The Analysis of the Form of Aspheric Lens Surfaces J.W. Mcbride, M. Hill, M. Jung Department of Mechanical Engineering, Universi

The analysis of the form of aspheric lens surfaces J.W. McBride, M. Hill, M. Jung

Department ofMechanica l Engineering, University of Southampton,

E-mail: [email protected], [email protected], [email protected]

Abstract: Recent developments in high precision machining have allowed for the design and manufacture of precision aspheric lenses. These have the advantage over conventional spherical lenses in that a multiple element lens can be replaced by a single aspheric lens. There are however difficulties both in the measurement of the lens surface after machining and the assessment of the quality of the lens. In this paper an aspheric lens surface is measured using a 3D Form Talysurf. The data are then analysed using an analytical least-squares technique to fit the points to an exponential function. The methods developed show that the machining methods can now be complemented by an analysis technique.

1. Introduction

When measuring an object in 3-dimensions a huge amount of data is produced and methods need to be developed to analyse these data-sets. The most straightforward technique is to fit all data points to one surface and then to analyse the deviations from the geometrical required shape. The method of least squares has a dominant place in assessing the deviation of form and is often recommended in standards (e.g. BS7172). However, depending on the nature of the noise included in the data other techniques could obtain a result more quickly and accurately, since the least squares technique is optimal only for

measurements with random, normally distributed error. In a previous investigation into contact lens measurement the fitting of spherical data-sets was investigated and some new techniques were developed

[1], [2] Currently lens designers mostly specify spherical surfaces, because their performance-to-cost ratio is higher than that of lenses with other contours. But, if aspherical lenses could be manufactured as easily as spherical ones they would replace them in most applications. In order to simplify the manufacture and to

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guarantee high quality it is necessary to develop techniques which allow a fit of the complex asphericai data-set to a surface describable by a few parameters.

2. Asphericai lenses

In the past lens designers tried to reduce the effects of aberration by combining several spherical lenses to one lens system or group. Each additional element gives some more degrees of freedom in design. Spherical aberration for example can be reduced by replacing a singlet with a cemented doublet lenses [3]. On the other hand light is reflected by each element, causing flare and reducing the intensity of the resulting light. About 4% of the light hitting the surface of a more dense media, the lens, is reflected on an uncoated glass interface (this can be calculated with the law of reflection and depends on the index of refraction of the optical glass [4]). Using asphericai elements in the design of lens groups has both advantages, more degrees of freedom and less elements with less potential for internal reflection. An example how spherical aberration can be corrected with an aspheric lens element is given in FIGURE 1.

Light rays passing through the edge of the aspheric lens will focus in the same point as light rays passing through the centre. For a conventional spherical lens the focus will change (dashed lines).

FIGURE 1: spherical aberration and its correction with an asphere

The positive effects of asphericai lenses are well understood however, their application is not as common as it might be expected. This is for the reason that it is difficult to manufacture high quality lenses at reasonable prices. Also the quality control is problematic since adequate measurement and analysis techniques are rare. There follows a brief summary of the more common manufacturing methods.

2.1 Manufacturing techniques - diamond turning

An economic way to manufacture asphericai lenses in small quantities is by single point diamond turning on a high precision lathe. The manufacturer is also able to react quickly to changing market demands since tooling costs are low

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[5]. The lenses for this study are manufactured using this diamond turning method.

2.2 Manufacturing techniques - moulding

The moulding technique is a very efficient way for producing aspherical lenses in large quantities. In the 1930's attempts were made to reduce costs of camera lenses by the use of plastic moulded lenses, however it took until the

1950's before acceptable quality of those lenses was achieved [6]. Male and female inserts are manufactured from steel or ceramic and the optical plastic is injected in-between. Once the lens material is cured the mould halves are separated and the lens extracted. Depending on the materials, the inserts can last several hundred thousand cycles before reconditioning is required which makes the production very cost efficient [7], [8]. The moulding technique is also used in the production of soft, nominally spherical contact lenses [1], [9].

