Profile and deformation coefficients measurement of fast optical surfaces.

M. F. González-Cardel; R. Díaz-Uribe

doi:10.1364/OE.14.009917

1. Introduction

In recent years the use of aspherical optical surfaces in the design and fabrication of optical systems has increased considerably. The reason for this is the advantages that they offer for improving image quality with fewer surfaces when compared to systems with spherical surfaces; they have, however, as a drawback difficulty of fabrication and testing.

To have a description of the surface implies knowledge of the coordinates of each point of the surface, and/or the determination of all its parameters, such as paraxial radius of curvature, conic constant and all the deformation coefficients that permit calculations of these coordinates. To determine the quality of an optical surface implies the measurement of the differences between the ideal and the real surfaces; this is called optical testing [1]. The procedures to perform this task, can be classified as geometrical or interferometric methods. In general, the geometrical methods are simpler while the interferometric ones are more accurate.

Fast aspherical surfaces (in this work we consider f/#≤1), are particularly difficult to test by conventional methods, and it is necessary to modify them or, in some cases, to develop new methods. R. Díaz-Uribe, et al developed [2] a novel geometrical method consisting of measurement of the angle of the normal to the surface and its longitudinal aberration, when a laser beam is incident on the test surface and the reflected is in the retroreflection direction. Doing this for a set of points on the surface, the profile of the surface can be obtained.

The angle θ and the intersection X, of the normal line and the optical axis, for a set of points on a radial profile of the surface, give relevant information about the shape of the surface; however, they are not easy to interpret so a procedure is required to obtain the Cartesian coordinates (the semidiameter s and the sagitta z) for each point, starting from the X and θ data. A solution to this problem was previously proposed by means of a polynomial reversion [3, 4], but González-Cardel [3] demonstrated that this solution is valid only near the vertex, or to fast surfaces that do not differ much from a parabola.

A more general method to find s and z for each measured point is proposed in this work. The method is based on an iterative numerical integration that is applied to the experimental X and θ data set. In addition, because of the symmetry of revolution, an even degree polynomial is fitted to the obtained set of points by the least square method, to obtain the deformation coefficients.

In order to show the feasibility of the method, it was applied to find the deformation coefficients of two commercial surfaces. The results obtained with this method are contrasted with direct measurements obtained with a Coordinate Measuring Machine.

In section 2 the problem is described in detail together with the proposed solution. Then, in section 3 the iterative method is described for obtaining successive approximations. In section 4 results obtained with experimental data for two commercial fast optical surfaces are shown as well as the comparison with the CMM data; finally, in section 5 additional remarks about the results and on the proposed method, and the conclusions are presented.

2. Theory

According to Malacara [1] the equations that relate to the Cartesian coordinates (s, z) with the parameters (X, θ) are

X = z + \frac{s}{\tan θ},

and

\frac{dz}{ds} = \tan θ .

where X is the distance from the vertex to the intersection of the normal line with the optical axis (see Fig. 1), Eq. (2) can be rewritten as

z = \int \tan θ ds,

whose solution z(s) would be obvious for each couple of points (X, θ), if we knew the dependence of θ as function of s. For fast surfaces is difficult to measure θ as a function of s, this is why it was proposed a different measuring method by Díaz-Uribe, et al [2], in that paper however a complete solution was found only for conic surfaces. For more general aspherical surfaces there is not a simple analytical solution, so the next approximate numerical method is proposed. In Appendix A, a brief outline of the experimental measuring method described in Ref [2] is given.

Fig. 1. The relationship between the parameters X and θ with the Cartesian coordinates s and z is shown.

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2.1 Initial approximation

As in any numerical method it is convenient that the initial approach is near the real solution, then in this method we propose as initial approximation a surface that is the most similar possible to the test surface, this it could be a parabola S_A described by

z = \frac{s^{2}}{2 r},

where r is its paraxial radius of curvature.

