0.094	0.018	0.3
4	Satisfactory ?	0.13	0.03	0.03
5	Useful ?	1.2	0.2	2.0
6	6 in F:8 Sph	0.26	0.08	0.6
7	12 in F:4 Sph	4.2	1.2	2.1

* The LA or Longitudinal Spherical Aberration is the actual displacement measured with a Foucault Test. It is very dependent on the angle of the light cone which explains apparent inconsistencies in the table.

Clearly the second, in practice, is essentially perfect. Errors such as spacings, auxiliary optics accuracy, null test interpretation etc. will be the practical limitation.

Tests 3 and 4, labelled satisfactory, may be sufficient to assure very good performance if the null wavefront error is fully transferred to the optic under test, and there are no others errors in the final system. Results may be acceptable. Certainly we would like better but if the alternatives are simply out of reach we certainly should not despise a null of this quality.

Test number 5 is labelled as 'Useful ????'. It certainly will not produce a sufficiently accurate figured surface directly, but if we calculate the aberrations present it is not too difficult to deliberately introduce opposite compensating errors into the null tests. This is identical to the testing of Paraboloids by most ATM's. We know they produce aberrations when source and Knife Edge (KE) distance is equal. By deliberately matching the theoretical amount of Longitudinal Aberration with predicted Foucault readings, a very useful final figure is possible.

The last two examples show how bad a 'Null' we are prepared to use when figuring Parabolas.

Sensitivity

A null test is of little use unless the almost perfect wavefront error is transposed to the optical surface under test.

We all know that a reflective surface must be roughly 4 times as accurate as a refractive one since a slope error on the surface has far more effect on the final wavefront error. A 1/4 wave (PV) lens surface is excellent but as a reflective surface is quite mediocre.

Even more sensitive is a surface traversed twice or even more times by the light.

A good example is the Hindle test which can be classed as 'double pass'.

Here the Source and Knife Edge are artificially separated to make ray paths more obvious. Note the rays reflecting twice from the Hyperbola under test.

This makes for doubled sensitivity compared to a single reflection.

In contrast, sensitivity of refractive surfaces depends on the angle of incidence.

Refractive surfaces are extremely sensitive to slope errors when close to total internal reflection.

The more usual refractive case delivers about one quarter the sensitivity to slope errors compared to reflective surfaces. If double pass is used, this becomes about one half.

Review of possible Convex Hyperboloid tests.

There are only six basic schemes available. For completeness, all are listed.

1. Star tests.

2. Autocollimation of the entire telescope.

3. Hindle test in one of its forms.

4. Produce a concave hyperboloidal testplate and fringe test the convex mirror against it.

5. Gaviola test using a large auxiliary test lens.

6. Other null (or close null) tests which involve passing light through the secondary mirror material.

The last strategy is the main topic of this discussion.

We will consider tests where the secondary mirror is the only refractive element in the test since for one off secondaries this removes the necessity of producing an auxiliary large refracting lens.

Testplate Method

In general, there are only two really accurate methods of making the concave test plate.

1. The concave hyperboloid test plate may either be tested by measuring and matching the required zonal Longitudinal Spherical Aberration in a similar way to making a Parabola using the Foucault or Caustic test.

2. Or wemay use a near perfect null with some auxiliary optical element such as a lens or spherical mirror. The test now resembles either a Ross Null or Waineo or Jones setup.

Even when the testplate has been produced, the prospect of using it on a very large and heavy convex mirror is not trivial.

Gravitational deformation and scratching are obvious problems.

Since the departure from spherical will amount to many microns the test must be applied often and many fringes counted at the beginning of figuring. Centring and elimination of fringe viewing errors must be rigorous. It is interesting that some very large convex hyperboloidal secondaries for the Keck telescope were pneumatically floated during fitting to testplates. This method was particularly useful in this case because many identical secondaries were needed.

It is certainly a route an amateur could take. There are few other options when using a non homogenous material for a large convex mirror without a huge Hindle Sphere available.

Since this approach is well known it will not be discussed in any more detail.

Size range of convex secondaries considered?

The secondary size determines the form and material used and hence the testing method.

Mechanical problems with respect to cassegeain secondaries are due to Gravitational deformation and thermal expansion, which are exacerbated by heat capacity and thermal inertia.

Pyrex has been very successfully used in large primary mirrors where these problems are mitigated by using a thin mirror with reduced thermal capacity and thermal inertia and a sophisticated mechanical flotation system.

