Dual-paraboloidal mirror

The measurement geometry of a dual-paraboloidal mirror reflectometer is shown in Fig. 2 in cross section. The orientation of the top (sample) paraboloid is similar to that of the hemispheroidal mirror with its open side lying in the X-Y plane. Because of circular symmetry about the Z axis, only the F-Z cross section is shown. The axes of the two paraboloids are collinear, with the sample centered on the axes. The sample and detector (or source) are located at Z = 0 and Z = — Lp, respectively. The radius of the sample paraboloidal mirror is given by 2f, where is its focal length. For comparison purposes, a quantity s" = rJ2f, analogous to the eccentricity, is defined for the sample paraboloid, where is the sample radius. The lower paraboloid has a focal length and radius 2fo- The separation between the focal points of the two paraboloids is denoted by Lp. 

Fig. 2. Dual-paraboloidal mirror reflectometer geometric parameters. The upper paraboloid has a focal length /, and radius 2L The lower paraboloid has a focal length /p and radius 2/ .

The image size on the detector (0/2jr), and hence the magnification of the mirror, depends on the BRDF of the sample. For Lambertian samples, the magnification will be a maximum. Knowledge of the maximum linear magnification of a hemispheroidal or dual-paraboloidal mirror is required to... 

The maximum magnification of a dual-paraboloid system has been evaluated by Hanssen (46) for the case of equal focal-length parabolas and coincident foci. If we define a quantity s" = rJ2f, then the maximum linear magnification of a dual-paraboloidal mirror withy = /d and Lp = 0 is given exactly by... 

Fig. 3. Maximum linear magnification from Ref. (46) for a hemispherical, hemiellipsoidal, and dual-paraboloidal mirror for the geometries discussed in the text. For the hemisphere, the ratio of the sample to hemisphere radius was set at 0.05.

Fig. 7. Normalized throughput of a dual-paraboloid, hemiellipsoid, and hemispherical mirror reflectoraeter versus the sample position.

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