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BACKGROUND OF THE INVENTION
The present invention relates to method and apparatus for measuring a
minute displacement of an object, using a diffraction grating.
Heretofore, in the manufacture of a semiconductor device there has been
used a stepper for performing a reduced projection exposure while moving
wafers successively step by step. The semiconductor device is manufactured
by putting circuit patterns formed on each wafer and mask one upon another
followed by exposure. Recently, such circuit patterns have become finer
and higher in density, so it is necessary that the wafer and the mask be
aligned in higher accuracy. To this end it is first required to measure
the position of the circuit pattern on the wafer more accurately. As one
method for this measurement there is known, for example, such a method
using heterodyne interference as is disclosed in Japanese Patent Laid-Open
Nos. 215905/86 and 274216/87. Now, an example of a minute displacement
measuring apparatus using such known method will be described below with
reference to FIG. 1.
FIG. 1 is a schematic construction diagram showing an example of
conventional method and apparatus for measuring a minute displacement. In
FIG. 1, the reference numeral 1 denotes a two-wavelength orthogonal
polarizing laser; the numeral 2 denotes a half mirror; numeral 3 denotes a
polarized beam splitter; numerals 4a, 4b and 4c denote mirrors; numeral 5
denotes a substrate; numeral 6 denotes a diffraction grating; numerals 8a
and 8b denote polarizing plates; numerals 9a and 9b denote photo
detectors; numerals 9a' and 9b' denote amplifiers; and numeral 10 denotes
a detected signal processing portion.
In the above construction, coherent light beams whose wavelengths are
slightly different from each other and whose polarization directions are
orthogonal to each other, are emitted from the two-wavelength orthogonal
polarizing laser 1. These light beams pass through the half mirror 2 and
are separated into light beams wavelengths .lambda..sub.1 and
.lambda..sub.2, respectively, by the polarized beam splitter 3. The
thus-split light beams are allowed to travel through optical paths 111 and
112 by the mirrors 4a, 4b and 4c and enter the diffraction grating 6 as
parallel beams. Primary diffracted light beams of wavelengths
.lambda..sub.1 and .lambda..sub.2, respectively, travel along optical
paths 121 and 122 which are perpendicular to the surface of the
diffraction grating 6. The optical paths 121 and 122 are substantially the
same, and a heterodyne interference signal I.sub.m is detected by the
photo detector 9a through the polarizing plate 8a. This heterodyne
interference signal is represented by the following equation:
I.sub.m =A.sub.m cos {(.omega..sub.I -.omega..sub.2)t+4.pi..epsilon./P}(1)
wherein A.sub.m represents the amplitude of the heterodyne interference
signal I.sub.m ; .omega..sub.I and .omega..sub.2 represent angular
frequencies of the wavelengths .lambda..sub.1 and .lambda..sub.2,
respectively; t is time; P represents the pitch of the diffraction grating
6; and .epsilon. represents the amount of movement of the diffraction
grating 6. Since the heterodyne interference signal I.sub.m contains
information on the amount of movement .epsilon. of the diffraction grating
6, it will hereinafter be referred to as the measurement signal. The
amount of movement of the diffraction grating 6 ca be determined by
subtracting the time term (.omega..sub.1 -.omega..sub.2) from the phase of
the measurement signal I.sub.m in the equation (1).
The said time term is measured separately as a reference signal. More
specifically, the light beam emitted from the two-wavelength orthogonal
polarizing laser 1 and reflected by the half mirror 2 is detected as a
heterodyne interference signal I.sub.s by the photo detector 9b through
the polarizing plate 8b. This heterodyne interference signal I.sub.s,
which is represented by the following equation, serves as a reference
signal I.sub.s :
I.sub.s =A.sub.s cos {(.OMEGA..sub.1 -.omega..sub.2)t} (2)
wherein A.sub.s represents the amplitude of the reference signal I.sub.s.
Therefore, once a phase difference between the heterodyne interference
signals I.sub.m and I.sub.s in the equations (1) and (2) detected by the
photo detectors 9a and 9b is determined by the detected signal processing
portion 10, the amount of movement .epsilon. of the diffraction grating is
obtained from the pitch P of the diffraction grating 6. According to this
method, since the phase of light is detected, the detection of position
can be done in high accuracy independently of the distribution of
illumination light and the resolution of an optical system in comparison
with a conventional method in which the light intensity distribution of an
alignment mark image is detected.
