Exposure method and apparatus

The present invention relates to a projection exposure method and an exposure apparatus for projecting a pattern on a photo-mask through a projection optical system onto a predetermined photosensitive material. The photosensitive material is a nonlinear photosensitive material in which the effective light intensity distribution is nonlinear to the intensity of incident exposure light. Further, multiple exposure is made with plural patterns having a certain lateral shift therebetween, or multiple exposure is made with shifting the position of an identical pattern, whereby multiple exposure is effected with patterns different in intensity distribution on the photosensitive material. A high-resolution pattern can be formed over the resolution limit of projection optical system by the above arrangement.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
The present invention relates to an exposure apparatus for producing 
semiconductor devices or liquid crystal display devices and so on, 
applying the photolithography technology, and more particularly to a 
projection exposure method therefor and a projection-type exposure 
apparatus. 
2. Related Background Art 
Conventional exposure methods employ such a process, for example, as 
disclosed in Japanese Patent Publication No. 62-50811 or EP. 90924, that 
predetermined transfer patterns such as circuit patterns are provided on a 
photo-mask or reticle and each transfer pattern is printed on a substrate 
by single exposure. In the process, the substrate is coated with a 
photosensitive material (photoresist) which has such a proportional 
relation that a latent image produced according to irradiated exposure 
light (hereinafter referred to as latent image .xi.) is proportional to 
exposure intensity I. For example, conventionally generally used positive 
photoresists have a property as represented by the following equation. 
EQU .xi.=exp (-CD) and D=I.multidot.t (1) 
In the above equation, I is a light intensity at each point of the exposure 
pattern, t is an exposure time, and C is a characteristic constant 
depending upon the type of photoresist. 
The Equation (1) can be generalized by the following equation. 
EQU .xi.=exp (-CD) and D=J.multidot.t=I.sup.m .multidot.t (2) 
In the above equation, m is an index for indicating the linearity of 
photoresist. Namely, a photoresist with m=1 has a linear characteristic 
satisfying Equation (1), while a photoresist with m.noteq.1 shows a 
nonlinear characteristic. Further, the m-th power of light intensity I, 
that is, J (=I.sup.m) is defined as effective light intensity peculiar to 
a photoresist for producing the latent image .xi. by light intensity I of 
exposure light irradiated onto the photoresist (hereinafter referred to as 
effective light intensity J). 
Furthermore, assuming that image formation is completely incoherent, 
spectrum i of intensity distribution I(x) (hereinafter referred to as 
light intensity distribution) of exposure light for producing a latent 
image in the photoresist is given as follows with object spectrum i.sub.0, 
OTF (Optical Transfer Function) of projection optical system in exposure 
apparatus being f, and spatial frequency .nu.. 
EQU i(.nu.)=i.sub.0 (.nu.).multidot.f(.nu.) (3) 
Now, a spatial frequency .nu..sub.0, which is a limit frequency at which 
the OTF (i.e., f) becomes not significant in respect of process, is given 
as follows with wavelength .lambda. of exposure light, numerical aperture 
NA of projection optical system on the photosensitive material side (in 
the image space), and process constant K.sub.1. 
EQU .nu..sub.0 =0.5NA/(K.sub.1 .multidot..lambda.) (4) 
Further, the resolution limit of a projection optical system is 
theoretically determined by numerical aperture NA. In this case, K.sub.1 
=0.25, and therefore a cutoff frequency .nu..sub.c of projection optical 
system is given as follows. 
EQU .nu..sub.c =2NA/.lambda. (5) 
Accordingly, in order to achieve high-resolution exposure by the 
conventional technique, the wavelength .lambda. of exposure light must be 
decreased or the numerical aperture NA must be increased. 
As described above, the conventional exposure methods and exposure 
apparatus had no other means to achieve high-resolution exposure than 
increasing the numerical aperture NA or decreasing the wavelength .lambda. 
of exposure light. However, since the depth of focus F.sub.d of a 
projection optical system is proportional to the wavelength .lambda. and 
to the inverse of the second power of numerical aperture NA, as shown by 
the following Equation (6), the depth of focus F.sub.d becomes shallower 
in both cases, which results in failing to secure the process latitude of 
the exposure apparatus. In addition, the scale of projection optical 
system must be increased or a special structure must be employed therefor, 
which makes it difficult to achieve a practical exposure apparatus. 
Further, the final resolution limit of the photosensitive material 
(photoresist) could not surpass (or go beyond) the resolution limit 
determined by the projection optical system. 
EQU Fd=K.sub.2 .multidot..lambda./(NA).sup.2 ( 6) 
(where K.sub.2 is the process constant.) 
SUMMARY OF THE INVENTION 
The present invention has been accomplished taking the above points in the 
conventional technology into consideration. It is an object of the present 
invention to provide a novel exposure method and a novel exposure 
apparatus which can achieve high-resolution exposure surpassing (beyond) 
the resolution limit of projection optical system with neither employing 
means to change the wavelength of exposure light nor greatly changing the 
structure of projection optical system (with neither increase in scale nor 
complexity of the system). 
To achieve the above object, the present invention is directed to an 
exposure method and exposure apparatus for projecting patterns on a 
photo-mask (reticle) onto photosensitive material coated on a substrate 
through a projection optical system, in which the photosensitive material 
has nonlinear sensitivity characteristic by which a latent image is formed 
nonlinearly to the light intensity of incident exposure light and multiple 
exposure processes different in light intensity distribution are 
performed, whereby patterns in the photosensitive material are formed with 
higher resolution than the resolution limit of the projection optical 
system. 
The photosensitive material with the nonlinear characteristic as described 
may be any with nonlinear characteristic, for example, those with such a 
nonlinear relation that the latent image (.xi.) is emphasized according to 
the m-th power of incident light intensity (I) (i.e., I.sup.m where m&gt;1), 
or conversely, those with such a nonlinear relation that the latent image 
(.xi.) is relaxed according to the m-th power of incident light intensity 
(I) (i.e., I.sup.m where m&lt;1). 
An embodiment of the exposure method and exposure apparatus according to 
the present invention is arranged as follows. Plural types of patterns 
mutually spatially shifted are preliminarily prepared independent of each 
other on plural photo-masks, in accordance with the predetermined pattern 
to be formed in nonlinear photosensitive material. The exposure processes 
are successively carried out to effect multiple exposure with plural light 
intensity distributions having certain spatial shift therebetween (which 
are light intensity distributions corresponding to the plural types of 
patterns as described above). Namely, the plural patterns are 
preliminarily prepared on the plural photo-masks in the relation of 
predetermined spatial shift therebetween, and the multiple exposure is 
effected based on these photo-masks to form a substantially 
higher-resolution exposure pattern in photosensitive material than the 
resolution limit of projection optical system. 
Another embodiment of the exposure method and exposure apparatus according 
to the present invention is so arranged that a single photo-mask is 
prepared with a pattern corresponding to an exposure pattern of 
predetermined pattern to be formed in nonlinear photosensitive material 
and the multiple exposure is effected by shifting the photo-mask, whereby 
the exposure pattern is formed in the photosensitive material, with a 
substantially higher resolution than the resolution limit of projection 
optical system. 
Still another embodiment of the exposure method and exposure apparatus 
according to the present invention is so arranged that exposure is 
effected with plural light intensity distributions spatially shifted each 
other by an arbitrary combination of the exposure processes as employed in 
the above two embodiments, i.e., by multiple exposure in an arbitrary 
combination of the multiple exposure using the plural types of photo-mask 
and the multiple exposure using a single specific photo-mask, whereby an 
exposure pattern is formed on the photosensitive material, with 
substantially higher resolution than the resolution limit of projection 
optical system. 
It should be noted that the present invention is not limited to the above 
three embodiments described as relatively specific examples. The essence 
of the present invention is that multiple-exposure is effected onto 
nonlinear photosensitive material with two or more exposure patterns 
mutually spatially shifted, whereby an exposed pattern is formed in 
photosensitive material, with higher resolution than the resolution limit 
of the projection optical system. 
An example of the above nonlinear photosensitive material is a resist 
having the two-photon absorption property as observed with aromatic 
materials or polystyrene materials (hereinafter referred to as a 
two-photon absorption resist). The nonlinearity-of such a two-photon 
absorption resist is described, for example, in "NIKKEI MICRODEVICES, 
FEBRUARY, 1987, pp. 91-101" or in "En S. Wu, et al, "TWO-PHOTON 
LITHOGRAPHY FOR MICROELECTRONIC APPLICATION," SPIE vol. 1674 Optical/Laser 
Microlithography V (1992), pp. 776-782." 
Further, the above nonlinear photosensitivity may be obtained by coating 
the substrate with plural photosensitive materials having mutually 
different latent image depending upon the light intensity. 
The present invention will become more fully understood from the detailed 
description given hereinbelow and the accompanying drawings which are 
given by way of illustration only, and thus are not to be considered as 
limiting the present invention. 
Further scope of applicability of the present invention will become 
apparent from the detailed description given hereinafter. However, it 
should be understood that the detailed description and specific examples, 
while indicating preferred embodiments of the invention, are given by way 
of illustration only, since various changes and modifications within the 
spirit and scope of the invention will become apparent to those skilled in 
the art from this detailed description.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
The basic principle of the present invention is first described in detail. 
The following description concerns only the most fundamental case, that 
is, a case in which double exposure is effected with two photo-masks, 
which are a photo-mask having a first pattern and a photo-mask having a 
second pattern, so as to achieve a resolution over the resolution limit of 
the projection optical system. A photosensitive material with nonlinear 
sensitivity property as stated herein means a material in which the latent 
image .xi. changes according to the m-th power (where m.noteq.1) of light 
intensity I of incident exposure light. Further, an effective light 
intensity J is defined as an effective light intensity contributing to 
cause the latent image .xi. in the photosensitive material for light 
intensity I of exposure light irradiated from the projection optical 
system onto the photosensitive material. Then, an effective light 
intensity distribution J(x) is defined as the m-th power of the light 
intensity distribution I(x). 
First described are various examples of studies conducted by Inventors 
before reaching the present invention. Then the principle and effect of 
the present invention will be described to clarify the concept of the 
present invention. 
First considered is a first study example, in which only one exposure 
process is carried out with linear photosensitive material in the 
conventional exposure method and exposure apparatus. FIG. 12A shows an 
effective light intensity distribution J(x) of the linear photosensitive 
material in the case that an exposure pattern on the image plane in the 
photosensitive material takes a sinusoidal light intensity distribution as 
one-dimensionally seen. When single exposure is effected under incoherent 
illumination light, the light intensity distribution I(x) of the exposure 
pattern imaged on the photosensitive material through the projection 
optical system is given by the following equation, if the light intensity 
distribution of the object is expressed by I.sub.0 (x) and the point 
spread function of projection optical system by F(x). 
EQU I(x)=I.sub.0 (x)*F(x) (7) 
In the above equation, x is a positional coordinate on the photosensitive 
material and the symbol * is an operator representing the convolution. 
Then, spectrum i of intensity on the image plane in the linear 
photosensitive material is given as follows by the convolution theorem of 
Fourier transformation. 
EQU i(.nu.)=i.sub.0 (.nu.).multidot.f(.nu.) (8) 
In the above equation, .nu. is a spatial frequency, i.sub.0 a spectrum of 
intensity of the object, and f corresponds to the so-called OTF of the 
projection optical system. Accordingly, as apparent from FIG. 12A and 
Equation (8), the spectrum of effective light intensity distribution J(x) 
is not formed in a single exposure process over the cutoff frequency 
(2NA/.nu.) of projection optical system and it is, therefore, impossible 
to surpass the resolution limit of projection optical system. 
Let us consider a second study example, in which only one exposure process 
is carried out using photosensitive material with nonlinear sensitivity 
characteristic applied in the present invention, for example, two-photon 
absorption resist. Reference should be made as to the two-photon 
absorption resist, for example, to Proceedings of SPIE vol. 1674 (1992) 
pp. 776-778. The two-photon absorption resist has a property in which the 
effective light intensity distribution J(x) is proportional to the square 
(m=2) of light intensity distribution I(x). Then, if the intensity 
distribution of the exposure pattern is sinusoidal on the image plane in 
the two-photon absorption resist, the resultant effective light intensity 
distribution is sharper in some portions but duller in some other portions 
as shown in FIG. 12B. Although such a pattern of sharp peaks is partially 
with high contrast, the pitch of the pattern as shown in FIG. 12B never 
becomes smaller than that of light intensity distribution I(x) of the 
pattern, because Equation (8) is satisfied. Thus, the resolution cannot 
surpass the resolution limit of projection optical system. Therefore, it 
is understood that the higher resolution cannot be achieved simply by 
applying the nonlinear photosensitive material to the conventional 
exposure apparatus in place of the linear photosensitive material which 
has been generally used in the conventional apparatus. 
Let us now consider a third study example, in which a nonlinear 
photosensitive material applied in the present invention is employed and 
double exposure is made at a same position onto the photosensitive 
material with a single pattern in a superimposed manner. Assuming the 
exposure is carried out under incoherent illumination light, the effective 
light intensity distribution J(x) is as follows with the intensity 
distribution of object being expressed by I.sub.0 (x) and the point spread 
function of projection optical system by F(x). 
EQU J(x)={I(x)}.sup.2 ={I.sub.0 (x)*F(x)}.sup.2 (9) 
From this, the spectrum j of effective light intensity distribution is as 
follows by the convolution theorem of Fourier transformation. 
EQU j(.nu.)={i.sub.0 (.nu.).multidot.f(.nu.)}*{i.sub.0 
(.nu.).multidot.f(.nu.)}(10) 
Since the effective light intensity distribution is given by Equation (9) 
in case of the two-photon absorption resist, the effective light intensity 
distribution is further sharper when compared with Equation (7) in case of 
the first study examples (see FIG. 12A and FIG. 12B). 
While the effective light intensity distribution is sharper, the resolution 
is not raised in respect of the spatial frequency. Therefore, if such 
exposure is simply repeated twice, the resolution cannot surpass the 
resolution limit of projection optical system as in the second study 
result. 
Now considered is a fourth study example, in which a linear photosensitive 
material which is conventionally generally used is employed and exposure 
is carried out twice with two types of intensity distributions mutually 
spatially shifted. For example, as shown in FIG. 13, first exposure 
irradiates the linear photosensitive material with an exposure pattern of 
a sinusoidal intensity distribution I.sub.1 (x) as shown by the solid line 
(which is shown in one dimension for convenience), and second exposure 
irradiates the linear photosensitive material with an exposure pattern of 
a sinusoidal intensity distribution I.sub.2 (x) shifted .pi. in phase as 
shown by the dotted line in FIG. 13. FIG. 14A shows the properties of the 
linear material about development rate against exposure, in which the 
horizontal axis represents the product of intensity I and exposure time t, 
which means exposure E. Since the exposure E corresponds to energy, the 
effective light intensity J and the exposure E are shown as substantially 
equivalent to each other. 
Let us consider two different points X.sub.A and X.sub.B on the linear 
photosensitive material with these drawings. The intensity distribution 
I.sub.1 (x) provides I.sub.1 (X.sub.A)=I.sub.MAX as the intensity of light 
incident into a certain point X.sub.A on the linear photosensitive 
material, while the intensity distribution I.sub.2 (x) provides I.sub.2 
(X.sub.2)=0 as the intensity of light incident into the certain point 
X.sub.A on the linear photosensitive material. Similarly, the intensity of 
light incident into another point X.sub.B spatially shifted by .pi. in 
phase from the point X.sub.A is obtained as I.sub.1 (X.sub.B)=I.sub.MAX /2 
and I.sub.2 (X.sub.B)=I.sub.MAX /2. Accordingly, the exposure E.sub.A at 
point X.sub.A is calculated as follows. 
##EQU1## 
Also, exposure E.sub.B at point X.sub.B is as follows. 
##EQU2## 
Consequently, in case of the linear photosensitive material being used, 
development rate G.sub.A and G.sub.B at two different points X.sub.A and 
X.sub.B are equal to each other as shown in FIG. 14A. This means that when 
the exposures are carried out twice onto the linear photosensitive 
material with sinusaidal intensity distributions I.sub.1 (x) and I.sub.2 
(x) spatially shifted by half period, no contrast appears after 
development (that is, the line-and-space pattern cannot be separated), 
failing to achieve higher resolution. Explaining this by the effective 
light intensity distribution, the distribution becomes uniform as shown in 
FIG. 15, which raises a problem that the desired exposure cannot be 
achieved to transfer the mask pattern. 
The principle and effect of the present invention is next described taking 
the above studies into account. The present invention enables formation of 
finer patterns over the resolution limit of projection optical system by 
using a photosensitive material with nonlinear sensitivity characteristic 
and by carrying out multiple exposure processes with exposure patterns 
being mutually spatially shifted. To describe the basic concept of the 
present invention, let us consider an isolated pattern (point-like 
pattern) focused by the projection optical system in the case in which 
exposure is made twice onto two-photon absorption resist. In this case, 
the two-photon absorption resist makes the effective light intensity 
distribution of point image sharper. In this case, consideration suffices 
on the intensity distribution F(x) of point-like pattern by the projection 
optical system irrespective of the state of illumination light. Then, 
supposing the desired intensity distribution I.sub.0 (x) is formed by 
superposing the images of point-like pattern, the effective light 
intensity distribution J(x) will be a superposition of intensities of 
light images obtained by focusing the point images, which is given by the 
following equation. 
EQU J(x)=I.sub.0 (x)*{F(x)}.sup.2 (11) 
Since the effective light intensity distribution of the image of point-like 
pattern is expressed by {F(x)}.sup.2, it is sharper than the point spread 
distribution F(x) of the projection optical system, and is higher in 
resolution. By the Fourier transformation of the above Equation (11), the 
spectrum of the effective light intensity distribution is given by the 
following equation. 
EQU j(.nu.)=i.sub.0 (.nu.).multidot.{f(.nu.)*f(.nu.)} (12) 
Accordingly, f*f can be considered as substantial OTF of projection optical 
system to obtain the effective light intensity distribution in the 
principle of the present invention. This means that the cutoff frequency 
is 2NA/.lambda. in OTF (i.e., f) in the conventional technology as shown 
by Equation (8), whilst the cutoff frequency is 4NA/.lambda. in the 
present invention, obtaining a doubled resolution. FIG. 16A and FIG. 16B 
schematically show the comparison of OTF. FIG. 16A shows the OTF of the 
conventional exposure method, while FIG. 16B shows the effective OTF in 
the case that the two-photon absorption resist is used. This suggests that 
if a latent image is formed by multiple exposure scanning such an isolated 
pattern and using the two-photon absorption resist, a finer pattern can be 
formed with a resolution beyond the resolution limit of projection optical 
system. As described above, the pattern can be formed over the resolution 
limit of optical system by the combination of multiple exposure of 
isolated pattern with the photosensitive material having the nonlinear 
sensitivity characteristic. 
Further, in the case that multiple exposure is made with a pattern which is 
not perfectly isolated but nearly isolated, a resultant pattern can be 
also formed over the resolution limit of projection optical system in the 
photosensitive material similarly as in the case of completely isolated 
pattern. In this case, the spectrum j of effective light intensity 
distribution is as follows. 
EQU j(.nu.)=.SIGMA.i.sub.oj (.nu.).multidot.{f(.nu.)*f(.nu.)} (13) 
In the above equation, i.sub.oj is an object spectrum of the j-th nearly 
isolated pattern. Also, expressing I'(.nu.)=.SIGMA.i.sub.oj (.nu.) in 
Equation (13), I'(.nu.) can be deemed as an object spectrum of an 
imaginary pattern constituted by superposing isolated patterns. The cutoff 
frequency of OTF (f) never exceeds 2NA/.lambda. as shown by Equation (8) 
in the conventional spectrum of effective light intensity distribution, 
but in the present invention the cutoff frequency of the effective OTF of 
{f(.nu.)*f(.nu.)} is 4NA/.lambda. as shown by Equation (13), including a 
spectrum of effective light intensity distribution up to the cutoff 
frequency (4NA/.lambda.). 
As described above, even with use of the nonlinear photosensitive material 
in the conventional exposure method, the pitch in the pattern formed in 
the photosensitive material can never surpass the resolution limit of 
projection optical system. In the present invention, in contrast, multiple 
exposure is effected onto the nonlinear photosensitive material with 
intensity distributions mutually spatially shifted so as to properly 
provide the object spectrum I', whereby the effective light intensity 
distribution can be formed with the pitch over the resolution limit of the 
projection optical system. The present invention is further described as 
to a relation between exposure and development rate with reference to FIG. 
14B and FIG. 14C. The two types of intensity distributions with a mutual 
shift are those as shown in FIG. 13. In more detail, in the first exposure 
the nonlinear photosensitive material is irradiated by the exposure 
pattern of sinusoidal intensity distribution I'(x) as shown by the solid 
line in FIG. 13 and in the second exposure the nonlinear photosensitive 
material is irradiated by the exposure pattern of sinusoidal intensity 
distribution I.sub.2 (x) spacially shifted by .pi. in phase as shown by 
the dotted line in FIG. 13. FIG. 14B shows the characteristics of the 
nonlinear photosensitive material about the development rate against 
exposure, in which the horizontal axis represents the product of intensity 
I and exposure time t, i.e., exposure E. Since the exposure E corresponds 
to energy, the effective light intensity J and the exposure E are shown as 
equivalent to each other. 
