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BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a method for measuring relative
displacements among a plurality of objects with a high degree of accuracy
by utilizing diffraction and interference phenomena of waves through
diffraction gratings formed on the objects.
2. Description of the Prior Art
As a prior art, a method for measuring a relative displacement by utilizing
diffraction gratings disclosed by Flanders et al. in Applied Phisics
Letters 31,426 (1977) will be described with reference to FIG. 1.
Diffraction gratings G1 and G2 having the same period d are formed on two
objects 1 and 2, respectively, and are spaced apart from each other in
parallel with each other. When the wave I is vertically incident to the
diffraction grating G1, it is diffracted by an angle .theta. (d sin
.theta.=n.lambda., where n is an integer representing the order of
diffraction) which is dependent upon the period d and the wavelength
.lambda. of the wave I. An intensity of the diffracted waves varies in
response to a relative position between the diffraction gratings G1 and G2
so that the relative displacement between the objects 1 and 2 or between
the diffraction gratings G1 and G2 can be measured by measuring the
intensity of the diffracted waves.
Especially, since intensities of the diffracted waves D+ in the +.theta.
direction and those D- in the -.theta. direction varies in the opposite
directions in response to the relative displacement in the direction (X
direction) perpendicular to the grating within the surface thereof, so
that in principle it becomes possible to measure the relative displacement
in the X direction by measuring the difference in intensity between two
diffracted waves I(D+)-I(D-).
However, the method described above with reference to FIG. 1 has defects so
that it cannot be satisfactorily applied in practice because of the
reasons described below. First, D+ and D- consist of many diffracted waves
as shown in FIG. 1. The wave which is diffracted at the i-th order by the
diffraction grating G1, diffracted at the j-th order by the diffraction
grating G2 and further diffracted at the k-th order by the diffraction
grating G1 is designated by D(i,j,k) while the wave which is reflected and
diffracted at the i-th order on the upper surface of the diffraction
grating G1 is designated by R(i). Then in the case of the first order
diffraction, D+ is a mixed wave of diffracted waves R(1) and D(i,j,k)
(where i+j+k=1) such as D(0,0,1), D(0,1,0), D(1,0,0), D(0,-1,2), D(-1,0,2)
and so on. For the sake of simplicity, FIG. 1 only shows R(1), D(-1,1,1),
D(0,0,1), D(1,-1,1) and D(1,0,0). The dependence of these diffracted waves
on the relative displacement in the X direction of the gratings and on the
distance S between the diffraction gratings G1 and G2 vary depending upon
their diffraction orders. Thus, the intensities of D+ and D- are a
complicated function of the relative displacement in the X direction and
the distance S so that the measurement of the relative displacement in the
X direction by the measurement of the difference between I(D+) and I(D-)
is limited within an extremely limited range of the distance S.
Furthermore, if there exist differences in characteristics between the
instruments for measuring the intensities of D+ and D-, the measured
relative displacement includes errors. It follows therefore that in order
to improve the accuracy of measurement, the characteristics of the
instruments for measuring the intensities of D+ and D- must be made to
strictly coincide with each other. As a result, the measurement with a
high degree of accuracy becomes difficult.
FIG. 2 shows a measurement method capable of measuring the relative
displacement without depending on the distance S, which is proposed by the
present inventor and is disclosed in the Japanese Patent Application No.
60-165231. According to this method, two diffraction gratings G1 and G1'
are formed on the object 1 and are spaced apart from each other by a
suitable distance so that only the diffracted waves in the specific orders
are incident on the diffraction grating G2 of the object 2, whereby the
strong dependence of the intensity of the diffracted wave D from the
diffraction grating G2 on the distance S between the diffraction gratings
G1 and G2 is eliminated. Therefore, the relative displacement in the X
direction can be measured by measuring the intensity of the diffracted
light D by a detector 3 without being limited by the range of the distance
S.
However, the above-described method also has the following defects so that
it can not be satisfactorily used in practice. First, the intensity I(D)
of the diffracted wave D depends on the displacement x in the x direction
in proportion to cos.sup.2 (2.pi.x/d) (where d is the period of G1). But,
the absolute value of I(D) is influenced by various factors so that it is
impossible to theoretically estimate. It follows therefore in order to
obtain x from the intensity I(D), the variation in I(D) must be measured
while X is varied in the range of about d/4 in practice. Another defect of
the above-described method is that since the intensity of the diffracted
wave is used as a signal representative of a displacement (a displacement
signal), the measurement result is easily adversely affected very often by
the variations of characteristics of the measurement instrument.
