Grazing incidence x-ray telescopes consist of surfaces which are nearly cylindrical in shape. The abrasive figuring of these surfaces is accomplished by moving a grinding tool along a helical path on this almost cylindrical surface. The measurement of the surface is, however, performed along "axial" scan lines which intercept this helical path. This approach to figuring and measuring permits a relatively simple scheme to be implemented for the determination of the optimal dwell times of the figuring tool. These optimal dwell times are determined by a deconvolution which approaches the problem in a linear programming context and uses the Simplex Method. The approach maximizes the amount of material removed at any point subject to inequality constraints. The effect of using these ''optimum" dwell times is to significantly improve the tools effectiveness at removing the higher spatial frequencies while staying (strictly) within the bounds and constraints imposed by the hardware. In addition, the ringing at the edges of the optic, frequently present in deconvolution problems, is completely eliminated

Waluschka, Eugene

NASA Technical Reports Server

I 
I Source of Acquisition 
I NASA Goddard Space Flight Center 
Cylindrical Optic Figuring & Dwell Time Optimization 
Eugene Waluschka 
NASA Goddard Space Flight Center 
Optics Branch/% 1.0 
Greenbelt, Maryland 2077 1 
ABSTRACT 
Grazing incidence x-ray telescopes consist of surfaces which are nearly cylindrical in shape. The abrasive figuring of these 
surfaces is accomplished by moving a grinding tool along a helical path on this almost cylindrical surface. The measurement of 
the surface is, however, performed along "axial" scan lines which intercept this helical path. This approach to figuring and 
measuring permits a relatively simple scheme to be implemented for the determination of the optimal dwell times of the figuring 
tool. These optimal dwell times are determined by a deconvolution which approaches the problem in a linear programming 
context and uses the Simplex Method. The approach maximizes the amount of material removed at any point subject to inequality 
constraints. The effect of using these ''optimum" dwell times is to significantly improve the tools effectiveness at removing the 
higher spatial frequencies while staying (strictly) within the bounds and constraints imposed by the hardware. In addition, the 
ringing at the edges of the optic, frequently present in deconvolution problems, is completely eliminated. 
Subject Terms: X-ray telescopes, grazing incidence optics, linear programming, Simplex method, deconvolution 
1. INTRODUCTION 
A typical x-ray telescope consists of nested shells'". Each shell is part of a long conical surface of revolution as shown in figure 
1. To simplify the discussion we will assume that the part we are making is basically a cylinder. When fabricating these surfaces, 
Figure 1: Conical Parent Surface and Mandrel Portion 
the axis of symmetry is of course put to good use. This usually results in surfaces which have longer period surface errors 
azimuthally (about the axis of symmetry) and shorter period errors axially (along the axis where there is no symmetry). This is 
why spherical surfaces, where a symmetry about a point exists. are easier to make than aspheres. The fact that surface errors tend 
https://ntrs.nasa.gov/search.jsp?R=20090013857 2019-08-30T06:30:03+00:00Z
to have longer periods azimuthally means that the separation of measurement points is much smaller axially, along "scan" lines, 
than azimuthally. This note will describe a relatively simple procedure for the considerable reduction of these axial errors by 
moving an abrasive (wet) grinding tool along a helical path about the cylinder. A considerable simplification results from the 
fact that we need only to consider one axial scan line at a time. We do not need to view the surface as two dimensional, just as 
a collection of one dimensional lines, one at a time. As the tool moves in its helical path and crosses each scan line the material 
removal is again viewed one dimensionally. This one dimensional approachreduces the amount of data needed to deconvolve 
dwell times and permits the determination of optimal dwell times over pny scan line, one line at a time. 
Optimal in our case will mean the set of dwell times determined by the Linear Programming Simplex Method 43 which maximizes 
the amount of material removed, along any scan line, subject to inequality constraints. It should be noted that without constraints 
maximal removal means all the material (so that one is digging holes), which is definitely not optimal and not desirable. Once 
the dwell times and hence the speed of the tool at each scan line crossing are determined the speed between the lines is determined 
by linear interpolation. The result of using these optimal dwell times is to considerably reduce the axial errors, as will be shown 
below. 