Now let us consider two different points X.sub.A and X.sub.B on the 
nonlinear photosensitive material with these drawings. The intensity 
distribution I.sub.1 (x) gives I.sub.1 (X.sub.A)=I.sub.MAX as the 
intensity of light incident into a certain point X.sub.A on the nonlinear 
photosensitive material, and the intensity distribution I.sub.2 (x) gives 
I.sub.1 (X.sub.A)=0 as the intensity of light incident into the certain 
point X.sub.A on the nonlinear photosensitive material. Similarly, the 
intensity of light incident onto another point X.sub.B shifted .pi. in 
phase from the point X.sub.A is obtained as I.sub.1 (X.sub.B)=I.sub.MAX /2 
and I.sub.2 (X.sub.B)=I.sub.MAX /2. Since the effective light intensity is 
proportional to the square of intensity in case of the two-photon 
absorption resist, the exposure E.sub.A at the point X.sub.A is obtained 
as follows. 
##EQU3## 
Also, the exposure E.sub.B at point X.sub.B is as follows. 
##EQU4## 
Accordingly, in case of the nonlinear photosensitive material being used, 
the development rate G.sub.A and G.sub.B at two different points X.sub.A 
and X.sub.B are different from each other as shown in FIG. 14B. This means 
that if exposure is carried out twice using the nonlinear photosensitive 
material with two different intensity distributions I.sub.1 (x) and 
I.sub.2 (x) which are spatially shifted each other, a finer pattern can be 
developed with the nonlinear photosensitive material after development 
than the first intensity distribution I.sub.1 (x) and the second intensity 
distribution I.sub.2 (x). In the described case, the density of 
line-and-space pattern can be doubled. As described above, the present 
invention can achieve the high-resolution exposure over the limit 
resolution of projection optical system. 
Although the above description concerns the case in which the effective 
light intensity J is proportional to the square of light intensity I, 
which is the case of employing the two-photon absorption resist in which 
the latent image reaction intensity (.xi.) is formed in proportion to the 
square of light intensity (I), the present invention is not limited to 
this example but may be applied to cases employing other types of 
photosensitive material having nonlinear sensitivity characteristic in 
which the effective light intensity distribution J is formed in proportion 
with the m-th power (m&gt;1) of light intensity (I). In this case, the 
effective light intensity distribution of point-like pattern is expressed 
by the m-th power of the point spread function F(x). Therefore, the 
effective light intensity distribution of point-like pattern is sharper 
than the point spread function F(x). Equation (11) should be replaced by 
the following Equation (14). 
EQU J(x)=I.sub.0 (x)*{F(x)}.sup.m (14) 
The state of illumination light is not limited to the incoherent 
illumination. Finer patterns can be formed in nonlinear photosensitive 
material with oblique illumination or with various types of other modified 
illumination. The nonlinear photosensitive material may be irradiated by a 
beam emitted from a self-emitting object of course. 
Further, by the convolution theorem of Fourier transformation of Equation 
(14), it is seen that a pattern (effective light intensity distribution) 
can be formed up to a frequency m times higher than the cutoff frequency 
of projection optical system. A further finer pattern may be formed by 
multiple exposure of imperfectly isolated patterns. 
Although the above description concerns the case in which the power m in 
Equation (14) satisfies the relation of m&gt;1, that is, the case that the 
effective light intensity distribution J is emphasized to the intensity I, 
the invention can be applied to cases in which a nonlinear photosensitive 
material with the relation of power m as m&lt;1 is employed. With the 
nonlinear photosensitive material with m&lt;1, fine patterns can be formed 
substantially over the resolution limit of projection optical system. The 
power m in Equation (14) may vary. For example, as shown in FIG. 14C, 
nonlinear photosensitive material may be employed with characteristics in 
which the latent image reaction changes in a nonlinear manner depending 
upon the intensity I. 
If patterns are formed with high resolution and high contrast in nonlinear 
photosensitive material using a phase shift mask or employing the modified 
illumination technique during each exposure in multiple exposure 
processes, the effective light intensity distribution over the resolution 
limit of imaging optical system can be formed with high contrast. 
As described above, semiconductor devices or liquid crystal devices can be 
attained with high-resolution patterns over the resolution limit of 
projection optical system by such an arrangement that a photosensitive 
material is employed with a nonlinear sensitivity characteristic in which 
the effective light intensity distribution J is nonlinear to the incident 
light intensity (I) and that multiple exposure processes with different 
light intensity distribution are carried out, for example, by shifting the 
pattern spatially from each other on the nonlinear photosensitive 
material. 
Embodiment 1 
The first embodiment according to the present invention is described with 
FIGS. 1A and 1B to FIG. 6 and FIGS. 15 and 16B. FIG. 1A and FIG. 1B are 
vertical sectional views to show reticles as plates to be projected, 
employed in the present embodiment. Double exposure is effected on 
nonlinear photosensitive material in such a manner that after the first 
exposure is made with the first pattern on the reticle shown in FIG. 1A, 
second exposure is carried out with the second pattern on the reticle 
shown in FIG. 1B. 
The first pattern in FIG. 1A has opaque portions 2a provided on an 
optically transparent reticle substrate 1a, and open portions 4a in which 
no opaque film is formed. A phase film 3a having the function of so-called 
phase-shifter is provided over either one of two mutually adjacent open 
portions 4a. The phase shifting mask is described in Japanese Patent 
Publication No. 62-50811. 
The second pattern shown in FIG. 1B similarly has opaque portions 2b 
provided on an optically transparent reticle substrate 1b, and open 
portions 4b in which no oblique film is formed. Also, phase films 3b are 
arranged over alternate open portions 4b similarly as in the first 
pattern. 
Exposure is made twice onto the identical nonlinear photosensitive material 
consecutively (separately) using the first and second patterns in such a 
positional relation that the open portions 4a in the first pattern 
spatially correspond to the opaque portions 2b in the second pattern and 
the open portions 4b of the second pattern to the opaque portions 2a in 
the first pattern. 
FIG. 2A and FIG. 2B show intensity distributions on the nonlinear 
photosensitive material, obtained upon exposure with the first and second 
patterns, respectively. In the present embodiment, sinusoidal intensity 
distributions Ia(x) and Ib(x) are produced upon respective exposure 
processes by coherent illumination and only.+-.first-order diffracted 
light, as shown in FIG. 2A and FIG. 2B. By the double exposure processes 
the intensity distributions Ia(x) and Ib(x) have a relative spatial shift 
on the nonlinear photosensitive material. 
To consider the highest possible resolution, suppose the two exposure 
processes produced the respective intensity distributions Ia(x) and Ib(x) 
having the frequency equal to the resolution limit of projection optical 
system. Namely, the numerical aperture NA is fully effectively used as 
the.+-.first-order diffracted light passes through the periphery of the 
aperture of projection optical system. Then, the pitch of the pattern 
produced on the nonlinear photosensitive material in each exposure process 
is within the resolution limit (.lambda./2NA). Further, the intensity 
distributions are respectively expressed as follows. 
EQU Ia(x)=1+cos (2.pi..multidot.2NA.multidot.x/.lambda.) (15) 
EQU Ib(x)=1+cos (2.pi..multidot.2NA.multidot.x/.lambda.+.pi.) (16) 
Since, in case of the two-photon absorption resist being used as the 
nonlinear photosensitive material, the effective light intensity 
distribution J in the resist is given by the square of the intensity 
distributions, effective light intensity distributions upon the two 
exposure processes are given by the following Equations (17) and (18), 
which are shown in FIG. 3A and FIG. 3B. 
EQU Ja(x)={Ia(x)}.sup.2 =3/2+2 cos (2.pi..multidot.2NA.multidot.x/.lambda.)+cos 
(4.pi..multidot.2NA.multidot.x/.lambda.)/2 (17) 
EQU Jb(x)={Ib(x)}.sup.2 =3/2+2 cos 
(2.pi..multidot.2NA.multidot.x/.lambda.+.pi.)+cos 
(4.pi..multidot.2NA.multidot.x/.lambda.)/2 (18) 
Then, the finally obtainable effective light intensity distribution J.sub.T 
(x) after the double exposure processes is a sum of Equations (17) and 
(18) as follows. 
##EQU5## 
As seen from the above Equation (19), the effective light intensity 
distribution J.sub.T (x) in this embodiment has a period of pitch 
(.lambda./4NA), which is half of the limit resolution (.lambda./2NA) of 
projection optical system. The effective light intensity distribution 
J.sub.T (x) is shown in FIG. 4. When development is carried out after the 
multiple (double in this embodiment) exposure processes, finer resist 
patterns will appear. 
As apparent from Equation (12) and FIG. 16B, superposition of perfectly 
isolated patterns (point objects) forms a latent image containing the 
frequency corresponding to the pitch of .lambda./4NA. However, the profile 
of developed patterns could not be so good in this case because OTF 
(4NA/.lambda.) is nearly zero. Thus, the present embodiment employs the 
phase shift masks as shown in FIG. 1A and FIG. 1B and the exposure under 
coherent illumination. Such an arrangement can permit formation of an 
effective light intensity distribution which can realize high contrast and 
high resolution. 
On the other hand, in case of the conventional linear photosensitive 
material being used, the exposure is effected by the intensity 
distribution Ia(x)+Ib(x), which is a simple sum of two Equations (15) and 
(16), i.e., a simple sum of the intensity distributions shown in FIG. 2A 
and FIG. 2B. In that case, the effective light intensity distribution for 
the first and second patterns becomes constant as shown in FIG. 15, 
failing to achieve the principal purpose of exposure. 
The present embodiment showed an example in which the two-photon absorption 
resist was used to obtain the effective light intensity distribution by 
squaring (m=2) the intensity, but high resolution can also be expected 
with a nonlinear photosensitive material in which the effective light 
intensity distribution is obtained by nonlinearity of over the square of 
intensity (i.e., m&gt;2). For example, FIG. 5 shows characteristics of a 
nonlinear photosensitive material in which the effective light intensity 
distribution is obtained by the third power (m=3) of light intensity 
distribution. FIG. 5 shows the effective light intensity distribution 
obtained after triple exposure processes with the transfer pattern shown 
in FIG. 1A being shifted by one third (.lambda./6NA) of the pitch 
(.lambda./2NA) between the processes. As shown in FIG. 5, this example 
provides such a periodic pattern that the pitch of lines and spaces is 
.lambda./6NA, which is three times finer than the resolution limit 
(.lambda./2NA) of projection optical system. 
Similarly, high resolution can be attained with a nonlinear photosensitive 
material having nonlinearity in the range of 1&lt;m&lt;2. FIG. 6 shows an 
effective light intensity distribution obtained with a nonlinear 
photosensitive material in which the latent image is obtained according to 
the 1.5 (m=1.5) power of the light intensity and after double exposure 
processes using the patterns shown in FIG. 1A and FIG. 1B. This example 
can also provide the effective light intensity distribution two times 
finer than the resolution limit of projection optical system. Further, the 
contrast can be improved by employing phase shifting masks and coherent 
illumination. 
Although the present embodiment was described as an example using the 
two-photon absorption resist as nonlinear photosensitive material, the 
present invention is not limited to this example. The nonlinear 
photosensitive material may be one of photosensitive materials which can 
enhance the contrast by another method, for example, by the CEL (Contrast 
Enhanced Lithography) method (Griffing, B. F. and West, P. R., IEEE, EDL 
Vol 1 p 14 (1983)). Further, the resist with nonlinear sensitivity 
characteristic may be employed as the upper layer resist in the so-called 
multi-layered resist method. 
Embodiment 2 
Another embodiment according to the present invention is next described. 
The first embodiment was an example in which the phase shift masks were 
employed with the patterns shown in FIG. 1A and FIG. 1B and the exposure 
was effected under coherent illumination. In contrast, the present 
embodiment shows an example in which an ordinary reticle without 
phase-shifter is employed and partially coherent illumination is used. 
The conditions of projection optical system are set as operating wavelength 
.lambda.=0.365 .mu.m, numerical aperture NA=0.5, coherence factor 
.sigma.=0.6, and two-photon absorption resist being used as the nonlinear 
photosensitive material. FIG. 7A shows an effective light intensity 
distribution in the present embodiment, in which three lines of 0.25 .mu.m 
wide are formed by three exposure processes by shifting a pattern of 
isolated line of 0.25 .mu.m wide. FIG. 7B is given for comparison with the 
present embodiment and shows an effective light intensity distribution in 
a comparative example in which a two-photon absorption resist is used and 
three lines are formed in the photosensitive material by single exposure 
of a pattern of three lines of 0.25 .mu.m wide without spatial shifting. 
FIG. 7C is given for further comparison with the present embodiment and 
shows an effective light intensity distribution in another comparative 
example in which a conventional linear photosensitive material is used and 
three lines are formed on the photosensitive material by single exposure 
of a pattern of three lines of 0.25 .mu.m wide without spatial shifting. 
The present embodiment can better provide a fine effective light intensity 
distribution as shown in FIG. 7A, so that fine patterns may appear after 
development. Though resist patterns after development are formed 
approximately in proportion to the latent image, high-contrast and 
high-resolution resist patterns can be formed by effecting an enhancement 
treatment upon development. 
As described above, the present embodiment can provide high-contrast and 
high-resolution resist patterns even without using phase shifting mask. 
Embodiment 3 
Still another embodiment is now described. The third embodiment shows an 
example in which nonlinear photosensitive material employed has the power 
m of intensity smaller than 1 (m&lt;1), i.e., the photosensitive material has 
m=0.5 for example. The effective light intensity distribution J(x) is 
given as follows for the intensity distribution I(x) of the nonlinear 
photosensitive material with m=0.5. 
EQU J(x)={I(x)}.sup.0.5 (20) 
In the above equation, x represents an x-coordinate position. In the 
present embodiment, the phase shifting mask as shown in FIG. 1A is used 
with coherent illumination. 
As described above, the period of the pattern on this reticle is the 
resolution limit (.lambda./2NA) of projection optical system. FIG. 8 shows 
an intensity distribution I(x) on the image plane in the nonlinear 
photosensitive material, which is a sinusoidal distribution as defined by 
the following Equation (21). 
EQU I(x)=1+cos (2.pi..multidot.2NA.multidot.x/.lambda.) (21) 
Also, with the intensity distribution I(x), the effective light intensity 
distribution Ja(x) of the nonlinear photosensitive material is as shown in 
FIG. 9A and defined by the following Equation (22). 
EQU Ja(x)={1+cos (2.pi..multidot.2NA.multidot.x/.lambda.)}.sup.0.5(22) 
Single exposure would result in making the curve of J(x) gentler in bright 
portions and steeper in dark portions. But the resolution is kept equal to 
that of intensity distribution I(x), thus never surpasses the resolution 
limit of projection optical system. 
On the other hand, the present embodiment employs double exposure with 
patterns having lateral shift as shown in FIG. 1A and FIG. 1B, in the 
present embodiment, the first exposure is effected with the intensity 
distribution I(x) shown in FIG. 8 on the nonlinear photosensitive material 
and the second exposure with the intensity distribution (referred to as 
Ib(x)) corresponding to a half-period-shifted pattern from the intensity 
distribution I(x) shown in FIG. 8. Letting Jb(x) be the effective light 
intensity distribution for the second intensity distribution Ib(x), the 
effective light intensity distribution Jb(x) shown in FIG. 9B can be 
deemed as a half-period-shifted distribution from the effective light 
intensity distribution Ja(x) shown in FIG. 9A accordingly. 
The actual effective light intensity distribution J.sub.T (x) is obtained 
by superimposing the first and second effective light intensity 
distributions on each other. FIG. 9C shows the effective light intensity 
distribution J.sub.T (x). As seen from FIG. 9C, the effective light 
intensity distribution J.sub.T is obtained with a resolution 
(.lambda./4NA) which is the double of the resolution limit (.lambda./2NA) 
of projection optical system, forming fine developed patterns. 
Let us compare the present embodiment with the case of linear 
photosensitive material with m=1. As shown in Embodiment 1, the effective 
light intensity distribution J.sub.T (x) becomes constant (=2). In other 
words, no lines and spaces pattern as shown in FIG. 9C appears but a 
uniform distribution is obtained, failing to achieve the main purpose of 
exposure. 
As described above, the present embodiment can achieve the high resolution 
over the resolution limit of projection optical system, by employing the 
nonlinear material with m&lt;1 and the multiple exposure with lateral shift. 
It should be noted that in the above-described embodiments the nonlinear 
photosensitive material may be either positive resist or negative resist. 
However, the positive resist is more preferable as the nonlinear 
photosensitive material having the relation of m&lt;1 to realize the fine 
patterns shown in FIG. 9C. 
Embodiment 4 
A further embodiment shows an example of two-dimensional pattern in a more 
practical semiconductor element. FIG. 10A shows a desired pattern to be 
finally obtained, FIG. 10B a pattern of the first exposure reticle, and 
FIG. 10C a pattern on the second exposure reticle. Portions 51, 52, 53 and 
54 transmit light, among which the light transmitting portions 52 and 54 
are provided with phase film 52s, 54s. In the pattern shown in FIG. 10A, 
the smallest gap between the portions 51, 52, 53, 54 is the resolution 
limit of projection optical system, and an image with sufficient contrast 
can be formed by single exposure using the phase shifting mask for 
example. With the two-dimensional pattern as shown in FIG. 10A, some 
portions would remain unresolved even with any arrangement of phase 
shifter of 0.degree. or 180.degree.. In contrast, the present invention 
can permit a semiconductor element to have a very fine pattern as shown in 
FIG. 10A, which it has been impossible to resolve, by using the two-photon 
absorption resist and effecting such exposures that the first exposure is 
made with the pattern of FIG. 10B and thereafter the second exposure with 
the pattern of FIG. 10C. 
The patterns shown in FIG. 10B and FIG. 10C are formed on separate reticles 
in the present embodiment, but the reticles may be replaced by an 
electrooptic element such as a liquid crystal plate, in which the transfer 
patterns in FIG. 10B and FIG. 10C can be formed by electrically changing 
the pattern transparent portions, substantially acting as a photo-mask. 
Incidentally, it is effective in the present invention to use the phase 
shift pattern (phase film) in order to form a high-resolution pattern as 
in the first embodiment. It is also effective to use the annular 
illumination as proposed in Japanese Laid-open Patent Application No. 
61-91662 or the so-called SHRINC illumination as proposed, for example, in 
Japanese Laid-open Patent Application No. 4-225358. 
Embodiment 5 
An embodiment of exposure apparatus according to the present invention is 
next described with the scheme of FIG. 11. The structure of the apparatus 
is first described referring to FIG. 11. A light source 100 emits an 
illumination beam for exposure. The illumination beam is collected by an 
elliptic mirror 110. Then the illumination beam is guided via a plane 
mirror 120 to a collimator lens 130. The collimator lens 130 forms a beam 
of nearly parallel rays. Further, the parallel beam is guided via a plane 
mirror 140 to enter a fly's eye integrator 150. A beam emerging from the 
fly's eye integrator 150 irradiates the exposure position 160 with uniform 
intensity. Plural reticles 170 are conveyed by an automatic conveying 
mechanism (not shown) to be set at or removed from the exposure position 
160. When a reticle 170 is set at the exposure position 160, patterns 
formed thereon is illuminated by the illumination beam to transmit 
diffracted beams. The diffracted beam is focused by a projection optical 
system 180 on the surface of nonlinear photosensitive material 200 mounted 
on a semiconductor wafer 190 to effect exposure thereon. 
A control unit 210 has a built-in microprocessor (CPU) having an 
operational function, which controls a drive circuit 220 for adjusting an 
angle of elevation of the elliptic mirror 110 to the light source 100, 
controls the drive of the three-dimensionally movable stage 230 for 
mounting of the semiconductor wafer 190 thereon, and gives a command for 
automatic mounting or dismounting of reticle 170 to the automatic 
conveying mechanism (not shown). The control unit 210 executes a process 
according to an instructed order input by an operator by means of an 
inputting device 240 such as a keyboard. Also, the control unit 210 is 
provided with a memory device 250 storing data and processing programs for 
executing various processes. Since the exposure is carried out with plural 
reticles 170, the automatic conveying mechanism has an optical reading 
device 260 for selecting one out of the reticles. Data read by the reading 
device 260 is transferred to the control unit 210 to be analyzed. 
Next described is an example of series operations in the present invention. 
Under control of the control unit 210, the automatic conveying mechanism 
(not shown) conveys the first reticle out of two types of reticles for 
obtaining high resolution by double exposure and sets it at the exposure 
position 160. Patterns on the first reticle is projected onto the 
nonlinear photosensitive material 200. Then, the first reticle is removed 
and the second reticle is set at the exposure position 160. The first and 
second reticles are exchanged for each other as described, in case the 
patterns on the first and second reticles are preliminarily given a 
predetermined lateral shift to get the resolution over the limit of 
projection optical system 180. Then, the pattern on the second reticle is 
projected onto the nonlinear photosensitive material 200. Thus, a 
high-resolution pattern is obtained by the double exposure. Similar 
processes are carried out if multiple exposure is more than two processes. 
Then, the same exposure process is repeated for another type of reticle 
for projecting a next pattern onto the wafer. 
