Serif mask design for correcting severe corner rounding and line end shortening in lithography

A photolithographic mask for conducting illumination from a light source onto a semiconductor surface during a microlithographic manufacturing process. The mask includes a line end portion of a width w and including two corners, each corner defining a respective region for locating one or more serifs for correcting severe corner rounding and line end foreshortening effects caused by the optical diffraction during the optical imaging process. For aerial image/resist pattern modeled as a convolution or the square of a convolution between the photomask and an intensity/amplitude kernel function having an effective range r in x and y directions, and under a condition that w<r<2w, a hanging square serif of a size w.times.w is provided attached to a respective corner within a corner region. For the condition of 2nw<r.ltoreq.2(n+1)w, with n=1, 2, . . . , each corner region includes an associated (n+1) serifs, each being linearly aligned along line-end extension line and spaced apart from an adjacent serif by a distance w, with each of the first n serifs being square and of a width w, and the (n+1).sup.th serif being a rectangle of a length w and a width min(r-2nw, w). When the intensity/amplitude kernel function is modeled as being azimuthal-angle independent and being non-zero over a circular area of radius r, which is typical for usual circular aperture, the serif size in the corner regions may be optimized. For a hanging square serif of size w.times.w under the condition w<r.ltoreq.2w, if further w<r<.sqroot.2 w, then the hanging serif in each corner region may be reduced in size without altering the aerial image/resist pattern intensity at its respective corner. The portion of the square serif to be removed is that portion of the square which is outside the circular intensity/amplitude kernel function of radius r centered at its respective corner. For a set of associated (n+1) serifs under a condition 2nw<r.ltoreq.2(n+1)w, if further [(2n+1).sup.2 +1].sup.1/2 w.ltoreq.r .ltoreq.2(n +1)w, with n=1, 2, . . . , then the (n+1).sup.th serif in each corner region may be reduced in size without altering the aerial image/resist pattern intensity at its respective corner. The portion of the (n+1).sup.th serif to be removed is that portion of the rectangle which is outside the circular intensity/amplitude kernel function of radius r centered at its respective corner.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates for microlithographic technology and, 
particularly, to techniques implementing serifs for correcting severe 
optical proximity effects in microlithography. 
2. Discussion of the Prior Art 
Photolithography is the technology of reproducing patterns using light. As 
presently used in semiconductor industry, a photomask pattern for a 
desired circuit is transferred to a wafer through light exposure, 
development, etch, and resist strip, etc. As feature sizes on a circuit 
become smaller and smaller, the circuit shape on the wafer differs from 
the original mask pattern more and more. In particular, optical proximity 
effect phenomena such as corner rounding, line end foreshortening, 
iso-dense print bias, etc., are typically observed. 
FIG. 1 illustrates a rectangular mask 10 and corresponding aerial image 
(after light exposure) of the final photoresist pattern 10' (indicated as 
broken lines) after further development, etch, and resist strip, etc. As 
shown in FIG. 1, the optical proximity effect of corner rounding is 
clearly exhibited as indicated at corners 12a-12d. 
A simple geometric picture for understanding the optical proximity effects, 
such as the corner rounding, is additionally shown in FIG. 1. For 
definiteness, it is assumed that the clear region 10 of the mask portion 8 
is inside the mask and serif boundaries. 
For the case of incoherent light illumination (using either circular or 
rectangular aperture), the aerial image intensity at a point "E" is given 
by the convolution between the intensity kernel function and the 
transmitted light intensity around the point, and is proportional to the 
volume of a truncated cone-type 3D structure 15a (for the case of a square 
or rectangular aperture), and at a point "E'" by a truncated cone-type 3D 
structure 15b (for the case of a circular aperture), as shown in FIG. 1. 