In the above prior art, since the phase of light is detected according to
heterodyne interference, the phase of a detected interference signal
involves an error due to the difference in density of the air present in
the optical paths. More specifically, in the optical system of FIG. 1, the
light beams of wavelengths .lambda..sub.1 and .lambda..sub.2 are spaced
farthest from each other when passing through the optical paths 111 and
112, respectively. In this case, if the density of air differs depending
on places, the refractive index also differs, so a phase difference occurs
between the light beams of wavelengths .lambda..sub.1 and .lambda..sub.2
which have passed through the optical paths 111 and 112, and the
heterodyne interference signal or measurement signal I.sub.m detected by
the photo detector 9a is as follows:
I.sub.m =A.sub.m cos {(.OMEGA..sub.1 -.omega..sub.2)t+4
.pi..epsilon./P+.gamma.} (3)
wherein .gamma. represents a phase difference based on the density
distribution of air and it is an error in obtaining the amount of movement
.epsilon. of the diffraction grating 6 on the basis of the phase
difference from the reference signal I.sub.s in the equation (2). The
stepper is disposed within a chamber where there is the flow of air for
keeping the temperature constant. If the flow of air goes away from an
object, a vortex of a different density will be formed, and if this vortex
crosses the optical paths 111 and 112, there will occur a measurement
error. This error has been a problem in measuring a circuit pattern
position on each wafer always in good reproducibility.
SUMMARY OF THE INVENTION
It is the object of the present invention to solve the above-mentioned
problem of the prior art and provide method and apparatus for measuring a
minute displacement capable of reducing the influence of the density
distribution of air.
To this end, according to the present invention there are provided a method
for measuring a minute displacement of an object whose position is to be
detected, comprising applying a light beam of a first wavelength at a
predetermined angle to a diffraction grating formed on the object,
subjecting each of the resulting diffracted light beam and regular
reflected light beam to heterodyne interference with a light beam of a
second wavelength different from the first wavelength to generate a
measurement signal and a reference signal, and then measuring a phase
difference between the measurement signal and the reference signal to
thereby measure a minute displacement of the object; as well as an
apparatus for measuring a minute displacement of an object, including a
diffraction grating fixed onto the object, a light source for generating a
light beam of a first wavelength and a light beam of a second wavelength
which are slightly different in frequency from each other, means for
applying the light beam of the first wavelength to the diffraction
grating, means for subjecting a diffracted light beam generated from the
diffraction grating to heterodyne interference with the light beam of the
second wavelength and producing a measurement signal, means for subjecting
a regular reflected light beam generated from the diffraction grating to
heterodyne interference with the light beam of the second wavelength and
producing a reference signal, a photo detector means for detecting time
variations of the measurement signal and the reference signal, and a
signal processing circuit for calculating a phase difference between the
measurement signal detected by the photo detector means and the reference
signal and converting it into a displacement of the object.
According to the minute displacement measuring method and apparatus of the
present invention, a coherent light beam of wavelength .lambda..sub.1 is
directed to a diffraction grating at an angle of about a half of a first
order diffraction angle, then a first order diffracted light beam
travelling back along the incident side optical path in the reverse
direction and a coherent light beam of wavelength .lambda..sub.2 are
combined to produce a measurement signal, while a regular reflected light
beam generated at the time of the light incidence and the coherent light
beam of wavelength .lambda..sub.2 are combined to produce a reference
signal, and a displacement of the diffraction grating is determined from a
phase difference between the measurement signal and the reference signal.