2. GEOMETRY 
The clear aperture of a x-ray optic is annular and somewhat distant from the parent optic vertex as shown in figure 1. Because 
of this large distance between the vertex of the parent optic and the working $portion of the grazing incidence optic it is convenient 
to move the origin of the coordinate 
frame from the vertex to roughly the 
center of the used surface as shown in 
figure 2. i rx 
The equation of the surface6 in this 1 / 
cylindrical coordinate frame is given by / 
equation 1. To simplify the discussion, 
p = (1) z 
but, without loss of generality, we will 
assume that the surface is a cylinder, in 
which case K=P=O, and the equation of 
the surface is then just given by 
equation 2. 
/ I 
P = Po (2) Figure 2: Body Centered Coordinate Frame 
Equations 1 and 2 represent mathematically ideal surfaces. Actual surfaces will have errors or deviations (hopefully excess 
material), from these ideal surfaces. In the case of our cylindrical surface this will be represented by an additional (small) term 
as in equation 3. 
p= po + E(~ ,z )  (3) 
In the following we will assume, to simplify the discussion, that ~ (8 ,z )  z Oand that the desired surface is attained when 
~(8 .z)  = 0,  that is we want to remove the maximum amount of excess material without digging holes. 
3. SURFACE ERRORS 
Determining the function ~(0 ,z )  through measurements can be, depending on the accuracies required, not a trivial task. 
However difficult. the end result of all the measurements is (usually) some agreed to table of values of ~ ( 0 ~ .  2,) for i=l, ..., 1 
(scan line index) and j=I, ..., J (the axial index) and just to be specific we typically have 8,+, - 8, = 10 "and z,,, - zj = I mm. The 
actual spacing and total number of data points, of course, depends on the surface knowledge requirements. 
A typical scan line profile is shown in figure 3 and the goal is to reduce the error, as nearly as possible, to zero. 
1 +  
4. HELICAL PATH 
0.6 
,0.57585, 
0.4 - - 
B 
.: P 
- 
0.2 - - 
,0.06887, O I I I I I 1 A 
0 20 40 60 80 100 1 20 140 
LO, z ,128.2375, 
mm 
Figure 3: Axial Scan Line Profile 
Reducing the figure error of our a cylinder, by moving a (wet) abrasive grinding (figuring) tool around it on a helical path is very 
similar to a raster pattern used for figuring "flat" optics. To see this all we need to do is imagine that we cut our cylinder along 
an axial scan (&constant) line and laying it flat. So that 
we get a rectangle, part of which is shown in figure 4. 
The pitch of the helix is determined by (among other I * 
things) the width and shape of the tool wear profile. The z 
helix crosses the iih axial scan line at the points Cu, given 
by equation 5, and where k=I,  ... .K. The magnitude of K Figure 4: Flat Cylinder and Helical Path 
depends on the pitch of the helix and is not necessarily the 
same for all the scan lines. It will be at these crossing points, C,, that we will determine the speed, v(B,z), of the tool along the 
helical path taking into account the tool wear profile and the excess material. In most of the discussion below the pitch of the 
The inclined line(s) in figure 4 is the helical path and it 0 
looks somewhat like a raster pattern. 'The equation of a A 
helical path on a cylinder is given by equation 4 
z =pitchxO . (4) 
0 ,  
0 ,I 
5;.& =pitchx(Oi + 
\ \ \ \ \ - 
27:k) (5) 
\ \ \ \ 
helix will be such that the tool advances one half its width, b , while going once around the cylinder. That is, the pitch in 
equations 4 and 5 will be b/4n. 
