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RELATED APPLICATIONS
This application claims the priorities of Japanese Patent Application No.
2001-23200 filed on Jan. 31, 2001 and Japanese Patent Application No.
2001-399179 filed on Dec. 28, 2001, which are incorporated herein by
reference.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a phase shift fringe analysis method using
Fourier transform when analyzing a fringe image by using a phase shift
method, and an apparatus using the same. In particular, the present
invention relates to a phase shift fringe analysis method comprising the
steps of obtaining image information of interference fringes and the like
while shifting a phase by using a phase shift device such as PZT
(piezoelectric device), and analyzing thus obtained plurality of image
data items having a fringe pattern of interference fringes and the like,
thereby attaining highly accurate phase information of an object to be
observed; and an apparatus using the same.
2. Description of the Prior Art
While light-wave interference method, for example, has conventionally been
known as important means concerning precise measurement of object
surfaces, there have recently been urgent needs for developing an
interferometry technique (sub-fringe interferometry) for reading out
information from a fraction of a single interference fringe (one fringe)
or less from the necessity to measure a surface or wavefront aberration of
1/10 wavelength or higher.
An example of typical techniques widely in practice as such sub-fringe
interferometry is a phase shift fringe analysis method (also known as
fringe scanning method or phase scanning method) disclosed in
"PHASE-MEASUREMENT INTERFEROMETRY TECHNIQUES," PROGRESS IN OPTICS, VOL.
XXVI (1988), pp. 349-393.
In the phase shift method, a phase shift device such as a PZT
(piezoelectric) device is used for phase-shifting the relative
relationship between an object to be observed and a reference,
interference fringe data is captured each time a predetermined step amount
shifts, so as to measure the interference fringe intensity of each point
of the object surface, and the phase of each point of the surface is
determined by using the results of measurement.
In the case of a 4-step phase shift method, for example, respective
interference fringe intensities I.sub.1, I.sub.2, I.sub.3, and I.sub.4 at
individual phase shift steps are expressed as follows:
I.sub.1 (x, y)=I.sub.0 (x, y){1+.gamma.(x, y)cos[.phi.(x, y)]}
I.sub.2 (x, y)=I.sub.0 (x, y){1+.gamma.(x, y)cos[.phi.(x, y)+.pi./2]} (2)
I.sub.3 (x, y)=I.sub.0 (x, y){1+.gamma.(x, y)cos[.phi.(x, y)+.pi.]}
I.sub.4 (x, y)=I.sub.0 (x, y){1+.gamma.(x, y)cos[.phi.(x, y)+3.pi./2]}
where
x and y are coordinates;
.phi. (x, y) is a phase;
I.sub.0 (x, y) is the average optical intensity at each point; and
.gamma. (x, y) is the modulation of interference fringes.
From these expressions, the phase .phi. (x, y) can be determined as
##EQU1##
Though the phase shift method enables measurement with a very high accuracy
if a predetermined step amount can shift accurately, it has been
problematic in terms of measurement errors occurring due to errors in step
amount and in that it is likely to be affected by disturbances during
measurement since it necessitates a plurality of interference fringe image
data items.
As a sub-fringe interferometry technique other than the phase shift method,
attention has been focused on one using a Fourier transform method, for
example, as disclosed in "Basics of Sub-fringe Interferometry," Kogaku,
Vol. 13, No. 1 (February 1984), pp. 55 to 65.
The Fourier transform fringe analysis method is a technique in which a
carrier frequency (caused by a relative tilt between an object surface to
be observed and a reference surface) is introduced, whereby the phase of
the object can be determined with a high accuracy from a single fringe
image. When a carrier frequency is introduced without the initial phase of
the object being taken into consideration, the interference fringe
intensity i(x, y) is represented by the following expression (4):
i(x, y)=a(x, y)+b(x, y)cos[2.pi..function..sub.x x+2.pi..function..sub.y
y+.phi.(x, y)] (4)
where
a(x, y) is the background of interference fringes;
b(x, y) is the visibility of fringes;
.phi. (x, y) is the phase of the object; and
f.sub.x and f.sub.y are respective carrier frequencies in x and y
directions expressed by:
##EQU2##
where .lambda. is the wavelength of light, and .theta..sub.x and
.theta..sub.y are respective inclinations of the object surface in x and y
directions.