The measurement signal, I.sub.1, and the reference signal, I.sub.o, both
generated in the above minute displacement measuring method and apparatus
are as follows:
I.sub.1 =A.sub.1 cos {(.omega..sub.1
-.omega..sub.2))t+2.pi..epsilon./P+.gamma..sub.1 } (4)
I.sub.0 =A.sub.0 cos {(.omega..sub.1 -.omega..sub.2)t+.gamma..sub.0 }(5)
where .gamma..sub.1 and .gamma..sub.0 represent phase terms based on
refractive indices of the air present in the optical paths which generate
the measurement signal and the reference signal, respectively. In this
method, since the angle between the optical path of the measurement signal
and that of the reference signal is about a half of the angle between two
first order diffracted light beams in the conventional method, the optical
path of wavelength .lambda..sub.1 for generating the measurement signal is
in proximity to the optical path of wavelength .lambda..sub.1 for
generating the reference signal. The optical path of wavelength
.lambda..sub.2 for generating the measurement signal and the optical path
of wavelength .lambda..sub.2 for generating the reference signal are also
close to each other. Therefore, it is considered that the refractive
indices of air in both optical paths for the measurement signal and the
reference signal are almost the same, and hence .gamma..sub.1 can be
considered almost equal to .gamma..sub.0. Accordingly, .gamma..sub.1 and
.gamma..sub.0 can be cancelled by taking the difference between the phase
terms in the equations (4) and (5), and it becomes possible to make a
minute displacement measurement which is difficult to be influenced by the
density distribution of air.
According to the present invention, the optical paths of wavelengths
.lambda..sub.1 and .lambda..sub.2 for producing a heterodyne interference
signal, or the measurement signal, and the optical paths of wavelengths
.lambda..sub.1 and .lambda..sub.2 for producing the reference signal,
using a diffraction grating, are close to each other, and this is
effective in reducing the influence of the density distribution of air in
the measurement of a minute displacement.
Moreover, the intensity of a first order diffracted light in a first state
and that in a second state can be measured separately by making
change-over of an AO modulator, and on the basis of the resulting measured
values it is possible to correct a measured displacement in the case of an
asymmetric resist distribution. Thus, a correct displacement can always be
measured.
According to the present invention there are further provided a method for
measuring a minute displacement of an object whose position is to be
detected, comprising subjecting a light beam of a first wavelength and a
light beam of a second wavelength which are slightly different in
wavelength from each other, to heterodyne interference to generate a
reference signal, further subjecting a diffracted light beam created when
the light beam of the first wavelength is directed at a predetermined
angle to a diffraction grating formed on the object, and a second regular
reflected light beam created when the light beam of the second wavelength
is made incident in a direction opposite to a reflective direction of a
first regular reflected light beam created upon the incidence of the light
beam of the first wavelength, to heterodyne interference to generate a
measurement signal, and measuring a minute displacement of the object from
a phase difference between the reference signal and the measurement
signal; as well as an apparatus for measuring a minute displacement of an
object, including a diffraction grating fixed onto the object, a light
source for generating a light beam of a first wavelength and a light beam
of a second wavelengths which are slightly different in wavelength from
each other, means for subjecting the light beams of the first and second
wavelengths to heterodyne interference and producing a reference signal,
means for applying the light beams of the first and second wavelengths to
the diffraction grating each at a predetermined angle, means for
subjecting a diffracted light beam of the first wavelength generated from
the diffraction grating and a regular reflected light beam of the second
wavelength obtained at the same time, to heterodyne interference and
producing a measurement signal, a first photo detector means for measuring
a time variation of the reference signal, a second photo detector means
for measuring a time variation of the measurement signal, and a signal
processing circuit for calculating a phase difference between the
reference signal and the measurement signal detected by the first and
second photo detector means and then converting it into a displacement of
the object.