Each whole cone-type structure 15a,15b represents the intensity kernel 
function on a respective 2D region centered at a point E (E') and has an 
effective range (radius) "r" which is basically the range of inference due 
to the optical proximity effects. In terms of the optical wavelength 
".lambda." and the numerical aperture NA, the effective range "r" is of 
the order .lambda./NA. The truncation is done according to the actual 
light transmission around that point, which may be blocked by any opaque 
region 20 in the photomask, e.g., resulting in a half-rectangle region or 
a half-circle region. That is, as shown in FIG. 1, for an edge point E 
(when its distance to its nearest corner is larger than r), after 
truncation, the 2D area becomes rectangle represented by the half-square 
region 16a, i.e., its volume is half of whole volume under the intensity 
kernel function (i.e., half of whole volume of 3D cone-type structure for 
the case of a square or rectangular aperture). Similarly, for an edge 
point E', after truncation, the 2D area becomes a half-circle region 16b 
(for the case of a circular aperture). Moreover, as shown in FIG. 1, for a 
corner point "C," after truncation, its volume 18a is one quarter (1/4) of 
whole volume under the intensity kernel function (for the case of a square 
or rectangular aperture) and similarly, for a corner point C', the volume 
18b is one quarter (1/4) of whole volume under the intensity kernel 
function (for the case of a circular aperture). Thus, I.sub.C =I.sub.E /2, 
with I.sub.C representing an illumination intensity at point C and I.sub.E 
representing an intensity at point E, independent of the range r and the 
form of the intensity kernel function. The aerial intensity contour line 
passing through the edge point E will not pass through the corner point C; 
rather, it passes through an inner point, resulting in the corner rounding 
effects. 
For the case of coherent light illumination (with either circular or 
rectangular aperture), the aerial image intensity at a point is given by 
the square of the convolution between the amplitude kernel function and 
the actual transmitted light amplitude, and is proportional to the square 
of the volume of a truncated cone-type 3D structure. The whole cone-type 
structure represents the amplitude kernel function on a 2D region and is 
centered at that point and has a horizontal range r. In this instance, the 
truncation is also done according to the actual light transmission around 
that point, which may be blocked by any opaque region in the photomask. 
For an edge point E (when its distance to its nearest corner is larger 
than r), after truncation, its volume is half of whole volume under the 
amplitude kernel function (i.e., one-half of whole volume 3D cone-type 
structure). For a corner point C, after cut, its volume is one quarter 
(1/4) of whole volume under the amplitude kernel function. Thus, it is the 
case that I.sub.C =I.sub.E /4, also independent of the range r and the 
form of the amplitude kernel function. Here, 4 comes from the square of 2. 
The aerial intensity contour curve passing through the edge point E will 
not pass through the corner point C; rather, it passes through an inner 
point. Consequently, the corner rounding also exists for the coherent 
illumination. 
For partially coherent light illumination, the corner rounding can be 
understood qualitatively: the light contribution to a corner point C' 
comes from within a quarter circle region of radius r (for circular 
aperture), e.g., at corner 12d (FIG. 1) which is less than the 
contribution of an edge point E coming from within a half-circle region of 
radius r; for a square aperture, it is from within a square region of 
length r, e.g., at corner 12b (FIG. 1) which is less than the contribution 
of an edge point E coming from within a rectangle of a size 2r.times.r. It 
is readily understood that a partial coherent illumination with large 
coherence factor ".sigma." may be approximated as an incoherent 
illumination. Likewise, a partial coherent illumination with small 
coherence factor ".sigma." may be treated as a coherent illumination. 
Line end shortening can be understood similarly based on the geometric 
representations depicted in FIG. 1. 
One of main reasons for optical proximity effects is light diffraction. 
Optical proximity effects coming from light diffraction may be overcome 
partly by using a shorter wavelength light source, and, with a projection 
system possessing a larger numerical aperture. In practice, the wavelength 
of an optical light source is typically fixed (365 nm, 248 nm, 193 nm, 
etc.) and there is a practical upper limit on numerical aperture. So other 
resolution enhancement methods, including the use of phase-shifting masks 
and masks with serifs, have been developed to correct optical proximity 
effects. 