2.2. First iteration

The first values in Cartesian coordinates for the measured points are obtained by finding the intersection between the normal line to the test surface (defined by X and θ) and the proposed surface S_A , as initial surface given by Eq. (4) (see Fig. 2). The family of normal lines to the test surface it is given by

z = - \frac{s}{\tan θ} + X .

Then, equaling the Eq. (4) and Eq. (5), and solving for s, the intersections are the solutions of the following second degree equation

\frac{s^{2}}{2 r} + \frac{s}{\tan θ} - X = 0,

that is,

s = \frac{- r \pm \sqrt{r^{2} + 2 Xr \tan^{2} θ}}{\tan θ} .

By considering only positive values of s, the plus sign in Eq. (7) must be used.

Fig. 2. The intersection point $P_{i}^{o}$ between the i-th normal line to the test surface S and the initial parabolic surface S_A are shown.

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Now we have an approximated value of s for each value of X and θ (Fig. 2), this set of values is a first approximation to the integration points with which a first approximation to the sagitta z can be evaluated by numerical integration, applying the trapezoid rule [6] as follows

z_{i + 1} = \int_{s_{0}}^{s_{i + 1}} \tan (θ) ds \approx \sum_{k = 0}^{i + 1} (\frac{\tan θ_{i + 1} + \tan θ_{i}}{2}) (s_{i + 1} - s_{i}) + z_{0},

Fig. 3. A first integration is using the trapezoid rule, to obtain approximation values, z₂ ,…, z_N .

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2.3 k-th iteration.

For the iterations 2, 3, and k, an analytic approximation, we have the individuals points obtained in the previous iteration instead. For the next approximations, the values of s are obtained from the intersection $s_{i}^{k}$ between the i-th normal and the straight line defined by the Cartesian coordinates (s, z) of the i-th and (i+1)-th points (Fig. 4). Using the Lagrange representation for a straight line [6], we have

z = \frac{s - s_{i}^{k}}{s_{i + 1}^{k} - s_{i}^{k}} z_{i + 1}^{k} + \frac{s - s_{i + 1}^{k}}{s_{i}^{k} - s_{i + 1}^{k}} z_{i}^{k} .

where the subscript i means that we are working with the i-th point and the superscript k means the k-th iteration. By equaling the Eqs. (5) and (9), an equation for $S_{i + 1}^{k + 1}$ as a function of the experimental data X_i+1 and θ_i+1 , and the k-th approximated coordinates ( $s_{i}^{k}$ ${, z}_{i}^{k}$ ), ( ${s_{i + 1}}^{k}$ , zi+1 k) is obtained as

s_{i + 1}^{k + 1} = \frac{{X_{i + 1} - \frac{s_{i + 1}^{k} z_{i}^{k} - s_{i}^{k} z_{i + 1}^{k}}{s_{i + 1}^{k} - s_{i}^{k}}}}{{\frac{z_{i + 1}^{k} - z_{i}^{k}}{s_{i + 1}^{k} - s_{i}^{k}} + \frac{1}{\tan θ_{i + 1}}}} .

The corresponding z_k+1 _i+1 is obtained through Eq. (8), where all the quantities have the same superscript k+1. Further approximations are obtained iteratively with Eqs. (8) and (10);

Fig. 4. Intersections between normal lines and the straight lines built with successive points (z₁, s₁) and (z₂, s₂); (z₂, s₂) and (z₃, s₃). The new values $s_{1}^{1}$ , $s_{2}^{1}$ , $s_{3}^{1}$ , … $s_{N}^{1}$ , are obtained. Further approximations are obtained in an iterative way.

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It can be demonstrated that the method converges [5] then it is possible to apply it successively until obtaining the desired approximation; i.e., until the next criteria is satisfied

max {∣ z_{i}^{k + 1} - z_{i}^{k} ∣, ∣ s_{j}^{k + 1} - s_{j}^{k} ∣} < ε i, j = 0, 1, \dots, n,

where ε it is the tolerance that must agree with the uncertainty of the experimental measurements

3. Application of the method

In order to show the feasibility of the method, it is applied to two real surfaces, the aspherical surfaces of two commercial single lenses.