Thus, coherent light beams of wavelengths .lambda..sub.1 and .lambda..sub.2
which are not made incident on the diffraction grating are combined to
generate a reference signal, further a first order diffracted light beam
created when the coherent light beam of wavelength .lambda..sub.1 is made
incident at an angle of about a half of the first order diffraction angle
of the diffraction grating, and a regular reflected light beam created
when the coherent light beam of wavelength .lambda..sub.2 is made incident
in a direction opposite to a reflective direction of a regular reflected
light beam generated at the same time, are combined together to produce a
measurement signal, and an amount of movement of the diffraction grating
is determined from a phase difference between those signals
In the above construction, the reference signal, I'.sub.0, and the
measurement signal, I'.sub.1 are as follows:
I'.sub.0 =A.sub.0 cos {(.omega..sub.1 -.omega..sub.2)t+.gamma..sub.01
-.gamma..sub.02 } (6)
I'.sub.1 =A.sub.1 cos {(.omega..sub.1
-.omega..sub.2)t+2.pi..gamma./P+.gamma..sub.11 -.gamma..sub.12 }(7)
wherein .gamma..sub.01 and .gamma..sub.02 represent phase terms based on
refractive indices of air present in the optical paths along which there
travel light beams of wavelengths .lambda..sub.1 and .lambda..sub.2 for
generating the reference signal, while .gamma..sub.11 and .gamma..sub.12
represent phase terms based on refractive indices of air present in the
optical paths along which there travel light beams of wavelengths
.lambda..sub.1 and .lambda..sub.2 for generating the measurement signal.
In the present invention, since the angle between the optical paths of
wavelengths .lambda..sub.1 and .lambda..sub.2 for generating the
measurement signal is about a half of the angle between two first order
diffracted light beams in the conventional method, those optical paths are
close to each other and it can be regarded that .gamma..sub.11 is equal to
.gamma..sub.12. Since the optical paths of wavelengths .lambda..sub.1 and
.lambda..sub.2 for generating the reference signal are also in close
proximity to each other, .gamma..sub.01 can be regarded as being equal to
.gamma..sub.02. In the equations (6) and (7), therefore, approximately
.gamma..sub.01 -.gamma..sub.02 =0 and .gamma..sub.11 -.gamma..sub.12 =0,
whereby there can be realized method and apparatus for measuring a minute
displacement which are difficult to be influenced by the density
distribution of air.
Thus, according to the present invention, in the measurement of a minute
displacement using heterodyne interference, since the angle between the
optical paths of wavelengths .lambda..sub.1 and .lambda..sub.2 for
generating the measurement signal is about a half of the angle between two
first order diffracted light beams in the conventional method, those
optical paths are in proximity to each other, and the optical paths for
generating the reference signal are also very close to each other, so it
is possible to diminish the influence of the density distribution of air
in the minute displacement measurement. It is also possible to detect
first and second measurement signals simultaneously, and in this case the
reference signal can be omitted. Further, since it is possible to measure
the intensities of the first and second light beams of first order
diffraction separately, a measured displacement in the case of an
asymmetric resist distribution on the diffraction grating can be corrected
on the basis of such measured values, thus permitting constant measurement
of an exact displacement.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a schematic construction diagram showing an example of
conventional method and apparatus for measuring a minute displacement;
FIG. 2 is a schematic construction diagram showing method and apparatus for
measuring a minute displacement according to a first embodiment of the
present invention;
FIG. 3(a) is a detail view showing a displacement of a diffraction grating
illustrated in FIG. 2 and also showing optical paths;
FIG. 3(b) shows a displacement of the diffraction grating illustrated in
FIG. 2 and also shows optical paths;
FIG. 4 is a section view showing a symmetric resist distribution on the
diffraction grating illustrated in FIG. 2;
FIG. 5 is a sectional view showing an asymmetric resist distribution on the
diffraction grating illustrated in FIG. 2;
FIG. 6 is a calculation example diagram showing a relation between the
amount of resist shift and phases of measurement signals in first and
second states;
FIG. 7 is calculation example diagram showing a relation between the amount
of resist shift and intensities of first order diffracted light beams in
the first and second states;
FIG. 8 is a calculation example diagram showing a relation between the
difference in intensity of the first order diffracted light beams in the
first and second states in FIGS. 6 and 7 and the difference in phase of
the measurement signals in the two states, in the first embodiment of the
present invention;
FIG. 9 is a schematic construction diagram showing method and apparatus for
measuring a minute displacement according to a second embodiment of the
present invention;
FIG. 10 is a detail view showing a diffraction grating optical paths
illustrated in FIG. 9; and
FIG. 11 is a graph showing an intensity difference - phase difference
relation between the first- and second-state first order diffracted light
beams in FIGS. 6 and 7 in the second embodiment of the present invention.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
A first embodiment of the present invention will be described below with
reference to FIGS. 2 to 8.
