4.5.4. Waineo null test

Probably the simplest way to null paraboloid is to use another reflecting surface. By placing point source of light at the infinity focus of a paraboloid, the collimated light output can be reflected back to it, to produce focus that can be nulled. The disadvantage of this setup is that it requires a flat as large as the paraboloid.

A setup with the flat replaced by a smaller, easy to make reflecting surface, is more practical. And this is what Waineo null test offers: spherical aberration of a concave paraboloid - or a conical surface in general - is compensated for by a smaller concave sphere (FIG. 36).

FIGURE 36: Waineo null test setup: the sphere is facing test surface, both centered around a common axis, with the light source placed inside the focus of the sphere. Diverging light reflected from the sphere is focused by the test surface; if the aberration contribution of a sphere is at the level of mirror tested, and of opposite sign, the final focus will be aberration-free, and the surface tested - if sufficiently accurate - will null when the converging light is intercepted at the focus.

While the setup is seemingly simple, it is fairly complex optically, with a number of interrelated factors affecting the aberration level: source to sphere separation is the object distance (LS) for the sphere, the images formed by it determines object distance (LT=IS+M, IS being the sphere-to-image separation and M the mirror separation) for the test surface, which in turn determines distance from test surface to its focus (IT), when the null focus (F) is formed. The two surfaces have to produce the same amount of aberration of the opposite sign and, at the same time, to be properly aligned in order for the diverging cone from the sphere match nearly exactly the test surface.

Fortunately, Waineo2006 freeware program (Richard, UK) makes determining Waineo null test setup parameters a breeze ("knife position" in the input refers to the back focus distance B on the above illustration). The program balances lower- and higher-order aberration components, making the null focus near perfect. Large, fast paraboloid, say, 400mm ƒ/4, can be null tested with less than half its diameter high-quality sphere, say ƒ/4, to a 1/175 wave RMS setup accuracy; a 500mm ƒ/3 paraboloid can be nulled with a 9-inch ƒ/3.6 sphere to better than 1/40 wave RMS accuracy, and so on.

Additional advantage of the test vs. regular tests at the center of curvature is that the length of the setup is smaller than the radius of curvature of test surface, by nearly 1/3 on the average.

Sensitivity of the test to setup errors is moderately high but, similarly to Hindle sphere test, can be rather easily controlled by keeping the final focus within several mm of its specified distance from the test surface. This is due to magnification of the secondary mirror - i.e. test surface - being very sensitive to the changes in its object distance, which is the image of the light source projected by the sphere (FS on the above illustration). Specific change of the test surface magnification mT is given by:

ƒT being the test surface focal length. Change in magnification is caused by the test surface object distance LT=IS-M (M being numerically negative) changes, resulting primarily from spacing error between the source and the sphere, as a result of the change in the image distance projected by the sphere, IS=RS/(2-RS/LS), RS being the sphere radius of curvature. The effect of change between the two mirrors is comparatively negligible.

For illustration, in the above Waineo setup for 400mm ƒ/4 paraboloid, 1mm spacing error between the sphere and the source induces ~1/125 wave RMS error, and nearly 3.5mm shift in the location of final focus. For the 500mm ƒ/3 mirror, the setup is, as expected, more sensitive: 1mm spacing error between the source and sphere induces about ~1/80 wave RMS error, with the final focus shift of nearly 4mm.

The P-V wavefront error of lower-order spherical aberration at best focus resulting from spacing errors is given by:

which is negligibly small for the correct setup (k=DS/DT, DS being the effective sphere diameter, ρ=RS/RT, lS=RS/LS and lT=RT/LT). The left side in main brackets is aberration contribution of the sphere, and on the right side is aberration contribution of the surface on test.

◄ 4.5.3. Ronchi test ▐ 4.5.5. Hindle sphere test ►

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