Instead of exchanging the patterns on different reticles by an automatic 
conveying mechanism, a lateral shift may be produced by such an 
arrangement that after first exposure with a reticle, the identical 
reticle is moved by a predetermined amount in the direction Ay 
perpendicular to the optical axis Ax of projection optical system 180 then 
to carry out second exposure. The predetermined amount is (.lambda./4NA) 
as reduced coordinates on the wafer 190 in case of the example of the 
pattern in FIG. 1A as described above, in which the effective light 
intensity of photosensitive material is proportional to the square of 
intensity (m=2). Also, in case the effective light intensity is 
proportional to the cube of intensity (m=3), the predetermined amount may 
be effectively set to (.lambda./6NA) as reduced coordinates on the wafer 
190. 
For multiple exposure with a same reticle pattern, the wafer 190 itself can 
of course be arranged to move for each exposure process by a 
three-dimensionally movable stage 230 instead of moving the reticle. 
Effective alignment between the multiple exposure processes is the 
so-called latent image alignment in which alignment is made by observing 
latent image. 
Next, optimization of numerical aperture NA of the projection optical 
system in the case that photosensitive material having nonlinear 
sensitivity is used in exposure will be described. 
The present invention relates to an exposure method and an exposure 
apparatus, more particularly to a projection exposure method and a 
projection-type exposure apparatus, used in fabricating semiconductor 
devices or liquid crystal plates. 
There are conventionally used exposure methods in which, in fabricating 
semiconductor devices, liquid crystal display devices, thin-film magnetic 
heads, or the like, illumination light from a light source illuminates a 
photo-mask or reticle (hereinafter referred to totally as "reticle") with 
uniform illuminance and a pattern on the reticle is projected onto a wafer 
(or glass plate, or the like) coated with a photosensitive material to be 
printed there, and exposure apparatus for realizing the exposure methods. 
In the conventional exposure methods, all of patterns desired to be exposed 
are provided on a same reticle and are printed on the substrate by single 
exposure. On that occasion, a latent image reaction density .xi. according 
to exposure intensity I appears in the photoresist coating on the 
substrate. For example, for positive photoresists presently used in 
general, it can be represented by the following equation. 
(Eq 1-5) 
EQU .xi.=exp (-CD), D=I.multidot.t 
More generally, the following expression can be employed. 
(Eq 1-6) 
EQU .xi.=exp (-CD), D=J.multidot.t=I.sup.m .multidot.t 
In the equation, I represents light intensity, t an exposure time, and C a 
constant determined by the photosensitive material. Further, m is an 
exponent indicating linearity of the photosensitive material, which is 
linear with m=1 while nonlinear with m.noteq.1. For easy recognition, J 
replaces I.sup.m as in the above equation and J is called as a latent 
image density. Most of the photoresists presently used in general are 
linear with m =1. 
In the above methods, assuming for simplicity that image formation is 
completely incoherent, a spectrum i of exposure intensity distribution 
I(x) for forming a latent image in the photoresist is given by the below 
equation with an object spectrum being i.sub.0 and an OTF (Optical 
Transfer Function) of an optical system being f. 
(Eq 1-7) 
EQU i(.nu.)=i.sub.0 (.nu.).multidot.f(.nu.) 
.nu.: spatial frequency 
Now, a spatial frequency .nu..sub.0, which is a limit spatial frequency at 
which the OTF or f becomes not significant in respect of process, is given 
by the below equation with an exposure wavelength being .lambda. and a 
numerical aperture of projection optical system on the photosensitive 
material side being NA. 
(Eq 1-8) 
EQU .nu..sub.0 =0.5NA/(K.sub.1 .multidot..lambda.) 
K.sub.1 : process constant 
Further, the resolution limit of the optical system is theoretically 
determined by the numerical aperture NA, and in that case K.sub.1 =0.25, 
whereby a cutoff frequency .nu..sub.c of the optical system is given by 
the following equation. 
(Eq 1-9) 
EQU .nu..sub.c =2NA/.lambda. 
Accordingly, in order to achieve high resolution, either a decrease of 
wavelength or an increase of numerical aperture NA was needed. 
Further, there are recent demands to further enhance the resolution of the 
pattern projected onto the wafer with an increase in degree of integration 
for semiconductor devices, etc. Where the resolving power of the 
projection optical system is simply increased as described above in order 
to respond to such demands, there is a limit of the increase. In addition, 
the simple increase of the resolving power of the projection optical 
system would result in making the depth of focus shallower, which is not 
suitable for practical use. Thus, there have been various suggestions of 
techniques to improve the resolution as a whole in combination of the 
projection optical system with modification of the construction of the 
illumination optical system or with modification of the construction of 
the pattern on the reticle. 
First, as a technique to devise the pattern arrangement of the reticle, 
there is a so-called phase shift method, as proposed by the present 
applicant in Japanese Patent Publication No. 62-50811, in which phase 
shifters are used to provide a phase difference between illumination beams 
passing through the pattern on the reticle. Further, as an example of a 
technique to devise the arrangement of the illumination optical system, 
there is a so-called modified light source method (also called as a 
modified illumination method or as an oblique illumination method), as 
proposed by the present applicant, in which illumination light from a 
light source is arranged to form a plurality of secondary light sources 
arranged at equal angular intervals around the optical axis and the 
reticle is illuminated with illumination beams from the plurality of (for 
example, four) secondary light sources. 
The final resolution limit, however, has never surpassed the optically 
determined resolution limit by simply devising the arrangement of the 
illumination optical system or the arrangement of the pattern on the 
reticle in combination with the conventional exposure method as described 
above. 
Also, conventionally, precise calculation based on the Fourier 
image-formation theory was applied or actual exposure was carried out in 
order to obtain best conditions for exposure, but analytic conditions for 
obtaining the best resolution inconveniently have been unclear. 
It is, therefore, an object of the present invention, in view of the above 
points, to provide an exposure method and an exposure apparatus by which a 
pattern with high resolution surpassing the optically determined 
resolution limit is formed with little modifying the conventional exposure 
wavelength and optical system and further by which, with use of the phase 
shift method or the modified light source method, the pattern on the 
reticle can be printed with nearly best resolution on the wafer under 
given conditions. 
To solve the above problems, the present invention provides an exposure 
method in which a mask is so arranged that a line-and-space pattern for 
transfer is formed of adjacent space portions having mutually inverse 
phases and in which the mask is illuminated with illumination light from a 
light source whereby an image of the pattern on said mask is projected 
onto a photosensitive material to e focused thereon: 
wherein said photosensitive material has a nonlinear sensitivity property 
of being sensitive in proportion with the square of intensity of incident 
light, and a plurality of exposure processes different in light intensity 
distribution on the photosensitive material are carried out to form a 
pattern with high resolution surpassing a resolution limit of a projection 
optical system; 
wherein a value of numerical aperture NA of said projection optical system 
is set within .+-.20% with respect to a value determined by the below 
equation, where .sigma. is a coherence factor of the illumination light, 
which corresponds to a value obtained by dividing a numerical aperture of 
an illumination optical system for performing said illumination by a 
numerical aperture of the projection optical system for performing said 
projection focusing, C' a contrast necessary on said photosensitive 
material, of the image of the pattern on said mask, d a depth of focus of 
said projection optical system, and .lambda. a wavelength of said 
illumination light. 
##EQU6## 
Also, the present invention provides an exposure method in which 
illumination light from a light source forms four secondary light sources 
arranged at equal angular intervals around the optical axis and a mask on 
which a pattern for transfer is formed is illuminated with illumination 
light from said four secondary light sources whereby an image of the 
pattern on said mask is projected onto a photosensitive material to be 
focused thereon: 
wherein said photosensitive material has a nonlinear sensitivity property 
of being sensitive in proportion with the square of intensity of incident 
light, and a plurality of exposure processes different in light intensity 
distribution on the photosensitive material are carried out to form a 
pattern with high resolution surpassing a resolution limit of a projection 
optical system; 
wherein a value of numerical aperture NA of said projection optical system 
is set within the range of -20% to +10% with respect to a value determined 
by the below equation, where .sigma..sub.0 is a coherence factor of the 
illumination light, which corresponds to a value obtained by dividing a 
numerical aperture of an illumination optical system for performing said 
illumination defined by light from the centers of said four secondary 
light sources by 2.sup.0.5 times a numerical aperture of the projection 
optical system for performing said focusing projection, .sigma. a 
coherence factor of each illumination light from said four secondary light 
sources, which corresponds to a value obtained by dividing a numerical 
aperture for each illumination light from said four secondary light 
sources where each illumination light from said four secondary light 
sources is outgoing from said illumination light source, by the numerical 
aperture of said projection optical system, C' a contrast necessary on 
said photosensitive material, of the image of the pattern on said mask, d 
a depth of focus of said projection optical system, .lambda. a wavelength 
of said illumination light, and a a constant of 2/.pi.. 
##EQU7## 
Further, the present invention provides an exposure apparatus having an 
illumination optical system for illuminating a mask on which a 
line-and-space pattern for transfer is formed of adjacent space portions 
having mutually inverse phases, with illumination light from a light 
source, and a projection optical system for projecting an image of the 
pattern on said mask onto a photosensitive material to focus the image 
thereon: 
wherein said photosensitive material has a nonlinear sensitivity property 
of being sensitive in proportion with the square of intensity of incident 
light, and a plurality of exposure processes different in light intensity 
distribution on the photosensitive material are carried out to form a 
pattern with high resolution surpassing a resolution limit of the 
projection optical system; 
wherein a value of numerical aperture NA of said projection optical system 
is set within .+-.10% with respect to a value determined by the below 
equation, where .sigma. is a coherence factor of the illumination light, 
which corresponds to a value obtained by dividing a numerical aperture of 
said illumination optical system by a numerical aperture of said 
projection optical system, C' a contrast necessary on said photosensitive 
material, of the image of the pattern on said mask, d a depth of focus of 
said projection optical system, and .lambda. a wavelength of said 
illumination light. 
##EQU8## 
Yet further, the present invention provides an exposure apparatus having an 
illumination optical system for arranging illumination light from a light 
source to form four secondary light sources arranged at equal angular 
intervals around the optical axis and illuminating a mask on which a 
pattern for transfer is formed, with illumination light beams from said 
four secondary light sources, and a projection optical system for 
projecting an image of the pattern on said mask onto a photosensitive 
material to focus the image thereon: 
wherein said photosensitive material has a nonlinear sensitivity property 
of being sensitive in proportion with the square of intensity of incident 
light, and a plurality of exposure processes different in light intensity 
distribution on the photosensitive material are carried out to form a 
pattern with high resolution surpassing a resolution limit of the 
projection optical system; 
wherein a value of numerical aperture NA of said projection optical system 
is set within the range of -20% to +10% with respect to a value determined 
by the below equation, where .sigma..sub.0 is a coherence factor of the 
illumination light, which corresponds to a value obtained by dividing a 
numerical aperture of said illumination optical system defined by light 
from the centers of said four secondary light sources by 2.sup.0.5 times a 
numerical aperture of said projection optical system, .sigma. a coherence 
factor of each illumination light from said four secondary light sources, 
which corresponds to a value obtained by dividing a numerical aperture for 
each illumination light from said four secondary light sources where each 
illumination light from said four secondary light sources is outgoing from 
said illumination light source, by the numerical aperture of said 
projection optical system, C' a contrast necessary on said photosensitive 
material, of the image of the pattern on said mask, d a depth of focus of 
said projection optical system, .lambda. a wavelength of said illumination 
light, and a constant of 2/.pi.. 
##EQU9## 
We proposed exposure methods using a so-called two-photon absorption 
photoresist, which is a photosensitive material having a nonlinear 
sensitivity property. The two-photon absorption photoresist is a 
photoresist which forms a latent image nucleus when absorbing two photons 
and which is represented as m=2 in (Eq 1-6), as detailed in Proceedings of 
SPIE, vol. 1674, pp 776-778, 1992. In the case of this photoresist, a 
latent image density distribution J(x) is formed in accordance with the 
square of an exposure intensity distribution I(x). Namely, for incoherent 
illumination, the following relation holds with a point spread function 
F(x) of an optical system and a light intensity distribution I.sub.0 (x) 
of object. 
(Eq 1-14) 
EQU J(x)=I(x).sup.2 {I.sub.0 (x)*F(x)}.sup.2 
Then, a spectrum j of the latent image density distribution is similarly 
given by the following equation from the convolution theorem of the 
Fourier transformation. 
(Eq 1-15) 
EQU j(.nu.)={i.sub.0 (.nu.).multidot.f(.nu.)}*{i.sub.0 (.nu.).multidot.f(.nu.)} 
Since the latent image density distribution is given in accordance with (Eq 
1-14) for the two-photon absorption photoresist, the distribution is 
sharper than those for conventional photoresists. This will be 
specifically described with an example shown in FIG. 17A and FIG. 17B for 
sinusoidal exposure intensity distribution. 
FIG. 17A shows a latent image density distribution in a normal photoresist, 
which is sinusoidal, similarly as the exposure intensity distribution. 
FIG. 17B shows a latent image density distribution in a two-photon 
absorption photoresist. Comparing FIG. 17A with FIG. 17B, the contrast of 
latent image in FIG. 17B is higher than that in FIG. 17A, but the pitch of 
the formed pattern in FIG. 17B is equal to that in FIG. 17A. Thus, simple 
use only of the two-photon absorption photoresist cannot make the pitch of 
the latent image density distribution formed finer than the pitch of an 
image formed by the optical system, thus never permitting the resolution 
to surpass the resolution limit of the optical system. Although there are 
components with frequencies surpassing the resolution limit of the optical 
system in the latent image density distribution from (Eq 1-15), the pitch 
of the formed pattern is still kept below the resolution limit of the 
optical system. 
Next, let us consider image formation of point image by the optical system 
with the two-photon absorption photoresist. In this case, the two-photon 
absorption photoresist makes the latent image density distribution of 
point image sharper. It is sufficient for this case taking the point 
spread distribution F(x) by the optical system into consideration 
irrespective of the illumination state. Then, supposing a desired object 
intensity distribution I.sub.0 (x) is formed by superposition of point 
images and a latent image density distribution J(x) is formed thereby, the 
density distribution can basically be expressed by the following equation, 
because it is superposition of light intensities given by image formation 
of respective point images. 
(Eq 1-16) 
EQU J(x)=I.sub.0 (x)*{F(x)}.sup.2 
Since the latent image density distribution of point image involves 
{F(x)}.sup.2, it is sharper than the point spread distribution F(x) by the 
optical system, thus realizing high resolution. By Fourier transformation 
of (Eq 1-16), the following equation is obtained. 
(Eq 1-17) 
EQU j(.nu.)=i.sub.0 (.nu.).multidot.{f(.nu.)*f(.nu.)} 
Then, f*f can be interpreted as OTF of the optical system to obtain the 
latent image density distribution by this method. This means that the 
cutoff frequency (4NA/.lambda.) can be achieved against the cutoff 
frequency (2NA/.lambda.) of the conventional OTF or f, thus doubling the 
resolution. 
FIG. 18A and FIG. 18B diagrammatically show this comparison. FIG. 18A 
indicates the OTF in the conventional method, while FIG. 18B the OTF for 
the case where a two-photon absorption photoresist is exposed with 
isolated patterns. Therefore, a fine pattern can be formed with resolution 
over the resolution limit of the optical system when a latent image is 
formed using the two-photon absorption photoresist and performing a 
plurality of exposure processes based on isolated patterns. Accordingly, 
formation of a pattern with resolution over the resolution limit of the 
optical system becomes possible by the combination of the two-photon 
absorption photoresist with the plural exposure processes by isolated 
patterns. 
Further, a pattern with resolution over the resolution limit of the optical 
system can also be formed, similarly as in the case of the isolated 
patterns, where a plurality of exposure processes are carried out using 
patterns which are not perfectly isolated but can be considered as nearly 
isolated. In this case, a spectrum j of latent image density distribution 
is defined by the following equation. 
##EQU10## 
In the above equation, i.sub.0j represents object spectra of mutually 
nearly isolated patterns, and i' is considered as an object spectrum of an 
imaginary pattern constructed by superposition of the nearly isolated 
patterns. No spectra of conventional latent image density distributions 
can surpass the cutoff frequency (2NA/.lambda.) of f, while the present 
invention permits a spectrum up to the cutoff frequency (4NA/.lambda.) of 
{f(.nu.)*f(.nu.)} to be formed as a latent image density distribution, as 
indicated by (Eq 1-13). 
As described above, the pitch of formed pattern never surpasses the 
resolution limit of the optical system by the single use of the two-photon 
absorption photoresist in the conventional exposure method, while a latent 
image density distribution of pattern with a pitch surpassing the 
resolution limit of the optical system can be formed by performing a 
plurality of exposure processes different in light intensity distribution 
on such a photosensitive material so as to provide appropriate i'. 
As described above, semiconductor devices can be obtained with patterns of 
high resolution surpassing the resolution limit of the projection optical 
system, using the photosensitive material having such a property that the 
latent image density, that is, the latent image reaction density is in 
proportion with the square of the intensity of incident light, and 
performing the plurality of exposure processes different in light 
intensity distribution on the photosensitive material. 
For effecting exposure of reticle pattern with nearly best resolution on 
the wafer, the present invention employs such parameters for obtaining 
best resolution for either case of the phase shift method and the modified 
light source method, as the wavelength .lambda. of illumination light, the 
coherence factor of illumination light (so-called .sigma. value, and 
.sigma..sub.0 as well for the modified light source method), which is a 
ratio between the reticle-side numerical aperture of illumination optical 
system and the reticle-side numerical aperture of projection optical 
system, the contrast C' necessary for image on the substrate, and the 
depth of focus d of the projected image (a depth of focus required by the 
projection optical system, taking into consideration factors such as the 
thickness of photoresist, warpage and bending of the substrate such as the 
wafer or the like, deviation of focus adjustment, etc.). 
Then, either one of (Eq 1-10) to (Eq 1-13) is used to set a value of 
numerical aperture NA of the projection optical system to give best 
resolution when these parameters are given. As will be described in detail 
in the following embodiments, nearly best resolution can be obtained 
within .+-.20% for (Eq 1-10) and (Eq 1-12), but within the range of from 
-20% to +10% for (Eq 1-11) and (Eq 1-13) with respect to the setting 
value. Considerably good resolution can be obtained within .+-.10% for (Eq 
1-10) and (Eq 1-12) with respect to the above setting value, and within 
the range of from -10% to +5% for (Eq 1-11) and (Eq 1-13) with respect to 
the above setting value. 
Embodiments 
First described is an embodiment in which a plurality of exposure processes 
different in light intensity distribution are carried out using the phase 
shift method and the two-photon absorption photoresist. Here is also 
described how to obtain an optimum value of numerical aperture NA of the 
projection optical system. 
First described is a relation between the numerical aperture NA of 
projection optical system, the resolving power R, the wavelength .lambda., 
the coherence factor .sigma., and the defocus d, where the conventional 
exposure method is executed employing the phase shift method and a 
conventional photoresist. 
The phase shift method is a technique for forming a high-resolution and 
high-contrast image by making a predetermined phase difference between 
opening portions in the reticle and utilizing the interference effect 
between the opening portions. (See Japanese Patent Publication No. 
62-50811.) For example, if a phase difference of .lambda./2 (180.degree.) 
is given between two adjacent opening portions in a line-and-space pattern 
on the reticle, the interference effect between them basically nullifies 
the zeroth-order diffracted light, and, therefore, consideration is 
necessary only for .+-.first-order diffracted light as shown in FIG. 19. 
In FIG. 19, there are line portions of light shielding film formed at the 
pitch 2R in the direction perpendicular to the plane of FIG. 19, and phase 
shifters 1017 are provided over every other space portion (at the pitch 
4R) between the line portions. Since the fundamental frequency of an image 
focused on the wafer W is determined by an angle of divergence of two 
diffracted beams from the reticle 1016 in this case, a finer pattern can 
be expected to be formed than those by the conventional methods. The 
exposure intensity distribution I(x) on the wafer W is given by the below 
equation. It is noted that the following equations 1-19, 1-20, 1-21, and 
1-22 are described in Optics, vol. 23, No. 1, pp 29-37, 1994; and these 
equations are described in SPIE, vol.1780 (1992) pp117-131 in detail. 
##EQU11## 
Also, the contrast C of the image is given by the following equation. 
##EQU12## 
Further, expanding and approximating the Bessel function J.sub.1 in (Eq 
1-20), the following relation is obtained. 
##EQU13## 
With a large coherence factor .sigma. in the case of the phase shift 
method, (Eq 1-19) needs to be rewritten taking into consideration an 
eclipse due to the pupil of the projection optical system as in the case 
of the conventional method. However, no problem will result with (Eq 1-19) 
because the coherence factor .sigma. in the actual phase shift method is 
small. Although it is assumed in (Eq 1-20) that all the.+-.first-order 
diffracted light passes through the pupil, the contrast on the Gauss plane 
(i.e., at a defocus amount d=0) is determined by the eclipse 
of.+-.first-order diffracted light. The following equation provides a 
relation between the resolving power R and the contrast C for that case. 
##EQU14## 
Now, with exposure using the two-photon absorption photoresist instead of 
the conventional photoresist, a latent image density distribution as shown 
in FIG. 20A is obtained in the two-photon absorption photoresist coating 
the wafer. When second exposure is then performed after moving the reticle 
1016 shown in FIG. 19 by a half pitch R in the direction perpendicular to 
the optical axis, a latent image density distribution as shown in FIG. 20B 
is obtained in the two-photon absorption photoresist coating the wafer. 