2.3 Manufacturing techniques - hybrid

This is a new technique. An epoxy or acrylic layer, mounted on a spherical lens, is pressed into aspherical form using a 'stamp' [10], [11]. Once the aspheric surface is hardened it gets coated to improve durability and endurance against shock and climatic changes.

2.4 Other manufacturing techniques

A similar method to the hybrid technique is the Bright Press Method. A high temperature (400°C) glass block is pressed into aspherical form. This technique is only economic for large quantities [10]. A related technique is the Stressed Mirror Polishing Method. The elastic deformation of material under load is used to apply forces to a mirror blank so

that after a sphere has been ground the forces can be removed, and the polished spherical surface deforms elastically into the desired nonaxisymmetric surface [12].

A further method uses ultraviolet-light-hardening resin to form an aspherical surface layer on a glass lens [13].

3. Measurement

Initial studies have revealed problems in measuring the curved surface of contact lens moulds to the required accuracy [9]. A further project resulted in

the development of a number of software techniques and an initial comparison of measurement methods. The researcher investigated and compared 2D and 3D measurements taken on a laser radius-scope, a Rodenstock laser probe and a Form Talysurf. Although the stylus based measurement technique is seen as the

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current standard, a number of improvements were required, including the use of ball stylus, the reduction of start position errors, and the elimination of some system errors [1], [2]. Optical, high precision surfaces and in particular those used in the manufacture of contact lenses are difficult to measure with contact methods due to the mechanical properties of the materials used in manufacture. Existing non- contact methods on the other hand are often difficult or inaccurate when applied to aspherical, optical surfaces. Research in Southampton University is working on the development of new, 3D metrology instrumentation systems for rapid surface analysis of curved, optical objects.

4. Parametrization and pre-processing of the data

The aim in the parametrization of aspherical surfaces is to define mathematical functions, capable of describing a wide range of surfaces with a few parameters. This will enable the fitting of the measured data to a general form, which can then be compared with the required form. To have only a few parameters is also advantageous when the manufacturer needs to archive surface details since the five or so parameters will be representative of the whole data- set. In the ideal case the functions are modelled such that with some mathematical manipulation an algebraic solution can be obtained. This is advantageous in time and accuracy since iterative non-linear least squares techniques are not required. To describe points, graphs and surfaces a Cartesian co-ordinate system will be used throughout the paper The z-axis always represents the variation in height, consequently the xz-plane is used to display a 2-dimensional graph. Many forms of equations may be used including the following example which has advantages when used with a least squares fit.

^ [1]

To allow a linear least squaresfi tth e parameters are obtained in two steps:

i) find the offset in x, y and z, i.e. XQ, yo and ZQ and pre-process the data such that the vertex is in the origin of the global co-ordinate

system ii) manipulate the pre-processed data such that a linear least squares solution is possible and find the other parameters, i.e. a, n, b and c

Fitting aspherical surfaces is more complex than fitting a sphere as roll, pitch and possibly yaw must be taken into account. The three rotations are accommodated by three angles and in some cases it is necessary to extend step i) to find these angles prior to fitting.

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5. Form fitting of an aspheric lens

The data-set to be fitted is shown in FIGURE 2 and FIGURE 6 shows a 3D plot of the same surface. The measurement was taken on the Rank Taylor Hobson Talysurf at the University of Southampton. The form is described by 135 data-points, equally spaced along the x-axis and with a maximum resolution of lOnm (lnrn=10~^metre) in the z-axis. The original data-set contains

information about the lens in three-dimensions, however the numerical example will be shown in 2D only so that the actions can be visualised easily.

The Original Data-Set (all dimensions are in millimetre)

FIGURE 2: The original data-set as measured with the Talysurf. The data-set includes 135 points equally spaced on the x-axis, the dimensions show the real surface (no magnification or manipulation of the data).