3.1 Cinephore lens

The first one is the Cinephore lens, plane-convex lens distributed by Bausch & Lomb. The aspherical surface is convex, has a diameter of 100 mm and a paraxial radius of curvature of 44.95 mm; i.e, the focal ratio is f/0.225 when considered as a convex mirror. The experimentally measured parameters θ and X for that surface are shown on the Table 1. The experimental procedure proposed in Ref. [2], was used.

According to Diaz-Uribe [7], a way for determining the paraxial radius of curvature of the surface consists on fitting a straight line to the data on a plot of tan²θ vs X; the slope of the best fit equals a half of the paraxial radius of curvature and its ordinate intercept is the distance between the origin and the paraxial center of curvature. These values are used to get the first approximation.

Table 1. Data (θ±0.002°, X±0.01 mm) for the aspheric surface of the Cinephore lens.

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In Cartesian coordinates, a parabola with a paraxial radius of curvature of 45.00 mm is described by the equation

z = \frac{1}{2 (45.00)} s^{2} .

The Eq. (12) will be considered as the initial surface to apply the proposed method. After 11 iterations the maximum numerical error between iterations is 1.73×10^-13 mm. The differences between successive iterations are shown in Fig. (5), so the process of convergence of the method can be observed.

Fig. 5. Differences between successive iterations, a) in a linear scale and b) in logarithmic scale to shown better the variation of last iterations

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The obtained data for the Cartesian coordinates s and z for each of the measured points on the surface are listed in Table 2 and plotted in Fig. 6.

According to Ref. [1], z can be approximated by a polynomial with only even terms as

z = \sum_{j = 0}^{5} D_{2 j} s^{2 j},

Using the least square method to fit the polynomial in Eq. (13) to the data of the Table 2, the D coefficients turn out to be those listed in Table 3.

The resulting polynomial approximates the obtained data with a maximum deviation in z of 0.0085 mm.

The zero order term accounts for a displacement of the vertex from the origin of coordinates (defocusing), in this case is of only one micrometer; this result shows that the initial alignment was done quite well.

Table 2. Data of the lens Cinephore of Bausch & Lomb obtained by the method of the numerical integration.

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Table 3. Deformation coefficients for the Cinephore lens

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Starting from Eqs. (2) and (13) a relationship for θ can be obtained as

\tan θ = \sum_{j = 1}^{5} 2 j D_{2 j} s^{2 j - 1}

X can be obtained from Eq. (6), so values of θ and X for each value of s can also be recovered. Doing this, a maximum difference of 0.24 degrees for the angle θ and 0.38 mm for X, were obtained for the measured point farthest from the vertex; the rms difference in the angle is 0.045 degrees and for the longitudinal aberration X the rms difference is 0.082 mm. The corresponding differences for each experimental point are shown in the Figs 8 and 9.

Fig. 6. Profile of the lens Cinephore obtained by numerical integration

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The paraxial radius of curvature is related to the second order coefficient of the polynomial in Eq. (13) by

r = \frac{1}{2 D_{2}}

being 43.975 mm the numerical result for the measured surface; this result has a difference of about 2 %, regarding the value obtained in Díaz-Uribe [2], however, the Numerical Integration method (NIM) can deal with more general aspherics, not only with conics.

3.2 Lens LAG 017

The second surface evaluated is the aspheric convex surface of the LAG 017 condenser lens distributed by Melles Griot [8], whose back surface is spherical. The θ and X data obtained for the first surface are listed in the Table 4:

Table 4. Obtained experimental data of the lens LAG 017 (θ±0.002°, X±0.01 mm).

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Again, a linear Least Square fit for the tan2θ vs X plot gives a first approximation to the paraxial radius of curvature; the obtained value is r=21.35 mm. Then, the initial surface is described by the equation:

z = \frac{s^{2}}{2 (21.35)}

After eleven iterations, the obtained data set for the Cartesian coordinates of the surface are plotted in Fig. 7, the numerical error between the last two iterations is less than 4×10^-10 mm. The data for the plot in the Fig. 7 are shown in the Table 5.