When optical proximity effects are not severe, both corner rounding and 
line end shortening can be corrected completely with the use of hanging 
serifs. This has been disclosed in detail in commonly-owned, co-pending 
U.S. patent application Ser. No. 09/167,948 entitled SERIF MASK DESIGN 
METHODOLOGY BASED ON ENHANCING HIGH SPATIAL FREQUENCY CONTRIBUTION FOR 
IMPROVED PRINTABILITY" the contents and drawings of which are wholly 
incorporated by reference as if fully set forth herein. U.S. Pat. No. 
5,707,765 describes further techniques implementing serifs for correcting 
optical proximity effects in microlithographic circuits. 
FIG. 2(a) depicts a serif design 28 in a photomask line end portion 21 for 
illumination by means including a square aperture and when the kernel 
function's range r is less than or equal to the pattern line width w, 
i.e., r.ltoreq.w. Within the intensity/amplitude kernel function 
representation at respective square regions 22a, . . . ,22d in FIG. 2(a), 
different truncations still lead to the same total volume of 3D cone 
structure resulting in I.sub.E =I.sub.C =I.sub.S =I.sub.N where I.sub.S is 
the illumination (aerial) intensity at point S and .sup.1 N is the 
illumination intensity at point N for both incoherent and coherent light 
illuminations. Thus, hanging square serifs 23a, . . . ,23d each of size 
r.times.r work ideally when r.ltoreq.w. 
FIG. 2(b) similarly depicts a serif design 29 in a photomask line end 
portion 27 for illumination by means including a circular aperture and 
when the kernel function's radius r is not larger than the wire width "w", 
i.e., r.ltoreq.w. Within the intensity/amplitude kernel function 
representations at respective circular regions 24a, . . . ,24d in FIG. 
2(b), different truncations still lead to the same total volume of 3D cone 
structure, i.e., I.sub.E =I.sub.C =I.sub.S =I.sub.N for both incoherent 
and coherent light illuminations. Thus, hanging quarter-circle serifs 25a, 
. . . ,25d of radius r serifs work ideally when r.ltoreq.w. 
When optical proximity effects are severe (i.e., big corner rounding and 
large amount of foreshortening), the prior art serif mask designs (shown 
in FIGS. 2(a) and 2(b)) no longer work satisfactorily. These severe 
optical proximity effects occur when the kernel function's range r is 
larger than wire width w, which represents the typical feature size, i.e., 
patterned line width. 
FIG. 3(a) depicts a serif design 30 in a photomask line end portion 31 for 
illumination by means including a square aperture when the kernel 
function's range r becomes larger than the line width w, i.e., r&gt;w. Now 
the masked areas (or unmasked areas) within 2D kernel representations 
32a,32b in FIG. 3(a) are no longer equal, resulting in an unequal 
intensity relation I.sub.E .noteq.I.sub.C. Thus, hanging square serifs 
33a,b, of size r.times.r no longer provides desired results on aerial 
image/resist pattern when r&lt;w. 
FIG. 3(b) depicts a serif design 35 in a photomask line end portion 38 for 
illumination by means including a circular aperture when the kernel 
function's radius r becomes larger than the line width w, i.e., r&gt;w. Here, 
the mask areas (or unmasked areas) within 2D kernel representations 36a, 
36b in FIG. 3(b) becomes unequal, resulting in an unequal intensity 
relation I.sub.E .noteq.I.sub.C. Consequently, hanging quarter-circle 
serifs 37a, 37b of radius r no longer provides desired results for optical 
proximity correction (OPC) when r&gt;w. 
It would be highly desirable to provide a serif mask design for 
photolithographic mask that sufficiently corrects severe line end 
foreshortening and corner rounding effects for situations whereby the 
illumination kernel function's effective range (or radius) r is larger 
than the patterned line width w. 