The latent image density distribution obtained by the second exposure is 
one resulting from shift of the latent image density distribution obtained 
by the first exposure by the half cycle. On this occasion, the respective 
latent image densities are emphasized by the two-photon absorption 
photoresist, as shown in FIG. 17B, so that portions with high light 
intensity become much higher while portions with low light intensity much 
lower. Adding the latent image density distribution obtained by the first 
exposure onto the latent image density distribution obtained by the second 
exposure, a latent image density distribution shown in FIG. 21 is 
obtained, thus enabling to form a pattern with resolution over the 
resolution limit of the optical system. 
Considered as a method for the second exposure is either a method in which 
the second exposure is carried out after moving the wafer by an amount 
corresponding to movement of reticle 1016 by the half pitch R on the wafer 
W, or a method using another reticle having a pattern obtained by moving 
the pattern of reticle 1016 by the half pitch R in the direction of the 
optical axis. Further, conducting exposure twice as described will be 
hereinafter called as double exposure. 
Here, considering the sensitivity property of photoresist, some 
photoresists could have slightly different effects as to the latent image 
density between the first exposure and the second exposure. It is 
preferred in that case that exposure amounts in the respective exposure 
processes be properly adjusted. 
Incidentally, where the contrast C of the exposure intensity distribution 
I(x) formed in the first exposure is sinusoidal, the latent image density 
J is given by the following equation for double exposure using the 
two-photon absorption photoresist. 
##EQU15## 
In the above equation, C is the contrast of the exposure intensity 
distribution I(x) in the first exposure, C' the contrast of the latent 
image after exposure, and R' the line width of the latent image. There are 
the following relations between C, C', R, and R'. 
##EQU16## 
Substituting the relations of (Eq 1-24) into (Eq 1-21) and (Eq 1-22), the 
following relations are obtained. 
##EQU17## 
(Eq 1-25) and (Eq 1-26) have the relation shown in FIG. 22. An intersection 
between (Eq 1-25) and (Eq 1-26) gives an optimum NA for double exposure 
using the two-photon absorption photoresist. Namely, eliminating R', the 
following relation is obtained. 
##EQU18## 
It is seen from FIG. 22 that the resolution is not so degraded within 
.+-.20% from the optimum NA value. 
Next described is an embodiment in which double exposure is carried out 
using the modified light source method and the two-photon absorption 
photoresist. Here is also described how to obtain an optimum value of 
numerical aperture NA of the projection optical system. 
First described is how to obtain an optimum value of numerical aperture NA 
of the projection optical system for the case where the conventional 
exposure method is executed employing the modified light source method and 
the conventional photoresist. 
In the modified light source method an image is formed with the 
zeroth-order diffracted light and .+-.first-order diffracted light. FIG. 
23 shows the shape of secondary light sources on the pupil of the 
projection optical system in the modified light source method. The 
secondary light sources are constructed of four small light sources 1018A 
to 1018D arranged at intervals of 90.degree. around the optical axis. The 
small light sources 1018A to 1018D each have a radius of 
.sigma..multidot.NA, and a distance of each center of the small light 
source 1018A to 1018D from the .xi. axis or from the .eta. axis of the 
pupil coordinate system is .sigma..sub.0 .multidot.NA. This means that the 
coherence factor of illumination light from each of the four secondary 
light sources in the illumination optical system is .sigma. and that 
(1/2).sup.1/2 of the coherence factor by the illumination light from each 
center of the four secondary light sources is .sigma..sub.0. In this case, 
the exposure intensity I(x) of a projected image on the wafer, excluding a 
proportional constant, is expressed by the below equation. It is noted 
that the following equations 1-28, 1-29, 1-30, 1-31, 1-32, and 1-33 are 
described in Optics, vol. 23, No. 1, pp 29-37, 1994; and these equations 
are described in SPIE, vol.1780 (1992) pp117-131 in detail. 
##EQU19## 
In the above equation, a is a ratio of amplitude between the first-order 
diffracted light and the zeroth-order diffracted light, which is 2/.pi. 
for a normal mask in which the width of transmitting portions is equal to 
that of shielding portions. 
Also, the contrast C of the projected image is given by the following 
equation. 
##EQU20## 
Further, expanding and approximating the cosine function and Bessel 
function, the contrast C becomes as follows. 
##EQU21## 
Then the resolving power R on the plane of Gaussian image for fine line 
width is given by the following equation. 
##EQU22## 
FIG. 24 shows a relation between the numerical aperture NA of the 
projection optical system and the resolving power R for the case using the 
modified light source method. In FIG. 24, the solid line represents the 
results by (Eq 1-30) and (Eq 1-31) and solid dots by precise diffraction 
integration, from which it is understood that (Eq 1-30) is a good 
approximation. As the pattern size becomes rougher with an increase in 
defocus, the contrast rather tends to decrease. This is because a phase 
shift between image intensity distributions formed by the small light 
sources paired becomes larger. It is also given as a solution of (Eq 
1-30), which is represented by the dot line in FIG. 24. Since this dot 
line is nearly vertical, an optimum numerical aperture NA in the modified 
light source method can be considered as an intersection between the solid 
line and the dot line. The point representing the optimum numerical 
aperture NA is indicated by a triangle in FIG. 24. The resolving power R 
is given by the following equation. 
##EQU23## 
In FIG. 24, the numerical aperture at the intersection between the solid 
line and the dot line is the optimum numerical aperture NA. From (Eq 1-30) 
and (Eq 1-32), the optimum numerical aperture NA is given by the following 
equation. 
##EQU24## 
Here, formation of a pattern with resolution over the resolution limit of 
the optical system becomes possible by the present invention using the 
two-photon absorption photoresist and performing the double exposure, 
similarly as in the previous embodiment. 
For the case where the nonlinear photoresist (m=2) is used and double 
exposure is carried out by the modified light source method, substituting 
the following into (Eq 1-33), 
##EQU25## 
The optimum NA is given by the following equation. 
##EQU26## 
In the present embodiment the resolution is not so degraded within -20% 
from the optimum NA value, similarly as in the embodiment using the phase 
shift method. However, if the upper limit exceeds +10%, the resolution is 
degraded, as seen from FIG. 24. 
If the phase shift mask as disclosed in Japanese Laid-open Patent 
Application No. 4-162039 (generally called as a halftone phase mask) is 
used with the above modified light source method using the two-photon 
absorption photoresist and performing the double exposure, the ratio 
between the zeroth-order diffracted light and the first-order diffracted 
light becomes 1, thus enabling to achieve an excellent effect of improved 
contrast. 
The above two embodiments described only the examples about the two-photon 
absorption photoresist, but, for cases with m=3 or m=0.5, the best 
numerical aperture NA of the projection optical system can similarly be 
obtained by first attaining a general way of obtaining the optimum value 
of numerical aperture NA of the projection optical system and next 
substituting an equation taking the nonlinearity of the photoresist into 
consideration, into it. 
Further, an embodiment of the apparatus of the present invention will be 
described below with reference to the drawing. 
FIG. 25 shows schematic structure of a projection exposure apparatus of the 
present embodiment. In FIG. 25, illumination light emitted from a mercury 
lamp 1001 is reflected by an elliptic mirror 1002, and thereafter is led 
through an input lens 1004 and then through a first filter plate 1005 and 
a second filter plate 1006 for letting light in a predetermined wavelength 
band pass, then to enter a fly's eye lens 1007. A shutter 1003 for 
switching between transmission and interruption of illumination light is 
provided near the second focal point of the elliptic mirror 1003, and a 
main control system 1023 for controlling operations of the entire 
apparatus controls the switching of the shutter 1003 through a drive unit 
1024. 
A variable aperture stop for illumination light (hereinafter referred to as 
"variable .sigma. stop") 1008 is located on the rear (reticle-side) focal 
plane of the fly's eye lens 1007. The illumination light from numerous 
secondary light sources in the variable .sigma. stop 1008 is reflected by 
a mirror 1009, and thereafter is led via a first relay lens 1010, a 
variable field stop (reticle blind) 1011, a second relay lens 1012, a 
mirror 1013, and a condenser lens 1014 then to illuminate a pattern area 
of the photo-mask with uniform illuminance. In this case, the plane on 
which the variable field stop 1011 is located is conjugate with the plane 
on which the pattern of reticle 1016 is formed, and the plane on which the 
variable .sigma. stop 1008 is located is conjugate with the plane of the 
pupil of the projection optical system PL (Fourier transform plane of the 
pattern area of the reticle 1016). The main control system 1023 sets 
apertures of the variable o stop 1008 and variable field stop 1011 to 
respectively predetermined shapes through the drive unit 1025. 
Specifically, when the aperture of the variable .sigma. stop 1008 is set as 
circular about the optical axis, the conventional illumination method 
results, whereby the coherence factor (.sigma. value) of the illumination 
light can be adjusted by controlling the diameter of the aperture. Where 
four apertures are provided in the variable .sigma. stop 1008 about the 
optical axis, the modified light source method results. Further, exposure 
by the phase shift method becomes possible by providing some parts of the 
pattern on the reticle 1016 with phase shifters. 
Also, the reticle 1016 is held on a reticle stage RST and a reticle reader 
1026 is located near the reticle stage RST. When the reticle 1016 is 
loaded onto the reticle stage RST by an unrepresented reticle loader 
system, the reticle reader 1026 reads reticle information (bar code or the 
like) formed on the reticle 1016 and supplies the thus read reticle 
information to the main control system 1023. By this, the main control 
system 1023 can identify the contents of the reticle 1016 (type of 
pattern, minimum line width of pattern, etc.) currently held on the 
reticle stage RST, and optimizes the type of aperture and the size of 
aperture in the variable .sigma. stop 1008 depending upon the result of 
identification. Further, depending upon the contents of the reticle 1016, 
the reticle 1016 is finely moved by a reticle moving unit 1027 in a 
direction perpendicular to the optical axis of the projection optical 
system PL. 
Then, under the illumination light ILB emergent from the condenser lens 
1014, an image of the pattern on the reticle 1016 is projected through the 
projection optical system PL onto a wafer W coated with a nonlinear 
photoresist. The wafer W is mounted on a wafer stage WST, which is 
composed of an XY stage for positioning the wafer W in the XY plane 
perpendicular to the optical axis AX of the projection optical system PL, 
a Z stage for positioning the wafer W in the Z-direction parallel to the 
optical axis AX, and so on. Depending upon the contents of the reticle 
1016, the XY stage is very finely moved in the X direction or in the Y 
direction. The main control system 1023 controls the motion of the wafer 
stage WST through a wafer drive unit 1022, whereby a desired shot area on 
the wafer W is positioned in the image field of the projection optical 
system PL and a position of the shot area (focus position) along the 
optical axis of the projection optical system PL is set at the position of 
the image plane of the projection optical system PL. 
In this example, a variable aperture stop 1015 is provided on the pupil 
plane of the projection optical system PL (i.e., on the Fourier transform 
plane with respect to the plane on which the pattern of photo-mask R is 
formed). The drive unit 1021 sets the diameter of aperture in the variable 
aperture stop 1015 to an optimum value in accordance with an instruction 
from the main control system 1023. The present embodiment can employ 
either one of the conventional illumination method and the modified 
illumination method by the aperture(s) set in the variable .sigma. stop 
1008. The phase shift method can also be used according to the type of 
reticle 1016 mounted on the reticle stage RST. In each of the cases using 
the respective methods, exposure is conducted while the coherence factor 
.sigma. of the illumination light and the numerical aperture NA of the 
projection optical system PL are set as respective optimum values to 
maximize the resolution of the pattern image projected onto the wafer W. 
Further, where the intensity of light from the light source changes with 
time, an appropriate exposure amount can be obtained by properly changing 
the exposure time as to match with an output signal from an integral 
exposure monitor 1029. Generally, the integral exposure monitor 1029 
measures the intensity of a very small portion of beam between the 
variable field stop 1011 and the second relay lens 1012. A reflective 
mirror 1028 is set between the variable field stop 1011 and the second 
relay lens 1012 so that it guides the beam to the integral exposure 
monitor 1028. An output from the integral exposure monitor 1029 is 
transmitted to a driving device and further the driving device moves a 
shutter 1003. The integral exposure monitor 1029 can an exposure time in 
proper even when the first exposure is different from the second exposure 
in a light intensity. 
The exposure method and exposure apparatus of the present invention can 
form a pattern with resolution surpassing the resolution limit which is 
optically given, and, where the phase shift method or the modified light 
source method is employed, they show an advantage that exposure can be 
made with nearly best resolution by setting the numerical aperture of the 
projection optical system within a predetermined range. The exposure 
method of using the phase shift patterns or mask is described in Optics, 
vol.23, No,1, pp 29-37, 1994; and in SPIE vol.1780, Lens and Optical 
Systems Design, pp 117-131, 1992. 
Next, an exposure method according to present invention using a reticle 
having separated line patterns and the photosensitive material having a 
nonlinear sensitivity will be described. 
The present invention relates to an exposure method, particularly to a 
projection-type exposure method, used in fabricating semiconductor devices 
or liquid crystal plates. 
In the conventional exposure methods, all of patterns desired to be exposed 
are provided on a same reticle and are printed on the substrate by single 
exposure. On that occasion, a latent image reaction density .xi. according 
to exposure intensity I appears in the photoresist coating on the 
substrate. 
EQU .xi.=exp (-CD), D=J.multidot.t=I.sup.m .multidot.t (2-1) 
In the equation, I represents light intensity, t an exposure time, and C a 
constant determined by the photosensitive material. Further, m is an 
exponent indicating linearity of the photosensitive material, which is 
linear with m=1 while nonlinear with m.noteq.1. For easy recognition, J 
replaces I.sup.m as in the above equation and J is called as a latent 
image density. Most of the photoresists presently used in general are 
linear with m=1. 
Using a general linear photoresist and assuming for simplicity that image 
formation is completely incoherent, a spectrum i of exposure intensity 
distribution I (x) for forming a latent image in the photoresist is given 
by the below equation with an object spectrum being i.sub.0 and an OTF 
(Optical Transfer Function) of an optical system being f. 
EQU i(.nu.)=i.sub.0 (.nu.).multidot.f(.nu.) (2-2) 
.nu.: spatial frequency 
Now, a spatial frequency .nu..sub.0, which is a limit spatial frequency at 
which the OTF or f becomes not significant in respect of process, is given 
by the below equation with an exposure wavelength being .lambda. and a 
numerical aperture of projection optical system on the photosensitive 
material side being NA. 
EQU .nu..sub.0 =0.5NA/(K.sub.1 .multidot..lambda.) (2-3) 
K.sub.1 : process constant 
Further, the resolution limit of the optical system is theoretically 
determined by the numerical aperture NA, and in that case K.sub.1 =0.25, 
whereby a cutoff frequency .nu..sub.c of the optical system is given by 
the following equation. 
EQU .nu..sub.c =2NA/.lambda. (2-4) 
Accordingly, in order to achieve high resolution, either a decrease of 
wavelength or an increase of numerical aperture NA was needed. 
In conventionally general exposure methods for two-dimensional pattern, a 
pattern, for example as shown in FIG. 26, was used without modification as 
a reticle pattern and was subjected to full exposure on a linear 
photoresist through the projection optical system. 
The above conventional exposure methods require either an increase in 
numerical aperture NA or a decrease in wavelength .lambda. in order to 
achieve high resolution. However, since the depth of focus Fd of the 
projection optical system is proportional to the wavelength .lambda. and 
inversely proportional to the square of NA, as expressed by the following 
equation, the depth of focus becomes shallower in either case. 
EQU Fd=K.sub.2 .multidot..lambda./NA.sup.2 (2-5) 
K.sub.2 : process constant 
Also, the optical system becomes larger and specialized, which is not 
suitable for practical use. Further, the final resolution limit on the 
photosensitive material has never surpassed the resolution limit 
determined by the projection optical system. 
The present invention has been accomplished in view of the above problems, 
and an object of the present invention is to provide an exposure method 
which can form a two-dimensional pattern with high resolution surpassing 
the resolution limit of projection optical system with little modification 
of the conventional exposure wavelength and optical system. 
To solve the above problems, the present invention provides a projection 
exposure method for projecting a pattern on a photo-mask onto a 
predetermined photosensitive material through a projection optical system, 
wherein the pattern on said photo-mask is separated into line patterns in 
each of which lines extend only in the longitudinal direction, the thus 
separated line patterns each are further separated into new line patterns 
with a pitch arranged every several said lines, said photosensitive 
material is one a latent image reaction density of which has a nonlinear 
sensitivity property against intensity of incident light, and a plurality 
of exposure processes using said new line patterns thus separated are 
carried out, thereby enabling to form a two-dimensional pattern with high 
resolution surpassing the resolution limit of the projection of optical 
system. 
In the above exposure method, the pitch of the new line patterns separated 
is preferably set as to be not smaller than the resolution limit of the 
projection optical system. 
Further, the above exposure method is preferably so arranged that where one 
of the new line patterns separated includes separate lines on a same line 
and a space between the lines separately formed is narrower than the 
resolution limit of the projection optical system, one of said separate 
lines is further separated as a further new line pattern from the other, 
and the further new line patterns separated should be used. 
First, in order to describe the most basic concept of the present 
invention, here is considered formation of point image by the optical 
system in the use of a so-called two-photon absorption photoresist with 
m=2. In this case, the two-photon absorption photoresist makes the latent 
image density distribution of point image sharper. It is sufficient for 
this case taking the point spread distribution F(x) by the optical system 
into consideration irrespective of the illumination state. Then, supposing 
a desired object intensity distribution I.sub.0 (x) is formed by 
superposition of point images and a latent image density distribution J(x) 
is formed thereby, the density distribution can basically be expressed by 
the following equation, because it is superposition of light intensities 
given by image formation of respective point images. 
EQU J(x)=I.sub.0 (x)*{F(x)}.sup.2 (2-6) 
Since the latent image density distribution of point image involves 
{F(x)}.sup.2, it is sharper than the point spread distribution F(x) by the 
optical system, thus realizing high resolution. By Fourier transformation 
of Equation (2-6), the following equation is obtained. 
EQU j(.nu.)=i.sub.0 (.nu.).multidot.{f(.nu.)*f(.nu.)} (2-7) 
Then, f*f can be interpreted as OTF of the optical system to obtain the 
latent image density distribution by this method. This means that the 
cutoff frequency (4NA/.lambda.) can be achieved against the cutoff 
frequency (2NA/.lambda.) of the conventional OTF or f, thus doubling the 
resolution. 
FIG. 36A and FIG. 36B diagrammatically show this comparison. FIG. 36A 
indicates the OTF in the conventional method, while FIG. 36B the OTF for 
the case where a two-photon absorption photoresist is exposed to isolated 
patterns. Therefore, a fine pattern can be formed with resolution over the 
resolution limit of the optical system when a latent image is formed using 
the two-photon absorption photoresist and performing a plurality of 
exposure processes based on isolated patterns. Accordingly, formation of a 
pattern with resolution over the resolution limit of the optical system 
becomes possible by the combination of the photosensitive material having 
the nonlinear sensitivity property with the plural exposure processes by 
isolated patterns. 
Further, a pattern with resolution over the resolution limit of the optical 
system can also be formed, similarly as in the case of the isolated 
patterns, where a plurality of exposure processes are carried out using 
patterns which are not perfectly isolated but can be considered as nearly 
isolated. In this case, a spectrum j of latent image density distribution 
is defined by the following equation. 
##EQU27## 
In the above equation, i.sub.0j represents object spectra of mutually 
nearly isolated patterns, and i' is considered as an object spectrum of an 
imaginary pattern constructed by superposition of the nearly isolated 
patterns. No spectra of conventional latent image density distributions 
can surpass the cutoff frequency (2NA/.lambda.) of f, while the present 
invention permits a spectrum up to the cutoff frequency (4NA/.lambda.) of 
{f(.nu.)f(.nu.)} to be formed as a latent image density distribution, as 
indicated by Equation (2-8). 
As described above, the pitch of formed pattern never surpasses the 
resolution limit of the optical system by the single use of the 
photosensitive material having the nonlinear sensitivity property in the 
conventional exposure method, while in the present invention a latent 
image density distribution of pattern with a pitch surpassing the 
resolution limit of the optical system can be formed by performing a 
plurality of exposure processes different in light intensity distribution 
on such a photosensitive material so as to provide appropriate i'. 
Although the above description is given with the so-called two-photon 
absorption photoresist in which the latent image density J is in 
proportion to the square of the exposure intensity I, that is, in which 
the latent image reaction density .xi. is formed according to the square 
of the exposure intensity i, the present invention is by no means limited 
to it. The present invention may employ any photosensitive materials 
having the nonlinear sensitivity property as long as the latent image 
reaction density .xi. is formed according to the m-th power (m&gt;1) of the 
exposure intensity I. In this case, the latent image density distribution 
is expressed by the m-th power of the light intensity distribution F(x) of 
point image, so that it is a sharper distribution than the light intensity 
distribution F(x) of point image and the above equation (2-6) can be 
rewritten as follows. 
