Testing

Testing your mirror with a 'knife-edge' type of tester:

NOTE: This section is based on the use of a tester as described in the "Knife-edge tester section" This tester uses a fixed light source and a moving knife-edge (see note under 'calculations' below)

If you find this method of testing too complicated, you could use the "Ronchi test" - this uses a lined 'screen' placed into the path of the reflected image from the mirror to evaluate its form.click here

This file prints about 8 pages letter size - portrait orientation.

Page updated March 8th 2003

The diagram below shows the slit in the tester's lamp housing is set at the distance of the radius of the mirror, but is offset a small distance to one side of the mirror's axis. An image of the slit will appear at the same distance, but will be offset in the opposite direction.

This is the principle used in most of the popular tests which can easily be afforded by the amateur. This type of tester is easy to make and use - and can give results with enough accuracy to satisfy the need to attain a 'perfect' mirror. With practice, careful measuring can allow one to determine the profile of the mirror to better than a millionth of an inch.

Using the tester described in this web-page, the tester and mirror are to be set up a distance apart equal to the radius, previously measured. They should be set so that the center of the mirror is at the same height from the floor as the illuminated slit in the testers lamp-housing. The mirror needs to be supported on some type of stand which allows it to stand vertically and can be adjusted in tilt so as to project the reflected image of the source to a point close to the knife-edge.

The setting up needs to be done in a darkened, or dimly lit room. When the mirror and tester are placed in position - the slit-plate should be removed. This allows the larger 3/16" hole in the lamp housing to provide a larger, brighter image.

Set the knife-edge carriage about half-way along its travel with the lead-screw.Place a letter-sized card in the area of the knife edge and try to find the reflected image of the light-source. Unless the mirror is exactly perpendicular to the source it could be above, below, or to either side of the tester.

This is the reason for having an adjustable support for the mirror. Once you have located the image you can then move the mirror to bring the image close to the knife-edge position at the edge of the lamp housing. The image should be between the housing and level with the center of the knife-edge. You may also need to move the tester or the mirror closer together, or further apart until the image appears the same size as the hole in front of the lamp.

Once they are set up, replace the slit-plate. Now look at the mirror, with your eye aligned with the edge of the knife, from behind the tester. You will be looking along the edge of the lamp-housing - in the space between it and the knife-edge. Your eye should be very close to the lamp housing. You should see the mirror disc illuminated across its surface.

Now adjust the position of the knife edge towards the lamp-housing. The mirror should start to darken, until with the knife completely across the image, the mirror will darken and appear black. Backing off the knife again should make the image reappear and the mirror brighten again. You may see a vertical shadow of the knife-edge move across the disc, and as the knife moves across the mirror you may see various kinds of 'shadows' form on the mirror's surface.

NOTE: If you see only shadows across the mirror -then push the knife-edge carriage further towards the mirror, so that when you bring in the knife-edge you see a vertical shadow crossing the mirror - this is the dark edge of the knife, and it should be vertical upand down the mirror when it is positioned so that the edge is exactly half-way across the mirror. If it is not vertical adjust the knife-edge orientation until it is vertical. It is important that the knife-edge is parallel to the reflected image of the vertical slit on the lamp housing. If this is not so, then the readings you get may not be accurate.

If you only see the shadow of the knife cross the mirror, then you should move the knife carriage closer to, or further from, the mirror until you reach a point where shadows appear on the mirror - as if it was moulded in relief.

These shadows appear whether the mirror is 'parabolic,' ' hyperbolic,' ' elliptical ' or anything other than perfectly spherical in cross-section. If it is perfectly spherical, the whole surface of the mirror will darken uniformly when the knife- edge crosses the point at which all the reflected rays cross the axis. Inside and outside of that point the mirror will appear to darken first at one side or the other. (This will be a gradual darkening and not the same as seeing a sharp shadow of the knife crossing the mirror.) The density or 'contrast' of these shadows will vary with the 'F-number' of the mirror (an F6 will give darker shadows than an F8.) and with the degree of deviation from a true spherical surface.