The best tenth degree, even polynomial fit to the data of the Fig. 7 (Table 5) is given by the coefficients in Table 6.

Fig. 7. Profile of the surface LAG 017, obtained by the proposed numerical method

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Table 5. Data of the lens LAG 017 obtained by the method of the numerical integration

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Table 6. Deformation coefficients for the LAG017 lens

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The maximum difference in z for the polynomial that is fitting to the data have a of 0.0055 mm. and a value rms of .0019 mm; the defocusing (zero order term) is less than two tenths of a micrometer.

As in the previous case, the values of θ y X for each of the measured points can be recovered by using the coefficients in Table 6, Eqs. (5) and (13). The resulting differences for θ, between the values obtained with the best fit polynomial and the experimentally measured values, are shown in Fig. 8, from which a maximum difference of 0.34 degrees (0.0059 radians) and a rms value of 0.074 degrees (0.0012 radians) can be observed.

For the variable X, the resulting differences between the values obtained with Eq. (5) and the experimentally measured values, are shown in Fig. 9. The maximum difference is 0.21 mm (0.66%) for the farthest point from the vertex, and the rms value is 0.072 mm.

Fig. 8. Differences in θ angle associated to each point experimentally measured on the aspheric surfaces of the Cinephore and LAG 017 lenses.

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We can also recover the paraxial radius of curvature through Eq. (15), being 21.52 mm. This value has a difference smaller to 0.8 % with the paraxial radius of curvature reported by the manufacturer that is of 21.36 mm [9].

Fig. 9. Differences in X for each experimentally measured point on the aspheric surfaces of the Cinephore and LAG 017 lenses.

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4. Comparison with CMM measurements

A Coordinate Measuring Machine (CMM; Carl Zeiss MC850 operated in the absolute mode) was used as an alternative method to measure the Cartesian coordinates for each surface; the uncertainty for each of the three coordinates x,y,z, of the center of the stylus position is ±0.001 mm; after processing the data, however, the uncertainty varies from 0.003 to 0.021 mm, from the center to the edge of the surface. For comparison, the coefficients of best fit polynomial obtained with the Numerical Integration Method (NIM) and the coefficients obtained by fitting the CMM data are shown in Table 7. In spite of the differences between corresponding coefficients increase with the order of the coefficients, their contribution to the sagitta is decreasing, so the differences in sagitta are not so high; this is clearly shown in Fig. 10. For the Cinephore lens the maximum difference in sagitta is |Δz|=0.35 mm (2 %), while for the LAG 017 lens this value reduces to |Δz|=0.26 mm (1.9 %). In addition, from table 7 it can be observed that the paraxial radius of curvature obtained with the NIM differ slightly from the values obtained with the CMM (0.11% for the cinephore lens, while for the LAG lens this is 1.0 %). It is well worth to remark here that for the very central part of the surfaces (s≤10 mm), the differences |Δz|≤0.010 mm, as is clearly shown in detail in the small plot of Fig. 10. Then, the large differences are for the points far from the vertex. Presently it is not clear why this happens, so we have to do a further analysis in future work.

Fig. 10. Sagitta differences between the best fit polynomial obtained for the NIM and CMM data for the Cinephore lens

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5. Analysis of the method

5.1 The initial approximation

In order to know the influence of the initial approximation on the results obtained with the NIM, other simpler surfaces, such as spheres or axicons, were proposed; it was found that they can be used, provided that their intersections with the normals to the test surface can be obtained; in such a case, Eqs. (4), (6), and (7) must be properly changed, in the Appendix B the modified equations are deduced. For an approximating sphere, its radius of curvature must be long enough to fulfill that condition; in this case, the Cartesian coordinates obtained for the test surface, after eleven iterations, differs less than 10^–17 mm from the data in Table 5 NIM with an initial approximation given by a parabolic surface described by Eq. (4). For an axicon, on the other hand, the angle of the generating straight line must be less than 45 degrees; in this case, the differences are smaller than 10^-14 mm from the data in Table 6, also after eleven iterations.