EQU J(x)=I.sub.0 (x)*{F(x)}.sup.m (2-9) 
Further, the illumination state is not limited to the incoherent 
illumination, but may be any of the oblique illumination and various 
modified illumination methods, similarly forming a very fine pattern. Of 
course, a self-emitting object can be used. 
With the Fourier transformation of Equation (2-9), it is seen from the 
convolution theorem of the Fourier transformation that a pattern (latent 
image density distribution) can be formed up to a frequency of m times the 
cutoff frequency of the optical system. There is a possibility of forming 
a further finer pattern by a plurality of exposure processes with patterns 
which are not completely isolated from each other in each exposure. 
Although the above description concerned the cases where the power m was 
greater than 1 (m&gt;1) in the above Equation (2-9), that is, where the 
latent image density J was emphasized more than the light intensity I, 
simulation results verified that a fine pattern with resolution over the 
resolution limit of the projection optical system could substantially be 
formed where the power m was smaller than 1 (m&lt;1). Such a sensitivity 
property is also effective that the power m in Equation (2-9) is not 
constant but changes depending upon the light intensity I. 
Next described is a method for forming a high-resolution pattern surpassing 
the resolution limit of the projection optical system in the case of a 
one-dimensional pattern (line pattern) using the two-photon absorption 
photoresist and performing a plurality of exposure processes. 
In this case, exposure is carried out using reticles shown in FIG. 32A and 
FIG. 32B. The reticle shown in FIG. 32A consists of a transparent 
substrate 2001a, open portions 2004a, opaque portions 2002a, and phase 
shifters 2003a. Also, a distance between a certain open portion and a next 
open portion is .lambda./2NA, which is the resolution limit of the optical 
system on the image plane, and a phase shifter 2003a is provided for every 
other open portion 2004a. When exposure is made on the two-photon 
absorption photoresist through the projection optical system, using this 
reticle, the exposure intensity distribution shown in FIG. 33A is given on 
the two-photon absorption photoresist. 
Then the reticle shown in FIG. 32A is exchanged for the reticle shown in 
FIG. 32B, and exposure is effected therewith. The reticle shown in FIG. 
32B consists of a transparent substrate 2001b, open portions 2004b, opaque 
portions 2002b, and phase shifters 2003b, arranged as if the reticle shown 
in FIG. 32A and FIG. 32B is shifted by a half cycle of the pitch. Such 
second exposure gives the exposure intensity distribution shown in FIG. 
33B on the two-photon absorption photoresist. The exposure intensity 
distribution shown in FIG. 33B corresponds to one obtained by shifting the 
exposure intensity distribution shown in FIG. 33A by a half cycle of the 
pitch. 
With the two exposure intensity distributions different in pattern as 
described above, from Equation (2-1), the first exposure intensity 
distribution gives the latent image density distribution shown in FIG. 34A 
in the two-photon absorption photoresist and the second exposure intensity 
distribution gives the latent image density distribution shown in FIG. 34B 
in the two-photon absorption photoresist. Since these two latent image 
density distributions are formed in the same two-photon absorption 
photoresist, addition of these two latent image density distributions 
results in the latent image density distribution shown in FIG. 35, in 
which a fine pattern is formed at the pitch of a half of the resolution 
limit of the optical system. 
Next described is a method for forming a high-resolution pattern surpassing 
the resolution limit of the projection optical system for the cases of 
two-dimensional pattern. 
For example, let us consider a case where a high-resolution pattern 
surpassing the resolution limit of the projection optical system is formed 
using the pattern shown in FIG. 26 and the two-photon absorption 
photoresist. If the pattern is separated every open portion, any way of 
separation cannot form a pattern with high resolution being a half of that 
of the conventional projection optical system. On the other hand, 
one-dimensional patterns permit a pattern with high resolution being a 
half of that of the conventional projection optical system to be formed as 
described above. 
Therefore, the two-dimensional pattern is first decomposed into 
one-dimensional patterns in the present invention, as shown in FIG. 27A 
and FIG. 27B. After the decomposition into one-dimensional patterns, the 
one-dimensional patterns decomposed are separated, as shown in FIGS. 28A 
to 28D or in FIGS. 29A to 29D, so as to enable to use the above technique 
for forming a fine pattern with the pitch being a half of the resolution 
limit of the optical system with one-dimensional patterns. The formation 
of fine pattern with resolution surpassing the resolution limit of the 
optical system becomes possible by exposure using the two-photon 
absorption photoresist and the thus separated one-dimensional patterns 
shown in FIGS. 28A to 28D or in FIGS. 29A to 29D. 
The embodiment is described below. Here is described a method for forming 
an image in the size of a half of the resolution limit by the conventional 
projection optical system from the pattern shown in FIG. 26 and with 
illumination of coherent light, using the two-photon absorption 
photoresist (m=2) and the phase shifters. 
In the pattern shown in FIG. 26 there are lines extending in the vertical 
direction and in the horizontal direction in the plane of the drawing. 
Then the pattern shown in FIG. 26 is decomposed as shown in FIG. 27A and 
27B, thus obtaining a line pattern having only lines horizontally 
extending in the plane of the drawing as shown in FIG. 27A and a line 
pattern having only lines vertically extending as shown in FIG. 27B. 
The line pattern having the lines horizontally extending in the plane of 
the drawing as shown in FIG. 27A consists of elements 2111, 2112, 2113, 
2114, 2115, and 2116 of open portions, which will be open portions in a 
reticle pattern, arranged in the named order from the top of the drawing. 
Also, the line pattern having the lines vertically extending as shown in 
FIG. 27B consists of elements 2121, 2122, 2123, and 2124 of open portions, 
which will be open portions in a reticle, arranged in the named order from 
the left of the drawing. 
Further, the line pattern shown in FIG. 27A is separated into a pattern 
shown in FIG. 28A and a pattern shown in FIG. 28B so that the decomposed 
line pattern with the horizontally long lines shown in FIG. 27A can permit 
a plurality of exposure processes using the two-photon absorption 
photoresist. In th present embodiment, the elements of open portions are 
separated into the pattern shown in of FIG. 28A and the pattern shown in 
FIG. 28B every other element. In more detail, the pattern shown in FIG. 
28A has open portions 2011, 2013, and 2015 while the pattern shown in FIG. 
28B open portions 2012, 2014, and 2016. The open portions 2011, 2013, and 
2015 are based on the elements 2111, 2113, and 2115 of open portions in 
the line pattern shown in FIG. 27A. Similarly, the open portions 2012, 
2014, and 2016 are based on the elements 2112, 2114, and 2116 of open 
portions in the line pattern shown in FIG. 27B. 
The decomposed line pattern with the vertically long lines shown in FIG. 
27B is also separated into line patterns shown in FIG. 28C and FIG. 28D, 
similarly as the horizontally decomposed line pattern was separated. In 
the present embodiment, the line pattern is separated into the pattern 
shown in FIG. 28C and the pattern shown in FIG. 28D every other open 
portion. The pattern shown in FIG. 28C has open portions 2021 and 202 
pattern the pattern shown in FIG. 28D has open portions 2022 and 2024. The 
open portions 2021 and 2023 are based on the elements 2121 and 2123 of 
open portions in the line pattern shown in FIG. 27B. Also, the open 
portions 2022 and 2024 are based on the elements 2122 and 2124 of open 
portions in the line pattern shown in FIG. 27B. 
Here, the resolution limit by incoherent light is the spacing between the 
open portions shown in FIG. 28A and FIG. 28B on the image plane. However, 
where the patterns shown in FIG. 28A and FIG. 28B are actually used as 
reticles in an exposure apparatus, the patterns cannot be resolved, 
because the light source emits coherent light. Therefore, when the 
patterns shown in FIG. 28A and FIG. 28B are actually used as reticles in 
the exposure apparatus, a phase shifter is provided every other open 
portion as shown in FIG. 29A and FIG. 29B. As shown in FIG. 29A and FIG. 
29B, the present embodiment is so arranged that among the three open 
portions in each pattern, the middle open portions 2013 and 2014 are 
provided with respective phase shifters 2013S and 2014S. Such a 
modification is of course possible that a phase shifter is given for each 
of open portions 2011, 2012, 2015, 2016 located on the sides in the 
patterns shown in FIG. 28A and FIG. 28B and no phase shifter is given for 
the middle open portions 2013 and 2014. 
The patterns shown in FIG. 28C and FIG. 28D are also arranged, similarly as 
the patterns shown in FIG. 28A and FIG. 28B, so that a phase shifter 2023S 
or 2024S is given for every other open portion, i.e., for each of open 
portions 2023 and 2024. Here, the case of the present embodiment does not 
necessarily need the phase shifters, because the line patterns with 
vertically extending lines shown in FIG. 28C and FIG. 28D include lines 
apart more than the resolution limit by coherent light in the plane of the 
drawing. In the case of the present embodiment the phase shifters 2023S 
and 2024S are provided in order to obtain an image with better contrast. 
In actual exposure, the line patterns shown in FIG. 29A to FIG. 29D, 
produced as described above, are successively used. Using the two-photon 
absorption photoresist, the reticle patterns are projected one by one 
through the projection optical system, whereby a two-dimensional pattern 
which is complex and very fine can be formed with resolution being a half 
of that of the conventional projection optical system. On this occasion, 
there is no specific order for projection of the reticle patterns shown in 
FIG. 29A to FIG. 29D. Also, the two-photon absorption photoresist may be 
one of those described in Proceeding of SPIE, vol. 1674, pp 776-778, 1992. 
The second embodiment is next described. Here is described a method for 
forming an image with resolution being one third of the resolution limit 
by the conventional projection optical system from a pattern shown in FIG. 
30A and with illumination of coherent light, using a nonlinear photoresist 
of m=3 (hereinafter referred to as a three-photon absorption photoresist 
for convenience' sake) and the phase shifters. 
The pattern shown in FIG. 30A includes lines extending in the vertical 
direction and in the horizontal direction in the plane of the drawing. 
Then the pattern shown in FIG. 30A is decomposed into a line pattern 
having only lines horizontally extending in the plane of the drawing as 
shown in FIG. 31A and a line pattern having only lines vertically 
extending in the plane of the drawing as shown in FIG. 31B, similarly as 
in the first embodiment. 
The line pattern having the lines horizontally extending in the plane of 
the drawing as shown in FIG. 31A consists of elements 2231, 2232, 2233, 
2234, 2235, 2236, and 2237 of open portions, which will be open portions 
in a reticle, arranged in the named order from the top of the drawing. 
Also, the line pattern having the lines vertically extending in the plane 
of the drawing as shown in FIG. 31B consists of an element 2241 of open 
portion, which will be an open portion in a reticle, in the left upper 
portion of the drawing, an element 2242 of open portion on the left lower 
portion of the drawing, and an element 2243 of open portion on the right 
side of the drawing. 
Further, the decomposed line pattern with the lines horizontally extending 
in the plane of the drawing as shown in FIG. 31A is separated similarly as 
in the first embodiment so as to enable to perform a plurality of exposure 
processes using the three-photon absorption photoresist. Since the present 
embodiment employs the three-photon absorption photoresist, the lines are 
separated every three lines. 
Further, similarly as in the first embodiment, a phase shifter is provided 
every other open portion in order to realize resolution up to the 
resolution limit under illumination of incoherent light. Through the above 
steps the reticle patterns shown in FIG. 30B, FIG. 30C, and FIG. 30D are 
obtained. The reticle pattern shown in FIG. 30B consists of an open 
portion 2031, an open portion 2034 on which a phase shifter 2034S is 
provided, and an open portion 2037 arranged in the named order from the 
top of the drawing. The reticle pattern shown in FIG. 30C consists of an 
open portion 2032 on which a phase shifter 2032S is provided, and an open 
portion 2035 arranged in the named order from the top of the drawing. The 
reticle pattern shown in of FIG. 30D consists of an open portion 2033 on 
which a phase shifter 2033S is provided, and an open portion 2037 arranged 
in the named order from the top of the drawing. The open portions 2031, 
2032, 2033, 2034, 2035, 2036, and 2037 shown in FIGS. 30A to 30F are based 
on the elements 2231, 2232, 2233, 2234, 2235, 2236, and 2237 of open 
portions, respectively, shown in FIG. 31A and FIG. 31B. Here, the 
resolution limit of the optical system under incoherent illumination is a 
spacing between the open portions shown in FIG. 30B, FIG. 30C, and FIG. 
30D. 
In the case of the decomposed line pattern with the lines vertically 
extending in the plane of the drawing as shown in FIG. 31B, the elements 
of open portions present on the left side of the drawing are separate from 
each other on a same line. In addition, a spacing between the element of 
open portion and the element of open portion separate from each other is 
narrower than the resolution limit of the optical system. Therefore, the 
line on the left lower side of the drawing is separated out as another 
line pattern. Then, a phase shifter is provided every other open portion 
for the patterns having the vertically extending lines as thus separated, 
similarly as for the patterns shown in FIG. 30B, FIG. 30C, and FIG. 30D. 
The thus obtained reticle patterns are shown in FIG. 30E and FIG. 30F. The 
reticle pattern shown in FIG. 30E consists of an open portion 204I on 
which a phase shifter 2041S is provided, and an open portion 2043 arranged 
in the named order from the left of the drawing. Also, the reticle pattern 
shown in FIG. 30F consists only of an open portion 2042 on the left lower 
side of the drawing. In the case of the present embodiment the phase 
shifter 2041S can be omitted, because the lines in the line pattern having 
the vertically extending lines as shown in FIG. 30E are separated from 
each other more than the resolution limit by coherent light. It should be, 
however, noted that the light intensity will not be zero at the center 
between the lines if the phase shifter 2041S is omitted in contrast, with 
provision of the phase shifter 2041S the light intensity becomes zero at 
the center between the lines, thus providing an image with better 
contrast. 
In actual exposure, the thus produced line patterns shown in FIG. 30B to 
FIG. 30F are successively used, so that, using the three-photon absorption 
photoresist, the reticle patterns are projected through the projection 
optical system to form a two-dimensional pattern which is complex and very 
fine, with the pitch being one third of the resolution limit of the 
conventional projection optical system. On this occasion, there is no 
specific order for projection of the reticle patterns shown in FIGS. 30B 
to 30F. 
Although the above description showed the method for forming a 
two-dimensional pattern surpassing the resolution limit of the optical 
system with the phase shifters and coherent light source, the present 
invention is also effective to cases using an incoherent light source. For 
example, using the patterns shown in FIGS. 28A to 28D as reticles without 
modification, a two-dimensional pattern can be obtained with the pitch 
being three quarters of the resolution limit of the optical system. 
Further, the above description showed only the examples where the pattern 
had the lines vertically and horizontally extending in the plane of the 
drawing, but the present invention is also effective to other cases, for 
example, to a case of hexagonal pattern which has lines vertically 
extending in the plane of the drawing, lines inclined 60.degree. to the 
right from the vertical direction in the plane of the drawing, and lines 
inclined 60.degree. to the left from the vertical direction in the plane 
of the drawing. 
As described above, the present invention realized formation of 
two-dimensional pattern with high resolution surpassing the resolution 
limit of the projection optical system by using a photosensitive material 
having the nonlinear sensitivity property of the latent image reaction 
density against the intensity of incident light and performing a plurality 
of exposure processes using decomposed and separated line patterns. 
Next, an exposure method or apparatus using two or more exposure wavelength 
will be described. 
The present invention relates to an exposure apparatus, particularly to a 
projection-type exposure apparatus and a projection exposure method, used 
in fabricating semiconductor devices or liquid crystal plates. 
The present invention is related to the technique for enabling to form a 
finer pattern by using a photosensitive material with nonlinear 
sensitivity property to intensity of incident light and performing a 
plurality of exposure processes with a change of light intensity 
distribution on the photosensitive material. 
To improve the exposure technique, two or more exposure wavelengths are 
used and a pattern on a reticle is constructed using optical filters 
having different transmittances for the two or more different exposure 
wavelengths. Also, some cases employ a photosensitive material consisting 
of two or more types of photoresists having different sensitivity 
properties for the two or more different wavelengths. 
Specifically, an exposure method employs a photosensitive material having a 
nonlinear sensitivity property of latent image reaction density against 
the intensity of incident light and performs a plurality of exposure 
processes different in light intensity distribution on the photosensitive 
material, thereby enabling to form a pattern with high resolution 
surpassing the resolution limit of the projection optical system, wherein 
light beams having two or more different wavelengths are used as exposure 
light to illuminate a photo-mask in which members for selectively 
transmitting the two or more different wavelengths are provided. 
Also, the photosensitive material may be two or more photosensitive 
materials having respective sensitivity properties different from each 
other. 
Further, a projection exposure apparatus is so arranged that a 
photosensitive material has a nonlinear sensitivity property of latent 
image reaction density against intensity of incident light and exposure is 
repeated plural times with relative movement by a predetermined amount 
between the photosensitive material and a photo-mask for each exposure, 
thereby forming a latent image density distribution of a finer pattern 
than a pattern on the photo-mask, which has a light source for emitting 
light beams having two or more different wavelengths as exposure light, in 
which said photo-mask has selectively transmitting members having 
different transmittances for said two or more wavelengths, and in which 
said photo-mask is illuminated with rays emitted from said light source, 
whereby a latent image density distribution of a fine pattern is formed 
without relative movement by a predetermined amount between said 
photosensitive material and said photo-mask for each exposure. 
The operation of the present invention will be described based on the first 
embodiment according to the present invention as shown in FIG. 38 and FIG. 
39. Different exposure wavelengths .lambda.1, .lambda.2 are used, and a 
one-dimensional periodic pattern is used as a photo-mask 3030 as shown in 
FIG. 38. Further, the periodic pattern is constructed in such a manner 
that portions 3031 transmitting the exposure wavelength .lambda.1 and 
non-transmitting the exposure wavelength .lambda.2 and portions 3032 
non-transmitting the exposure wavelength .lambda.1 and transmitting the 
exposure wavelength .lambda.2 are alternately arranged. Using this 
photo-mask, a light intensity distribution upon projection with exposure 
wavelength .lambda.1 is inverted 180.degree. relative to a light intensity 
distribution upon projection with exposure wavelength .lambda.2 on the 
image plane, as shown in FIG. 39. Namely, the light intensity 
distributions inverted in bright and dark portions can be formed without 
necessitating any mechanical shift. Under the above, the resolution 
limitation in the projecting optical system correspond to the light 
intensity distribution in each wavelength thereof. In the above method, it 
is possible to form a line-and-space pattern with superpassing the 
resolution limitation of the optical system. 
A generally effective technique to increase degrees of freedom of the 
pattern on the photo-mask is addition of portions transmitting the both 
wavelengths .lambda.1 and .lambda.2 and portions transmitting neither 
.lambda.1 nor .lambda.2 to the portions transmitting .lambda.1 and 
non-transmitting .lambda.2 and the portions non-transmitting .lambda.1 and 
transmitting .lambda.2. 
Also, for exposure of complex pattern, it is generally required to form a 
light intensity distribution completely independent of the wavelengths 
.lambda.1 and .lambda.2. It can be considered in that case that formation 
of a pattern with a plurality of portions could be complicated as shown in 
FIG. 40. The complexity of pattern is rather less for two wavelengths, but 
it becomes greater and greater for three or more wavelengths. Then, if a 
synthetic photosensitive material comprising two or more types of 
photosensitive materials having different sensitivity properties for two 
wavelengths is used as the photosensitive material, a pattern can be 
formed more easily. 
Employed is a synthetic photosensitive material consisting of two or more 
types of photosensitive materials having different sensitivity properties, 
for example, a photosensitive material sensitive to the wavelength 
.lambda.1 but not sensitive to the other wavelengths and a photosensitive 
material sensitive to the wavelength .lambda.2 but not sensitive to the 
other wavelengths. In this case, illumination with wavelength .lambda.1 
causes only the .lambda.1 transmitting portions on the photo-mask to 
transmit the light, which forms a first light intensity distribution. Next 
illumination with wavelength .lambda.2 causes only the .lambda.2 
transmitting portions to transmit the light, independent of transmission 
or non-transmission of .lambda.1, and the light forms a second light 
intensity distribution. These first and second light intensity 
distributions make two components contained in the synthetic 
photosensitive material exposed independently of each other. 
Embodiments 
FIG. 37 is a drawing to show schematic entire structure of an exposure 
apparatus in the present invention. A beam of illumination light from a 
light source 3011 which can emit beams of two or more different 
wavelengths is collected by an elliptic mirror 3012 and then guided by a 
mirror 3013 into a collimator lens 3014, where the beam becomes a bundle 
of nearly parallel rays. The beam then passes through an interference 
filter 3021, which is a wavelength selecting means detachably arranged, 
then to enter a fly's eye integrator 3015. Beams emergent from the fly's 
eye integrator 3015 are guided by a mirror 3016 into a main condenser 3017 
to uniformly illuminate a reticle 3018 as a photo-mask. A predetermined 
pattern on the photo-mask 3018 is projected by a projection optical system 
3019 onto a wafer 3020 coated with a photosensitive material. Exchanging 
the interference filter 3021 for another filter having a different 
wavelength property, as shown by the arrows in the drawing, the wavelength 
.lambda.1 is selected for first exposure and the wavelength .lambda.2 for 
second exposure as exposure wavelength. Here, it is of course that the 
projection optical system 3019 is arranged to have no chromatic aberration 
for the two wavelengths .lambda.1, .lambda.2. 