FIG. 2 is a schematic construction diagram showing method and apparatus for
measuring a minute displacement according to a first embodiment of the
present invention. In FIG. 2, the numeral 11 denotes a linear polarization
laser having a vertical polarization; numerals 21, 22 and 23 denote half
mirrors; numerals 24 and 25 denote mirrors; numerals 31, 32 and 33 denote
AO (acousto-optical) modulators; numerals 41 and 42 denote polarized beam
splitters; numeral 43 denotes a 1/4 wavelength plate; numeral 51 denotes a
collimator lens; numeral 52 denotes a mirror; numeral 6 denotes a
reduction lens; numeral 7 denotes a diffraction grating; numeral 81
denotes a polarizing plate; numerals 91 and 92 denote photo detectors; and
numeral 100 denotes processing/control circuit.
In the above construction, light emitted from the linear polarization laser
11 is divided into three light beams by the half mirrors 21 and 22, of
which two light beams are incident on the AO modulators 31 and 32, which
in turn shift the frequency of the incident light beams by a drive
frequency f.sub.1 which is fed from the processing/control circuit 100.
When the AO modulator 31 is turned ON and the AO modulator 32 OFF, as the
first state, a light beam LBl of wavelength .lambda..sub.1 is emitted from
the AO modulator 31. The light beam LBl is applied as S polarized light to
the polarized beam splitter 41, so is mostly reflected, travels along an
optical path 130 and is incident on the 1/4 wavelength plate 43, whereby
the light beam LBl is made into a circularly polarized light. The
circularly polarized light travels along an optical path 141 through the
collimator lens 52 and the mirror 52, then travels along an optical path
151 through the reduction lens 6 and is incident on the diffraction
grating 7 on the wafer at a predetermined angle which is about a half of
the first order diffraction angle. A first order diffracted light from the
diffraction grating 7 travels back through the optical path 151, while a
regular reflected light travels along an optical path 152. Then, both
travels back along optical paths 141 and 142, respectively, through the
reduction lens 6, then further through the mirror 52 and the collimator
lens 51 back to the 1/4 wavelength plate 43. By the 1/4 wavelength plate
43 these light beams are converted into linearly polarized light beams
orthogonal to their incident direction, which beams are then incident on
the polarized beam splitter 41 as P polarized beams. After passing through
the splitter 41, the first order diffracted light beam and the regular
reflected light beam travel along the optical paths 131 and 132,
respectively, then are incident on the polarized beam splitter 42 and pass
through it.
On the other hand, out of the light beams emitted from the linear
polarization laser 11, one light beam which has been reflected by the half
mirror 21 and passed through the half mirror 22 enters the AO modulator
33. Since the AO modulator 33 is driven at a frequency f.sub.2, it emits a
light beam LB2 of wavelength .lambda..sub.2 which is slightly different
from the wavelength .lambda..sub.1. The light beam LB2 travels through the
mirror 25 and is split into two light beams by the half mirrors 23 and 24.
The two light beams are incident on the polarized beam splitter 42 as S
polarized light beams, reflected thereby, then travel along optical paths
161 and 162. The optical paths along which the light beams after the
bisplitting of the light beam LB2 of wavelength .lambda..sub.2 travels are
close to each other. If the polarizing plate 81 positioned ahead of the
optical paths 161 and 162 is inclined at 45.degree. with respect to the
horizontal direction, then in the photo detector 91 positioned ahead of
the polarizing direction, then in the photo detector 91 positioned ahead
of the polarizing plate 81, a heterodyne interference signal produced
through the polarized beam splitter 42 from both the first order
diffracted light of wavelength .lambda..sub.1 travelling along the optical
path 131 and the light of wavelength .lambda..sub.2 which has passed
through the half mirror 23 is detected as a measurement signal, while in
the photo detector 92, a heterodyne interference signal produced through
the polarized beam splitter 42 through both the regular reflected light of
wavelength .lambda..sub.1 travelling along the optical path 132 and the
light of wavelength .lambda..sub.2 which has passed through the mirror 24
is detected as a reference signal. The heterodyne interference signals
thus detected as a measurement signal and a reference signal,
respectively, by the photo detectors 91 and 92 are fed to the
processing/control circuit 100, which in turn calculates a phase
difference between the measurement signal and the reference signal, then
converts it into a displacement of the diffraction grating 7 and outputs a
minute displacement of the same grating.
