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4.8.3. Ronchi test   ▐    4.8.5. Hindle sphere test

4.8.4. 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 surface with a negative conic in general - is compensated for by a smaller concave sphere (FIG. 55).

FIGURE 55: Waineo null test setup: the sphere is facing test surface, both centered around 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, with the image formed by it being object for the test surface (with its corresponding object distance LT=M+IS, IS being the sphere-to-image separation and M the mirror separation), 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 (centered, and at the proper separation ensuing that the beam width nearly coincides with surface) in order for the diverging cone from the sphere to match nearly exactly the test surface.

According to Eq. 9, the combined primary spherical aberration contributions of the two surfaces as the P-V error at paraxial focus is given by:

W(S+T)= [(ΩS-1)2(kd)4/4RS3] + [K+(ΩT-1)2d4/4RT3]

= [(ΩS-1)2(k43)+K+(ΩT-1)2](d4/4RT3)

where Ω is the inverse object (point source for the sphere, and image formed by the sphere for the test surface) distance (LS and LT) in units of the radius of curvature R (subscripts S and T refer to the sphere and test surface, respectively), k is the ratio of sphere vs. test surface effective diameter, d is the effective pupil radius of the test surface, K is the test surface conic and ρ is the ratio of curvature radii RS/RT (since the reflective index n and radius R both change sign going from one surface to another, the index from Eq. 9 can be omitted and R can be assumed numerically positive for both surfaces). 

The left side of the sum represents the wavefront error of the sphere, and the right side that of the test surface. Since all parameters except the test surface conic K are numerically positive, the condition for zero sum is [K+(ΩT-1)2]<0, i.e. K<0 and |K|>(ΩT-1)2. Since the object for the test surface is the imaginary focus of the sphere FS, and the requirement for a real final focus means that distance from this imaginary focus to the test surface has to be larger than its focal length, a two alternatives for the sphere's imaginary focus location are either somewhat inside test surface's center of curvature, or somewhat outside of it. Since the latter allows for much smaller mirror separation (in part due to the need to avoid too large obstruction of the converging beam by the sphere when it is inside the beam), it is the more practical one. The final focus forms at relatively short distance behind the sphere, which requires either a central hole on it, or diverging mirror in front of it.

Consequently, object separation LT for the test surface is larger than its radius of curvature RT, and ΩT<1. For paraboloid, LT is closer to 2RT, closer to RT (larger) for prolate ellipsoids and around 2RT for mild to moderate hyperboloids for hyperboloids. Toward a mild prolate ellipsoid test surface, sphere's imaginary focus and the final focus are shifting closer to the test surface's center of curvature, as the result of the sphere becoming smaller in order to match lower spherical aberration of such surface (and opposite for more strongly aspherized surfaces). With LT in the 1.1-2RT range from weak ellipsoid to weak hyperboloid test surface, ΩT ranges from 0.9 to 0.5, gravitating toward 0.7.

Consequently, (ΩT-1)2 ranges from 1/100 to 1/4 from weak prolate ellipsoid to a mild hyperboloid as the test surface.

Similarly, since the sphere forms a diverging beam, its object - the point source - is inside its focus, hence LS is more than twice smaller than RS, and ΩS>2. Roughly, LS ranges between RS/5 for weak prolate ellipsoids to RS/3 for paraboloids (and larger for hyperboloids), with ΩS in the 5 to 3 range, and, correspondingly, (ΩS-1)2 is in the 16 to 4 range, respetively. Since the sum [(ΩS-1)2(k43)+K+(ΩT-1)2] has to be zero for a null, the much larger value of (ΩS-1)2 has to be offset by the correspondingly smaller value of k4 (for given sphere radius, i.e. radii ratio ρ), in order to match numerically opposite in sign value of the test mirror contribution, proportional to
ΩT-1)2. In other words, as the test mirror conic increases toward zero, and its aberration contribution diminishes, the effective aperture of the sphere DS=kDT also decreases, theoretically to zero for K=0.

Calculating needed setup parameters is a pretty involved procedure, since the radii, separations and effective aperture values need to result in near zero spherical aberration sum for the given test surface conic. 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 f/4, can be null tested with less than half its diameter high-quality sphere, say f/4, to a 1/175 wave RMS setup accuracy; a 500mm f/3 paraboloid can be nulled with a 9-inch f/3.6 sphere to better than 1/40 wave RMS accuracy, and so on.

Additional advantage of the test vs. regular tests for paraboloid 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:

 mT = -fT/(LT-fT)

fT being the test surface focal length. Change in the magnification is caused by the change in test surface object distance LT=IS+M, due to the change in separation of the image formed by the sphere, IS=RS/[2-(RS/LS)]=RS/(ΩS-2), with LS and RS being object separation (i.e. source-to-sphere separation) and curvature radius for the sphere, respectively. The effect of change in separation between the two mirrors is comparatively negligible. 

    Note that mirror-to-image separation I in terms of mirror radius of curvature R and object distance o is given by I=R/[2-(R/o)] - or by I=f/[1-(f/o)] in terms of the focal length f - where a negative I value for R and o of the same sign indicates object location inside the focal point and diverging rays (i.e. forming virtual image). As the setup illustration indicates, this is the manner in which the Waineo sphere forms its image of the source. In this configuration, according to the sign convention, mirror-to-image separation is numerically positive. Since n/R is positive for the sphere, making RS effectively positive, IS is also made positive by expressing it as I=RS/(ΩS-2).

For illustration, in the above Waineo setup for 400mm f/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 f/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.

According to the wavefront error equation above, for given curvature radii and test surface conic, the only spacing-related variables are the effective mirror apertures and the relative inverse object distance Ω. Changes in the effective aperture at the two surfaces are in the same direction, thus relatively insignificant. Changes in the Ω factor, with ΩS=RS/LS and ΩT=RT/LT=RT/(M+IS)=RT/{M+[RS/(ΩS-2)]}, are directly dependant on the source-to-sphere separation LS in the case of ΩS, and less directly but significantly in the case of ΩT.

Denoting the change in (ΩS-1)2 due to the change in the source-to-sphere separation LS by ΔS, and the corresponding change in K+(ΩT-1)2 as ΔT, the resulting P-V wavefront error of primary spherical aberration at the best focus is:

WΔ= SDS4/256RS3)+(ΔTDT4/256RT3)

= [(ΔSk43)+ΔT]DT4/256RT3

with the parameters needed for calculation given by the specifics of the particular setup. In the above setup for a 400mm f/4 paraboloid (DT=400, RT=3200,  K=-1), with the test sphere parameters DS=85mm (k=0.425), Rs=1600mm (ρ=0.5), LS=552mm, M=2431mm, and ΩS=RS/LS=2.89855, -1mm change in LS results in ΩS change of +0.0053 and (QS-1)2 change of +0.02, which is the value of ΔS. At the test surface, the +0.0053 change in ΩS and -1mm change in mirror separation M result in ΩT change from 0.7598  to 0.7254, i.e. by -0.034. It causes (ΩT-1)2 change by -0.0177, which is the value of change at the test surface, ΔT. The corresponding P-V wavefront error of primary spherical aberration (best focus) induced is W=(0.26ΔS+ΔT)DT4/256RT3=-0.000038mm, or 1/14.4 wave in units of 550nm wavelength. Raytrace gives about twice smaller error, the difference probably coming mainly from the best focus being optimally balanced with the higher-order spherical aberration.

4.8.3. Ronchi test   ▐    4.8.5. Hindle sphere test

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