Improving the quantitative testing of fast aspherics surfaces with null screen using Dijkstra algorithm Víctor Iván Moreno Oliva a *, Álvaro Castañeda Mendozaa, Manuel Campos García b, Rufino Díaz Uribe b a Universidad del Istmo, Campus Tehuantepec, C.P. 70760, Oax., México ; b Universidad Nacional Autónoma de México, Centro de Ciencias Aplicadas y Desarrollo Tecnológico, Apdo. Postal 70-186, C.P. 04510, México, D.F. ABSTRACT The null screen is a geometric method that allows the testing of fast aspherical surfaces, this method measured the local slope at the surface and by numerical integration the shape of the surface is measured. The usual technique for the numerical evaluation of the surface is the trapezoidal rule, is well-known fact that the truncation error increases with the second power of the spacing between spots of the integration path. Those paths are constructed following spots reflected on the surface and starting in an initial select spot. To reduce the numerical errors in this work we propose the use of the Dijkstra algorithm. 1 This algorithm can find the shortest path from one spot (or vertex) to another spot in a weighted connex graph. Using a modification of the algorithm it is possible to find the minimal path from one select spot to all others ones. This automates and simplifies the integration process in the test with null screens. In this work is shown the efficient proposed evaluating a previously surface with a traditional process. Keywords: Optical Testing, Aspherics, Dijkstra algoritm. 1. INTRODUCTION In the last years the null screen method has been used to testing very fast aspheric convex, 2 concave, 3 and off axis surfaces. 4 This geometrical method measures the slope of the test surface and by a numerical integration procedure the shape of the test surface can be obtained. 5 Usually the select paths of numerical integration are proposed manually starting of an initial spot a 0 (is the sagitta for one point of the surface that must be known in advance) to all the spots, each selected path is integrated numerically using approximated values for the sagitta in order to get the shape of the surface. The common method for the numerical evaluation is the trapezoidal rule for nonequally spaced data. One of the main reasons for the popularity of Dijkstra s Algorithm is that it is one of the most important and useful algorithms available for generating (exact) optimal solutions to a large class of shortest path problems. The point being that this class of problems is extremely important theoretically, practically, as well as educationally. The shortest path problem is one of the most important generic problem in such fields as Operation Research, Computer Science and Artificial Intelligence. One of the reasons for this is that essentially any combinatorial optimization problem can be formulated as a shortest path problem. In this paper we report how by means of the Dijkstra algorithm allows us to automate the process of selecting paths for the numerical integration. In this paper we report the test of a hyperbolic concave surface previously tested in another paper, 6 showing an accuracy improvement with our method. Further author information: Send correspondence to Víctor Iván Moreno Oliva E-mail: vmorenofcfm@hotmail.com. 22nd Congress of the International Commission for Optics: Light for the Development of the World, edited by Ramón Rodríguez-Vera, Rufino Díaz-Uribe, Proc. of SPIE Vol. 8011, 801125 2011 SPIE CCC code: 0277-786X/11/$18 doi: 10.1117/12.903399 Proc. of SPIE Vol. 8011 801125-1
2. PRELIMINARIES The Dijkstra algorithm is an algorithm to find the shortest paths from a single source vertex to any other vertex in a weighted directed connex graph. A graph is called connex if for any two of its vertices, v i, v j there is at least one route or path which connects them. One classical problem is to find the shortest path between any two vertexes (i.e. a path that has minimum length), and the algorithm proposed by Edgar W. Dijkstra (1930-2002) efficiently solves this problem. Many practical problems in various fields can be modeled using graph theory. A transport network is an example of an oriented weighted directed graph G =(V,E), where V = v 1,v 2,...,v n is the vertexes collection and E is the collection of edges that connect pairs of vertexes. Here V can be stations or locations on the network, and E routes or paths going from station x to station y. Each edge has a number assigned, w ij for the edge s weight. For this example w ij can be the distance (miles) or time (minutes). Description of the Dijkstra algorithm: a) Begin with the source node (initial vertex), and call this the current node. Set its value to 0. Set the value of all other nodes to infinity. Mark all nodes as unvisited. b) For each unvisited node that is adjacent to the current node, do the following. If the value of the current node plus the value of the edge is less than the value of the adjacent node, change the alue of the adjacent node to this value. Otherwise leave the value as is. c) Set the current node to visited. If there are still some unvisited nodes, set the unvisited node with the smallest value as the new current node, and go to step b). if there are no unvisited nodes, then we are done. 200 100 200 100 100 200 100 200 Figure 1. a) (left) Centroids of the resultant image of the screen after reflection on the test surface. b) (right)the weighted connex graph resultant, shown the possible edges for find the shortest path In the qualitative evaluation of the shape of the optical surfaces with null screens method, we have a graph with the positions of the centroids of the spots (vertexes) by the resultant image of the null screen after reflection on the test surface (Fig. 1 left). In order to implement the Dijkstra algorithm we need a weighted connex graph. For these, we calculate the distances between adjacent vertexes; this distance is the weight (w ij )ofeachedgein the graph. The Fig. 1 (left) shows the centroids positions of the spots obtained in a previous test of a hyperbolic surface by null screens method. 