SUMMARY OF THE INVENTION 
It is an object of the present invention to provide a photomask structure 
for correcting severe corner rounding and severe line end shortening 
effects in microlithography. Preferably, the photomask of the invention is 
designed to address the situation of r&gt;w, where w is the line or wire 
width (typical feature size), and r is the kernel function's effective 
range. Namely, in a local coordinate system which centers at a point under 
consideration for calculating its aerial image, the kernel function K is 
substantially zero when its argument is larger than r: K(x,y)=0 when 
.vertline.x.vertline.&gt;r and/or .vertline.y.vertline.&gt;r, for a square 
aperture and, in terms of polar coordinates (.rho., .phi.), and K(.rho.)=0 
when .rho.&gt;r for a circular aperture. For incoherent light illumination, r 
may be the range of the combined kernel function for both optical imaging 
process (which leads to aerial image) and subsequent etch and resist 
development processes (assumed to be a convolution process, e.g., a simple 
diffusion process), in which case the intensities I.sub.E and I.sub.C 
represent final resist pattern. 
According to the principles of the invention, there is provided a 
photolithographic mask for conducting a light source onto a semiconductor 
surface during a photolithographic imaging process, the mask including a 
line end portion for illumination by the light source, the line end 
portion of a width w and including two corners each defining a respective 
region for locating one or more serifs therein for correcting severe 
corner rounding and line end foreshortening effects caused by optical 
diffraction during the photolithographic process, wherein the aerial 
image/resist pattern of an imaging system having a square/circular 
aperture is modeled as a convolution or the square of a convolution 
between the photomask and an intensity/amplitude kernel function having an 
effective range r in x and y directions, a design whereby under a 
condition that w&lt;r&lt;=2w, a hanging square serif of a size w.times.w is 
provided attached to a respective corner within a corner region. For the 
condition of 2nw 21 r.ltoreq.2(n+1)w, with n=1, 2, . . . , each corner 
region includes an associated (n+1) serifs, each being linearly aligned 
along line-end extension line and spaced apart from an adjacent serif by a 
distance w, with each of the first n serifs being square and of a width w, 
and the (n+1).sup.th serif being a rectangle of a length w and a width 
min(r-2nw, w). 
When an imaging system including a circular aperture is implemented during 
photolithographic process, it may be represented as providing an effective 
intensity/amplitude kernel function represented as a circle of radius r, 
enabling the serif size in the corner regions to be optimized. For a 
hanging square serif of size w.times.w under the condition w&lt;r.ltoreq.2w 
mentioned above, if further w&lt;r&lt;.sqroot.2 w, then circular 
intensity/amplitude kernel function of radius r centered at its respective 
wire corner intersects the hanging square serif of size w.times.w. 
Consequently, the hanging serif in each corner region may be reduced in 
size without altering the aerial image/resist pattern intensity at its 
respective corner. The portion of the square serif to be removed is that 
portion of the square which are outside the circular intensity/amplitude 
kernel function of radius r centered at its respective corner. For a set 
of associated (n+1) serifs under a condition 2nw&lt;r.ltoreq.2(n+1)w 
mentioned above, if further 2nw&lt;r&lt;[(2n+1).sup.2 +1].sup.1/2 w, with n=1, 
2, . . . , then circular intensity/amplitude kernel function of radius r 
centered at its respective wire corner intersects the (n+1).sup.th 
rectangular serif, which is the most outside one among (n+1) serifs. Thus, 
the (n+1).sup.th serif in each corner region may be reduced in size 
without altering the aerial image/resist pattern intensity at its 
respective corner. The portion of the (n+1).sup.th serif to be removed is 
that portion of the rectangle which are outside the circular 
intensity/amplitude kernel function of radius r centered at its respective 
corner. On the other hand, if further [(2n+1).sup.2 +1].sup.1/2 
w.ltoreq.r.ltoreq.2(n+1)w, with n=1, 2, . . . , then the (n+1).sup.th 
serif is not intersected by the circular intensity/amplitude kernel 
function and thus there is no removal of part of the (n+1).sup.th serif.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
For the case of a square aperture, the following symmetry properties 
usually exist: 
EQU K(x,y)=K(.vertline.x.vertline.,.vertline.y.vertline.)=K(y,x) 
where K is the kernel function defined herein, and x=y=0 is the center of 
the kernel function. 