The obvious material for an amateur to use for a cassegrain secondary is Pyrex. It has considerably better thermal properties than common plate (soda) glass.

Another approach is to use a sufficiently rigid thick low expansion material such as Zerodur or Fused Quartz. The cost may not be prohibitive in small cassegrains.

But, if the size of amateur cassegrains is to grow hand in hand with the large Newronians being constructed, one of the refractive nulls described later may be required. Pyrex is not optically homogenous enough for this test. Nor are some even more superior and exotic materials as homogenous as good optical glass.

Hindle Test

This test was a major breakthrough for the production of Hyperboloidal Cassegrain secondaries. It handles non homogenous glass, is extremely sensitive because of the double pass on the test surface, and uses only one easily tested auxiliary optical element. Compared to autocollimation and star testing of the entire telescope it is a relatively easy shop test to set up.

If the surface under test is unsilvered, the double pass reduces illumination somewhat.

The elegant principle is easily understood from the following diagrams.

When the centre of the test sphere exactly overlays the focus of the hyperboloid under test, light strikes the Hindle Sphere normal to the surface. Thus, it is reflected back to exactly retrace its path.

Any sphere within sensible limits may be used.

So why is this test not the answer to all problems ??.

Unfortunately, the Hindle sphere must be nearly the same diameter as the finisher telescope, and as it often must be about F:1, it is excessively deep to make.

Professionals may have this facility for testing up to maybe a 1/2 m secondary although in some really large sizes they have used alternative methods.

While for a professional the maximum size before turning to other methods might be 1/2 m, for an amateur it probably occurs at about 4 -6 inches.

Other problems are obvious from the following diagrams.

1. The source may obstruct some central rays which makes examination of the central area impossible. This can be eliminated by using a very small source such as an illuminated optical fibre, artificial star, or a beam splitter.

2. Any cutout in the spherical mirror also eliminates some central rays again rendering examination of the centre impossible. The only way to reduce this is to move the source and Knife edge closer to the sphere.

Unfortunately, moving the source and knife edge forward has the unfortunate effect of increasing the diameter of the spherical mirror needed as the above diagrams indicate.

It is interesting that, when carried to extremes, the diameter of the testing mirror is only slightly larger than the convex under test. This seemingly useless variation has been utilised by making the nulling sphere in the form of a meniscus lens. Partially silvering the reflecting front helps.

While this is a possible solution, there are other schemes presented later which do not require the manufacture of a large auxiliary lens and make use of more common optical components. Unfortunately, most require optical glass rather than Pyrex be used for the secondary. Top optical quality fused quartz is also an option.

The other main restraint involves holding the elements aligned and correctly spaced during testing. This is a problem common to all null tests, especially complex ones.

Units and Conventions

In the examples to follow, conic deformation as referred to is the Schwarzschild constant unless specifically stated otherwise.

'Null' test quality is usually related to wavefront error. To avoid confusion, both RMS and PV are given. Without being drawn into a discussion of the merits of these different criteria, we should realise that either may be useful but care is needed to avoid confusion.

Wavefront errors have not been quoted to meaningless accuracy but are usually rounded off. Sometimes roundoff error may produce slight anomalies.

One or at the most 2 significant figures are perfectly adequate for comparisons here.

Optical glass is always ubiquitous BK7.

Well known but very restricted method.

The Simple Reverse Foucault Test.

One often mentioned method is simply to do a standard Foucault test from behind through optical glass.

The following is typical but not to scale. Thickness, curves and angles are exaggerated.

A guaranteed accurate surface must first be generated on the back. This rules out a convex rear surface.

4000 mm concave is a reasonable limit for this concave surface. Unfortunately, only a small range of hyperboloids are now possible unless very exotic glass is used

The limitations of this test are probably best illustrated by actual figures. These are for a 300 mm diameter surface of Radius of Curvature of 3112mm. . Only BK7 will be considered.

K (Conic deformation)	PV wavefront error	Longitudinal Aberration (mm)

-0.0365	0	0
-1	2.8	8
-2	5.6	18
-4	11.4	35
-8	23	70

While it would be possible to figure until zones match the required LA, the test is far from a null. When the value of K is large (K = -8.1 is required in the design example), it would be nearly impossible to examine the entire surface for irregularities with a remaining PV error of 23 waves.

As mentioned previously, if the rear concave surface is made convex most of the problems disappear. For example, if the rear surface takes a Radius of Curvature of 3200mm convex, a nearly perfect null occurs when K = -8. Unfortunately this convex surface is nearly as difficult to produce as the target Hyperbola.