Although this example is so arranged that the interference filter 3021 is 
located in the illumination optical system between the collimator lens 
3014 and the fly's eye integrator 3015, the interference filter 3021 can 
be located at any position in the exposure apparatus as long as it does 
not affect the imaging performance there. Further, it is of course 
possible that, instead of the interference filter 3021 located in the 
exposure apparatus, means such as a dichroic mirror or a glass filter 
having absorption property can be used to select the exposure wavelength 
.lambda.1 for first exposure and the exposure wavelength .lambda.2 for 
second exposure. 
As described previously, FIG. 38 is a drawing to show a schematic cross 
section of construction of reticle as the first embodiment, which is an 
enlarged drawing of reticle 3018 as the photo-mask in FIG. 37. On a 
reticle substrate 3033 a one-dimensional periodic pattern is formed of an 
optical thin film constituting an interference filter. As shown in FIG. 
38, portions 3031 (hatched portions) having characteristics of 
transmitting .lambda.1 and non-transmitting .lambda.2 and portions 3032 
(dotted portions) having opposite characteristics of non-transmitting 
.lambda.1 and transmitting .lambda.2 are alternately arranged in the same 
width. When this reticle pattern is illuminated first with illumination 
light of wavelength .lambda.1 and then with illumination light of 
.lambda.2, periodic patterns are formed in light intensity distributions 
with bright and dark portions inverted between the two wavelengths, as 
shown in FIG. 39. Under such circumstances, a fine pattern surpassing the 
resolution limit can be obtained by using a photosensitive material having 
the non-linear sensitivity property of latent image reaction density 
against intensity of incident light with the wavelengths .lambda.1, 
.lambda.2. 
The above embodiment showed an example where the reticle as the photo-mask 
had the one-dimensional periodic pattern. However, normal reticles rarely 
have such a simple arrangement of one-dimensional periodic pattern, but 
often have complex patterns. 
Thus, the second embodiment is given as an example where degrees of freedom 
of usable patterns are increased by adding portions transmitting both the 
wavelengths and portions transmitting neither of the two wavelengths on 
the pattern. FIG. 40 shows a cross section of a reticle employed in the 
second embodiment. The reticle 3040 has portions 3041 (dotted portions) 
having characteristics of transmitting .lambda.1 and non-transmitting 
.lambda.2, and portions 3042 (hatched portions) having opposite 
characteristics of non-transmitting .lambda.1 and transmitting .lambda.2, 
which are alternately arranged, and further has additional portions which 
are a perfectly opaque portion 3043 (black portion) non-transmitting 
either .lambda.1 or .lambda.2 on the left end side and a perfectly 
transparent portion 3044 (white portion) transmitting both .lambda.1 and 
.lambda.2 on the right end side. Patterns of light intensity distributions 
as shown in FIG. 41 are obtained upon image formation with respective 
illumination light of wavelengths .lambda.1, .lambda.2 using the reticle. 
Under such circumstances, a fine pattern surpassing the resolution limit 
can be obtained using a photosensitive material having the non-linear 
sensitivity property of latent image reaction density against intensity of 
incident light of wavelengths .lambda.1, .lambda.2. 
Next described is the third embodiment employing a photoresist which is a 
synthetic photosensitive material obtained by mixing a photosensitive 
component 1 sensitive only to the wavelength .lambda.1 with a 
photosensitive component 2 sensitive only to the wavelength .lambda.2. 
FIG. 42 is a drawing to show a schematic cross section of construction of 
reticle used in the third embodiment. The reticle 3060 is so arranged that 
a first pattern is formed of an optical thin film 3062 having 
characteristics of non-transmitting only the wavelength .lambda.1 and 
transmitting wavelengths other than the wavelength .lambda.1 on a reticle 
substrate 3061, that a second pattern is formed of an optical thin film 
3063 having characteristics of non-transmitting only the wavelength 
.lambda.2 and transmitting wavelengths other than the wavelength .lambda.2 
on the substrate, and that with superposition of the two patterns the 
photosensitive component 1 in the photoresist forms only a latent image of 
the first pattern while the photosensitive component 2 forms only a latent 
image of the second pattern. Since the photoresist has the nonlinear 
sensitivity property of latent image reaction density against the 
intensity of incident light, a fine pattern surpassing the resolution 
limit can be obtained. 
It is clear that this technique can be applied to cases where three or more 
types of exposure wavelengths are employed. It is preferred that the two 
reticle patterns 3062 and 3063 shown in FIG. 42 be formed at a suitable 
clearance with a transparent member 3064 inbetween. The clearance is 
preferably set as small as possible if the spectral characteristics 
necessary for the reticle 3060 are satisfied. More preferably, the 
clearance is within the depth of focus of the projection optical system. 
Even if it is out of the depth of focus, exposure is possible with slight 
correction of the focus position. This case also keeps the features of the 
present invention. 
Further described in the following is the fourth embodiment using two 
different monochromatic light sources and a photoresist which is a 
synthetic photosensitive material obtained by mixing two photosensitive 
components different in wavelength sensitivity property with each other. 
Here, let .lambda.1, .lambda.2, .lambda.3, .lambda.4, .lambda.5, and 
.lambda.6 be wavelengths increasing in the named order and different from 
each other. 
The light sources employed in the present embodiment are a laser L1 
emitting monochromatic light of wavelength .lambda.2 and a laser L2 
emitting monochromatic light of wavelength .lambda.5, which are arranged 
to emit rays of mixture of these chromatic light beams onto a reticle. 
The reticle consists of a one-dimensional periodic pattern in which 
portions having characteristics of transmitting only wavelengths near the 
wavelength .lambda.2 and non-transmitting the wavelengths other than the 
wavelength .lambda.2 and portions having characteristics of transmitting 
the wavelengths near the wavelength .lambda.5 and non-transmitting 
wavelengths other than the wavelength .lambda.5 are alternately formed in 
a same width on a reticle substrate. 
A photoresist employed herein is a synthetic photosensitive material 
consisting of a photosensitive component 1 having a sensitivity to the 
range of from wavelength .lambda.1 to .lambda.3 and a photosensitive 
component 2 having a sensitivity to the range of from wavelength .lambda.4 
to .lambda.6, formed on the substrate as shown in FIG. 43. 
Upon full exposure using the photoresist and the two monochromatic light 
sources, the photosensitive component 1 is sensitive to the wavelength 
.lambda.2 but not sensitive to the wavelength .lambda.5, and, therefore, 
it is irrespective of a light intensity distribution formed by the 
wavelength .lambda.5. Thus, a latent image is an image of a light 
intensity distribution formed by the wavelength .lambda.2. Conversely, the 
photosensitive component 2 is sensitive to the wavelength .lambda.5 but 
not sensitive to the wavelength .lambda.2, and, therefore, it is 
irrespective of a light intensity distribution formed by the wavelength 
.lambda.2. Thus, a latent image is an image of a light intensity 
distribution formed by the wavelength .lambda.5. Since the photoresist has 
the nonlinear sensitivity property of the latent image reaction density 
against the intensity of incident light, a fine pattern surpassing the 
resolution limit can be obtained. 
Although the full exposure was carried out at the same time using the 
different wavelengths in the fourth embodiment, it is of course possible 
that exposure can be made for each wavelength in a non-simultaneous 
manner. As well as the photoresist with photosensitive component 1 and 
photosensitive component 2 having the wavelength sensitivity 
characteristics in which the wavelengths increase in the above order of 
.lambda.1, .lambda.2, .lambda.3, .lambda.4, .lambda.5, and .lambda.6, the 
same results as the fourth embodiment can be obtained using a photoresist 
where .lambda.4 (the short wavelength edge of the photosensitive component 
2)&lt;.lambda.3 (the long wavelength edge of the photosensitive component 1), 
if .lambda.4 (the short wavelength edge of photosensitive component 
2)&gt;.lambda.2 (the exposure wavelength from laser L1) and if .lambda.5 (the 
exposure wavelength from laser L2)&gt;.lambda.3 (the long wavelength edge of 
photosensitive component 1). 
According to the present invention, two or more independent images can be 
thus formed on a photosensitive material without necessity of performing 
either of horizontal shift of reticle or wafer and exchange of reticles 
which require accurate positioning, thus considerably decreasing necessity 
of exchange of photo-masks. 
Next, an exposure method or apparatus using polarizing plate or film will 
be described. 
According to the method or apparatus using polarizing plate or film, where 
exposure is made using the photosensitive material having the nonlinear 
sensitivity property that the latent image reaction density is so formed 
as to be emphasized in accordance with the m-th power (m&gt;1) of the 
intensity of incident light and using the linearly polarized light 
parallel on the image plane with the longitudinal direction of the pattern 
formed on the image plane, a pattern can be formed with resolution 
surpassing the resolution limit of the projection optical system, the 
reason of which will be described in the following. 
First described is the effect due to the nonlinear sensitivity property. 
In the conventional exposure methods, a light intensity distribution I(x) 
on the image plane after passage through the total system under incoherent 
illumination is given by the below equation, where I.sub.0 (x) is a light 
intensity distribution of object and F(x) is a point spread function of 
the optical system. 
(Eq 4-7) 
EQU I(x)=I.sub.0 (x)*F(x) 
Here, x is a coordinate of position on the photosensitive material and * 
means the convolution. Then, a spectrum i of light intensity on the image 
plane is given by the following equation from the convolution theorem of 
Fourier transformation. 
(Eq 4-8) 
EQU i(.nu.)=i.sub.0 (.nu.).multidot.f(.nu.) 
Here, .nu. is a spatial frequency, i.sub.0 a spectrum of light intensity of 
object, and f corresponds to a so-called OTF of the optical system. Thus, 
the spectrum of latent image density cannot be formed over the cutoff 
frequency (2NA/.lambda.) of optical system. 
Also, there are suggestions to use a so-called two-photon absorption 
photoresist as a photosensitive material having the nonlinear sensitivity 
property in the conventional exposure methods. The two-photon absorption 
photoresist is a photoresist which forms a latent image nucleus when 
absorbing two photons. This is described, for example, in Proceedings of 
SPIE, vol. 1674, pp 776-778, 1992. In this case, the latent image density 
distribution J(x) is formed according to the square of the exposure 
intensity distribution I(x). Namely, the latent image density distribution 
is given by the following equation under incoherent illumination, where 
I.sub.0 (x) is the light intensity distribution of object and F(x) is the 
point spread function of optical system. 
(Eq 4-9) 
EQU J(x)=I(x).sup.2 ={I.sub.0 (x)*F(x)}.sup.2 
Then, a spectrum j of latent image density distribution is given by the 
following equation from the convolution theorem of Fourier transformation, 
similarly. 
(Eq 4-10) 
EQU j(.nu.)={i.sub.0 (.nu.).multidot.f(.nu.)}*{i.sub.0 (.nu.).multidot.f(.nu.)} 
Since the latent image density distribution is given by Equation (4-9) for 
the two-photon absorption photoresist, the latent image density 
distribution is sharper than those for conventional photoresists. This 
will be specifically described with examples shown in FIG. 52A and FIG. 
52B for sinusoidal exposure intensity distribution. 
FIG. 52A shows a latent image density distribution in a normal photoresist, 
which is sinusoidal similarly as the exposure intensity distribution is. 
FIG. 52B shows a latent image density distribution in a two-photon 
absorption photoresist. Comparing FIG. 52A with FIG. 52B, the contrast of 
latent image in FIG. 52B is higher than that in FIG. 52A, but the pitch of 
the formed pattern in FIG. 52B is equal to that in FIG. 52A. Thus, simple 
use only of the two-photon absorption photoresist cannot make the pitch of 
the latent image density distribution formed finer than the pitch of an 
image formed by the optical system, thus never permitting the resolution 
to surpass the resolution limit of the optical system. Although in the 
latent image density distribution there are components with frequencies 
surpassing the resolution limit of optical system from Equation (4-10), 
the pitch of the formed pattern is still kept not surpassing the 
resolution limit of the optical system. 
Accordingly, it is impossible to form a finer pattern than the resolution 
limit determined by the optical system, simply by using the photosensitive 
material having the nonlinear sensitivity property. However, the present 
invention permits a finer pattern surpassing the resolution limit of the 
optical system to be formed by using the photosensitive material having 
the nonlinear sensitivity property and further performing a plurality of 
separate exposure processes. 
In order to describe the most basic concept of the present invention, here 
is considered formation of point image by the optical system in the use of 
a two-photon absorption photoresist with m=2, similarly as above. In this 
case, the two-photon absorption photoresist makes the latent image density 
distribution of point image sharper. It is sufficient for this case taking 
the point spread distribution F(x) by the optical system into 
consideration irrespective of the illumination state. Then, supposing a 
desired object intensity distribution I.sub.0 (x) is formed by 
superposition of point images and a latent image density distribution J(x) 
is formed thereby, the density distribution can basically be expressed by 
the following equation, because it is superposition of light intensities 
given by image formation of respective point images. 
(Eq 4-11) 
EQU J(x)=I.sub.0 (x)*{F(x)}.sup.2 
Since the latent image density distribution of point image involves 
{F(x)}.sup.2, it is sharper than the point spread distribution F(x) by the 
optical system, thus realizing high resolution. By Fourier transformation 
of Equation (4-11), the following equation is obtained. 
(Eq 4-12) 
EQU j(.nu.)=i.sub.0 (.nu.).multidot.{f(.nu.)*f(.nu.)} 
Then, f*f can be interpreted as OTF of the optical system to obtain the 
latent image density distribution by this method. This means that the 
cutoff frequency (4NA/.lambda.) can be achieved against the cutoff 
frequency (2NA/.lambda.) of the conventional OTF or f as expressed by 
Equation (4-8), thus doubling the resolution. 
FIG. 53A and FIG. 53B diagrammatically show this comparison. FIG. 53A 
indicates the OTF in the conventional method, while FIG. 53B the OTF for 
the case where a two-photon absorption photoresist is exposed to isolated 
patterns. Therefore, a fine pattern can be formed with resolution over the 
resolution limit of the optical system when a latent image is formed using 
the two-photon absorption photoresist and performing a plurality of 
exposure processes based on isolated patterns. Accordingly, formation of a 
pattern with resolution over the resolution limit of the optical system 
becomes possible by the combination of the photosensitive material having 
the nonlinear sensitivity property with the plural exposure processes by 
isolated patterns. 
Further, a pattern with resolution over the resolution limit of the optical 
system can also be formed, similarly as in the case of the isolated 
patterns, where a plurality of exposure processes are carried out using 
patterns which are not perfectly isolated but can be considered as nearly 
isolated. In this case, a spectrum j of latent image density distribution 
is defined by the following equation. 
##EQU28## 
In the above equation, i.sub.0j represents object spectra of mutually 
nearly isolated patterns, and i' is considered as an object spectrum of an 
imaginary pattern constructed by superposition of the nearly isolated 
patterns. No spectra of conventional latent image density distributions 
can surpass the cutoff frequency (2NA/.lambda.) of f, as shown by Equation 
(4-8), while the present invention permits a spectrum up to the cutoff 
frequency (4NA/.lambda.) of {f(.nu.)*f(.nu.)} to be formed as a latent 
image density distribution, as indicated by Equation (4-13). 
As described above, the pitch of formed pattern never surpasses the 
resolution limit of the optical system by the single use of the 
photosensitive material having the nonlinear sensitivity property in the 
conventional exposure method, while in the present invention a latent 
image density distribution of pattern with a pitch surpassing the 
resolution limit of the optical system can be formed by performing a 
plurality of exposure processes different in light intensity distribution 
on such a photosensitive material so as to provide appropriate i'. 
Although the above description is given with the so-called two-photon 
absorption photoresist in which the latent image density J is in 
proportion to the square of the exposure intensity I, that is, in which 
the latent image reaction density .xi. is formed according to the square 
of the exposure intensity I, the present invention is by no means limited 
to it. The present invention may employ any photosensitive materials 
having the nonlinear sensitivity property as long as the latent image 
reaction density .xi. is formed according to the m-th power (m&gt;1) of the 
exposure intensity I. In this cased the latent image density distribution 
is expressed by the m-th power of the light intensity distribution F(x) of 
point image, so that it is a sharper distribution than the light intensity 
distribution F(x) of point image and the above equation (4-11) can be 
rewritten as follows. 
(Eq 4-14) 
EQU J(x)=I.sub.0 (x)*{F(x)}.sup.m 
Further, the illumination state is not limited to the incoherent 
illumination, but may be any of the oblique illumination and various 
modified illumination methods, similarly forming a very fine pattern. Of 
course, a self-emitting object can be used. 
With the Fourier transformation of Equation (4-14), it is seen from the 
convolution theorem of the Fourier transformation that a pattern (latent 
image density distribution) can be formed up to a frequency of m times the 
cutoff frequency of the optical system. There is a possibility of forming 
a further finer pattern by a plurality of exposure processes with patterns 
which are not completely isolated from each other in each exposure. 
Although the above description concerned the cases where the power m was 
greater than 1 (m&gt;1) in the above Equation (4-14), that is, where the 
latent image density J was emphasized more than the light intensity I, 
simulation results verified that a fine pattern with resolution over the 
resolution limit of the projection optical system could substantially be 
formed where the power m was smaller than 1 (m&lt;1). Such a sensitivity 
property is also effective that the power m in Equation (4-14) is not 
constant but changes depending upon the light intensity I. 
The latent image density distribution surpassing the resolution limit of 
optical system can be formed with further higher contrast and by a smaller 
number of exposure processes, if a pattern is formed with high resolution 
and with high contrast using the phase shift mask or using the modified 
illumination method in each of the plurality of exposure processes. 
In addition to the above arrangement, the present invention further employs 
the linearly polarized light with the direction of polarization arranged 
as parallel on the image plane with the longitudinal direction of a 
pattern formed on the image plane. FIG. 57 to FIG. 59 show an example in 
which an image intensity is formed by double beam interference. FIG. 57 is 
a drawing to show states of rays on a plane perpendicular to the 
longitudinal direction of the pattern and including the optical axis. FIG. 
58 is a drawing to show a plane with its normal line on the optical axis, 
which is a top plan view of the states of FIG. 57 and which includes a 
linear pattern formed as the hatched portion at the center on the image 
plane. In FIG. 57 and FIG. 58 double-sided arrows represent directions of 
polarization of rays (directions of vibration of the electric field), 
which are in-plane directions. In FIG. 57 each symbol indicated by a cross 
in a circle also represents a direction of polarization of rays (direction 
of vibration of the electric field), which indicates that the electric 
field vibrates in the direction normal to the plane of the drawing. 
FIG. 59 shows image intensity distributions for a case where the direction 
of polarization is perpendicular on the image plane to the longitudinal 
direction of the pattern formed on the image plane (hereinafter referred 
to as p-polarization in this specification for convenience' sake) and for 
a case where the direction of polarization is parallel on the image plane 
with the longitudinal direction of the pattern formed on the image plane 
(hereinafter referred to as s-polarization in the specification for 
convenience sake), from which it is seen that the s-polarization has a 
greater difference between the maximum and the minimum of light intensity 
I than the p-polarization has. Thus, the s-polarization has higher 
contrast than the p-polarization. 
With mathematical expression, the light intensity I for the p-polarization 
is given by the following equation. 
##EQU29## 
For the s-polarization, 
##EQU30## 
Here, .theta. is an angle of the rays incident onto the image plane with 
the optical axis in FIG. 57. 
As described above; from Equation (4-15) and Equation (4-16), the light 
intensity of p-polarization decreases by the rate of cos2.theta. 
(0.degree..ltoreq..theta..ltoreq.90.degree.) while the s-polarization has 
no such factor of decrease, thus indicating that the s-polarization can 
produce higher contrast. Therefore, a beam is preferably of s-polarized 
light in each exposure. 
Use of s-polarization is also effective for general exposure as well as the 
double beam interference. 
As described above, using the photosensitive material having the nonlinear 
sensitivity property of latent image density, in other words, latent image 
reaction density, against intensity of incident light, using the 
s-polarized light with the direction of polarization arranged as parallel 
on the image plane with the longitudinal direction of the pattern formed 
on the image plane, and performing a plurality of exposure processes 
different in light intensity distribution on the photosensitive material, 
semiconductor devices can be obtained with a pattern having high 
resolution surpassing the resolution limit of projection optical system 
and high contrast. 
The present invention will be described below based on embodiments thereof. 
FIG. 44A and FIG. 44B show cross-sectional views of reticle patterns as 
photo-masks according to the present invention. Here the direction 
perpendicular to the plane of the drawing is the longitudinal direction of 
the pattern formed on the image plane. First exposure is made with a 
pattern shown in FIG. 44A of pattern and thereafter second exposure with a 
pattern of FIG. 44B. In the first pattern of FIG. 44A, an opaque film 
4002a provided on a substrate 4001a forms open portions 4004a. A phase 
film 4003a is provided over either one of two mutually adjacent open 
portions 4004a, thus constituting a so-called phase shift mask. In the 
second pattern of FIG. 44B, an opaque film 4002b and a phase film 4003b 
are similarly provided on a substrate 4001b, thus also constituting a 
phase shift mask. The open portions 4004a in the first pattern are 
arranged to overlap with positions of the opaque film 4002b in the second 
pattern while the open portions 4004b in the second pattern are arranged 
to overlap with positions of the opaque film 4002a in the first pattern. 