Table 7. Paraxial curvature radius and coefficients of deformation for the surfaces characterized by the method of numerical Integration.

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5.2 Convergence

It is possible to show [5] that the method is convergent whenever next condition is satisfied

\frac{∣ z_{i}^{k} - z_{i}^{k - 1} ∣}{∣ s_{i}^{k} - s_{i}^{k - 1} ∣} < \frac{1}{\tan θ_{i}} .

Several numerical trials show that choosing a parabola as initial approximation meets that condition.

5.3 Stability and precision

The NIM is very stable, because when random deviations are introduced in the experimental data, similar to the variations that commonly are involved in the experimental work, the method converges to the same surface with a maximum difference of 6 µm. In order to find this result, some numerical simulations were conducted for maximum deviations of 10 millidegrees (36 arcsecs) in the angle θ, and 10 µm in X; these are typical uncertainties for the rotatory and linear stages used for measuring those quantities. This 6 µm stability can be considered a representative measure for the precision of the method.

6. Conclusion

A numerical method that allows to find the Cartesian coordinates from measurements of the angle θ and the intersection point X of the normal with the optical axis has been proposed, the method is relatively simple and easy to be applied, it gives good resolution and is susceptible to be improved. The precision obtained with our measurements for the tested surfaces are about of ten micrometers in z and s.

APPENDIX A: Experimental method for measuring θ and X.

An unexpanded laser beam is aligned with the rotation axis of a nodal slide. The test surface is placed on the slide; initially, its optical axis must be aligned with the laser beam in such a way that the beam is retroreflected from the vertex of the surface Fig. A.1). Then, using the rotary stage of the nodal slide, the surface is rotated an angle θ, so the beam is not retroreflected anymore (Fig. A.2). Now, the surface is displaced along its optical axis until the retroreflection condition is attained again; the laser beam is now directed along the normal direction of the surface on the point of incidence, so the reading of the linear stage scale gives the X parameter (Fig. A.3). Repeating the same procedure for different values of θ, corresponding values of X are obtained, for several points along a profile of the surface. For a detailed explanation of the practical issues involved in this measurement procedure, please see Ref. [2]

Fig. A.1 .1 Initial alignment of the test surface with the laser

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Fig. A.2 .2 After rotation of the test surface the laser beam is not retroreflected

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Fig. A.3 .3 Linear displacement of the surface gives again the retroreflection condition.

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APPENDIX B: Other initial approximations

a) Spherical surface

The equation that describes a sphere, (a indeed, circumference in the plane of the measured profile) is

z = R \pm \sqrt{R^{2} - s^{2}},

where R is the radius of the proposed circumference, and it should be the sufficiently large as for intersect to all the normal lines.

In this case, the intersections of the normals with the surfaces can be obtained when equaling the Eqs. (4) and (18), and solving for s, these intersections are given by the following expression:

s = \frac{\tan θ}{\tan^{2} θ + 1} [X - R + \sqrt{R^{2} + \tan^{2} θ (2 XR - X^{2})}],

where R is the radius of the proposed circumference.

b) An axicon

An axicon is a straight line in the plane of the measured profile, it can be used as initial approach, whenever its slope is such that the intersections with the normal ones are in the same quadrant that the testing surface. It is described by the equation

z = ms,

where m is the slope of the straight line.

Now the intersections are obtained when equaling the Eqs. (4) and (20), and solving for s, these intersections are given by the following expression:

s = \frac{X \tan θ}{m \tan θ + 1},

Eq. (19) or Eq. (21) can be used instead of Eq. (7) in order to calculate the initial approximation for s, the rest of the numerical procedure is the same.

Acknowledgments

The authors of this paper are indebt with Benjamin Valera and Sergio Padilla of the Laboratorio de Metrología of CCADET-UNAM, for his help in measuring the surfaces with the CMM. This research was partially sponsored by Consejo Nacional de Ciencia y Tecnología under grant 37077-E.