Furthermore, the intensity of the diffracted wave incident to the
diffraction grating G2 from the diffraction gratings G1 and G1' varies in
response to the variation in S so that the above-described measurement
method has a further defect that the variation in S during the measurement
process cannot be permitted, if it is desired to measure x with a high
degree of accuracy.
SUMMARY OF THE INVENTION
In view of the above, the primary object of the present invention is to
provide a method for substantially eliminating the above and other
defects, whereby the relative displacement can be measured with a high
degree of accuracy.
To the above and other ends, according to the present invention, at least
one of a plurality of objects is formed with a diffraction grating so that
the relative displacements among a plurality of objects can be measured by
measuring the phase of the diffracted wave obtained by the diffraction
through the diffraction grating in reference to another diffracted wave or
an incident wave.
The present invention is based upon the underlying principle to be
described below. The phase of the wave which is diffracted by the
diffraction grating formed on the first object and then reflected from the
second object or diffracted by the diffraction grating formed thereon, and
the phase of the wave which is reflected by the first object and then
diffracted by the diffraction grating formed on the second object vary
with the relative position between the first and second objects. It
follows therefore that the relative displacement between the two objects
can be obtained by measuring the phase of the diffracted wave in reference
to an incident wave or another diffracted wave having different dependence
of the phase on the relative displacement.
The prior art measuring methods measure the relative displacement between
two objects based on the fact that the intensity of the diffracted wave
varies in response to the relative displacement between the two objects.
However, the intensity of the diffracted wave is easily influenced not
only by the relative displacement to be measured but also by other various
factors such as the condition under which two diffraction gratings are
registered, the angle between the two diffraction gratings, the angle
between the detector and the diffraction grating, the intensity of
incident wave and so on. On the other hand, the phase of the diffracted
wave is a quantity which is fundamentally determined by the distance of
the passage of the wave so that it is hardly influenced by the
above-described factors influencing the intensity of the diffracted wave.
Therefore, according to the present invention, the relative displacement
can be measured in a stable manner essentially with a high degree of
accuracy without being affected by the external disturbances. Because of
the same reason the conditions for measurements can be relaxed and the
measurable range can be increased. Furthermore, when the frequency is less
than tens GHz, it becomes easy to measure the phase of the wave at an
accuracy higher than 1.degree. so that the phase measurement can be made
with a higher degree of accuracy than the intensity measurement.
However, when the frequency of the wave used for measurement becomes in
excess of tens GHz, it becomes difficult to measure the phase with a high
degree of accuracy. In order to solve this problem, the wave whose
frequency is slightly different from that of the wave used for measurement
and which can interfere with the latter measuring wave is provided, and
these two waves are interfered with each other so that resulting beats are
obtained a phase difference between the bear is measured. Such heterodyne
measurement in which the beat between two waves having different
wavelengthes is especially effective in the case of the relative
displacement measurement utilizing the diffraction gratings because when
the method for making two waves incident to the diffraction grating is
suitably devised, the beat signal can be obtained without requiring
additional component parts. When the measurement instrument is designed
and constructed in the manner described above, the phase of the beat
signal is hardly dependent upon the distance between the diffraction
grating and the detector. As a result, the measurement system can be
adjusted very easily and the stability against the external disturbances
can be improved.
When two different waves for the measurement, which have different
frequencies, are the electromagnetic waves polarized in different
directions respectively, the incidence method can be remarkably simplified
by utilizing the fact that diffraction efficiency attained by the
diffraction grating and the reflectivity at the object are different
depending upon the direction of polarization of the electromagnetic wave.
In this case, it is not necessary to separate the two electromagnetic
waves from each other but they may be focused as a single beam so that the
beat signal required for the measurement can be obtained. An intense beat
signal can be obtained when the polarization of the wave incident to the
detector is controlled by a suitable polarizer. In this method, two waves
used in the heterodyne measurement travel the completely same path so that
the phase of the beat signal is almost independent from the distance
between the wave source and the diffraction grating, and the distance
between the diffraction grating and the detector As a result, the
adjustment of the measuring system can be remarkably facilitated and the
stability against the external disturbances can be significantly improved,
whereby a high degree of reliability is ensured.