Exposure processes of the two patterns are carried out separately from 
each other on a photosensitive material. 
In order to control the direction of polarization, a polarizing plate 4022 
is set between an illumination optical system for illuminating a reticle, 
and the reticle in an apparatus shown in FIG. 51, thus obtaining linearly 
polarized light in the direction normal to the plane of the drawing. The 
apparatus shown in FIG. 51 will be detailed later. FIG. 45A and FIG. 45B 
show light quantity distributions on the photosensitive material, obtained 
by the exposure processes with the first and second patterns. In the 
present embodiment, each exposure produces a sinusoidal light intensity Ia 
or Ib, as shown in FIG. 45A or FIG. 45B, by only.+-.first-order diffracted 
light under coherent illumination. In the two exposure processes, 
positions of peaks in the light intensity distributions are shifted to 
each other by a half cycle in phase on the photosensitive material. 
Now, considering cases of high resolution, let us suppose that a light 
intensity distribution having the frequency equal to the resolution limit 
of optical system is produced in each exposure. Namely, the numerical 
aperture is fully effectively used so that the.+-.first-order diffracted 
light passes the periphery of the aperture of optical system, and then the 
produced pitch in each exposure is the resolution limit .lambda./2NA, 
providing light intensity distributions expressed by the following 
equations. 
(Eq 4-17) 
EQU Ia(x)=1+cos (2.pi..multidot.2NA.multidot.x/.lambda.) 
(Eq 4-18) 
EQU Ib(x)=1+cos (2.pi..multidot.2NA.multidot.x/.lambda.+.pi.) 
Since the latent image density in the photoresist is given by the square of 
the light intensity if the photoresist is a two-photon absorption 
photoresist, respective latent image density distributions are given by 
the following equations, which are shown in FIG. 46A and FIG. 46B. 
(Eq 4-19) 
EQU Ja(x)=Ia(x).sup.2 =3/2+2 cos (2.pi..multidot.2NA.multidot.x/.lambda.)+cos 
(4.pi..multidot.2NA.multidot.x/.lambda.)/2 
(Eq 4-20) 
EQU Jb(x)=Ib(x).sup.2 =3/2+2 cos 
(2.pi..multidot.2NA.multidot.x/.lambda.+.pi.)+cos 
((4.pi..multidot.2NA.multidot.x/.lambda.)/2 
A finally obtainable latent image density distribution after a plurality of 
exposure processes is a sum of Equations (4-19) and (4-20), as given by 
the following equation. 
(Eq 4-21) 
EQU J(x)=Ja(x)+Jb(x)=3+cos (4.pi..multidot.2NA.multidot.x/.lambda.) 
As seen from Equation (4-21), the latent image density distribution J(x) in 
the present embodiment has a periodic arrangement of pitch (.lambda./4NA), 
which is two times finer than the limit resolution (.lambda./2NA) of 
optical system. This latent image density distribution J(x) is shown in 
FIG. 47. A fine photoresist pattern is formed by development after 
completion of the plurality of (two in this embodiment) exposure 
processes. 
As apparent from Equation (4-12) and FIG. 53B, if a latent image is formed 
by superposition of perfectly isolated patterns (point objects), the 
latent image can be formed at the pitch (.lambda./4NA). However, the 
contrast is not so high in this case. Thus, the above embodiment employs 
the coherent illumination using the phase shift masks, whereby the latent 
image density distribution is formed with high contrast. 
If the photoresist is a conventional or linear photosensitive material, it 
is exposed in a simple sum of the above equations (4-17) and (4-18), i.e., 
in a simple sum of FIG. 45A and FIG. 45B, thus forming no pattern at all. 
In the present embodiment the linearly polarized light is obtained by 
setting the polarizing plate between the illumination optical system for 
illuminating the reticle, and the reticle. Because of this arrangement, if 
a new reticle pattern has a pattern as obtained by rotating the original 
reticle pattern by 90.degree. on the plane of pattern and if the 
polarizing means is kept unchanged as it is, the new reticle pattern is 
illuminated with p-polarized light. As described previously, the contrast 
with p-polarized light is lower than that with s-polarized light, and 
therefore some means is necessary for the new reticle pattern to be 
illuminated with s-polarized light. To realize it, the apparatus shown in 
FIG. 51 and detailed later is so arranged in the present embodiment that 
where the new reticle pattern has a pattern obtained by rotating the 
original reticle pattern 90.degree. on the plane of pattern, the 
polarizing plate 4022 is rotated 90.degree. about the optical axis Ax by a 
drive unit 4023, thus solving the problem. Although the description 
concerned the example of rotation of 90.degree., it is of course that the 
arrangement allows rotation of arbitrary angle. 
The above embodiment showed a case in which the two-photon absorption 
photoresist was used to obtain the latent image density according to the 
square (m=2) of the light intensity, but much higher resolution can be 
expected if the latent image density is obtained in nonlinearity of the 
third power, the fourth power, or the higher power (m=3, 4, . . . ) of the 
light intensity. For example, a latent image density distribution shown in 
FIG. 48 shows an example in which the latent image density distribution is 
obtained according to the cube (m=3) of the light intensity distribution, 
which was obtained by performing three exposure processes while shifting 
the pattern on the photo-mask shown in FIG. 44A by one third pitch each, 
i.e., by (.lambda./6NA) each in this example, a periodic arrangement of 
pitch (.lambda./6NA) is obtained as shown, which is three times finer than 
the resolution limit (.lambda./2NA) of optical system. 
Also,the present invention allows use of photosensitive materials in which 
the latent image density is obtained according to the power 1.5 (m=1.5) of 
the light intensity. FIG. 49 shows a latent image density distribution 
obtained by separate exposure processes for the respective reticles shown 
in FIG. 44A and FIG. 44B in the case of m=1.5, indicating that the latent 
image density distribution is obtained two times finer than the resolution 
limit of the optical system. The contrast in this case can also be 
enhanced by coherent illumination on the phase shift masks. 
It is general that the resist pattern after development is formed 
approximately in proportion with the latent image reaction density. 
However, further emphasizing the contrast in the development process, a 
resist pattern can be formed with much higher contrast. 
Next, FIG. 50A and FIG. 50B show an embodiment in which a polarizing film 
is directly set on a photo-mask, saving trouble to rotate the polarizing 
plate depending upon the pattern on the reticle. 
FIG. 50A and FIG. 50B show cross-sectional views of reticle patterns as 
photo-masks according to a present invention. First exposure is made with 
the pattern shown in FIG. 50A and thereafter second exposure with a 
pattern of FIG. 50B. In the first pattern of FIG. 50A, an opaque film 
4002a provided on a substrate 4001a forms open portions 4004a. Then, a 
phase film 4003a is provided over either one of two mutually adjacent open 
portions 4004a, thus constituting a so-called phase shift mask. Further, a 
polarization film 4005a is provided over the entire surface in the above 
arrangement so that the direction of polarization is parallel with the 
longitudinal direction of pattern (the direction normal to the plane of 
FIG. 50A and FIG. 50B). In the second pattern of FIG. 50B an opaque film 
4002b and a phase film 4003b are similarly provided on a substrate 4001b, 
also constituting a phase shift mask. Also, similarly as the first 
pattern, a polarization film 4005b is provided over the entire surface in 
the arrangement so that the direction of polarization is parallel with the 
longitudinal direction of pattern. Separate exposure processes are 
effected on a photosensitive material in such an arrangement that the open 
portions 4004a in the first pattern are located to overlap with positions 
of the opaque film 1b in the second pattern or that the open portions 
4004b in the second pattern are located to overlap with positions of the 
opaque film 1a in the first pattern. 
Using the above reticle patterns, a fine latent image can be formed over 
the limit resolution of projection optical system and with high contrast. 
Although the above embodiment was so arranged that the polarization films 
were provided on the open portion side of substrates 4001a and 4001b, the 
polarization films can be provided on the opposite lower side (where no 
open portion is provided) of substrates 4001a and 4001b. 
Further described is an embodiment where the power m is smaller than 1 
(m&lt;1). Described is an example where a photosensitive material with m=0.5 
is used. Using the photosensitive material with m=0.5, the latent image 
density is formed according to the power 0.5 of the light intensity. 
Namely, it is given by the following equation. 
(Eq 4-22) 
EQU J(x)=I(x).sup.0.5 
Here, x is a coordinate value. Described is a case where a line-and-space 
pattern is printed using the reticles with phase shifters shown in FIG. 44 
as described previously under coherent illumination and a polarizing plate 
is set between a projection lens and a photosensitive material to obtain 
linearly polarized light in the direction normal to the plane of the 
drawing. The period of the reticles is the resolution limit .lambda./2NA 
of projection optical system. FIG. 54 is a drawing to show a light 
intensity distribution formed on the image plane. The light intensity 
distribution I(x) has a sinusoidal distribution as shown. Namely, it is 
expressed as follows. 
(Eq 4-23) 
EQU I(x)=1+COS (2.pi..multidot.2NA.multidot.x/.lambda.) 
Further, FIG. 55A shows a latent image density distribution J(x) obtained 
from Equation (4-23). 
(Eq 4-24) 
EQU J(x)=(1+COS (2.pi..multidot.2NA.multidot.x/.lambda.).sup.0.5 
As shown in FIG. 55A, the latent image density distribution J(x) is more 
gently sloping near bright portions but becomes rapidly darker and very 
thinner in width in dark portions than the light intensity distribution 
I(x). However, it is clear that since the latent image density 
distribution J(x) is formed in the same period as the light intensity 
distribution I(x) is, a pattern cannot be formed exceeding the limit 
resolution of projection optical system in this state. 
On the other hand, since a pattern can be formed in finer structure as 
shown in FIG. 55C on the image plane by superimposing a latent image 
density distribution J(x) (FIG. 55B) obtained with a pattern shifted by a 
half period on FIG. 55A, the pattern will have a periodic arrangement two 
times finer than the limit resolution. 
In contrast, in case of a photosensitive material of m=1 being used, the 
latent image density distribution J(x) would be perfectly coincident with 
the light intensity distribution I(x) of FIG. 54 and in superimposing this 
on a half-shifted pattern, J(x) obtained would become flat as understood 
from the following equation, which is of no use (FIG. 56). 
(Eq 4-25) 
EQU J(x)=1+COS (2.pi..multidot.2NA.multidot.x/.lambda.)+1-COS 
(2.pi..multidot.2NA.multidot.x/.lambda.)=2 
As described, a finer latent image than the resolution limit of projection 
optical system can be formed even using photosensitive materials of m&lt;1. 
The present embodiment is also arranged in the same manner as the 
previously described embodiments, so that in order to control the 
direction of polarization in the apparatus shown in FIG. 51 a polarizing 
plate 4022 is set between the illumination optical system for illuminating 
the reticle and the reticle to obtain linearly polarized light. If a 
reticle has a pattern obtained by rotating the original reticle pattern 
90.degree. on the plane of pattern, the polarizing plate 4022 is rotated 
90.degree. about the optical axis Ax by the drive unit 4023. 
It is needless to mention that either a positive photoresist or a negative 
photoresist can be employed in the present invention. However especially, 
it is considered that the cases of m&lt;1 are advantageous for positive 
photoresists, and extremely fine left lines can be formed in the case of 
the example shown in FIG. 55A-55C. 
FIG. 51 shows schematic structure of an exposure apparatus for performing a 
plurality of exposure processes different in light intensity distribution 
on the photosensitive material as described above. An illumination beam 
from a light source 4011 is collected by an elliptic mirror 4012, is 
guided by a mirror 4013 into a collimator lens 4014 to become a bundle of 
nearly parallel rays, and is then incident into a fly's eye integrator 
4015. Beams emergent from the fly's eye integrator 4015 are guided by a 
mirror 4016 into a main condenser 4017 to uniformly illuminate a reticle 
4018a as a photo-mask. A predetermined pattern on the photo-mask 4018a is 
projected through a projection optical system 4019 onto a wafer 4020 
coated with a photosensitive material to effect exposure thereon. Here, 
after completion of exposure, the reticle 4018a is exchanged for another 
reticle 4018b having a different pattern by a reticle loader 4021, and 
second exposure is then conducted. 
Instead of the exchange of different patterns by the reticle loader 4021, 
the apparatus may be so arranged that after the first exposure with the 
reticle 4018a, the reticle 4018a is moved by a predetermined amount in the 
direction perpendicular to the optical axis Ax of projection optical 
system 4019 to then effect second exposure. This predetermined amount is 
(.lambda./4NA) when converted as a coordinate value on the wafer for 
example in the case using the pattern of FIG. 44A as described previously 
where the latent image density of photosensitive material is proportional 
to the square of the light intensity. Where the latent image density of 
photosensitive material is proportional to the cube of the light 
intensity, it is effective to set the predetermined value as 
(.lambda./6NA) when converted as a coordinate value on the wafer. 
Unless the reticle in which the polarization film is directly provided on 
the photo-mask is used, the polarizing plate 4022 is inserted into the 
above optical apparatus. In the embodiment shown in FIG. 51, it is 
inserted between the main condenser 4017 and the photo-mask 4018a. Here, 
the polarizing plate 4022 is so arranged that it can be rotated or 
reciprocated by the drive unit 4023 such as a motor, and is set so that 
the direction of polarization becomes parallel with the longitudinal 
direction of the pattern formed on the wafer 4020. Although the polarizing 
plate 4022 is inserted between the main condenser 4017 and the photo-mask 
4018a in this example, the polarizing plate 4022 can be so arranged as to 
be inserted between the photo-mask 4018a and the projection optical system 
4019, between the projection optical system 4019 and the wafer 4020, or at 
the position of aperture stop of the projection optical system 4019 in 
order to obtain a constant polarization state. 
Meanwhile, it is to be desired that an exposure amount in the second 
exposure is finely adjusted relative to that in the first exposure for 
photoresists not perfectly ideal. 
It is of course for cases where a same reticle pattern is used for plural 
exposure processes that the wafer itself can be arranged to move for each 
of plural exposure processes, instead of moving the reticle. 
Effective alignment between the plural exposure processes is the so-called 
latent image alignment in which alignment is performed observing the 
latent image. 
The embodiment of FIG. 51 was described keeping in mind so-called steppers 
for fabricating semiconductors in fine structure by successively repeating 
exposure processes, but the present invention can of course be applied to 
so-called slit scan exposure apparatus for scanning an exposure area, used 
when the screen size of optical system is small, instead of full exposure 
over the entire surface. 
Incidentally, as in the embodiment shown in FIG. 44A and FIG. 44B, it is 
effective in the present invention to use the phase shift patterns in 
order to form a high-resolution pattern. It is also effective to employ 
the annular illumination proposed in Japanese Laid-open Patent Application 
No. 61-91662 or to employ the so-called SHRINC illumination proposed in 
Japanese Laid-open Patent Application No. 4-225358, for example. 
The method of using the phase shift patterns or mask is described in 
Optics, vol.23, No,1, pp 29-37, 1994; and in SPIE vol.1780, Lens and 
Optical Systems Design, pp 117-131, 1992. The annular illumination is 
described in U.S. patent application Ser. No. 166,153. The SHRINC 
illumination is described in U.S. patent application Ser. No. 7,91138. 
Further, it is also effective to increase the depth of focus by provision 
of a filter giving a phase difference on the pupil of the imaging optical 
system. Although the above embodiments did not illustrate in particular, 
it is also effective to change the shape of the secondary light source, 
the wavelength band, etc. as illumination conditions in each exposure. 
The present invention has such a major feature that the plane of 
polarization is made parallel on the image plane with the longitudinal 
direction of the pattern formed on the image plane, but the invention is 
by no means limited to the strictly parallel arrangement. It is thus of 
course that the present invention involves modifications in which the 
plane of polarization is nearly parallel on the image plane with the 
longitudinal direction of the pattern formed on the image plane. 
As described above, the present invention enables a fine pattern surpassing 
the resolution limit of projection optical system to be formed, using the 
photosensitive material showing the nonlinear sensitivity property, using 
the s-polarization, and performing a plurality of exposure processes with 
different patterns. In addition, a high-resolution pattern can be formed 
with little changing the conventional exposure wavelength and optical 
system. 
With the exposure method according to the present invention, semiconductor 
devices can be fabricated with very fine circuit pattern which was 
impossible with the conventional projection exposure apparatus, thus 
achieving the great effect of highly increasing the degree of integration 
of integrated circuits. 
Next, an exposure method according to present invention using a 
transmitting film having nonlinear characteristics will be described. The 
photoresist as a photosensitive material is coated with the transmitting 
film having such a property that the intensity of outgoing light is equal 
to I.sup.n (n.noteq.1) where I is the intensity of incident light. This 
changes the light intensity distribution of incident light into the 
photoresist. 
When exposure light passes through the above transmitting film, the latent 
image reaction density .xi. appearing in the photoresist can be expressed 
by the following equation instead of Equation (4-1). 
(Eq 5-6) 
EQU .xi.=exp (-CD), D=H.multidot.t=I.sup.n .multidot.t 
Here, I represents exposure intensity, t an exposure time, and C a constant 
determined by the photosensitive material. Further, n is an exponent 
indicating linearity, which is defined as linear with n=1 and nonlinear 
with n.noteq.1. In the present invention, as described below, n.noteq.1 
can be achieved by changing the effective exposure intensity I for the 
photoresist. For easy recognition, H replaces I.sup.n as in the above 
equation and H is called as an effective exposure intensity. 
If the transmittance of the transmitting film is unchanged (or constant), 
the intensity of transmitted light will be proportional to the intensity 
of incident light. Namely, n=1. 
Next described in the following is the principle of capability of forming a 
pattern surpassing the resolution limit of projection optical system while 
using a transmitting film having such a nonlinear property that the 
intensity of transmitted light is formed so as to be emphasized in 
accordance with the power n (n&gt;1) of the intensity of incident light. 
In the conventional exposure methods, a light intensity distribution I(x) 
on the image plane after passage through the total system under incoherent 
illumination is given by the below equation, where I.sub.0 (x) is a light 
intensity distribution of object and F(x) is a point spread function of 
the optical system. 
(Eq 5-7) 
EQU I(x)=I.sub.0 (x)*F(x) 
Here, x is a coordinate of position on the photosensitive material and * 
means the convolution. Then, a spectrum i of light intensity on the image 
plane is given by the following equation from the convolution theorem of 
Fourier transformation. 
(Eq 5-8) 
EQU i(.nu.)=i.sub.0 (.nu.).multidot.f(.nu.) 
Here, .nu. is a spatial frequency, i.sub.0 a spectrum of light intensity of 
object, and f corresponds to a so-called OTF of the optical system. Thus, 
the spectrum of exposure intensity (light intensity on the image plane) 
cannot be formed over the cutoff frequency (2NA/.lambda.) of optical 
system. 
However, where the transmitting film having such a property that the 
intensity of transmitted light is emphasized in accordance with the power 
n (n&gt;1) of the intensity of incident light is formed on the photosensitive 
material, a fine pattern can be formed as described below. Described is an 
example of n=2, i.e., a case where the effective exposure intensity 
distribution H(x) is formed in accordance with the square of the 
transmitted light intensity distribution I(x). The effective exposure 
intensity distribution is given by the following equation under incoherent 
illumination, where I.sub.0 (x) is the light intensity distribution of 
object and F(x) is the point spread function of optical system. 
(Eq 5-9) 
EQU H(x)=I(x).sup.2 ={I.sub.0 (x)*F(x)}.sup.2 
Then, a spectrum h of effective exposure intensity distribution is given by 
the following equation from the convolution theorem of Fourier 
transformation, similarly. 
(Eq 5-10) 
EQU h(.nu.)={i.sub.0 (.nu.).multidot.f(.nu.)}*{i.sub.0 (.nu.).multidot.f(.nu.)} 
Since the effective exposure intensity distribution is given by Equation 
(5-9) for the cases of n=2, the effective light intensity distribution is 
sharper than those for conventional photoresists given by Equation (5-7). 
This will be specifically described with examples shown in FIG. 68A and 
FIG. 68B for sinusoidal exposure intensity distribution. 
FIG. 68A shows a light intensity distribution in normal exposure, which is 
sinusoidal similarly as the exposure intensity distribution is. FIG. 68B 
shows an effective exposure intensity distribution in a case of n=2. 
Comparing FIG. 68A with FIG. 68B, the contrast in FIG. 69B is higher than 
that in FIG. 68A, but the pitch of the formed pattern in FIG. 68B is equal 
to that in FIG. 68A. Thus, simple setting only of n=2 cannot make the 
pitch of the effective exposure intensity distribution formed finer than 
the pitch of an image formed by the optical system, thus never permitting 
the resolution to surpass the resolution limit of the optical system. 
Although in the effective exposure intensity distribution there are 
components with frequencies surpassing the resolution limit of optical 
system from Equation (5-10), the pitch of the formed pattern is still kept 
not surpassing the resolution limit of the optical system. 
Accordingly, it is impossible to form a finer pattern than the resolution 
limit determined by the optical system, simply by using the transmitting 
film having such a property that the intensity of transmitted light is 
emphasized in accordance with the intensity of incident light. However, 
the present invention permits a finer pattern surpassing the resolution 
limit of the optical system to be formed by using the transmitting film 
having such a property that the intensity of transmitted light is 
emphasized in accordance with the intensity of incident light and further 
performing a plurality of separate exposure processes. 