References and links

1. D. MalacaraOptical Shop Testing, 2nd Ed., (John Wiley & Sons, Inc., E.U.A, 1992).

2. R. Díaz-Uribe, A. Cornejo-Rodríguez, J. Pedraza-Contreras, O. Y. Cardona-Núñez, and A. Cordero-Davila, “Profile measurement of a conic surface, using a He-Ne laser and a nodal bench,” Appl. Opt. , 24, 2612–2615 (1985). [CrossRef] [PubMed]

3. R. Dìaz, R. Pastrana, and A. Cornejo, “Profile measurement of aspheric surfaces by laser beam reflection,” Proc. SPIE , 813, 355–356 (1987).

4. R. Díaz Uribe, Pruebas ópticas por deflectometría láser (Optical testing by laser deflectometry), Ph. D. dissertation (Facultad de Ciencias UNAM, México1990).

5. M. González Cardel, Determinación de los coeficientes de asfericidad de una superficie óptica rápida (Aspheric coefficients determination for a fast optical surface) M Sc. dissertation (Facultad de Ciencias, UNAM, México2003).

6. O. Patenleeva Vladimirova and M. González Cardel, Métodos Numéricos (Ed. Instituto de Investigación en Tecnología Educativa de la Universidad Tecnológica de México, México2002).

7. R. Díaz-Uribe and A. Cornejo-Rodríguez, “Conic constant and paraxial radius of curvature measurement for conic surfaces,” Appl. Opt. 25, 3731–3734 (1986). [CrossRef] [PubMed]

8. Melles — Griot, Product Catalog. http://shop.mellesgriot.com/products/optics/detail.asp?pf_id=01%20LAG%20017&plga=024192&mscssid=

9. ZEMAX, Optical Design Program, User’s Guide Version 7.0, Focus Software, Incorporated, USA, March 1998.

θ (°)	X (mm)	θ (°)	X (mm)	θ (°)	X (mm)	θ (°)	X (mm)
1.30	44.02	13.00	45.61	23.00	48.90	33.00	53.28
2.60	44.06	14.00	45.88	24.00	49.24	34.00	53.93
3.90	44.14	15.00	46.08	25.00	49.61	35.00	54.62
5.19	44.24	16.00	46.32	26.00	49.99	36.00	55.43
6.47	44.38	17.00	46.59	27.00	50.4	37.00	56.38
7.74	44.55	18.00	46.99	28.00	50.82	38.00	57.51
9.00	44.75	19.00	47.42	29.00	51.26	39.00	58.88
10.00	45.04	20.00	47.79	30.00	51.7	40.00	60.65
11.00	45.21	21.00	48.15	31.00	52.17	41.00	62.29
12.00	45.45	22.00	48.52	32.00	52.72	42.00	63.47

s(mm)	z (mm)	s(mm)	z (mm)	s(mm)	z (mm)	s(mm)	z (mm)
1.00	0.01	10.25	1.19	19.03	4.06	28.62	9.21
2.00	0.05	11.09	1.39	19.94	4.46	29.70	9.92
3.00	0.10	11.92	1.60	20.86	4.88	30.78	10.66
4.00	0.18	12.76	1.84	21.78	5.32	31.93	11.48
5.00	0.28	13.61	2.09	22.73	5.79	33.15	12.39
6.00	0.41	14.50	2.37	23.68	6.29	34.46	13.39
7.00	0.56	15.41	2.67	24.64	6.81	35.91	14.54
7.82	0.69	16.31	2.99	25.60	7.35	37.55	15.90
8.62	0.84	17.21	3.33	26.58	7.93	39.15	17.26
9.45	1.01	18.11	3.68	27.60	8.55	40.52	18.47