In the first aspect of the present invention, a relative-displacement
measurement method comprises the steps of:
forming a diffraction grating on at least one of a plurality of objects;
making a wave incident to the diffraction grating, thereby obtaining
diffracted waves;
measuring each of the diffracted waves;
a phase difference of the diffracted waves; and
measuring relative displacements among the plurality of objects in response
to the phase difference thus determined.
Here, diffraction gratings may be formed on each of the plurality of
objects.
At least one of a plurality of waves which can interfere with each other,
but are different in frequency from each other may be made incident to the
diffraction grating; and
phase difference in beats resulting from an interference among the
plurality of waves may be measured, thereby obtaining the relative
displacements among the plurality of objects based on the difference in
beat phases thus measured.
A plurality of waves which can interfere with each other, but are different
in frequency from each other may be made incident to the diffraction
gratings; and
phase difference in beats resulting from an interference among the
plurality of waves may be measured, thereby obtaining the relative
displacements among the plurality of objects based on the difference in
beat phases thus measured.
The plurality of waves may be electromagnetic waves which are different in
polarized state from each other.
The electromagnetic waves may be linearly polarized in different
directions, respectively.
Two of three diffraction gratings may be formed on one object, and one of
three diffraction gratings may be formed on the other object.
The three diffraction gratings may be arranged symmetrically.
A period of the diffraction grating formed on one object may be j/2i times
that of the diffraction grating formed on the other object, where i and j
are plus integers.
The period of the diffraction grating formed on one object may be 1.5 times
as that of the diffraction grating formed on the other object.
The electromagnetic waves may be transmitted through a polarizing beam
splitter in such a way that the transmitted electromagnetic waves are made
incident to the diffraction grating through a quarter-wave plate; a
reflected light beam is radiated through a quarter-wave plate and a
reflecting mirror to the polarizing beam splitter; and the diffracted
light beam from the diffraction grating and the reflected light beam
redirected to the polarizing beam splitter are combined and derived from
the polarizing beam splitter, thereby detecting the phase difference.
The obtained diffracted wave may be made to pass through a polarizer so as
to measure the phase difference in beats of the output light beam from the
polarizer.
An optical system may be interposed between at least two adjacent ones of
the plurality of objects.
The optical system may include a first lens, a second lens and a space
filter disposed at a focal plane of the first lens.
The plurality of objects may be disposed in parallel with each other.
A relative-displacement measurement method according to the present
invention may further comprise a step of measuring the phases difference
in the beats of the diffracted waves, thereby determining a verticality of
the wave incident in relation to the surface of the diffraction gratings.
In the second aspect of the present invention, a relative-displacement
measurement method comprises the steps of:
forming a diffraction grating on one of two objects which are overlapped in
parallel with each other;
causing the waves which are different in frequency and polarized state from
each other, incident to the diffraction grating, thereby obtaining firstly
diffracted waves;
making the firstly diffracted waves and/or waves passing through the
diffraction grating incident to the other object not formed with a
diffraction grating, thereby obtaining reflected light beams;
diffracting the reflected light beams by the diffraction grating, thereby
obtaining the secondary diffracted waves; and
measuring a phase of beat component of a combined wave of reflectively
diffracted light beam of the wave incident to the diffraction grating and
the firstly and secondary diffracted waves in reference to the beat formed
by interference between the incident waves, thereby obtaining a distance
between the two objects from the phase thus measured.
Here, the combined wave may be made to pass through a polarizer so as to
obtain the phase of the beat of the output light beam therefrom.
The above and other objects, effects, features and advantages of the
present invention will become more apparent from the following description
of preferred embodiments thereof taken in conjunction with the
accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
FIGS. 1 and 2 are schematic diagrams used to explain two conventional
methods, respectively;
FIGS. 3A and 3B are schematic diagrams of a first and a second embodiment
of the present invention, respectively; and
FIGS. 4-11 are schematic diagrams illustrating other embodiments of the
present invention, respectively.
DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS
(First Embodiment 1)
FIGS. 3A and 3B show first and second embodiments, respectively, in which
the method of the present invention is applied to the method shown in FIG.