If you were to draw a cross-section of these shapes you would have 1-an upside-down soup bowl, 2- a 'doughnut' shape but with the hole filled in a bit, and 3- a shallow dish, The shadows are interesting to see - and give a general idea of where the rays from various radii of the mirror cross the optical axis. They appear at different longitudinal positions of the knife-edge as it moves towards or away from the mirror. But they do not give you a clear idea of the accuracy of the mirror's figure.

The distance that the knife-edge travels to go from one to the other can be calculated - but without a better way of 'pinning' down the exact position of the knife at the center and edge, they only provide a general idea of its accuracy. To do this requires a better method, which we shall now explain. It is also the reason that we have placed scales on the tester's carriage and on the lead-screw knob.

Note IF YOU ARE USING METRIC NUMBERS - all the measurements you make should be made in millimeters. This includes the diameter of the mirror, the focal length and radius of curvature. Also the spacing of the windows in the Couder screen etc. All the calculations, and formulae such as hm^2/R remain the same....but you will just have metric numbers. Also note - the wavelength of light is 0.56 microns (in metric) so 1/4 wave will be 0.14 microns, you will need to remember this when assessing your graph using metric numbers.]



MEASURING:

Some Math and a bit more optics: The principle of testing is to use a method which can allow us to determine where the rays from each part of the mirror are crossing the axis, and to calculate whether they are doing so at the correct point along the axis. If they do not, we need to know how large the errors of position are and thereby be able to make corrections to the mirror surface to eliminate such errors.

A simple way to do this is to make some type of 'mask' or 'screen' which covers the mirror, with 'windows' cut into it at various radii of the mirror, on each side of center.

This will allow us to find the point at which the knife-edge causes an even darkening of each pair of windows, thereby allowing us to find exactly the point along the axis where this occurs. By doing this for at least 4 or 5 points across the diameter of the mirror - we can obtain a measurement of the spacing of these points along the axis and compare them with the calculated spacing which an accurately 'figured' mirror should have.

The screen which we use follows a design that was suggested ( a long time ago) by A. Couder, and is called appropriately enough - a Couder screen ! A thick paper or thin card circle is cut, the exact size of the mirror and 'windows' are drawn in a horizontal band about 1- 1/2" deep across the mask. Into this at intervals are cut pairs of windows on each side of center. They are 'staggered' above and below the center line of the horizontal band - so that we can easily identify them when looking from behind the tester, at a distance of some 10 feet from the mirror. There is a nice formula for calculating the width and position of these windows, based on the fact that the parabola (which is the shape we want for the mirror cross-section) deviates most from a sphere as we approach the outer areas of the disc.

In practice we select the width and the position of the windows so that none are too large nor too small to see easily and compare when performing our test. The outer window should be made about 3/4" wide to be easily seen, with the next inner window being perhap 1" the next 1-1/8" leaving 1-1/8" each side of center - giving 2-1/4" - for the central window. These are typical dimensions for an 8" disc, and would be easily viewed from behind the knife-edge. (For larger mirrors you can use 5 or 6 windows for a more accurate profile.)

The outer windows on this mask would give us readings at about 42%, 70%, and 92% of the mirror's radius - where the deviation from a sphere is greatest.

(In some cases you can if necessary make two masks - one with windows covering the inner half of the mirror with say two or three pairs of windows and another with perhaps 3 or 4 windows covering the outer radii of the mirror. When measuring you would use one mask and read the relative knife-edge positions. Then carefully changing over to the second mask make the readings for the outer zones. This needs to be done carefuly without disturbing the relative positions of the mirror and tester. Usually it is better to try to do this test with a single mask - 4 or 5 good readings made with a single mask are better than 6 or 7 doubtful readings made on two different masks if the set-up is even slightly disturbed during the switch from one mask to the other.)


MAKING THE TEST

The screen is placed on the front of the mirror, once the tester is set-up and the slit-plate has been replaced. A little masking-tape holds the top of the screen to the top edge of the mirror, but we try not to touch the mirror itself as our fingers will cause 'local heating' at its edge.