The null may be improved by separating the Source and Knife Edge. Apart from introducing more complexity with increased possibility of error due to the measurement of the positions of Source and Knife Edge, the main problem of a convex rear surface is only slightly mitigated.

While this simple reverse Foucault test may be used, better options are available. For small Cassegrain secondaries with extremely small Schwarzschild constants, it may occasionally be useful.

Double Pass Refraction Test by Autocollimation

This should not be confused with the Autocollimation test using a certified flat surface and optical setup of the finished telescope where the flat needs to be as large as the finished telescope.

In this test, the hyperboloidal surface refracts, not reflects light. Again, the following diagram is exaggerated.

A very useful Null is obtainable, but only with a convex rear reflecting back surface.

Although the test surface uses refraction, it is traversed twice by the rays. Thus, although sensitivity is reduced by refraction at the hyperboloidal surface, the test is still reasonable.

It would very nice to silver the rear surface but we manage to successfully test unsilvered Paraboloidal mirrors without silvering them.

Fortuitously, most of the problems may be eliminated at one stroke by air spacing a concave reflecting surface a short distance behind the Cassegrainian secondary under test.

This allows the following gains.

1. The Aluminised surface gives full reflection - thus a bright image results.

2. The Reflecting surface is typical of a deep but common primary used in some Newtonians. Since it may be either spherical or Parabolic , no special deep null mirror is required.

3. Only one other auxiliary surface need be crafted. This rear surface of the secondary is concave. Thus it may be rigorously tested.

4. The small spacing between mirror and secondary is easily guaranteed with a mounting collar and accurate spacer. Once the test optic is mounted, there are no other spacings to worry about. The null is then examined in exactly the same way as a standard Foucault test. Interpretation will be different but not complicated.

5. A large range of K values may be handled without any exotic glass types.

6. There should be no other surface reflections easily confused with the null.

The following example of a large extreme secondary illustrates the quality of null possible. The figures are for a secondary of diameter 300 mm, Radius of curvature 3112mm and Conic deformation K = -8.1802

RC Spherical Mirror	RC Rear Concave	Spacing	PV Error	RMS Error	LA mm.	F:NO Test Mirror

1500	4159	15	0.036	0.008	0.03	2.5
2000	4615	15	0.015	0.0035	0.01	3.33
3000	8270	15	0.002	0.001	0.005	5

As can be seen the range of nulls available with Spherical test mirrors up to F:5 is very useful. A paraboloidal null mirror unfortunately gives a longer Radius of Curvature on the rear of the secondary although the null is truly excellent.

If the Conic deformation (K) is less, the range of useful nulls is much increased.

If the diameter of the Hyperboloid is smaller, the range of excellent nulls is again much increased.

If necessary, better nulls are achievable by separating knife edge and source. This is unlikely to be needed unless the secondary is very large and has a very high K value.

All in all, when the alternatives are considered, the double pass refraction test by autocollimation looks very useful.

Nulling with spherical mirror by reverse reflection.

Another well known null test uses a spherical concave nulling mirror for testing concave Paraboloids. It may be extended to other Conic shapes. This method is often used to make concave Hyperboloidal testplates.

There is no reason this scheme cannot be used to test a convex Hyperboloid through the back of the cassegrain secondary.

In fact there are many variants possible.

We may avoid cutting holes in either Primary null mirror and Cassegrain Secondary by utilising both refraction through and reflection from the secondary. The refraction is not meant to make the test any more sensitive since it only uses the central portion. The reflected null should be visible over the entire surface.

The first has the advantage of not having the source facing the observer while the second allows easy measuring of the placement of the source.

Neither require an accurate figure on the back surface of the secondary out to the edge - thus turned edge will be of no consequence, nor will a polish which stops short of the edge matter. The surface accuracy in the central zone required is high but this is easily achievable.

The following example illustrates the quality of null possible. The figures are for a secondary of diameter 300 mm, Radius of curvature 3112mm and Conic deformation K = -8.1802 using the first arrangement.

RC Spherical Mirror	RC Rear Concave	Spacing	Source Dist.	Knife Edge Dist.	PV Error	RMS Error	LA mm.

1500	2948	626	247	459	0.09	0.017	0.02
2000	4507	705	162	647	0.018	0.004	0.005
3000	25,626	751	98	764	0.05	0.01	0.015

When the source is always on the spherical surface and the Knife Edge is a constant 100 mm from the rear of the secondary, there is a more limited range of useable nulls, This represents the second arrangement.