2. In the embodiment shown in FIG. 3A, two objects 1 and 2 are overlapped
to each other. The object 2 is formed with a diffraction grating G2, to
which a light wave I is incident vertically. The object 1 is formed with
two diffraction gratings G1 and G1' at the positions on which are incident
the diffracted waves from the diffraction grating G2. The diffraction
gratings G1 and G1' have the same period d as the diffraction grating G2
and are maintained in parallel with the diffraction grating G2. The phase
difference .phi. between the wave D which is diffracted by the diffraction
grating G2 and then by the diffraction grating G1 to travel in the
direction opposite to the direction of the vertically incident wave
(light) I and the wave D' which is diffracted by the diffraction grating
G2 and then by the diffraction grating G1' to travel in the direction
opposite to the direction of the wave I is measured by detecting the waves
D and D' with detectors 4 and 5. The phase difference can be measured with
various conventional devices such as a phase meter, a lock-in amplifier,
an impedance meter or a counter in the time interval measurement mode. The
phase difference .phi. is completely independent of the distance S between
the objects 1 and 2, but is in proportion to the relative displacement x
between the objects 1 and 2 in the direction (X direction) which is in
parallel with the surfaces of the diffraction gratings G1, G1' and G2 and
is perpendicular to the direction thereof. Thus, .phi.=4.pi. nx/d
(radian), where n is the diffraction order. This relation is very simple
as compared with the fact that the intensities of D and D' are dependent
on the distance S and the relative displacement x.
Therefore, the value of the relative displacement x can be obtained by the
measurement of the phase difference .phi. independently of the distance S.
Furthermore, as the proportional coefficient n/d can be determined easily
with a high degree of accuracy, the relative displacement x can be
measured in the stationary state without causing the displacement of the
objects in the X direction. When the frequency of the wave is less than
tens GHz, the phase difference between the waves can be detected at a
resolution less than 1.degree. in a simple manner. Therefore, it is
possible that the relative displacement x can be measured at a resolution
less than d/720 so that as compared with the conventional methods for
measuring the intensity of the wave, the measurement accuracy can be
significantly improved.
In the second embodiment as shown in FIG. 3B, unlike the embodiment as
shown in FIG. 3A, the diffraction grating G2 is formed on the object 1
while the diffraction gratings G1 and G1' are formed on the object 2. The
wave I is made to fall on the diffraction grating G2 and the waves D and
D' from the diffraction gratings G1 and G1', respectively, are received by
the detectors 4 and 5, respectively, so that the measurement of the
relative displacement x can be accomplished in a manner substantially
similar to that described above.
(Second Embodiment 2)
When the wave used for the measurement exceeds tens GHz, the direct
measurement of the phase difference described above with reference to
FIGS. 3A and 3B becomes difficult. Therefore, as shown in FIG. 4, two wave
sources 6 and 7 are provided. The wave source 7 produces the wave I.sub.1
(of a frequency f.sub.1) which is made incident on the diffraction
grating, while the wave source 6 produces the wave I.sub.2 which has a
frequency f.sub.2 and can interfere with the wave I.sub.1. The waves
I.sub.2 are made incident on the detectors 4 and 5 simultaneous with the
waves D and D' so that the phase difference between the beat obtained by
the superposition of the wave D and the wave I.sub.2 and the beat obtained
by the superposition of the wave D' and the wave I.sub.2 is detected. In
this case, the phase difference is obtained from the relation of
.phi.=4.pi.n.multidot.x/d as in the case of the embodiments described
above with reference to FIGS. 3A and 3B. When the frequency f.sub.2 is
suitably selected, the frequency .vertline.f.sub.1 -f.sub.2 .vertline. of
the beat can be dropped to a suitable value so that the value of the
relative displacement x can be measured with a high degree of accuracy.
This method is especially effective in the case of the measurement of an
extremely small displacement with an accuracy of the order of nanometer by
utilizing the light having the frequencies in the visible range.
Various conventional methods may be used to produce the wave I.sub.2. In
order to produce the wave up to a frequency of the order of hundreds GHz,
an electric mixer may be used. Furthermore, in the case of light, the
light which can interfere with the wave I.sub.1 and has a different
frequency can be produced by using electrooptical elements, accoustoptical
elements, oscillating mirrors and quater-wave plates. A Zeeman laser, in
which a laser medium is subjected to a magnetic field simultaneously
produces two beam components which are different in polarization and
frequency from each other but can interfere with each other and is very
advantageously used in the method shown in FIG. 4.