Now we look at the windows in the mask (screen) some of which will be shaded as we insert the knife edge towards the lamp housing. What we are looking for is to find a longitudinal position of the knife edge which will cause only the center window to darken evenly as the knife edge cuts off the rays reflected from that part of the mirror.

Do this a number of times - until you are sure that you have found the exact longitudinal position at which the window darkens evenly. Once you are sure - note the position of the knife edge from the scales on the top of the carriage ( in tenths of an inch) and the position of the pointer against the edge scale on the lead-screw knob (in thousandths of an inch) - Recall that one complete turn of the knob represents 50 thousandths of an inch. From these two observations you should be able to establish a (relative) reading of - for example - something like .328" which is noted down.

Now you repeat the same process but this time looking at the second window at each side of the central window. You now need to establish a reading at which the knife edge causes these two windows to darken at the same time as the knife-edge cuts across the axis of the reflected rays. To get an accurate reading - the knife- edge needs to be inserted very gradually the best reading is taken when the windows are just beginning to darken, as they start to become grey, rather than blacked out completely. Try to compare the center of each window so that they both appear to have the same shade of greyness.

The longitudinal position of the knife-edge is recorded for each pair of windows. They will then be compared with the calculated relative positions to determine if they are correct. The results can be drawn out as a graph - showing the 'profile' of the mirror surface. From this you will be able to decide what kind of correction may be required to reduce any errors to an acceptable level.


Calculations: The math involved

NOTE: There is another type of knife-edge tester on which the light-source and knife-edge are mounted together on the movable carriage. If you buy (or Build) this type of tester then the calculation of the different positions at which the reflected rays cross the axis - (the relative spacings) would be calculated using the formula hm^2 / 2R ,..... that is : they will be half the spacings when compared to the type of tester described here. That type of tester and the system of graphing results is fully explained in Richard Berry's book "Build your own Telescope" published by Willman-Bell. You could use the tester described by Richard Berry with the system described here as long as you remember to adjust the readings you obtain for the knife-edge positions and multiply them by a factor of 2. All of the following calculations would remain valid - but your knife-edge readings each would have to be adjusted as noted above. ( by a factor of two.)

Now for the numbers and stuff: When you make the mask and have cut out the windows, you need to make careful measurements (within a half-millimeter or 64th of an inch of the diameters of the inside and outside) of each pair of windows. The center window's inside diameter is, of course zero. Measure its outside diam, and divide it by 2. List this on the mask as hx1. Do the same with the 2nd windows, and the 3rd and 4th, dividing each by 2.(These numbers are of course the radius of the outside of each window.)List them on the mask as hx2, hx3, and hx4.(and h5, H6, H7 etc., if using 5 or more windows)

Then by adding the inside and outside diameter of each pair of windows(#1, #2, #3, and #4 (and #5 etc., if used)) dividing the result by 4 you will arrive at a radius of the mirror which coincides with the 'center' of each window. These will be listed as hm1, hm2, hm3, and hm4.etc. When testing it is most natural to compare the centers of the windows for shading.

As noted earlier, light coming from a source at the center of curvature of a spherical mirror, will reflect directly back to the source. When the source is at a very long distance the rays become parallel. They then meet at the 'focus' of the mirror where they will form what is termed a 'circle of confusion'. The image under these conditions will be fuzzy due to the longitudinal spreading of the reflected rays.

To correct this situation we need to change the mirror's cross section to that of a parabola. However - when using our tester - with the light source at the mirror's radius, if our section is parabolic the light reaching the mirror will be diverging and not parallel. This causes a "circle of confusion" once again where the image is formed close to our knife-edge.The diagram shows this effect but greatly exaggerated for illustration.

This is where our tester with its mask tells us whether we have the correct parabolic shape. If we do - then the degree of shift along the axis for rays coming from various radii of the mirror must agree with the positions (relative spacing along the axis) which we have calculated. Although this all may sound complicated - in practice it is not very hard to do. Once you have gone through the process it will not seem nearly as difficult as reading these details.

Calculations:

Note - in computerese the symbol ( ^ ) stands for 'to the power of' - so 2^2 means two-squared , 2^3 means two cubed etc., and of course 2/6 means two divided by 6. ( - I hope this doesn't sound too basic but some people are not very conversant with math symbols.)

