Methods and apparatus of aligning surfaces

Abstract

Ultra-precision freeform surfaces are important to the development of complex and micro-optical-electro-mechanical devices used in many photonics and telecommunication products such as F-theta lenses for laser printers. These surfaces are complex and large scale surface topologies with shapes that generally possesses no rotational symmetry. Due to the geometrical complexities of these ultra-precision freeform surfaces, it is difficult to characterize the form accuracy and surface quality of freeform optical surfaces. The method of this invention is based on feature-point pre-fixture, and iterative precision alignment algorithm, which can provide sufficient capability of form characterization for ultra-precision freeform surfaces with form accuracy down to below sub-micrometer range

Description

BRIEF DESCRIPTION OF THE DRAWINGS

Preferred embodiments of the present invention will now be explained by way of example and with reference to the accompanying drawings in which:

FIG. 1 shows a flow chart of typical form characterization for two surface patterns;

FIG. 2 shows exemplary feature points of a surface to minimize first-degree separation (pre-fixture);

FIG. 4 shows exemplary computation of projection point on a surface to minimize second-degree separation

FIG. 5 shows the flow chart of form characterization for freeform surfaces of this invention;

FIG. 6 shows the interface of a program written according to the form characterization for freeform surfaces of this invention;

FIG. 7 shows a theoretical freeform surface used in the example;

FIG. 8 shows matching error in the example with only x-shift;

FIG. 9 shows matching error in the example with only y-shift;

FIG. 10 shows matching error in the example with only z-shift;

FIG. 11 shows matching error in the example with x-y-z-shift;

FIG. 12 shows matching error in the example with only x-rotation;

FIG. 13 shows matching error in the example with only y-rotation;

FIG. 14 shows matching error in the example with only z-rotation;

FIG. 15 shows matching error in the example with x-y-z-rotation;

FIG. 16 shows matching error in the example with shift and rotation and 15×15-measured points;

FIG. 17 shows matching error in the example with shift and rotation and 50×50-measured points;

FIG. 18 shows exemplary design aspheric surface of cameral lens;

FIG. 19 shows exemplary measured surface of cameral lens;

FIG. 20 shows the form error surface obtained by the method of this invention when the measured surface in FIG. 19 is compared with the design surface in FIG. 18;

FIG. 21 shows the form error surface obtained by a traditional method (Talymap software);

FIG. 22 shows a workpiece of F-theta lens insert;

FIG. 23 shows the design surface of the insert of F-theta lens of FIG. 22;

FIG. 24 shows the measured surface of the workpiece shown in FIG. 22; and

FIG. 25 shows the form error surface of the measured F-theta surface comparing with the design surface of FIG. 23.

Claims

1. A method of aligning a first surface with a second surface, such that P(xi,yi,zi) is any point on the first surface and Q(x′i,y′i,z′i) is any point on the second surface, said first surface and said second surface being separated by a first-degree separation, a second-degree separation, and a third-degree separation, including the steps of: minimizing the first-degree separation by the steps of: selecting at least one feature point form each of the first surface and the second surface;adjusting the coordinate of the second surface to minimize the distance between said at least one feature point of each of the first surface and the second surface to be below a predetermined threshold value θ;minimizing the second-degree separation by the steps of dividing the first surface into a plurality of first zones, each first zone having at least one first coordinate defining said first zone;dividing the second surface into a plurality of second zones, each second zone having at least one second coordinate defining said second zone;for the plurality of first zones, repeating the steps of:a) obtaining distances between the at least one first coordinate and each of the plurality of second zones;b) comparing the distances obtained in step a) to determine a pair of corresponding zones on the first and the second surfaces having a minimum distance obtained in step a); until all zones on the on the first and the second surfaces are paired;determining the distances between each pair of corresponding zones on the first and the second surfaces;obtaining a summation of the distances between each pair of corresponding zones;adjusting the coordinate of the second surface to minimize the summation of the distances between each pair of corresponding zones to be below a predetermined threshold value ε;minimizing the third-degree separation by the steps of: minimizing a transfer matrix T to be below a predetermined threshold value λ by adjusting at least one pair of parameters along a direction, one of said parameters relates to translation and the other parameter relates to rotation along the direction.

2. The method of claim 1, wherein said at least one feature point includes five feature points.

3. The method of claim 2, wherein said five feature points includes a gravity center point G and four corner points

4. The method of claim 3, wherein the gravity center point G is identified by the equation

5. The method of claim 3, wherein the four corner points are identified as the four points that are farthest from the gravity center point G.

6. The method of claim 1, wherein Q(x′i,y′i,z′i) is obtained through a set of coordinates processed by a parameterization algorithm selected from the group consisting of Bezier surface algorithm, B-Spline algorithm, and Non-Uniform Rational B-Spline algorithm.