D₀	D₂	D₄	D₆	D₈	D₁₀
-9.62×10^-4	1.137×10^-2	-8.22×10^-7	1.33×10^-9	-7.63×10^-13	1.40×10^-16

θ(grad)	X (mm)	θ(grad)	X (mm)	θ(grad)	X (mm)	θ(grad)	X (mm)
0.80	21.36	14.00	21.86	28.00	23.22	42.00	26.79
1.60	21.37	15.00	21.90	29.00	23.40	43.00	27.07
2.50	21.38	16.00	21.93	30.00	23.65	44.00	27.41
3.30	21.40	17.00	21.97	31.00	23.85	45.00	27.75
4.10	21.43	18.00	22.05	32.00	24.16	46.00	28.15
4.30	21.46	19.00	22.12	33.00	24.41	47.00	28.54
5.70	21.51	20.00	22.17	34.00	24.63	48.00	29.00
7.00	21.51	21.00	22.23	35.00	24.94	49.00	29.45
8.00	21.57	22.00	22.33	36.00	25.13	50.00	29.87
9.00	21.64	23.00	22.43	37.00	25.52	51.00	30.30
10.00	21.68	24.00	22.54	38.00	25.76	52.00	30.84
11.00	21.73	25.00	22.65	39.00	26.00	53.00	31.38
12.00	21.78	26.00	22.85	40.00	26.25
13.00	21.82	27.00	23.04	41.00	26.50

s (mm)	z (mm)	s (mm)	z (mm)	s (mm)	z (mm)	s (mm)	z (mm)
0.00	0.00	4.91	0.56	10.42	2.60	16.92	7.03
0.30	0.00	5.29	0.66	10.85	2.82	17.40	7.46
0.60	0.01	5.67	0.75	11.28	3.06	17.88	7.90
0.93	0.02	6.04	0.86	11.74	3.32	18.38	8.37
1.23	0.04	6.42	0.97	12.18	3.58	18.88	8.87
1.53	0.05	6.81	1.09	12.67	3.88	19.41	9.40
1.61	0.06	7.20	1.22	13.14	4.18	19.93	9.95
2.14	0.11	7.58	1.36	13.59	4.48	20.49	10.55
2.62	0.16	7.96	1.50	14.09	4.82	21.03	11.17
3.00	0.21	8.35	1.65	14.53	5.13	21.55	11.78
3.38	0.27	8.75	1.82	15.06	5.53	22.08	12.42
3.76	0.33	9.15	1.99	15.53	5.88	22.65	13.14
4.15	0.40	9.55	2.17	15.99	6.25	23.22	13.88
4.53	0.48	9.98	2.38	16.46	6.64

Profile and deformation coefficients measurement of fast optical surfaces.

Abstract

1. Introduction

2. Theory

2.1 Initial approximation

2.2. First iteration

2.3 k-th iteration.

3. Application of the method

3.1 Cinephore lens

3.2 Lens LAG 017

4. Comparison with CMM measurements

5. Analysis of the method

5.1 The initial approximation

5.2 Convergence

5.3 Stability and precision

6. Conclusion

APPENDIX A: Experimental method for measuring θ and X.

APPENDIX B: Other initial approximations

Acknowledgments

References and links

Cited By

Figures (13)

Tables (7)

Equations (21)

Optics Express

Surface	Cinephore		LAG 017
Method	NIM	CMM	NIM	CMM
Paraxial radius of Curvature (mm)	43.975	44.025	21.524	21.737
D0 (10^-3 mm)	-0.962	0.160	-0.135	189.14
D2 (10^-2 mm^-1)	1.137	1.135	2.32	2.30
D4 (10^-7 mm^-3)	-8.22	-17.42	89.98	130
D6 (10^-9 mm^-5)	1.33	2.62	-31.27	-67.8
D8 (10^-12 mm^-7)	-0.763	-1.50	70.97	164
D10 (10^-15 mm^-9)	0.140	0.292	-0.5190	-127.57

D₀	D₂	D₄	D₆	D₈	D₁₀
-1.35×10^-4	2.32×10^-2	8.998×10^-6	-3.127×10^-8	7.097×10^-11	-5.19×10^-14

D₀	D₂	D₄	D₆	D₈	D₁₀
-1.35×10^-4	2.32×10^-2	8.998×10^-6	-3.127×10^-8	7.097×10^-11	-5.19×10^-14