5. ONE DIMENSIONAL TOOL WEAR FUNCTION 
During a figuring operation the instantaneous material removal occurs in the svall area where the tool is in contact with the part, 
as shown in figure 5a. The instantaneous wear is a two dimensional function. In our case determining this instantaneous removal 
profile with any certainty is difficult, the measured results are uncertain &d consequently the use of it in material removal 
predictions is avoided. However, because we measure the cylinder along scan lines it is natural to determine the tool wear 
function, also, along scan lines as shown in figures 5a and 5b. This one dimensional characterization of the material removal 
is adequate for our purposes and simplifies (because we need work only in one dimension and not two) the dwell time 
deconvolution algorithm. In a sense, we are treating the tool as if it were moving in a "straight" line on a (cylindrical) surface 
at constant speed and the only quantity of interest is the amount of material removed as the tool crosses the ifh scan line, as shown 
in figures 5a and 5b. 
Figure 5a: Tool moving along a cylinder Figure 5b: Triangular material removal profile 
The wear, w,(z), at any point z , along the ifh scan line from all of the (helical path) tool positions is given by equation 7 
To simplify the discussion, still further, we will 
assume that this one dimensional wear ~rofile is 
Figure 6 shows the wear pattern, along the ifh scan line when v=I and when the pitch of the helix is such that C, - C, = bn. 
exactly triangular in shape and given by 
equation 6. Here w( z - C, ) is the material 
removed, along the ifh scan line, at the point z w(Z-Cik)= 
when the tool is at Ci, and where b is the base 
v b  
and h/v is the height of the triangle which is 
inversely proportional to the speed, v, of the 
tool along its path. The constant, h, is a 
measured quantity and is the depth of wear for a unit speed. 
We see, in this case, that the material removal is (except for edge effects) just a uniform lowering of the surface, Ap=h. 
Now, another quantity that will be of interest is the wear, 
W, along the ith scan line given by equation 8 where the 
summation is over all of the (equispaced Az) data points, zj ,on 
a scan line. Equation 8 is just the total height reduction in,a 
@=constant plane. In the case of a cylinder @=constant) the 
volume of material removed in an angularly thin wedge , 68. 
is then just Wp68. Again, figure 6 shows the removal along a . 
scan line for three tool positions when all the h, are constant. 
An objective of a figuring operation is to vary the removal, by P A 
varying the h, , in such a way that we remove the maximum 
amount of material, but not "dig holes" by going below the 
desired surface. This maximum value of W can be found by 
means of the Linear Programming Simplex Method as Figure 6: Uniform removal 
described in the next section. 
6. MAXIMIZING THE TOTAL WEAR BY THE SIMPLEX METHOD 
The Linear Programming Simplex Method finds the maximum or minimum value of a linear functions taking into account 
inequality constraints. In our case, we want to find the values (for a given scan line, hence we drop the subscript i) 
4 for k=1 ... K 
which maximize the function W ,  given by equation 8 
subject to the, J, inequality constraints, given by equation 9, of not digging holes 
w(zj) s ~ ( 8  .zj) for j=l ..'. J 
and any additional constraints, given by equation 10, on the dwell times, which here are constraints on the heights of the 
triangles. 
h r i n  i hk s hkm for k=1 ... K (10) 
Again, we have dropped the scan line index, i , as we are discussing only one scan line at a time. 
7. OPTIMIZATION RESULTS - DECONVOLUTION EXAMPLE 
Figure 7 shows the results on this Simplex Method dwell time deconvolution applied to the axial scan profile shown in figure 
3. The tool wear is triangular, the tool spacing is, bE , one half the base of the triangle. The varying heights of the triangles are 
proportional to the dwell times. The amount of material removed at any tool position is equal to the area under the triangle. The 
tool, centered at the 93 mm point has a height of zero and none of the removals are negative. The predicted new surface is shown 
d 
Figure 7: triangular tool positions and wear heights 
in figure 8 as the bottom dashed curve. It should be noted that the amount of material removed never exceeds, because of the 
inequality constraints, the original residual ~(8,. zj) . 
8. FOURIER ANALYSIS OF SCAN LINE 
Equations 10 and 11 connect the scan line profiles, in figure 8, with the respective Fourier coefficients 
~(3) = A,, + $ [ A ~ c o s ( ~ ( z ~ ~ $ )  - B,sin(~nn$)] (10) 
0.6 
9.57585 , 
wik , . k . h r  
0.2 - 
;-. 