(Embodiment 3)
FIG. 5 shows a simplified construction of the embodiment described above
with reference to FIG. 4, when light beams which can intefere with each
other, are different in frequency and are linearly polarized in mutually
perpendicular directions are obtained in a single luminous flux (as in the
case of a lateral Zeeman laser). In FIG. 5, the light beam I.sub.1 at a
frequency f.sub.1 is permitted to pass through a polarized beam splitter 8
and passes on the diffraction grating G2 through a quater-wave plate 9
while the light beam I.sub.2 at a frequency f.sub.2 is reflected by the
polarized beam splitter 8 and reaches through a quarter-wave plate 10 to a
flat mirror 11. The diffracted waves D and D' from the diffraction grating
G2 are reflected by the polarized beam splitter 8 and falls on the
detectors 4 and 5 together with the light I.sub.2 reflected back from the
flat mirror 11. The adjustment of the optical system of the third
embodiment shown in FIG. 5 is easier than the second embodiment shown in
FIG. 4.
(Embodiment 4)
In the case of the embodiment as shown in FIG. 6, the arrangement of the
diffraction gratings G1, G1' and G2 is substantially similar to that shown
in FIG. 2. The wave I.sub.1 at a frequency f.sub.1 is made incident on the
diffraction grating G2 while the wave I.sub.2 at a frequency f.sub.2 is
made incident on the diffraction gratings G1 and G1'. It is not needed
that the diffraction gratings G1, G1' and G2 have the same period and it
suffices that the light beam I.sub.1 diffracted by the diffraction grating
G2 (period d.sub.2) includes components whose directions coincide with the
directions of the diffracted light beams I.sub.2 diffracted by the
diffraction gratings G1 and G1' (periods d.sub.1 and d.sub.1 ',
respectively). That is, it suffices to satisfy the relations d.sub.1
/d.sub.2 =m/n and d'/d.sub.2 =l/n (where l, m and n are integers).
However, the lower order diffractions have a high diffraction efficiency
so that it is preferable to set d.sub.1 =d.sub.2 =d.sub.1 ' in practice.
When the diffracted light beams D and D' are diffracted wave in which the
diffracted light beam I.sub.1 from the diffraction grating G2 and the
light beams I.sub.2 from the diffraction gratings G1 and G1',
respectively, are superimposed one upon another, the intensities of the
diffracted light beams D and D' produce the beat at a frequency
.vertline.f.sub.1 -f.sub.2 .vertline.. Thus, in this embodiment, the
diffracted light beams D and D' are detected by the detectors 4 and 5,
respectively, so that the phase difference .phi. between the beats of the
diffracted light beams D and D' is detected. The phase difference .phi.
and the relative displacement x in the X direction has the following
relationship:
.phi.=4.pi.nx/d.sub.2
so that the relative displacement x can be measured with a high degree of
accuracy by measuring the phase difference .phi. regardless of the
distance S between the diffraction gratings G1 and G1' and the diffraction
grating G2.
The embodiment shown in FIG. 6 has a feature that when .vertline.f.sub.1
-f.sub.2 .vertline. is selected significantly lower than f.sub.1 or
f.sub.2, the phase difference .phi. becomes completely independent of the
variations in distance between the diffraction grating G1 and the detector
4 and between the diffraction grating G1' and the detector 5. When the
difference between the distance between the diffraction grating G1 and the
detector 4 and the distance between the diffraction grating G1' and the
detector 5 is designated by .DELTA.L, the phase difference caused by
.DELTA.L becomes 2.pi..DELTA.L .vertline.f.sub.1 -f.sub.2 .vertline./c
(where c is the velocity of the wave) so that it becomes easy to make the
value negligible when .vertline.f.sub.1 -f.sub.2 .vertline. is suitably
selected. Therefore, the adjustment of the measuring system in this method
can be significantly simplified as compared with the embodiments 1-3 and
furthermore, this method is almost immune to the external disturbances.
In FIG. 6, when the light beam I.sub.2 is made uniformly incident over the
whole surface including the diffraction grating G2, the relative
displacement x in the X direction can be measured in a manner
substantially similar to that described above. However, the phase
difference .phi. is in proportion to the relative displacement x only when
x is considerably smaller than d.sub.2. But, since the illumination or
projection system can be made simple in construction, the above-described
method is advantageous for the purpose of detecting x=0 as in the case of
the registration or alignment between the objects 1 and 2. In order to
construct such illumination or light projection system, when, for
instance, the linearly polarized light beams I.sub.1 and I.sub.2 are
different in polarized direction, an optically anisotropic material is
inserted into a light beam focusing system so that there is difference in
optical path length between the light beams I.sub.1 and I.sub.2, whereby
only the light beam I.sub.1 is focused on the diffraction grating G2.
