pattern having ones and zeros and modifying the ideal pattern T. In order to determine x* values, x* may be treated as a
to approximate edge imperfections using numbers between perturbation around x, such that: zero and one.
In an embodiment, forming target intensity map 250 may T*=t+at (2)
include taking into account the Nyquist Sampling Theory, 5
which defines a limit resolution based on any given wave- which may give the vector:
length. With reference to FIG. 1, in an embodiment, forming _^ _^ _^
target intensity map 250 may depend on projection optics x *=x +A x (3) 150. In an embodiment, target intensity map 250 may include
a matrix. In an embodiment, target intensity map 250 may 10 which may be substituted into equation (1) to give: include a matrix designated by the symbol T, as is used in the
equations below. TtJ=~x/Kxy+A~x'/Kxy+x'/ka~xy+A~x'/KAx'tJ (4)
As shown in FIG. 2, complex x-map 240, target intensity
map 250, and system model 260 may be used to form optimal In an embodiment, equation (4) may be a representation of
x-map 290. The term optimal in reference optimal x-map 290 15 target intensity map 250.
to is used only in the sense that optimal x-map 290 has been r ~on , A-tr * . ^ , T ,
. . , /, J„„ • r , , Error 280 may be the difference between 1 and I, and may
optimized with respect to error 280, as is further discussed , , •
, , „j- , • , , s , be designated o. o may be given by:
below. Of course, the optimal x-map (or any x-map) may be ° J ° J
further optimized using the methods discussed. Optimal 8..=r.-/.. (5)
x-map 290 may therefore be generally referred to as an 20
improved x-map, an intermediate x-map, or as a x-map. More wmch, by substituting equations (1) and (4) may be written: generally, the term optimal is used throughout the specification only in the limited sense that indicates meeting a thresh- &..=a~x tk~x -+2a~x tk~x ■■ (6) old requirement.
In an embodiment, the following iterative method may be 25 Optimal x-map 290 may be approximated by determining
used to form optimal x-map 290: (a) forming x-map image an approximate solution to equation (6). In order to solve that
270 from complex x-map 240 and system model 260, (b) problem, the inventors have developed a method that they
calculating an error 280, (c) comparing error 280 to a thresh- have termed a method of proportional quadratics. The method
old level, and (d) if error 280 is below the threshold level, of proportional quadratics may include any suitable method
stopping the iteration or, if error 280 is above the threshold 30 that solves for the perturbation introduced in equation (3).
level, forming a new complex x-map. In aa embodiment5 the memod of proportional quadratics
In an embodiment, the iterative method may be repeated may include any or all of the following: diagonalizing the
until error 280 is below a chosen threshold level. In another system modd matrix using the eigen.values aad eigen-vec
embodiment, method (a)-(c) may be used only once to form tors of ^ system modd ... defining a new yector ^ a
optimal x-map 290. In an embodiment, the method discussed new ertalhationusi the eigen.vectors of the system modei
immediately below, either iteratively or singularly, may be , , . , , ,
. . J . , ° J J matrix and the x-value vector; re-wnting delta (o) based on
used to determine optimal x-map 290. . .. .. . . . . . .
. . . the diagonahzed system model matrix and the new vector and
A matrix, I, representing x-map image 270 may be formed , , .. , . , . , , , , ..
. , P • , • , ,. ,■ ■: , • new perturbation; expanding delta into independent quadratic
by the following technique. At any coordinate (i,i), the image- . . . n , , . , .; . „
, ,. . .. f, , , , , , ~ 40 equations; solving lor the new perturbation; and solving lor
level intensity I based on complex x-map 240 may be deter- , '. . . _ . ., ,
mined by perturbation using an inverse transformation of the second
perturbation using the eigen-vectors of the system model
iv='x/Kxvi=i^,...,'Ni=i^,...,'N (l) matrix.
In an embodiment, the method of proportional quadratics
where 7 may be a vector of selected x-values around the 45 may include all or portions of the following method. The K
pixel coordinate (i, j) of interest and K may be a matrix of matrix may be diagonahzed as:
system model 260, as discussed above. The number of x-val- K=vav (7)
ues around pixel coordinate x may be determined by the
—» 50 where V contains the eigen-vectors of K and A contains the
simulation size and the optical system. In an embodiment, x eigen-values along the principle diagonal. Also, a new vector
may include about 15 by 15 x-values around the pixel coor- may ^e degned
dinate. In another embodiment, x may include about 10 by _^ _^
10 x-values around the pixel coordinate. In another embodi- y i,=v'x y (8)
ment, x may include about 12 by 12 x-values around the Equation (6) may then be rewritten: pixel coordinate.
Givenx-map image 270, represented by matrix I, and target 6l7=A y /aa y J+2A y yTA y tJ (9) intensity map 250, represented by matrix T, error 280 may be
determined by taking the difference between I and T. In an 60 which may be expanded to give:
embodiment, if error 280 is below a threshold level, complex
x-map 240 may be optimal x-map 290 and if error 280 is
above the threshold level, method 200 may continue as is J J (10)
discussed immediately below to determine a new complex su =^^t^y\k ...
x-map. The threshold error level may be any chosen value. 65 'J 'J
Given target intensity map 250, matrix T, a suitable vector of x* values may be determined to approximately reproduce