# '. 
, . I .  
0 
Figure 8: Original and new residual 
where J is the total number data points and Nf= integer portion of JL?. Plotting the (un-normalized) power spectrum of the 
coefficients of each curve, as in figure 9, we see that the coefficients, in equation 11, are reduced out to slightly beyond n=IO 
or a spacial wavelength of about lOmm which is the width, b, of the triangular tool in this example. 
9. SPEED BETWEEN SCAN LINES 
Once we have determined the speed, v,, , of the tool at each scan line and helix crossing point we find the tool speed, v , between 
two consecutive scan line crossings by linear interpolation as in equation 13 
Where A v  = v,,,,, - v,, is the change in speed, A s  is the path length between two consecutive crossings. s and v i,k is the path 
length and speed, respectively, from the last crossing. 
10. CONCLUSION 
log( n) 
Figure 9: Power spectrum 
The end result of all of these operations is that we have determined the speed of motion of a figuring tool along a helical path 
and the speed is optimum at the scan line crossings. It should be noted that approach is very similar to the technique7.' used to 
determine the optimum dwell times for the polishing and figuring of the AXAF, now called Chandra, cylindrical optics. In the 
case of the Constellation-X mandrels it is hoped that this approach also improves the convergence of the figuring process. 
Deconvolution methods which attempt to reduce the rms deviation directly are simply solving sets of overdetermined linear 
equations for the unknown dwell times. This "traditional" approach certainly works very well when there are no constraints on 
the dwell times. However, even with the simple requirement that all dwell times be positive significantly, complicates finding 
an optimum solution. Other constraints further complicate the numerical search for the best solution. 
The Simplex, linear programming, approach is an alternative method for determining dwell times. This approach seeks to 
determine a set of dwell times which maximize the amount of material to be removed but, of course, subject to constraints. Its 
major advantage, is numerical stability, reduction in rms surface roughness which is comparable to or better than the direct rms 
reduction approach and a certain intuitive appeal. 
ACKNOWLEDGMENTS 
The development of the above algorithm is not a product of inspiration but of dogged work (at the then Perkin-Elmer 
Corporation) in support of the manufacture of (large) optics. It is in that setting that one is forced to seek optimum approaches 
to every step of a process as the alternatives are more costly. However it should be noted that (way back) just prior to the start 
of the AXAF polishing work the availability of the NIST FORTRAN computational libraries9 considerably simplified and made 
possible this particular, Simplex, approach. Now, the task is made still easier with the help of MathcadTM. 
Of course thanks go to Hughes Danbury Optical Systems (HDOS a.k.a. Perkin-Elmer) where the original development work 
occurred funded by various NASA contracts. The current work was performed at NASA and supported by the Constellation-X 
Soft X-ray Technology Program and the Cross Enterprise Technology Development Program. 
REFERENCES - 
1 
1. See the description of Constellation-X online at: http://constellation.gsfc.nasa.gov/bookletoc.ht. 
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Metrology on a Metal Mandrel and Replicated Segments for Constellation-X, Proc. SPIE, 3766.22, 1999. 
3. R. Petre, C. Chen, L. Cohen, D. Content, R.J. Harms, 0. Mongrard, G. Monnelly, T. Saha, M. Schattenburg, 
P. Serlernitsos, and W. Zhang, "Segmented x-ray mirror development for Constellation-X, Proc. SPIE, 3766, 11, 1999. 
4. G. Hadley, Linear Programming. Reading, Mass.: Addison-Wesley, 1961. 
5. A.P. Bogdanov, "Optimizing the t&chnological process of automated grinding and polishing of high-precision large 
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6. R.J. Noll, P.E. Glenn, J. Osantowski, "Optical surface analysis code (OSAC)", Proc. SPIE 362,78-85, 1983. 
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SPIE 2515,361,1995 
9. Now available at http://gams.nist.gov/serve.cgi/ModuldCMLIB/SPLP/2055 


Cylindrical Optic Figuring and Dwell Time Optimization

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