RC Spherical Mirror	RC Rear Concave	Spacing	Source Dist.	PV Error	RMS Error	LA mm.

1500	2641	901	on sphere	0.046	0.009	0.02
1713	4000	961	on sphere	0.027	0.006	0.01
2000	convex	993	on sphere	0.049	0.009	0.02
3000	convex	1044	on sphere	0.053	0.01	0.025

When the knife edge position is withdrawn far from the secondary, a greater range of concave values exist on the back of the secondary.

Another possibility is to move elements closer together superimposing Source and Knife Edge.

This restricts the degrees of freedom but an excellent null exists for the example.

Of particular attraction is the shortened spacing (approx 300 mm) which allows a precision spacing and aligning collar to be used. Since the source and Knife Edge are superimposed, no other measurement error exists. Some badly turned edge on the back of the secondary and unfinished edge polish is of no consequence.

The entire test now occupies no more than about 700 mm of bench space. A relatively wide ray cone is a potential source of error but may be handled by various methods.

Unfortunately, the null will not be as bright as for the Autcollimation method but should be little worse than testing an unsilvered paraboloid.

The following example are for a secondary of diameter 300 mm, Radius of curvature 3112mm and Conic deformation K = -8.1802

RC Spherical Mirror	RC Rear Concave	Spacing	Source Dist.	PV Error	RMS Error	LA mm.

1500	1934	261	379	0.25	0.05	0.05
1804	2427	304	401	0.033	0.007	0.01
2000	2896	310	422	0.10	0.025	0.03
3000	8341	324	496	0.18	0.037	0.06

While only one nulling mirror produces the very best results, the Radius of Curvature may still vary over useable limits.

If necessary, the knife edge and source may be separated to improve the null.

As the Radius of Curvature of the nulling sphere increases, so does the knife edge distance.

Suitable nulls will be found when Radius of curvature of nulling sphere roughly equals the knife edge distance.

For the secondary considered previously, with source distance of 199.95, KE of 2436, and spacing of 203.6, when tested against a sphere of Radius of Curvature of 2150, gives a null where PV wavefront error is 0.001. The concave on the back has Radius of Curvature of 2412 concave

This exquisite result uses a nulling mirror of about F:3 and is no larger than the secondary under test..

Other quite useable results are available with a nulling sphere OR parabola ranging all the way to F:8.

In general, a parabola gives better results in this test.

Analysing null position and quality

Many of these nulls represent very complex situations. In practice, if a beam splitter is used in the test, further complexity in introduced which precludes simple calculation of the null parameters. In all cases Zemax was used to analyse them.

Once a reasonably close estimate of the position of the null is known a raytracing program with a good optimiser may be used to fine tune and evaluate it. Unfortunately, optimising so close to a near perfect image position places severe demands on the optimising algorithm and some jiggling may be needed.

A range of examples has been given to provide many possible starting points.

Conclusions

1. A wide range of very useable nulls are available for testing Cassegrain Secondaries made of optical glass through the back. Recent availability of laser sources has made the method a more attractive option

2. There is little point in using the simple reverse Foucault test when much better and wider ranging tests are available.

3. Some tests use an auxiliary optic of smaller diameter than the secondary.

4. Auxiliary optics required are not as extreme in diameter or depth as a Hindle sphere.

5. It may be expedient to design a cassegrain telescope around the auxiliary testing optics one has on hand.

6. The ultimate limitation to this method lies in developing satisfactory methods of mechanical flotation or even load distribution using multiple Silastic or Urethane supports for the secondary.

7. Of crucial importance is the ability of raytracing software to determine the exact null parameters for specific cases.

8. From the testing point of view, we probably should re-examine the notion that a spherical secondary as in a Dall Kirkham is considerably easier to produce than other types. The simple truth is that ANY convex surface is difficult to test to a high accuracy.

Assistance

Crucial to using these methods is the ability to find and evaluate useable nulls. If anyone is really serious about fabricating a cassegrain secondary using these methods I would be willing to help.

Comment

In my own mind, I now feel confident in tackling a Cassegrain Secondary within the size range allowed by using optical glass. Amateurs have already pushed the envelope relating to large thin primary mirrors. A next step could be to develop methods of successfully using optical glass cassegrain secondaries.

Caveat

There may be errors present. Take care.

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