FIG. 4(a) depicts conceptually the serif mask design 34 for a 
photolithographic mask line end portion 39 of width w having corners C1 
and C2, with a corner C1 having extension lines a and b (indicated as 
broken lines) and defining a region Z1, and a corner C2 having extension 
lines c and d (indicated as broken lines) and defining a region Z2. Region 
Z1 is the corner region for corner C1, and region Z2 is the corner region 
for corner C2. The extension lines a and d are called line-end extension 
lines. As will be explained in greater detail hereinbelow, regions Z1 and 
Z2 include all the serifs according to the embodiments of invention. 
FIG. 4(b) depicts a serif design 40 in a photomask line end portion 41 for 
illumination by means including a square aperture for the case where 
w&lt;r.ltoreq.2w. Hanging square serifs 43a,43b are of size w.times.w. It is 
understood that the masked areas (or unmasked areas) within kernel 
function representations 42a,42b in FIG. 4 are the same. More importantly, 
total volumes of 3D cone structures represented within clear regions of 2D 
squares 42a, 42b are equal, independent of the specific shape of kernel 
function K(x,y). This means equal aerial intensity (i.e., I.sub.C 
=I.sub.E) for both incoherent and coherent light illuminations, 
independent of the shape of their kernel function. 
FIG. 5 depicts a serif design 45 in a photomask line end portion 47 for 
illumination by means including a square aperture for the case where 
2w&lt;r.ltoreq.4w. In this example, there is provided hanging square serifs 
43a, b, as in the example of FIG. 4(b), however, additionally there is 
provided respective disconnected serifs 48a, 48b, associated with a 
respective corner "C" of the wire 47. The first serifs 43a, 43b are the 
same as that in FIG. 4(b), i.e., squares of size w.times.w. The length of 
each disconnected serif 48a, 48b is w. The width of each disconnected 
serif 48a, 48b is the smaller of w and (r-2w), depending upon the kernel 
range r. As shown in FIG. 5, the distance between a hanging square serif, 
e.g., 43b, and its respective disconnected serif, e.g., 48b, is equal to 
the width w of the wire. It still holds true that the total volumes of 3D 
cone structures within clear regions of 2D squares 42a,42b are equal, 
independent of the specific shape of the kernel function K(x,y). Again, 
there is equal aerial intensity (I.sub.C =I.sub.E) for both incoherent and 
coherent light illuminations, independent of the shape of their kernel 
function. 
FIG. 6 depicts a serif design 50 in a photomask line end portion 44 for 
illumination by means including a square aperture for the case where 
4w&lt;r.ltoreq.6w. In this example, there are three serifs associated with 
each corner of the wire: a first hanging square serif 43a (43b) of width 
w, and two additional disconnected square serifs 48a (48b) each of width 
w, and disconnected serifs 49a (49b) each of length w and width being the 
smaller of w and (r-4w) depending upon the kernel function's range r. As 
shown in FIG. 6, the distance between a disconnected square serif, e.g., 
48b, and the second disconnected square serif, e.g., 49b, is equal to the 
width w of the wire. Once again, it still holds true that the total 
volumes of 3D cone structures within clear regions of 2D squares 42a,42b 
are equal and I.sub.C =I.sub.E for both incoherent and coherent light 
illuminations, independent of the shape of their kernel functions. 
In general, according to the invention, a serif design for a mask portion 
for illumination by means including a square aperture and wherein 
2nw&lt;r.ltoreq.2(n+1) w, with n=1, 2, . . . , each corner has (n+1) 
disconnected serifs. Each of the first n serifs being of size w.times.w, 
and the (n+1).sup.th serif being a rectangle of a length w and width 
min(r-2nw, w) and the distance between two adjacent serifs being w. 
For the case of a circular aperture, each of serif designs shown in FIGS. 