FIGS. 3(a) and 3(b) are detail views each showing a displacement of the
diffraction grating 7 used in the apparatus of FIG. 2 and the optical
paths 151, 152. With reference to these figures, the relation between a
displacement e of the diffraction grating 7 illustrated in FIG. 2 and a
phase difference .DELTA..phi. between the measurement signal and the
reference signal detected by the photo detectors 91 and 92 will now be
explained. First, a phase variation of the measurement signal detected by
the photo detector 91 will be explained with reference to FIG. 3(a). When
a coherent light of wavelength .lambda..sub.1 travels along an optical
path 151a and is incident on the diffraction grating 7 at an angle of
.theta.', a first order diffracted light travels back through the same
optical path 151a, provided it is assumed that the angle .theta.'
satisfies the following equations:
sin .theta.'=.lambda..sub.1 /2P=1/2sin .theta. (8)
.theta.'.apprxeq.1/2.theta. (8')
wherein P represents a pitch of the diffraction grating 7 and .theta.
represents a first order diffraction angle. The position of the
diffraction grating 7 after displacement .epsilon. is as indicated by a
broken line. In this case, the difference in length between an optical
path of the incident light and the primary diffracted light and the
original optical path 151a is 2DC, which can be expressed as follows:
2DC=2 .epsilon.sin .theta.'=2.epsilon..theta..sub.1
/2P=.epsilon..theta..sub.1 /P (9)
A phase variation .phi..sub.1 of the measurement signal at the displacement
.epsilon. of the diffraction grating 7 is:
.phi..sub.1 =2 .pi..epsilon./P (10)
Next, a phase variation of the reference signal detected by the photo
detector 92 will be explained with reference to FIG. 3(b). When a coherent
light of wavelength .lambda..sub.1 travels along the optical path 151a and
is incident at an angle of .theta.' relative to the diffraction grating 7,
a regular reflected light travels along an optical path 152a at a
reflection angle .theta.'. Like the foregoing, the position of the
diffraction grating 7 after displacement e is as indicated by a broken
line. In this case, the incident optical path and the reflective optical
path are as indicated by the reference numerals 151b and 152b,
respectively, and the difference between the optical path length before
the displacement and that after the displacement is GE-FE'. Since the
incidence angle .theta.' and the reflection angle .theta.' are equal to
each other, GE-FE'=0. Consequently, a phase variation .phi..sub.0 of the
reference signal at the displacement .gamma. of the diffraction grating 7
is also equal to 0. From this result, the phase difference .DELTA..phi.
between the measurement signal and the reference signal is as follows:
.DELTA..phi.=.phi..sub.1 -.phi..sub.0 =2 .pi..gamma./P (11)
Since the phase measuring range is -.pi.<.DELTA..phi.<.pi., the
displacement measuring range is -P/2<.gamma.<P/2. From the equation (1)
the conventional measuring range is-P/4<.gamma.<P/4 and thus this
embodiment of the present invention is advantageous in that the measuring
range widens twice.
FIG. 4 is a detail view showing a symmetric resist distribution on the
diffraction grating used in the apparatus of FIG. 2. The following
description is now provided about the method of measuring the displacement
.gamma. in the case where a resist is present on the diffraction grating
as an alignment mark. In this case, a phase difference .DELTA..phi..sub.a
between the measurement signal and the reference signal detected by the
photo detectors 91 and 92 is as follows:
.DELTA..phi..sub.a =2 .pi..gamma./P+a-c (12)
wherein a and c represent phases of the measurement signal and the
reference signal based on multiple reflection by the resist 71. More
particularly, when the resist 71 is present on the diffraction grating 7,
an offset of a-c is added to the phase difference. How to remove this
offset will be described below.
