Producing mask layouts with rounded corners
A two-dimensional representation of a polygon is converted to a parametric representation. A smoothing filter is applied to the parametric representation to produce corner rounding. In some embodiments, a polygon layout plus a model that specifies how much corner rounding should be applied are taken as inputs. The desired amount of rounding to the corners in the input polygons is applied and this produces a new polygon layout with corners that are properly rounded as its output. The process can be implemented so that it does not induce any pattern-size dependent bias. It also can be designed so that it does not induce line-end pullbacks. However, this feature can be turned off if line-end pullbacks are deemed appropriate for the specific application.
1. A method implemented on a computer system executing instructions to estimate corner rounding in lithographic masks, the method comprising:
accessing a design layout of a lithographic mask, the design layout comprising a plurality of polygons described by two-dimensional representations in (x,y) of boundaries of the polygons;
calculating parametric representations γ(t)=(x(t), y(t)) of the two-dimensional representations of the polygon boundaries, wherein the parametric representations are a function of the parameter t;
applying a smoothing filter f(t) to the parametric representations γ(t) to produce smoothed parametric representations {tilde over (γ)}(t);
calculating smoothed two-dimensional representations of the polygon boundaries from the smoothed parametric representations {tilde over (γ)}(t), wherein the smoothed two-dimensional representations of the polygon boundaries have rounded corners on the polygon boundaries; and
altering the design layout of the lithographic mask based on the smoothed two-dimensional representations of the polygon boundaries with rounded corners.
2. The computer-implemented method of claim 1 wherein the smoothing filter f(t) satisfies a normalization condition
∫
-
∞
+
∞
f
(
t
)
d
t
=
1
and a symmetrical condition
f (− t )= f ( t ).
3. The computer-implemented method of claim 2 wherein the smoothing filter f(t) further satisfies conditions
f
(
t
)
≥
0
,
f
(
t
)
=
0
for
t
>
T
,
m
n
=
∫
-
∞
+
∞
t
n
f
(
t
)
d
t
exists
for
any
n
≥
0
where T is a support boundary of the smoothing filter f(t).
4. The computer-implemented method of claim 1 wherein the smoothing filter f(t) is smooth and contains a single peak.
5. The computer-implemented method of claim 1 wherein the smoothing filter f(t) is given by
f
(
t
)
=
a
T
cos
2
[
π
2
p
(
t
T
)
]
-
T
≤
t
≤
T
where a is a normalization parameter, T is a support boundary of the smoothing filter f(t), and p(x) is a monotonically increasing function in [0,1].
6. The computer-implemented method of claim 5 wherein
p ( x )= x w
where w is a parameter.
7. The computer-implemented method of claim 1 wherein the polygon boundaries have rounded corners with different radii of curvature.
8. The computer-implemented method of claim 7 wherein concave corners have a first radius of curvature and convex corners have a different second radius of curvature.
9. The computer-implemented method of claim 7 wherein the radius of curvature for a corner is calculated based on the coordinates for the corner and for two vertices adjacent to the corner.
10. The computer-implemented method of claim 7 wherein the different radii of curvature are implemented by varying a scale factor R that relates distance in the two-dimensional (x,y) representation with the parameter t.
11. The computer-implemented method of claim 10 wherein an edge between a first corner and a second corner having different radii of curvature implemented by scale factors R 1 and R 2 , respectively, is parameterized using scale factor R 1 for a fraction [R 1 /(R 1 +R 2 )] of the edge adjacent to the first corner and using scale factor R 2 for a fraction [R 2 /(R 1 +R 2 )] of the edge adjacent to the second corner.
12. The computer-implemented method of claim 1 wherein calculating smoothed two-dimensional representations of the polygon boundaries comprises:
sampling the smoothed parametric representations {tilde over (γ)}(t) at parameter values t 1 wherein the sampling granularity is adjustable.
13. The computer-implemented method of claim 1 wherein the smoothed two-dimensional representations of the polygon boundaries comprise:
straight edges represented by the (x,y) coordinates of vertices of the edges; and
rounded corners represented by samples of the smoothed parametric representations {tilde over (γ)}(t) of the rounded corners.
14. The computer-implemented method of claim 1 further comprising:
grouping consecutive corners of polygon boundaries into corner clusters based on separation as measured by the parameter t, wherein applying the smoothing filter f(t) is applied to segments of the parametric representations γ(t) that contain all corners in the same corner cluster.
15. The computer-implemented method of claim 14 further comprising:
determining a pair of pivot points for each corner cluster, wherein the smoothing filter f(t) is applied only between the pivot points.
16. The computer-implemented method of claim 1 further comprising:
reducing line-end pullback.
17. The computer-implemented method of claim 1 wherein the smoothed two-dimensional representations are a continuous function of the design layout.
18. A system for estimating corner rounding in lithographic masks, the system comprising:
means for accessing a design layout of a lithographic mask, the design layout comprising a plurality of polygons described by two-dimensional representations in (x,y);
means for calculating parametric representations γ(t)=(x(t), y(t)) of the two-dimensional representations of the polygon boundaries, wherein the parametric representations are a function of the parameter t;
means for applying a smoothing filter f(t) to the parametric representations γ(t) to produce smoothed parametric representations {tilde over (γ)}(t);
means for calculating smoothed two-dimensional representations of the polygon boundaries from the smoothed parametric representations {tilde over (γ)}(t), wherein the smoothed two-dimensional representations have rounded corners on the polygon boundaries; and
means for altering the design layout of the lithographic mask based on the smoothed two-dimensional representations of the polygon boundaries with rounded corners.
19. A non-transitory computer-readable storage medium storing executable computer program instructions for estimating corner rounding in lithographic masks, the instructions executable by a processor and causing the processor to perform a method comprising:
accessing a design layout of a lithographic mask, the design layout comprising a plurality of polygons described by two-dimensional representations in (x,y) of boundaries of the polygons;
calculating parametric representations γ(t)=(x(t), y(t)) of the two-dimensional representations of the polygon boundaries, wherein the parametric representations are a function of the parameter t;
applying a smoothing filter f(t) to the parametric representations γ(t) to produce smoothed parametric representations {tilde over (γ)}(t); and
calculating smoothed two-dimensional representations of the polygon boundaries from the smoothed parametric representations {tilde over (γ)}(t), wherein the smoothed two-dimensional representations of the polygon boundaries have rounded corners on the polygon boundaries; and
altering the design layout of the lithographic mask based on the smoothed two-dimensional representations of the polygon boundaries with rounded corners.