Results

Slew Rate and Heading Characterization

Our hypothesis was that the source of inaccuracies in the goniometer arises from the variations in the slew rate as a function of the absolute position along a goniometer axis. In other words the goniometer does not move precisely 1nm/tick, except on average over long distances. We thus measured the piecewise slew rate over the range -800 mm to 800 mm for both the X and Y axis of the goniometer. Measurement of the full range of goniometer motion (+/- 1000 mm) was limited by the size of the viewable area of the grids. Figure 3 shows the slew rate of the goniometer as a function of the absolute position along the goniometer X and Y axis respectively. These data were acquired over a period of many weeks and the goniometer was reset several times over this period as a result of the instrument being turned off and on. The results show that there is a periodic variation away from the nominal slew rate of 1 nm/tick. The periodic variation is the same over the entire range of the goniometer movements although we have only plotted a small portion of the curve in the figure in order to emphasize the structure of the periodic variations. From these measurements we have determined the average measured slew rate for the x axis to be 1.03 nm/tick with a maximum departure from this value of +/- 0.18 nm/tick. Similarly, the average measured slew rate for the y axis is 1.04 nm/tick with a maximum departure from this value of +/- 0.18 nm/tick.

Figure 3: Characterizing the response of the goniometer. The piecewise slew rate for each axis of the goniometer plotted as a function of the absolute position along the goniometer X and Y axes. Starting with an estimate of the period of the function measured from the data, the best model was determined by performing a least squares fit using a Fourier Series expansion. These best fits to the data are shown as solid curves.

Similarly, we measured the heading, Q, of the goniometer axis as a function of the absolute position of the goniometer. Figures 4 show these as a function of the absolute position along the X and Y goniometer axes respectively. The heading for the X axis is fairly constant over most of the range of the goniometer and varies by only +/- 0.5^o from the average value of 35.6^o. The heading for the Y axis is less well behaved varying from the average of -53.4^o by +/- 1^o as a function of position along the axis. A more serious problem is that the Y axis heading measurements appear to vary as a function of the absolute position of the goniometer X axis. For example in figure 5 we show several plots of Y heading as a function of extremes in the absolute position of the X axis. A further problem is that the value of the heading will vary slightly with time. This is indicated by the multiple values of the heading that are measured for a set value of both X and Y goniometer axis.

Figure 4: Characterizing the response of the goniometer. The heading for each axis of the goniometer plotted as a function of the absolute position along the goniometer X and Y axes. In each case the perpendicular axis was held fixed at the zero position.

Figure 5: The Y heading varies as a function of the absolute position of the X axis of the goniometer. A few points along the Y axis were measured for a range of absolute X axis positions. Some variation with time has also been observed as evidenced by the variation of the measured values at fixed positions of the X and Y goniometer axes.

Modeling the Slew rate of the Goniometer

The characterization of the slew rate shown in figure 3 shows a distinct periodic behavior that can be modeled with a Fourier series. The period for the x axis slew rate is 61910 nm and the period for the y axis is 41590 nm. Starting with an estimate of the period of the function measured from the data, the best model was determined by performing a least squares fit using a Fourier Series:

where K = 2 p/ T

The best fit produces the following values:

X axis (n = 3):

T = 61910;
a0 = 1.029;
a1 = -0.01063; b1 = -0.0788;
a2 = 0.0451; b2 = 0.0635;
a3 = -0.0165; b3 = -0.0140.

Yaxis (n = 5):

T = 41590;
a0 = 1.040;
a1 = -0.0879; b1 = -0.0272;
a2 = 0.00661; b2 = -0.0839;
a3 = 0.0167; b3 = 0.00100;
a4 = -0.00668; b4 = 0.00142;
a5 = 0.00238; b5 = -0.00142.

These curves are plotted on figure 3 for both the X and Y goniometer axes.

In our model we have used constant values for the heading for both X and Y axes. This is a reasonable approximation for the X axis heading. It is clear that the dependence of the Y axis heading as a function of the position of the goniometer should be incorporated into the model to further improve the accuracy of targeting. This would require further investigation however because the measurements of Q_y appear to depend on the X goniometer position and to have some variation over time.

Integration of the Slew Rate Model into the specimen location algorithm

To accurately locate specimens we need to precisely determine the movement along the goniometer X and Y axes required to move a requested distance in the specimen (image) coordinate system. Using the modeled data, we calculate the required goniometer X and Y movements by integrating along the slew rate curve from the absolute starting goniometer position g0, to an end point, g1, which corresponds to the requested distance d on the specimen coordinate system

d = INTEGRAL[F(Kx)dx] from g0 to g1

Given a starting position, g0, and a requested distance, d, Newton-Raphson numerical integration methods were used to determine g1.

Validation of the Model

In order to validate the model we have compared the accuracy with which targets can be located using the modeled goniometer system to the results obtained using an unmodeled system that assumes a constant slew rate. First we consider the errors which result from motion along only one axis of the goniometer. Figure 6 shows these errors for movements along the X or Y goniometer axes. Cross correlation techniques were again used to measure the distance between a starting image and the image acquired after moving a specified distance along a goniometer axis. The error which is plotted is the difference between this measured distance and the distance predicted using either the modeled or unmodeled systems for several requested distances (from 1.5mm to 13mm, approximately every 1mm). In order to have the error measurement independent of the absolute starting position along the goniometer axis, approximately 50 measurements were made from a series of absolute starting points. The root mean square error of all the points are plotted in figure 6 for each requested distance for both the modeled and the unmodeled results.

Figure 6 - Error in repositioning goniometer

Figure 6: We have measured the errors resulting from a requested movement along each of the goniometer axes independently as well as along both axes simultaneously. The root mean square error for each requested distance is plotted for X, Y and both X and Y when using the modeled and unmodeled systems to determine goniometer movements.

Next we measured the errors which result when the requested distance requires a movement along both the X and the Y axis of the goniometer. Moving both axes of the goniometer would generally be required to center an identified target. These errors thus represent those found in a practical experiment. From a given starting position, the errors were calculated for several requested distances and directions of travel. The root mean square error for each requested distance was calculated over all of the directions and over several hundred absolute starting positions along the goniometer axis. Figure 6 also shows the plots of these errors for both the modeled and unmodeled systems.

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