As in the case of the above, when the light beams I.sub.1 and I.sub.2 are
the electromagnetic waves polarized in different directions, even when the
light beam I.sub.2 is made uniformly incident over the whole surface
including the diffraction grating G2, the same effect as that obtained
only when the diffraction gratings G1 and G1' are illuminated. To this
end, a quater-wave plate and a polarizer are disposed in front of each of
the detectors 4 and 5 so that the light beams I.sub.2 are eliminated and
does not reach the detectors 4 and 5. The polarized state of the
diffracted wave varies depending upon the polarized state of the incident
wave and the diffraction path so that only a specific diffracted component
can be eliminated as described above.
(Embodiment 5)
In the embodiment as shown in FIG. 7, the diffraction gratings G1 and G1'
having the same period d.sub.1 disposed on the object 1 and the wave
I.sub.1 at a frequency f.sub.1 is made incident on the diffraction grating
G1 while the wave I.sub.2 at a frequency f.sub.2 is made incident on the
diffraction grating G1'. The diffraction grating G2 is disposed in
parallel with the diffraction gratings G1 and G1' at a position where the
diffracted light beams I.sub.1 and I.sub.2 in the same order from the
diffraction gratings G1 and G1', respectively, strike. The period d.sub.2
of the diffraction grating G2 is so selected as to be d.sub.2 =d.sub.1 or
d.sub.2 =1.5 d.sub.1, whereby the wave which is diffracted first by the
diffraction grating G1 and then by the diffraction grating G2 and the wave
which is diffracted first by the diffraction grating G1' and then by the
diffraction grating G2 travel in the same direction. To this end, it is
preferable to select d.sub.2 /d.sub. 1 =j/i, where i and j are plus
integers. In the case of d.sub.2 =1.5 d.sub.1, the wave which is
diffracted by the diffraction grating G1 in the first order and then by
the diffraction grating G2 in the minus first order and the wave which is
diffracted in the minus first order by the diffraction grating G1' and
then by the diffraction grating G2 in the second order become the waves
which travel in the same direction. However, when d.sub.2 is an integer
multiple of d.sub.1, the diffracted wave from the diffraction grating G1
is diffracted by the diffraction grating G2 and then returned again to the
diffraction grating G1 so that the mutual dependence of the relative
displacement obtained by the measurement becomes complicated. Therefore,
it is preferable that d.sub.2 be a noninteger multiple of d.sub.1.
In this embodiment, as shown in FIG. 7, at least three diffracted waves
D.sub.1, D.sub.2 and D.sub.3 produce a beat at a frequency
.vertline.f.sub.1 -f.sub.2 .vertline. and the phase difference between the
beats is given by
.phi.(D.sub.2)-.phi.(D.sub.1)=1/2{.phi.(D.sub.3)-.phi.(D.sub.1)}+(4.pi.nx/d
.sub.1)
where .phi.(D.sub.i) is the phase of the beat of Di; n is the diffraction
order at G1 and G1'; and x is the relative displacement in the X
direction. Therefore, the value of the relative displacement x can be
obtained by the measurement of .phi.(D.sub.2)-.phi.(D.sub.1) by the
detectors 3 to 5. In this case, the detectors 3, 4 and 5 detect the
diffracted waves D.sub.1, D.sub.2 and D.sub.3, respectively.
The embodiment 5 has a feature that the construction of the illumination or
projection system is simpler than that of the embodiment 4. It should be
especially noted that when the light beams I.sub.1 and I.sub.2 are light
beams linearly polarized in the different directions, respectively, the
double luminous flux required in the embodiment 5 shown in FIG. 7 can be
obtained only by passing them through a suitable double refraction plate.
(Embodiment 6)
When the two waves I.sub.1 and I.sub.2 (at a frequency f.sub.1 and at a
frequency f.sub.2, respectively) which are used in the measurement are the
electromagnetic waves polarized in the different directions, respectively,
the illumination or light projection measurement system needed for the
measurement of the relative displacement can be significantly made simple
in construction by utilizing the fact that the diffraction efficiency of a
diffraction grating varies depending upon the polarized state of light.