4-6 still gives equal aerial intensity I.sub.C =I.sub.E for circular 
apertures under the same relationship between the kernel function's range 
r and wire width w. Under certain situations, the serif's shape may be 
optimized without changing aerial intensity I.sub.C at the corner (i.e., 
I.sub.C=I.sub.E is always maintained). That is, the serif's area may be 
reduced so that serif itself becomes less likely to show up in resist 
pattern. 
FIG. 7 depicts a serif design 60 in a photomask line end portion 61 for 
illumination by means including a circular aperture for the case where 
w&lt;r&lt;.sqroot.2 w. In this embodiment, each hanging serif 63a, 63b has a 
shape comprising the intersection (a Boolean operation) of the original 
square serif 43a,b, respectively, of size w.times.w, with the kernel 
function representation at circular region 62 of radius r centered at each 
line corner C and C' respectively. Thus, with circle region 62 centered at 
point C, hanging serif 63b is intersected, and with circle region 62 
centered at point C', hanging serif 63a is intersected, as shown in FIG. 
7. Each of the intersected portions 64 of each hanging serif 63a,b may be 
cut-out without changing the resultant aerial image intensity at corners C 
and C', and thus maintaining I.sub.E =I.sub.C. 
FIG. 8 depicts a serif design 70 in a photomask line end portion 71 for 
illumination by means including a circular aperture for the case where 
.sqroot.2 w.ltoreq.r.ltoreq.2w. In this embodiment, the 2D 
intensity/amplitude kernel function representation at circular region 72 
of radius r which centers at the corner C does not cut off the original 
square serif 63b, thus rendering the serif design identical to that 
depicted in FIG. 4(b). 
FIG. 9 depicts a serif design 80 in a photomask line end portion 81 for 
illumination by means including a circular aperture for the case where 
2w&lt;r&lt;.sqroot.10 w. In this embodiment, two hanging square serifs 63a, 63b, 
each of width w, are provided as in the case of FIG. 8. Additionally 
provided are disconnected serifs 83a, 83b aligned with respective hanging 
serifs 63a, 63b, and separated a distance w therefrom with each preferably 
of a length w and a width being the smaller of w and (r-2w) depending upon 
the kernel function's range r. Preferably, the shape of each outermost 
disconnected serif 83a, 83b comprises the intersection of the original 
rectangular serif of size w.times.min (r-2w,w) with the kernel function 
representation at circular region 82 of radius r centered at the corners 
C',C, respectively. Thus, as shown in FIG. 9, a portion 84 of each hanging 
serif 83a,b may be removed without changing the resultant aerial image 
intensity at corners C and C', and thus maintaining I.sub.E =I.sub.C. For 
the embodiment where .sqroot.10 w .ltoreq.r.ltoreq.4w, there is no cut-off 
(intersection of original rectangular serif of size w.times.min (r-2w,w) 
with the kernel function representation at circular region 82 of radius r) 
at the disconnected serif, and the serif design is exactly that as shown 
in FIG. 5. 
FIG. 10 depicts a serif design 90 in a photomask line end portion 91 for 
illumination by means including a circular aperture for the case where 
4w&lt;r&lt;.sqroot.26 w. In this embodiment, there are three serifs associated 
with each corner of the line end: a first hanging square serif 63a (63b) 
of width w, and two additional disconnected square serifs 93a (93b) each 
of width w, and separated from a respective hanging serif by a distance w, 
and disconnected serifs 94a (94b) each of length w and width being the 
smaller of w and (r-4w) depending upon the kernel function's range r. The 
ideal shape of each outermost disconnected serif 94a, 94b is the 
intersection of the original rectangular serif of size w.times.min 
(r-4w,w) with the kernel function representation comprising circular 
region 92 of radius r which centers at the corners C, C', respectively. 
For the embodiment where .sqroot.26 w.ltoreq.r.ltoreq.6w, there is no 
cut-off (intersection of original rectangular serif of size w.times.min 
(r-4w,w) with the kernel function representation at circular region 92 of 
radius r) at the disconnected serif, and the serif design is exactly that 
as shown in FIG. 6. 
