ductors,
which applies to surfaces containing corners. In our method, the calculation
of the gradients is performed by re-using results obtained in the calculation
of the eld scattered by the surface - so that an ultra-fast gradient calcula-
tion results. Interestingly, the total eciency of the method increases with
the number of gradients required.
Results.
We present solution times required by our method to produce
four digits of accuracy for the solution and gradients for test gratings of the
form A � f(x) with A = 0.5, 1, 1.5 and 2, where y = f(x) is the prole given
in Figure 1. (Here we are computing gradients with respect to the height
of the corner points points; the code necessary to compute gradients with
respect to the horizontal variations is now being written.) All calculations
were performed on an AMD 1700+ PC.
Tables 1 and 2 list solution times required to achieve 4 digits of accuracy
in the scattered eld together with a number of gradients of the eld with
respect to prole variations. Table 1 includes timings for evaluation of the
eld and the gradient with respect to the ve variables determining the
vertical position of the corners. To gain an idea of the relative cost of
evaluation of the main eld and the gradient, we include Table 2 which lists
solution times required to achieve 4 digits of accuracy in the scattered eld
and 25 gradients of that eld with respect to the heights of the prole at
the corner points. (The 25 elds amount to ve recomputations of the same
gradient mentioned in connection with Table 1.) We see that for proles
of the less than 2 periods in height, the evaluation of the main eld and
25 gradients with four digits of accuracy requires at most 2 seconds, or less
than 0.08 secs per solution.
1 1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
Figure 1: Two periods of the function y = f (x); the grating proles consid-
ered in this text are given by A � f(x) with A = 0.5, 1, 1.5 and 2.
a/P
time (s)
P/ = 1.0
P/ = 1.5
P/ = 2.0
0.50
0.237
0.259
0.265
1.00
0.351
0.378
0.406
1.50
0.580
0.614
0.658
2.00
0.881
0.933
0.981
Table 1: Computing times for various grating amplitudes/heights (a) and
incident wavelengths () to achieve 4 digits accuracy (P is the period) for
one calculation of the scattered eld, and 5 calculations of the gradient of
that eld with respect to the surface parameters. (TE case)
a/P
time (s)
P/ = 1.0
P/ = 1.5
P/ = 2.0
0.50
0.513
0.552
0.618
1.00
0.735
0.784
0.865
1.50
1.170
1.225
1.458
2.00
1.856
1.993
2.099
Table 2: Computing times for various grating amplitudes/heights (a) and
incident wavelengths () to achieve 4 digits accuracy for one calculation of
the scattered eld, and 25 calculations of the gradient of that eld with
respect to the surface parameters. (TE case)
2