In the foregoing description related to FIG. 2, the AO modulators 31 and 32
were turned ON and OFF, respectively, as the first state. This time,
conversely, the AO modulators 31 and 32 are turned OFF and ON,
respectively, as the second state. A light beam LB3 of wavelength
.lambda..sub.1 emitted from the AO modulator 32 travels along an optical
path which is symmetric with respect to the light beam LBl of wavelength
.lambda..sub.1 emitted from the AO modulator 31, relative to the optical
axis of the optical system illustrated in FIG. 2. More specifically, the
incident light is applied to the diffraction grating 7 through the optical
path 152, while a first order diffracted light travels back through the
optical path 152 and a regular reflected light travels along the optical
path 151. Therefore, reversely to the foregoing case, the photo detector
91 detects a heterodyne interference signal from both the regular
reflected light of wavelength .lambda..sub.1 as a reference signal and the
light of wavelength .lambda..sub.2, while the photo detector 92 detects a
heterodyne interference signal from both the first order diffracted light
of wavelength .lambda..sub.1 as a measurement signal and the light of
wavelength .lambda..sub.1. In this case, a phase difference
.DELTA..phi..sub.b between the measurement signal and the reference signal
is:
.DELTA..phi..sub.b =-2.pi..gamma./P+a-c (13)
Therefore, if the difference between the equations (12) and (13) is taken,
the result is:
.DELTA..phi..sub.a -.DELTA..phi..sub.b =4.pi..gamma./P (14)
Thus, the offset a-c can be cancelled. In this way, by changing over the AO
modulators 31 and 32 from the first to the second state with time, then
determining measurement signal - reference signal phase differences in
those states and taking the difference between those phase differences,
there can be realized a measurement of displacement .gamma. free of offset
error even in the presence of the symmetric resist 71.
FIG. 5 is a detail view showing an asymmetric resist distribution on the
diffraction grating 7 used in the apparatus of FIG. 2. The following
description is now provided about the case where the distribution of
resist 72 on the diffraction grating 7 is asymmetric. Since the resist 72
is applied rotatively, the distribution thereof is in many cases
asymmetric with respect to the diffraction grating 7. For example, a
central line of a concave portion of the resist 72 shifts by an amount
.delta. with respect to a central line of a concave portion of the
diffraction grating y. In this case, a phase difference .DELTA..phi..sub.e
between the measurement signal and the reference signal detected by the
photo detectors 91 and 92 in the first state of the AO modulators 31 and
32, and a phase difference .DELTA..phi..sub.f between both signals in the
second state of those AO modulators, are as follows:
.DELTA..phi..sub.e =2.pi..gamma./P+e-g (15)
.DELTA..phi..sub.f =-2.pi..gamma./P+f-g (16)
wherein e, f and g represent the phase of the measurement signal in the
first state, the phase of the measurement signal in the second state and
that of the reference signal, respectively. Like the above, if the
difference between the equations (15) and (16) is taken, the result is:
.DELTA..phi..sub.e -.DELTA..phi..sub.f =4.pi..gamma./P+e-f (17)
However, since e.noteq.f in the case where the distribution of the resist
72 is asymmetric as in FIG. 5, there remains an offset error of e-f.
The following description is now provided about the method of obtaining a
correct value of displacement .gamma. by measuring the offset error e-f
indirectly and substituting it into the equation (17), with reference to
FIGS. 6 to 8. It is here assumed that the asymmetry of the resist 72 can
be expressed in terms of the amount of shift .delta. of the central line
of the concave portion of the resist 72 with respect to the central line
of the concave portion of the diffraction grating 7, as in FIG. 5. The
values of the measurement signal phases e and f in the first and second
states relative to the amount of shift .delta. can be calculated by the
method described, for example, in J. Opt. Soc. Am. A, Vol. 5, No. 8
(1988), pp. 1270-1280.
FIG. 6 is a calculation example diagram showing a relation between the
amount of shift .delta. of the resist 72 in FIG. 5 and the measurement
signal phases e, f in the first and second states. An example of
calculation results based on the method described in the above literature
is shown therein, in which the amount of shift .delta. and the measurement
signal phases e, f are plotted along the axis of abscissa and the axis of
ordinate, respectively. The larger the amount of shift .delta. of the
resist 72, the larger becomes the difference between the measurement
signal phases e and f.
FIG. 7 is a calculation example diagram showing a relation between the
amount of shift .delta. of the resist 72 and the firs | | |