In general, when 2nw&lt;r&lt;[(2n+1).sup.2 +1].sup.1/2 w, with n=1, 2, . . . , 
the (n+1).sup.th serif will be intersected and cut by a circle of radius r 
which centers at the corner C; when [(2n+1).sup.2 +1].sup.1/2 
w.ltoreq.r.ltoreq.2(n+1)w, the (n+1).sup.th serif will not be cut-off by 
the circle. 
FIG. 11 illustrates a resultant aerial image/photoresist pattern 100 for 
the embodiment where w&lt;r.ltoreq.2w. In this embodiment, the serif mask 
design 110 including hanging serifs 102a,b and corresponding to the mask 
designs as depicted in FIG. 4(b) and FIG. 8, is employed. At one end 108 
of the pattern line 109, there is no serif, and severe corner rounding and 
line end shortening is apparent. At the opposite end 107, the serif design 
shown in either FIGS. 4(b) and 8 is used, and the aerial image/resist 
pattern 100 passes through both corner points C and C' exactly. 
FIG. 12 illustrates a resultant aerial image/photoresist pattern 120 for 
the embodiment where 3w&lt;r.ltoreq.4w. In this embodiment, the serif mask 
design 130 including hanging serifs 122a, . . . , 122d and corresponding 
to the mask designs as depicted in FIG. 5 and FIG. 9, is employed. At one 
end 128 of the pattern line 129, there is no serif, and very severe corner 
rounding and line end shortening is clearly apparent. At the opposite end 
127, the serif design shown in either FIG. 5 or 
FIG. 9 is used, and the aerial image/resist pattern 120 passes through both 
corner points C and C' exactly. 
FIG. 13 is an example of co-serif mask design 140 between two neighboring 
lines 149a, . . . , 149n for the case of w&lt;r.ltoreq.2w. In the 
minimum-pitch wire/space array illustrated in FIG. 13, each wire track is 
occupied. The corresponding resulting aerial image/photoresist patterns 
150a, . . . , 150n are also displayed. At respective first ends 148a, . . 
. , 148n of each the lines, there exists severe corner rounding and line 
end shortening of the resultant pattern 150. At each of the opposite ends 
147a, . . . , 147n of lines, both severe corner rounding and 
foreshortening are reduced significantly. Although I.sub.C is no longer 
equal to I.sub.E in FIG. 13, i.e., the neighboring line contributes more 
to an edge point than to a corner point, optimization of each serif's 
size, shape, and/or position may be carried out to achieve I.sub.C 
=I.sub.E. As shown in FIG. 13, each adjacent hanging serif 143 is 
separated by a distance w. 
FIG. 14 illustrates another co-serif mask design 160 between two 
neighboring lines 169a, . . . , 169n for the case of 2w&lt;r.ltoreq.4w. In 
this embodiment, wire tracks in the minimum-pitch wire/space array are 
occupied alternatively. In FIG. 14, the corresponding resulting aerial 
image/photoresist patterns 170a, . . . , 170n are also displayed. At 
respective first ends 168a, . . . , 168n of each the lines, there exist 
severe corner rounding and line end shortening of the resultant pattern 
170. At each of the opposite ends 167a, . . . , 167n of the lines, severe 
foreshortening is corrected completely (with some over-correction) due to 
serifs located at the end of the line and the serif located at the near 
end of the adjacent tracks (See FIG. 5). That is, serif 163c of line 169c 
additionally contributes to the correction of corner rounding for the 
pattern 170b at line 169b. If 3w&lt;r&lt;4w, then I.sub.C and I.sub.E are not 
exactly equal, but the difference between them is very small such that the 
curve of aerial image/resist pattern nearly passes through corner points C 
and C' at each opposite end 167. As shown in FIG. 14, each adjacent 
hanging serif is separated by a distance w. 
While the invention has been particularly shown and described with respect 
to illustrative and preferred embodiments thereof, it will be understood 
by those skilled in the art that the forgoing and other changes in form 
and detail may be made therein without departing from the spirit and scope 
of the invention which should be limited only by the scope of the appended 
claims.