One example of such system is shown in FIG. 8. The construction and
arrangement of the diffraction gratings G1, G1' and G2 are substantially
similar to those shown in FIG. 7. The period d.sub.2 of the diffraction
grating G2 is not an integer multiple of d.sub.1 and, for instance, is 1.5
d.sub.1. When the electromagnetic waves I.sub.1 and I.sub.2 which are
different in polarization and frequency from each other are combined to
form a combined wave I which in turn is made to strike the measurement
system, the combined wave Da obtained by the combination of the wave which
is diffracted in the minus first order by the diffraction grating G1 and
then in the second order by the diffraction grating G2 with the wave which
is diffracted in the first order by the diffraction grating G1' and then
in the minus first order by the diffraction grating G2, and the combined
wave Db diffracted in the orders of opposite sign in relation to the
combined wave Da are obtained. Da and Db thus obtained are made to strike
the detectors 4 and 5, respectively, through polarizers 21 and 22 which
polarize the light beams in suitable directions. Next, the phase
difference between the beats in Da and Db is measured so that the relative
displacement in the x direction between the objects 1 and 2 is measured.
No diffraction grating is formed between the diffraction gratings G1 and
G1' so that the major components of the diffracted wave I from the
diffraction grating G2 will not be made to strike the diffraction gratings
G1 and G1'. The embodiment as shown in FIG. 8 may be so modified that the
diffraction gratings G1 and G1' are formed on the object 2 while the
diffraction grating G2 is formed on the object 1 so that the light beam I
is made incident first on the diffraction grating G2.
The reason why the value x can be obtained by the above-described method
will be described. When the complex amplitude diffraction efficiency of
the light beam I.sub.1 at G1'.fwdarw.Da and the diffraction efficiency at
G1'.fwdarw.Da of the light beam I.sub.1 which are measured in the
direction determined by the polarizer 21 are designated by .gamma..sub.1
and .gamma..sub.1 .alpha., respectively, while those of the light beam
I.sub.2 measured by a manner similar to that described above are
designated by .gamma..sub.2 and .gamma..sub.2 .beta., the amplitude A(Da)
of Da is given by
A(Da)=.gamma..sub.1 (e.sup.-i.delta. +.alpha.e.sup.i.delta.)A.sub.1
+.gamma..sub.2 (e.sup.-i.delta. .beta.e.sup.i.delta.)A.sub.2
where A.sub.1 and A.sub.2 are the amplitudes, respectively, of I.sub.1 and
I.sub.2 on the diffraction gratings G1 and G1' and .delta.=2.pi.x/d.sub.1.
In general, the diffraction efficiency varies depending upon the polarized
state so that except the case of a high degree of symmetry such as both
I.sub.1, and I.sub.2 are circularly polarized light, .alpha..noteq..beta..
The amplitude of Db is equal to a value obtained when
.delta..fwdarw.-.delta. in A(Da). It follows therefore that the phases of
the beats of Da and Db vary depending upon the value of .delta. and the
phase difference is given by
##EQU1##
and the value x can be determined by this measurement.
In practice, it is highly practicable when the directions of polarization
of I.sub.1 and I.sub.2 are mutually perpendicular. In this case, for
instance, the direction of polarization of I.sub.1 is aligned with the
direction of the diffraction grating while I.sub.2 is polarized in the
direction perpendicular to the direction of polarization of I.sub.1. Then,
the phase difference .phi.(Db)-.phi.(Da) becomes independent of the
directions of the polarizers 21 and 22 disposed in front of the detectors
4 and 5. It is also easy to select the direction in which the highest
degree of measurement accuracy is obtained by rotating the directions of
polarization of I.sub.1 and I.sub.2.
The value x can be obtained with a high degree of accuracy independently of
the distance S between the objects 1 and 2 by the method of FIG. 8.
According to this method, the illumination or light projection system is
simple in construction so that its adjustment may be much facilitated. As
a result, the measurement is not adversely affected by external
disturbances and a high stability can be ensured. Furthermore, this method
has a feature that the adjustments required for measurement can be
accomplished by the measurements of the phase difference between the beats
of the diffracted waves D and D'. The phases difference between the beats
of D and D' are independent of the value x, but vary in response to the
parallelism between the diffraction gratings G1 and G1' on the one hand
and the diffraction grating G2 and the verticality of I in relation to the
surface of the diffraction grating. It follows therefore when the
parallelism between the diffraction gratings G1 and G1' on the one hand
and the diffraction grating G2 on the other hand and the inclination of I
are adjusted, so that the phase difference is minimized the adjustments
required for the measurement of x can be accomplished. These adjustments
are highly effective in the impr | | |