In order to describe the most basic concept of the present invention, here 
is considered formation of point image by the optical system in the case 
of n=2, similarly as above. In this case, the point-spread-distribution 
point image passing through the transmitting film becomes sharper. 
Further, it is sufficient for this case taking the point spread 
distribution F(x) by the optical system into consideration irrespective of 
the illumination state. Then, supposing a desired object intensity 
distribution I.sub.0 (x) is formed by superposition of point images and an 
effective exposure intensity distribution H(x) is formed thereby, the 
density distribution can basically be expressed by the following equation, 
because it is superposition of light intensities given by image formation 
of respective point images. 
(Eq 5-11) 
EQU H(x)=I.sub.0 (x)*{F(x)}.sup.2 
Since the effective exposure intensity distribution of point image involves 
{F(x)}.sup.2, it is sharper than the point spread distribution F(x) by the 
optical system, thus realizing high resolution. By Fourier transformation 
of Equation (5-11), the following equation is obtained. 
(Eq 5-12) 
EQU h(.nu.)=i.sub.0 (.nu.).multidot.{f(.nu.)*f(.nu.)} 
Then, f*f can be interpreted as OTF of the optical system to obtain the 
effective exposure intensity distribution by this method. This means that 
the cutoff frequency (4NA/.lambda.) can be achieved against the cutoff 
frequency (2NA/.lambda.) of the conventional OTF or f as expressed by 
Equation (5-8), thus doubling the resolution. 
FIG. 69A and 69B diagrammatically show this comparison. FIG. 69A indicates 
the OTF in the conventional method, while FIG. 69B the OTF for the case 
where a photoresist is exposed to isolated patterns, using the 
transmitting film of n=2. Therefore, a fine pattern can be formed with 
resolution over the resolution limit of the optical system when a latent 
image is formed using the transmitting film of n=2 and performing a 
plurality of exposure processes based on isolated patterns. Accordingly, 
formation of a pattern with resolution over the resolution limit of the 
optical system becomes possible by the combination of the transmitting 
film having the nonlinear property with the plural exposure processes by 
isolated patterns. 
Further, a pattern with resolution over the resolution limit of the optical 
system can also be formed, similarly as in the case of the isolated 
patterns, where a plurality of exposure processes are carried out using 
patterns which are not perfectly isolated but can be considered as nearly 
isolated. In this case, a spectrum h of effective exposure intensity 
distribution is defined by the following equation. 
##EQU31## 
In the above equation, i.sub.0j represents object spectra of mutually 
nearly isolated patterns, and i' is considered as an object spectrum of an 
imaginary pattern constructed by superposition of the nearly isolated 
patterns. No spectra of conventional effective exposure intensity 
distributions can surpass the cutoff frequency (2NA/.lambda.) of f, as 
shown by Equation (5-8), while the present invention permits a spectrum up 
to the cutoff frequency (4NA/.lambda.) of {f(.nu.)*f(.nu.)} to be formed 
as an effective exposure intensity distribution, as indicated by Equation 
(5-13). 
As described above, the pitch of formed pattern never surpasses the 
resolution limit of the optical system by the single use of the 
transmitting film having the nonlinear property in the conventional 
exposure method, while in the present invention an effective exposure 
intensity distribution of pattern with a pitch surpassing the resolution 
limit of the optical system can be formed by performing a plurality of 
exposure processes different in effective exposure intensity distribution 
on the photosensitive material so as to provide appropriate i'. 
Although the above description is given with the case in which the 
effective exposure intensity H is in proportion to the square of the 
exposure intensity I, the present invention is by no means limited to it. 
The present invention may employ any transmitting films having the 
nonlinear property as long as the effective exposure intensity H is formed 
according to the power n (n&gt;1) of the exposure intensity I. In this case, 
the effective exposure intensity distribution is expressed by the n-th 
power of the light intensity distribution F(x) of point image, so that it 
is a sharper distribution than the light intensity distribution F(x) of 
point image and the above equation (5-11) can be rewritten as follows. 
(Eq 5-14) 
EQU H(x)=I.sub.0 (x)*{F(x)}.sup.n 
Further, the illumination state is not limited to the incoherent 
illumination, but may be any of the oblique illumination and various 
modified illumination methods, similarly forming a very fine pattern. Of 
course, a self-emitting object can be used. 
With the Fourier transformation of Equation (5-14), it is seen from the 
convolution theorem of the Fourier transformation that a pattern 
(effective exposure intensity distribution) can be formed up to a 
frequency of n times the cutoff frequency of the optical system. There is 
a possibility of forming a further finer pattern by a plurality of 
exposure processes with patterns which are not completely isolated from 
each other in each exposure. 
Although the above description concerned the cases where the power n was 
greater than 1 (n&gt;1) in the above Equation (5-14), that is, where the 
effective exposure intensity H was emphasized more than the light 
intensity I, simulation results verified that a fine pattern with 
resolution over the resolution limit of the projection optical system 
could substantially be formed where the power n was smaller than 1 (n&lt;1). 
Such a transmission property is also effective that the power n in 
Equation (5-14) is not constant but changes depending upon the light 
intensity I. 
The effective exposure intensity distribution surpassing the resolution 
limit of optical system can be formed with further higher contrast, if a 
pattern is formed with high resolution and with high contrast using the 
phase shift mask or using the modified illumination method in each of the 
plurality of exposure processes. 
As described above, semiconductor devices having patterns with high 
resolution surpassing the resolution limit of projection optical system 
can be attained by using the transmitting film having the nonlinear 
transmission property for effective exposure intensity and performing a 
plurality of exposure processes different in light intensity distribution 
on the transmitting film. 
The present invention will be described below based on embodiments thereof. 
FIG. 1 shows cross-sectional views of reticle patterns as photo-masks 
according to the present invention. First exposure is made with a pattern 
shown in FIG. 60A of pattern and thereafter second exposure with a pattern 
of FIG. 60B. In the first pattern of FIG. 60A, an opaque film 5002a 
provided on a substrate 5001a forms open portions 5004a. A phase film 
5003a is provided over either one of two mutually adjacent open portions 
5004a, thus constituting a so-called phase shift mask. In the second 
pattern of FIG. 60B, an opaque film 5002b and a phase film 5003b are 
similarly provided on a substrate 5001b, thus also constituting a phase 
shift mask. The open portions 5004a in the first pattern are arranged to 
overlap with positions of the opaque film 5002b in the second pattern 
while the open portions 5004b in the second pattern are arranged to 
overlap with positions of the opaque film 5002a in the first pattern. 
Exposure processes of the two patterns are carried out separately from 
each other on a photosensitive material with a transmitting film provided 
thereon. 
FIG. 61A and FIG. 61B show light quantity distributions on the transmitting 
film, obtained by the exposure processes with the first and second 
patterns. In the present embodiment, each exposure produces a sinusoidal 
light intensity Ia or Ib, as shown in FIG. 61A or FIG. 61B, by 
only.+-.first-order diffracted light under coherent illumination. In the 
two exposure processes, positions of peaks in the light intensity 
distributions are shifted to each other by a half cycle in phase on the 
transmitting film. 
Now, considering cases of high resolution, let us suppose that a light 
intensity distribution having the frequency equal to the resolution limit 
of optical system is produced in each exposure. Namely, the numerical 
aperture is fully effectively used so that the.+-.first-order diffracted 
light passes the periphery of the aperture of optical system, and then the 
produced pitch in each exposure is the resolution limit .lambda./2NA, 
providing incident light intensity distributions expressed by the 
following equations. 
(Eq 5-15) 
EQU Ia(x)=1+cos (2.pi..multidot.2NA.multidot.x/.lambda.) 
(Eq 5-16) 
EQU Ib(x)=1+cos (2.pi..multidot.2NA.multidot.x/.lambda.+.pi.) 
Where the effective exposure intensity transmitted is given by the square 
of the light intensity because of the property of the coating film on the 
photoresist, respective effective exposure intensity distributions are 
given by the following equations, which are shown in FIG. 62A and FIG. 
62B. 
(Eq 5-17) 
EQU Ha(x)=Ia(x).sup.2 =3/2+2 cos (2.pi..multidot.2NA.multidot.x/.lambda.)+cos 
(4.pi..multidot.2NA.multidot.x/.lambda.)/2 
(EQ 5-18) 
EQU Hb(x)=Ib(x).sup.2 =3/2+2 cos(2 
.pi..multidot.2NA.multidot.x/.lambda.+.pi.)+cos 
(4.pi..multidot.2NA.multidot.x/.lambda.)/2 
A finally obtainable effective exposure intensity distribution after a 
plurality of exposure processes is a sum of Equations (5-17) and (5-18), 
as given by the following equation. 
(Eq 5-19) 
EQU H(x)=Ha(x)+Hb(x)=3+cos (4.pi..multidot.2NA.multidot.x/.lambda.) 
As seen from Equation (5-19), the effective exposure intensity distribution 
H(x) in the present embodiment has a periodic arrangement of pitch 
(.lambda./4NA), which is two times finer than the limit resolution 
(.lambda./2NA) of optical system. This effective exposure intensity 
distribution H(x) is shown in FIG. 63. A fine photoresist pattern is 
formed by development after completion of the plurality of (two in this 
embodiment) exposure processes. 
As apparent from Equation (5-12) and FIG. 69B, if a latent image is formed 
by superposition of perfectly isolated patterns (point objects), the 
latent image can be formed at the pitch (.lambda./4NA). However, the 
contrast is not so high in this case. Thus, the above embodiment employs 
the coherent illumination using the phase shift masks, whereby the 
effective exposure intensity distribution is formed with high contrast. 
If a transmitting film with constant transmittance which is not nonlinear 
is used, the photoresist is exposed in a simple sum of the above equations 
(5-15) and (5-16), i.e., in a simple sum of FIG. 61A and FIG. 61B, thus 
forming no pattern at all. 
The above embodiment showed a case in which the effective exposure 
intensity was obtained according to the square (n=2) of the light 
intensity, but much higher resolution can be expected if the effective 
exposure intensity is obtained in nonlinear property of the third power, 
the fourth power, or the higher power (n=3, 4, . . . ) of the light 
intensity. For example, an effective exposure intensity distribution shown 
in FIG. 64 shows an example in which the effective exposure intensity 
distribution is obtained according to the cube (n=3) of the light 
intensity distribution, which was obtained by performing three exposure 
processes while shifting the pattern on the photo-mask shown in FIG. 60A 
by one third pitch each, i.e., by (.lambda./6NA) each. In this example, a 
periodic arrangement of pitch (.lambda./6NA) is obtained as shown, which 
is three times finer than the resolution limit (.lambda./2NA) of optical 
system. 
Also, the present invention allows use of transmitting films in which the 
effective exposure intensity is obtained according to the power 1.5 
(n=1.5) of the light intensity. FIG. 65 shows an effective exposure 
intensity distribution obtained by separate exposure processes for the 
respective reticles shown in FIG. 60A and FIG. 60B in the case of n=1.5, 
indicating that the effective exposure intensity distribution is obtained 
two times finer than the resolution limit of the optical system. The 
contrast in this case can also be enhanced by coherent illumination on the 
phase shift masks. 
Next described is another embodiment according to the present invention. 
The above embodiment shown in FIG. 60A and 60B used the so-called phase 
shift method to form the light intensity distribution by coherent image 
formation, while this embodiment uses an ordinary reticle to perform 
partial coherent image formation. Conditions of optical system are 
determined as the operational wavelength X=0.365 .mu.m, the numerical 
aperture NA=0.5, and the coherence factor .sigma.=0.6. FIG. 67A shows a 
case of n=2, which is an effective exposure intensity distribution when 
three isolated lines of 0.25 .mu.m are formed by exposure while shifting 
isolated lines of 0.25 .mu.m in width three times. FIG. 67B also shows a 
case of n=2 which is a distribution when three lines of 0.25 .mu.m in 
width are subjected to full exposure. FIG. 67C shows an exposure intensity 
distribution of three lines of 0.25 .mu.m in width by the conventional 
method. 
It is apparent from the comparison between these FIG. 67A, FIG. 67B, and 
FIG. 67C that the effective exposure intensity distribution, FIG. 67A, 
obtained by the method of the present invention is effective to form a 
fine pattern far excellent as compared with the other methods. 
It is general that the resist pattern after development is formed 
approximately in proportion with the latent image reaction density. 
However, further emphasizing the contrast in the development process, a 
resist pattern can be formed with much higher contrast. 
Further described is an embodiment where the power n is smaller than 1 
(n&lt;1). Described is an example where a transmitting film with n=0.5 is 
used. Using the transmitting film with n=0.5, the effective exposure 
intensity is formed according to the power 0.5 of the light intensity. 
Namely, it is given by the following equation. 
(Eq 5-20) 
EQU H(x)=I(x).sup.0.5 
Here, x is a coordinate value. Described is a case where a line-and-space 
pattern is printed using the reticles with phase shifters shown in FIG. 
60A and 60B as described previously under coherent illumination. The 
period of the reticles is the resolution limit .lambda./2NA of projection 
optical system. FIG. 70 is a drawing to show an incident light intensity 
distribution formed on the transmitting film. The incident light intensity 
distribution I(x) has a sinusoidal distribution as shown. Namely, it is 
expressed as follows. 
(Eq 5-21) 
EQU I(x)=1+COS (2.pi..multidot.2NA.multidot.x/.lambda.) 
Further, FIG. 71A shows an effective exposure intensity distribution H(x) 
obtained from Equation (5-21). 
(Eq 5-22) 
EQU H(x)=(1+COS (2.pi..multidot.2NA.multidot.x/.lambda.).sup.0.5 
As shown in FIG. 71A, the effective exposure intensity distribution H(x) is 
more gently sloping near bright portions but becomes rapidly darker and 
very thinner in width in dark portions than the light intensity 
distribution I(x). However, it is clear that since the nonlinear exposure 
intensity distribution H(x) is formed in the same period as the light 
intensity distribution I(x) is, a pattern cannot be formed exceeding the 
limit resolution of projection optical system in this state. 
On the other hand, since a pattern can be formed in finer structure as 
shown in FIG. 71C on the image plane by superimposing an effective 
exposure intensity distribution H(x) (FIG. 71B) obtained with a pattern 
shifted by a half period on FIG. 71A, the pattern will have a periodic 
arrangement two times finer than the limit resolution. 
In contrast, in case of a transmitting film of n=1 being used, i.e., where 
a film with constant transmittance against exposure intensity is used, the 
effective exposure intensity distribution H(x) would be perfectly 
coincident with the light intensity distribution I(x) of FIG. 70 and in 
superimposing this on a half-shifted pattern, the effective exposure 
intensity distribution H(x) obtained would become flat as understood from 
the following equation, which is of no use (FIG. 72). 
(Eq 5-23) 
EQU H(x)=1+COS (2.pi..multidot.2NA.multidot.x/.lambda.)+1-COS 
(2.pi..multidot.2NA.multidot.x/.lambda.)=2 
As described, a finer latent image reaction density distribution .xi.(x) 
than the resolution limit of projection optical system can be formed even 
using transmitting films of n&lt;1. 
It is needless to mention that either a positive photoresist or a negative 
photoresist can be employed in the present invention. However especially, 
it is considered that the cases of n&lt;1 are advantageous for positive 
photoresists, and extremely fine left lines can be formed in the case of 
the example shown in FIG. 71C. 
FIG. 66 shows schematic structure of an exposure apparatus for performing a 
plurality of exposure processes different in light intensity distribution 
on the photosensitive material with the transmitting film as described 
above. An illumination beam from a light source 5011 is collected by an 
elliptic mirror 5012, is guided by a mirror 5013 into a collimator lens 
5014 to become a bundle of nearly parallel rays, and is then incident into 
a fly's eye integrator 5015. Beams emergent from the fly's eye integrator 
5015 are guided by a mirror 5016 into a main condenser 5017 to uniformly 
illuminate a reticle 5018a as a photo-mask. A predetermined pattern on the 
photo-mask 5018a is projected through a projection optical system 5019 
onto a wafer 5020 coated with a photosensitive material to effect exposure 
thereon. Here, after completion of exposure, the reticle 5018a is 
exchanged for another reticle 5018b having a different pattern by a 
reticle loader 5021, and second exposure is then conducted. 
Instead of the exchange of different patterns by the reticle loader 5021, 
the apparatus may be so arranged that after the first exposure with the 
reticle 5018a, the reticle 5018a is moved by a predetermined amount in the 
direction perpendicular to the optical axis Ax of projection optical 
system 5019 to then effect second exposure. This predetermined amount is 
(.lambda./4NA) when converted as a coordinate value on the wafer for 
example in the case using the pattern of FIG. 60A as described previously 
where the effective exposure intensity is proportional to the square of 
the light intensity. Where the effective exposure intensity is 
proportional to the cube of the light intensity, it is effective to set 
the predetermined value as (.lambda./6NA) when converted as a coordinate 
value on the wafer. 
It is of course for cases where a same reticle pattern is used for plural 
exposure processes that the wafer itself can be arranged to move for each 
of plural exposure processes, instead of moving the reticle. 
Also, before setting the wafer at a predetermined position for 
predetermined first exposure, a wafer substrate 5031 is coated with a 
photoresist 5032 as a photosensitive material and a transmitting film 5033 
having the nonlinear property, as shown in FIG. 73. First, the photoresist 
5032 as a photosensitive material is applied onto the wafer substrate 
5031. On this occasion, a rotation center portion of wafer is connected 
with the rotational shaft of a motor of an applicator, so that rotation of 
the motor of applicator 5022 uniformly spreads the photoresist 5032 over 
the wafer substrate 5031 into a thin film. Next, the transmitting film 
5033 having the nonlinear property is uniformly spread over the 
photoresist 5032 in a thin film by the same method as the previous step. 
Here, the vapor deposition can be used as the method for applying the 
transmitting film 5033 having the nonlinear property over the photoresist 
5032. 
Effective alignment between the plural exposure processes is the so-called 
latent image alignment in which alignment is performed observing the 
latent image. 
Incidentally, as in the embodiment shown in FIG. 60A and 60B, it is 
effective in the present invention to use the phase shift patterns in 
order to form a high-resolution pattern. It is also effective to employ 
the annular illumination proposed in Japanese Laid-open Patent Application 
No. 61-91662 or to employ the so-called SHRINC illumination proposed in 
Japanese Laid-open Patent Application No. 4-225358, for example. The 
exposure method of using the phase shift patterns or mask is described in 
Optics, vol.23, No,1, pp 29-37, 1994; and in SPIE vol.1780, Lens and 
Optical Systems Design, pp 117-131, 1992. The annular illumination is 
described in U.S. patent application Ser. No. 166,153. The SHRINC 
illumination is described in U.S. patent application Ser. No. 7,91138. 
At present, in the cases of n&lt;1, silver halides used for sunglasses are 
possibly available materials for the transmitting film with the intensity 
of transmitted light changing in accordance with the intensity of incident 
light, used in the present invention. Also, other possible materials are 
light-controlling materials such as spiro-oxazine, spiro-pyran, 
fluguimide, and chromel. The transmitting film having n less than 1 is 
described in Technique of Chemistry, vol. 3, PHOTOCHROMISM, edited by 
Glenn H. Brown, Wiley-Interscience, JOHN WILEY & SONS. INC. 
On the other hand, in the cases of n&gt;1, possibly available materials for 
the transmitting film with the intensity of transmitted light changing in 
accordance with the intensity of incident light, used in the present 
invention, are materials undergoing supersaturation absorption materials, 
such as fine particles of copper, cadmium sulfide (CdS), cadmium selenide 
(CdSe), phthalocyanine dyes, porphine dyes, etc., and materials undergoing 
the two-photon absorption, such as polysilicon, quinizarin, etc. Also, 
other possible materials are materials undergoing the three-photon 
absorption. The transmitting film having n more than 1 is described in The 
principles of Nonlinear Optics, edited by Y. R. SHEN, Wiley-Interscience 
Publication; and in Introduction to nonlinear laser spectroscopy, edited 
by M. D. Levenson, New York. Academic Press, 1988. 
As described above, the present invention enables a fine pattern surpassing 
the resolution limit of projection optical system to be formed by a 
plurality of exposure processes with different patterns using the 
transmitting film showing the nonlinear transmission property. In 
addition, a high-resolution pattern can be formed with little changing the 
conventional exposure wavelength and optical system. 
With the exposure method according to the present invention, semiconductor 
devices can be fabricated with very fine circuit pattern which was 
impossible with the conventional projection exposure apparatus, thus 
achieving the great effect of highly increasing the degree of integration 
of integrated circuits. 
From the invention thus described, it will be obvious that the invention 
may be varied in many ways. Such variations are not to be regarded as a 
departure from the spirit and scope of the invention, and all such 
modifications as would be obvious to one skilled in the art are intended 
to be included within the scope of the following claims. 
The basic Japanese Application Nos. 5-16267 filed on Feb. 3, 1993; 5-236031 
filed on Sep. 22, 1993; 6-36750 filed on Mar. 8, 1994; 6-38794 filed on 
Mar. 9, 1994; 6-118966 filed on May 31, 1994; 6-157389 filed on Jul. 8, 
1994; 6-177262 filed on Jul. 28, 1994; and 6-212440 filed on Sep. 6, 1994 
are hereby incorporated by reference.