The point at which rays from each 'zone' (centers of each pair of windows) should cross the axis, is found using the following formula : p= (hm^2 /R + {hm^4 /2R^3} ) The 2nd term in this can be ignored, for mirrors of moderate size and depth the difference is not measurable.

So in our case using the simpler formula (p = hm^2 /R ) where 'p' stands for the position where the rays converge, we will have for zone #1. (.5625)^2 /128 = .0025 : zone #2. (1.687)^2 /128 = .0222 : zone #3 (2.531)^2 /128 = .0590 and for zone #4 (3.625)^2 /128 = .1027

Let us assume that our measurements for each zone were : #1 = .302, #2 = .352, #3 = .372, and #4 = .425 Then we tabulate the results and check the errors by comparison to the calculated values. By subtracting a suitable constant from each of our numbers we can arrive at numbers closer to those given by our calculation, without changing their relative spacing.

If you are not sure about how to find a suitable constant refer to the link provided here How to find a suitable constant #howtofindconstant>

zone 1 zone 2 zone 3 zone 4
Measured readings .302 .352 .372 .425
Subtract constant -.315 -.0130 .0370 .0570 .1100
hm^2 / R calculated .0025 .0222 .0590 .1027
Error - .0155 + .0148 - .0020 + .0073

If our mirror surface had a perfectly parabolic cross-section we would not measure any errors. The readings would agree almost exactly with the points which we had calculated. (depending of course on the accuracy of our measurements) Our first try at doing these may not be accurate enough to be very exact. With practice they should improve, and we may be able to 'pin down' the readings to within a few thousandths of an inch. We can also repeat our measurements three or four times and average the results. This can help improve our accuracy. In any case if we can be fairly sure that we are within 3 or 4 thousandths, we can get an excellent evaluation of the mirror's curvature.

After we have determined the errors - we can now calculate what they mean in terms of the mirror 'profile.' We have been measuring the errors in the profile at a point where they show the greatest effect. A reflected ray doubles the longitudinal error at the center of curvature- but when the mirror is in use, the errors we measure here will be reduced by a fourth at the focus, and the lateral errors even more. (Consider the triangle made by the lateral error and the radius of curvature of the mirror -for an idea of how tiny the lateral errors could be.) So by making our measurement of longitudinal error at the point where the ray crosses the axis with our diverging light from the tester at the center of curvature (which is twice the focal-length) we are measuring errors which can be in the thousandths of an inch or even hundredths of an inch, even though at the surface of the mirror the errors are only millionths of an inch.

We can imagine the curve across the mirror to be made up of hundreds of 'flat' sections which follow the parabolic curve. Each of these has a specific 'slope' which coincides with the ideal parabola. These errors of 'slope' which are perhaps going towards the mirror's true curve, or may be sloping away from it, can be described as depressions or rises in the mirror profile.

Now for the math involved to turn these error readings into 'errors of slope' back on the mirror's surface:

First of all - these longitudinal errors which we have measured will be reduced to a quarter when we view distant objects at the focus rather than at the center of curvature. Also, the errors which interest us most will be the lateral errors at the focus. These errors are also going to be those in the reflected wavefront focussed by the mirror.

So our system measures the actual wavefront errors - the errors on the mirror will be only half as much. So we want our measured errors to be less than 5- 1/2 millionths of an inch - if we want our mirror to be "diffraction limited" (And half as much at the mirror's surface)That is, no error in our reflected image is going to exceed 1/4 wave.This is sometimes referred to as the 'Rayleigh tolerance.' If possible we would like to do much better than that. Well - let's see how our measured numbers turn out:

Before doing this, we should point out that errors in reflections from close to the center of the mirror are not as important as those of reflections from the outer areas of the mirror. We therefore need to be as accurate as possible when measuring the outer zones of the mirror.