7. The method of claim 1, wherein each zone has a square shape.

8. The method of claim 1, wherein three pairs of parameters are adjusted along respective three orthogonal directions.

9. The method of claim 8, wherein the transfer matrix T is defined as:

10. The method of claim 9, wherein T satisfying an expense function of

11. A method of determining deviation between a first surface with a second surface, such that P(xi,yi,zi) is any point on the first surface and Q(x′i,y′i,z′i) is any point on the second surface, said first surface and said second surface being separated by a first-degree separation, a second-degree separation, and a third-degree separation, including the steps of: aligning the first surface with the second surface according to the method of claim 1;calculating a distance di between P(xi,yi,zi) and Q(x′i,y′i,z′i), wherein di=±√{square root over ((xi−xi)2+(yi−yi)2+(zi−zi)2)}{square root over ((xi−xi)2+(yi−yi)2+(zi−zi)2)}{square root over ((xi−xi)2+(yi−yi)2+(zi−zi)2)}.

12. The method of claim 11 further including the step of calculating a profile error St, wherein St=|max(di)−min(di)|.

13. The method of claim 11 further including the step of calculating a root mean square deviation Sq wherein

14. An apparatus for aligning a first surface with a second surface, such that P(xi,yi,zi) is any point on the first surface and Q(x′i,y′i,z′i) is any point on the second surface, said first surface and said second surface being separated by a first-degree separation, a second-degree separation, and a third-degree separation, including a processor incorporating an aligning algorithm, said aligning algorithm aligns the first and second surfaces by the steps of: minimizing the first-degree separation by the steps of: selecting at least one feature point form each of the first surface and the second surface;adjusting coordinates of the second surface to minimize the distance between said at least one feature point of each of the first surface and the second surface to be below a predetermined threshold value θ;minimizing the second-degree separation by the steps of dividing the first surface into a plurality of first zones, each first zone having at least one first coordinate defining said first zone;dividing the second surface into a plurality of second zones, each second zone having at least one second coordinate defining said second zone;for the plurality of first zones, repeating the steps of:c) obtaining distances between the at least one first coordinate and each of the plurality of second zones;d) comparing the distances obtained in step a) to determine a pair of corresponding zones on the first and the second surfaces having a minimum distance obtained in step a); until all zones on the on the first and the second surfaces are paired;determining the distances between each pair of corresponding zones on the first and the second surfaces;obtaining a summation of the distances between each pair of corresponding zones;adjusting the coordinates of the second surface to minimize the summation of the distances between each pair of corresponding zones to be below a predetermined threshold value ε;minimizing the third-degree separation by the steps of: minimizing a transfer matrix T to be below a predetermined threshold value λ by adjusting at least one pair of parameters along a direction, one of said parameters relates to translation and the other parameter relates to rotation along the direction.

15. The apparatus of claim 14, wherein said at least one feature point includes five feature points.

16. The apparatus of claim 15, wherein said five feature points includes a gravity center point G and four corner points

17. The apparatus of claim 16, wherein the gravity center point G is identified by the equation

18. The apparatus of claim 16, wherein the four corner points are identified as the four points that are farthest from the gravity center point G.

19. The apparatus of claim 14, wherein Q(x′i,y′i,z′i) is obtained through a set of coordinates processed by a parameterization algorithm selected from the group consisting of Bezier surface algorithm, B-Spline algorithm, and Non-Uniform Rational B-Spline algorithm.

20. The apparatus of claim 14, wherein each zone has a square shape.

21. The apparatus of claim 14, wherein three pairs of parameters are adjusted along respective three orthogonal directions.

22. The apparatus of claim 21, wherein the transfer matrix T is defined as:

23. The apparatus of claim 22, wherein T satisfying an expense function of

24. An apparatus for determining deviation between a first surface with a second surface, such that P(xi,yi,zi) is any point on the first surface and Q(x′i,y′i,z′i) is any point on the second surface, said first surface and said second surface being separated by a first-degree separation, a second-degree separation, and a third-degree separation, including a processor incorporating a deviation determining algorithm, said deviation determining algorithm determine the deviation by the steps of: aligning the first surface with the second surface according to the method of claim 1;calculating a distance di between P(xi,yi,zi) and Q(x′i,y′i,z′i), wherein di=±√{square root over ((xi−xi)2+(yi−yi)2+(zi−zi)2)}{square root over ((xi−xi)2+(yi−yi)2+(zi−zi)2)}{square root over ((xi−xi)2+(yi−yi)2+(zi−zi)2)}.

25. apparatus of claim 24 further including the step of calculating a profile error St, wherein St=|max(di)−min(di)|.

26. The apparatus of claim 24 further including the step of calculating a root mean square deviation Sq wherein