From the general form of the data alternative functions to fit might be sinusoidal or a polynomial. The drawback with the sinusoidal curve is the non- linear nature of the function, hence an iterative non-linear least-squares technique would be required. The polynomial overcomes this problem, however

it involves more effort in fitting than the exponential function (equation 1) introduced in section 4. From FIGURE 2 it can be seen that the two valleys are not at the same height, indicating a rotation of the data. Prior to fitting, this rotation needs to be

eliminated. In order to correct for rotation the following equations can be used:

x = x • cos(/?) + y - sm(/?) [2]

y = -x • sin(/?) + y • cos(/?) [3]

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The 2 dimensional definition of the exponential surface (equation 1) is given by:

[4]

The offset parameters, i.e. XQ and ZQ have been found by pre-processing which simplifies equation [4] to:

r 2!" kx*>4-c(x*)*l r 2!" [&(or>c(x)<] = a\x •£"• J =a x •&• * [5]

Taking the logarithm of equation [5] gives:

\n(z) = l + n • b - x* 4- c • x* [6]

all the parameters which need to be found with a least squares technique, i.e. a, n, b, c are now linear. In its more general form, equation [6] can be written as:

where: z = ln(z), A = ln(a), fi(x) = In(x^), &(x) = x^ and fs(x) = x*. In FIGURE 3 the pre-processed and modified data are exhibited. The rotation and the offset are removed and it is also necessary to invert the data (z = -z) to take

logarithms of the z-values.

Rotated Modified Data Before & After Fitting (all dimensions are in millimetre)

1 --

-10 1]

FIGURE 3: The modified data, z , and the best fitting curve in a least squares sense. The figure does not show the two curves clearly. A better presentation is given in FIGURE 4.

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It can be shown that equation [14] solves the fitting problem in a least

squares sense [13]. Variable 'm' stands for the size of the data-set, in this case m=135, and functions fi(x) to fs(x) are as defined above.

Z/W ZAM

[8] /3/2

.'=1

FIGURE 4 shows that thefi ti s very close in the centre region but not so good in the valleys. This problem arises because the fitting is carried out on the logarithm of the data. Therefore a random error superimposed on the form (surface roughness from the manufacture) in areas closer to the origin (z-axis) is magnified while one further away is minimised. In order to compensate this a weighting function can be introduced into equation [8].

Rotated Original Data Before & After Fitting (all dimensions are in millimetre)

FIGURE 4: Afterfitting ,th e data can be re-transferred into original scale and the fitting curve can be compared with the original data.

The weighting function is derived from the gradient of the modified data. Plotting the unmodified data, z, over the modified data, ln(z), the gradient and hence the weighting function is given by:

— ln(z, / )\ = - weighting function. —: [9] dz z

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Multiplication of the error function, to be minimised by applying least squares, with the reciprocal of the gradient normalises the modified data, so that there is no maximisation or minimisation of the error at anyone point.

A calculation with the same data shows that it is advantageous to introduce such a weighting function (see FIGURE 5) since the overall representation is improved. It was possible to halve the standard deviation of the errors

(zmeasuremenL-Zfit) by using a weighted fit (unweighted: o=0.08 , weighted: 0=0.04).

Original Data Before & After Fitting (Weighted Errors) (all dimensions are in millimetre)

FIGURE 5: Introducing a \veightingfunction gives a better overall representation of the data.

With the assumption that the data used for fitting, which was only one trace of the 3D data-set, is representative not only for this trace but for the whole surface, a 3D diagram of the surface can be plotted (FIGURE 6).

FIGURE 6: On the left hand side the surface as 3D diagram, plotted with the parameters found in the calculation above. On the right hand side the same surface, cut through in the middle so that the cross-section can be seen. Please note that the diagram is plotted with no offset in x, y or z.

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6. Conclusion

The quality control of aspherical high precision lenses is still difficult and adequate assessment techniques need to be developed. When fitting the data

obtained from 3D measurement of a lens to a surface, it is important that no information gets lost and the representation is as accurate as possible. The introduction of a weighting function for a least squares fit was shown to be advantageous in the representation. Another feature which should be exhibited by thefittin g technique is numerical stability to random errors superimposed on the measured form. The exponential function introduced in this paper has some advantages over other possible functions which could be used for fitting with a linear least squares technique.

Acknowledgements

The authors would like to thank Ocular Sciences, Ltd. for the financial support of the project and Optics and Vision, Ltd. for providing aspherical lenses for

measurement proposes and lens surface details.

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390 Laser Metrology and Machine Performance

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