6 Fig. 1 (right) shows the graph with all the edges calculated between adjacent vertexes, some vertexes have more of two edges, this for an initial value condition of the distance. Now with the resultant graph (Fig. 1 right) it is possible implement the Dijkstra algorithm. In Campos-García 6 paper the numerical integration method is the trapezoidal rule, where the integrations paths were selected manually estimating shortest paths. This process is tired and the paths are not guaranteed to be the shortest ones, thus with the effect of an increase in the truncation error. Proc. of SPIE Vol. 8011 801125-2
The propose method in this paper basically consist in three steps: 1) make a connex graph with the centroids spots collections, 2) select an initial spot or vertex (z o ) by found the minimal path from zo to all the spots or vertex in the connex graph, and 3) each path find is a optimal trajectory by using the numerical integration by the trapezoidal rule. This minimizes the truncation error. A representative connex graph for the Fig. 1(right) is shown in Fig. 2, here we have the optimal path from the initial spot vi to another vertex v j. vj vi Figure 2. Representative graph for the weighted connex spots set. 3. RESULTS The results of the shortest path obtained are show in a graph in Fig. 3, starting at the vertex Z o,thedijkstra algorithm finds the optimal or minimal paths to all other vertexes in the graph. Finally a total of 1884 paths were found, i.e. 1884 vertexes or spots were used in the test of hyperbolic surface. The shape of the surface can be obtained through the formula 7 p ( nx z z 0 = dx + n ) y dy, (1) n z n z p 0 Where n x, n y,andn z are the Cartesian components of the normal vector N to the test surface, and z o is the sagitta for one point of the surface that is know in advance. All these parameters were previously calculate by Campos-García 5 and are used to perform the numerical integration with the paths foun by the algorithm. The method used for discrete evaluation of the integral is the trapezoidal rule. To analyze the details of the evaluation we fit the experimental data to an aspheric surface given by z = r [r2 (k +1)S 2 ] 1/2 k +1 + D 1 S 4 + D 2 S 6 + D 3 S 8 + D 4 S 10 + Ax + By, (2) by using the Levenberg-Marquart method 8 for nonlinear least square fitting that is suitable for this task. Where r is the radius of curvature at the vertex, k is the conic constant of the test surface, S 2 = x 2 + y 2, D 1, D 2, D 3 and D 4 are deformation coefficients; and A and B are the terms of tilt in x and y respectively. In Fig. 4 we show the evaluated surface and Fig. 5 are the differences in saggita between the measured surface and the best fitting aspheric. The design parameters for the test of the hyperbolic surface are: radii of Proc. of SPIE Vol. 8011 801125-3
z o Figure 3. Integration paths found by Dijkstra algorithm. All paths start at the same point z o. curvature r = 467mm and conic constant of k = 1.34514, the parameters resulting from the least square fitting of the sagitta data are shown in Table 1. In the same table are compared the results obtained in previously test of the surface. Figure 4. Evaluated surface The radius of curvature differs approximately by 1.498 mm or about 0.32% of the design value of r, andthe conic constant differs approximately 0.00337, about 0.25% of the design conic constant value. The P V differ- Proc. of SPIE Vol. 8011 801125-4
Figure 5. Sagitta differences between the measured surface and the best fitting aspheric. ences in sagitta between the evaluated surface and the best fit with the use the algorithm is Δz pv =0.078mm. Clearly shows that slightly better results are obtained using the optimal paths in the integration numerical process. 4. CONCLUSIONS We have shown a practical application of graph theory in the implementation of the Dijkstra algorithm in finding the optimal integration paths that minimize the truncation errors in the numerical integration process for the testing of fast aspheric surfaces with null screen. The Dijkstra algorithm allows to automates and speed up the numerical integration process. In previously works the selecting of integration paths are made manually by estimating the optimal paths from the initial vertex, this not guaranteed to be optimal path. in an arrangement of discrete points or vertex where there is not order or equal distances between points, it would be almost impossible to determine the optimal paths without the use of the algorithm. The results shown in comparison with the previous test, the repeatability of the method, and a slight improvement respect to the design parameters of the test surface. This ensures that the results are more reliable with the use of the algorithm. 5. ACKNOWLEDGMENTS This research was supported by DGAPA-UNAM under project PAPIIT under project No.IN114110. Table 1. Parameters resulting from the least square fitting of the sagitta data, using the new paths found by Dijkstra algorithm. With Dijkstra algorithm r(mm) k A B D 1 (mm 3 ) D 2 (mm 5 ) D 3 (mm 7 ) D 4 (mm 9 ) 465.502 1.34521 0.00027 0.00024 3.949 10 9 1.660 10 13 3.79010 10 18 3.113 10 23 Without Dijkstra algorithm (from Campos-García et al. 6 ) r(mm) k A B D 1 (mm 3 ) D 2 (mm 5 ) D 3 (mm 7 ) D 4 (mm 9 ) 465.178 1.34589 0.00026 0.0002 2.874 10 9 1.558 10 13 3.490 10 18 2.833 10 23 Proc. of SPIE Vol. 8011 801125-5
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