Taking our measured errors - we multiply them by what we term a 'zone factor.' (This factor is hm divided by 4 times the focal length, for each zone) This gives us the transverse aberration at the focus. This we multiply by a large factor 10^5 (100,000) Then we divide the answer by the focal length (64") and then multiply by 10 to arrive at our final figure for the 'slope of the mirror segment' (i.e. the center of the mask window) of each zone. We also change the 'sign' of the resultant answer. See table below:

zone1 zone 2 zone 3 zone 4
Error - .0155 + .0148 - .0020 + .0073
zone factor = hm/4f .0022 .0066 .0107 .0142
error x zone factor x 10^5 - 3.41 +9.77 - 2.14 +10.37
-mu (slope) = (error /64) x 10. + .53 - 1.5 + .33 -1.62

These finally, are the slopes in the wavefront which we would find at the focus of the mirror when in use. We can draw a graph of these - even though we have only 4 line segments to show the 'half profile' of the wavefront, as long as the shadows on the mirror show a smooth transition between our selected zones, we can have a useful measurement of the mirror's surface.(and that of the reflected rays)

The graph shows us the profile of the wave-front (and effectively also the shape of the mirror's surface) from the center to the edge. If we copy this on the opposite side, with the same slopes we would have an overall picture of the surface of the mirror. This shows us that the mirror slopes down from the center to the edges, with a couple of 'humps' or raised areas between zones 1 and 2, and also between zones 3 and 4. This would tell us exactly which areas of the mirror need to be polished down, locally, to correct the surface figure. A 'perfect mirror profile,' having no measured errors would follow the reference line 'x' - 'x' As it happens, in this case, if we consider the scale at which the graph is drawn these 'raised areas' are only some 1/18th of a wavelength high.

Despite the fact that the graph shows a gradual slope downwards, when in use we would focus on the point of the best image ! This would be represented on our graph, by another parabola of slightly different focus. Instead of the straight line representing our ideal parabola, this would be represented by a different line, a parabola starting at the center and crossing the highest points of the graph (this is illustrated as the 'reference parabola') in the diagram. If we compare the largest deviations of our graph from this line we see that the peaks are no more than 1.2 millionths of an inch or so - this would translate to about an eighteenth-wave accuracy.

This graph of only four lines may seem a little "crude" but it does show the slopes of the mirror. Obviously the jagged peaks and valleys (we hope) don't appear on the mirror's surface. They most certainly 'roll' into each other, so at least this is a very conservative measurement. Still, the actual slopes of these four 'zones' of the mirror profile are valid indications of the variations in the curve of its surface as one goes from the center to the edge.

A FEW FINAL WORDS

If this seems too complex and you prefer a simpler check : The length of the 'segment' of overlapping rays at the knife-edge position is a good indicator of how close the mirror is to being correct. You could make a mask with simply a central window - about 1 1/2" diam. and two windows about 3/4" wide at the edges. Make measurements of these two points of the radius, and from the length of this segment you can compare it with the length calculated - still using the formula - hm^2 /R - for the radius at say 3/8" and at 7 5/8" to get the length that the segment should be for the mirror to have the right curve. If according to the shadows seen (without any mask) the mirror appears to have a smooth curve, without any obvious raised or depressed zones, then this could suffice as a basic test.

Another suggestion - when testing and graphing the results the first time - you can scale your graph at 10^-5, rather than the 10^-6 shown above. When making the last calculation for the 'slopes' don't multiply the result by 10, just divide the adjusted errors by the focal length (64" in this case only.) This will 'keep the graph' on the page if the errors are fairly large.

Following your initial graph of results you do not need to graph the result after every 'figuring' (selective zonal polishing step) you need only compare the new knife-edge readings with the previous readings to see if you are going in the right direction. Only when the knife-edge readings appear to be very close to the calculated ideal numbers do you need to once again graph the results to assure yourself that the mirror profile is accurate enough. One further item : after even a short session of polishing - you must allow the mirror (especially Pyrex) to cool for several hours before measuring. The glass surface becomes locally heated during polishing and can take a few hours at least to settle to its final surface shape. So do not try to 'rush the process' patience is amongst the best tools available to the experienced mirror-maker.

AND - FINALLY: GOOD LUCK and happy stargazing.




Return to FVAS Home Page