8149 engineering & laboratory notes

Profile Identification of Aspheric Lenses Héctor Aceves-Campos, Augen-Wechen Labs, Ensenada, B.C. México.

Abstract A mathematical method to iden­tify aspheric surfaces of revolu­tion is described. A spherical mechanical probe contacts the surface to be measured in the profilometric apparatus used. The method corrects for tilt, decenter-ing, and the use of a non-point contact probe.

During the manufacturing process of an ophthalmic lens, measurement of a glass surface mold at different stages is required.1 A profilometric tech­nique is used to characterize a complete line of aspheric curves. The profilometer constructed uses a spherical probe for convenience. In practice, decentering and tilt are introduced when the aspheric mold is set on the instrument (see Fig. 1). This note

Figure 1 . The profilometric apparatus that was used.

describes the mathematics used to characterize the aspheric. The non-point character of the probe, as well as decentering and tilt, are con­sidered in the analysis.

Theoretical development Figure 2 (next page) shows the contact between the probe and the surface to be mea­sured. The profilometer actually registers coordinates {xm , zm }. To make a transforma-tion from micrometer coordinates to actual ones, the geometric derivative of the path of the probe at {xm , zm } is used. The transforma­tion equations are

xi = χm + Rssinθ, zi = zm + Rs(1- cosθ), and

where Rs is the radius of the sphere at the end-point of the probe. Since the data taken is noisy, a smooth­ing of the data is done to evaluate the required derivative.

engineering & laboratory notes 8150

In the ophthalmic industry, an asphere may have the form of a conicoid εz2 - 2z/c + x2 = 0, where c represents the curvature (c = 1/R) at the vertex and describes different kinds of aspheres (e.g., ε = 1, sphere; ε = 0, parabola; ε < - 1 , hyperbola). Varying ε, for example, one may diminish astig­matism. Consider for the moment no tilt nor decentering. We find the best parameters {c, ε} by fitting the data to the conicoid using a least-squares procedure. The resulting equations are

Now consider the vertex of the conicoid being decentered by amounts {x0 , z0 } , and tilted by an angle θ with respect to the measuring system. Let {X, Y} be the new measured coordinates after eliminating the non-point character of the profilometer's probe. The conicoid adopts the form of a general second order equation:

where the coefficients, in terms of the parameters {ε, R, x0, z0, θ} are

Figure 2. Schematic of the measuring method with a non-point probe.

with φ = εsin2θ +cos2θ, ß = 1 + ε - φ, and m = (ε - 1) cosθsinθ. Once the coefficients {B, C, D, E, F} are obtained—also from a least-squares procedure—we find {x0 , z0 , θ} by solving the equations in Eq. 4. Then, we can transform the measured data {X, Z} to the "normal" coordi­nates {x, z} by applying the following transformation

Afterward, we can obtain {c, ε} from Eq. 2. For a polynomi­al representation for aspheres, the tilt and decentering are eliminated assuming first the curve is a conicoid, with all coordinates {X, Z} transformed to {x, y}. Later, a least-squares polynomial fitting is done.

Results The previous method was used on synthetic computed gen­erated curves and exact results were acquired. The numeri­cal computations are easily coded, since they involve simple mathematics, and fast results are obtained. The procedure was applied to a conicoid glass mold (75 mm) of theoretical {c = 1.890D, ε = +30.0}, cut with an Augen surface genera-

Figure 3. Difference in microns between the fitted conicoid and the data taken with Augen's profilometer.

tor. In Figure 3, the difference in sag (in micrometers) is shown between the fitted conicoid and the data. The fitted values {c = 1.888D, ε = +29.76} were obtained. The differ­ence in curvature and epsilon is 0.1% and 0.8%, respective­ly. The discrepancy is attributed to the cutting process of the glass mold.

Conclusion Explicit formulae are given to identify parameters of a sym­metric aspheric. Correction for tilt and decentering is explained explicitly, a thing that is not shown in other relat­ed works.2 The formulae described can be applied to data taken with commmercial profilometers or other similar methods. Moreover, the procedure does not require know­ing a priori the asphere equation, thus yielding a method to identify unknown aspheric lenses.

References 1. R. Palum, "Surface Profile Error Measurement for Small Rota-

tionally Symmetric Surfaces," Proc. SPIE 966, (SPIE Press, Bellingham, Wash., 1988), pp. 138-149.

2. D.P. Hamblen and M.R. Jones, "Lens curvature measurements by shadow projection profilometry," Eng. Lab. Notes in Opt. & Phot. News 6 (2 ) , (1995).