The above-mentioned expression (4) can be deformed as the following
expression (5):
i(x, y)=a(x, y)+c(x, y)exp[i(2.pi..function..sub.x
+2.pi..function..sub.y)]+c.sup.* (x, y)exp[-i(2.pi..function..sub.x
+2.pi..function..sub.y)] (5)
where c(x, y) is the complex amplitude of interference fringes, and c.sup.*
(x, y) is the complex conjugate of c(x, y).
Here, c(x, y) is represented as the following expression (6):
##EQU3##
When the above-mentioned expression (5) is Fourier-transformed, the
following expression (7) is obtained:
I(.eta.,.zeta.)=A(.eta.,.zeta.)+C(.eta.-.function..sub.x,.zeta.-.function..
sub.y)+C.sup.* (.eta.+.function..sub.x,.zeta.+.function..sub.y) (7)
where
A(.eta., .zeta.) is the Fourier transform of a(x, y);
C(.eta.-f.sub.x, .zeta.-f.sub.y) is the Fourier transform of
c(x,y)exp[i(2.pi..function..sub.x +2.pi..function..sub.y)]; and
C.sup.* (.eta.+f.sub.x,.zeta.+f.sub.y) is the Fourier transform of c.sup.*
(x,y)exp[-i(2.pi..function..sub.x +2.pi..function..sub.y)].
Subsequently, C(.eta.-f.sub.x, .zeta.-f.sub.y) is taken out by filtering,
and the peak of a spectrum positioned at coordinates (f.sub.x, f.sub.y) is
moved to the origin of a frequency coordinate system (also known as a
Fourier spectrum coordinate system; see FIG. 8), so as to eliminate the
carrier frequency. Then, inverse Fourier transform is carried out, so as
to determine c(x, y), and the wrapped phase .phi. (x, y) is determined by
the following expression (8):
##EQU4##
where Im[c(x, y)] is the imaginary part of c(x, y), and Re[c(x, y)] is the
real part of c(x, y).
Finally, unwrapping is carried out, so as to determine the phase .PHI.(x,
y) of the object to be measured.
In the Fourier transform analysis method explained in the foregoing, the
fringe image data modulated by the carrier frequency is subjected to
Fourier transform as stated above.
In general, as mentioned above, the phase shift method captures the
brightness of an image while imparting a phase difference between object
light of an interferometer and reference light by a phase angle which is
an integral fraction 2.pi. and analyzes thus captured brightness, thereby
theoretically enabling a highly accurate phase analysis.
For securing highly accurate phase analysis, however, it is necessary that
the relative relationship between a sample and a reference be displaced at
a high accuracy by a predetermined phase amount (very short distance).
When the phase shift method is carried out by physically moving a
reference surface or the like by using a PZT (piezoelectric device), for
example, it is necessary that the amount of displacement of the PZT
(piezoelectric) device be controlled highly accurately. However, the
displacement error of the phase shift device or the tilt error of the
object surface is hard to eliminate completely, whereby controlling the
amount of phase shift or tilt with a high accuracy is a difficult task in
practice.
From such a viewpoint, the assignee has obtained an excellent result with a
technique for detecting the above-mentioned error resulting from the phase
shift device and correcting the fringe image analysis according to the
detection value upon carrying out the analysis. In this technique, the
fringe image data obtained by use of the phase shift method is subjected
to Fourier transform, the carrier frequency and complex amplitude caused
by fluctuations in wavefront occurring between an object to be observed
and a reference, and the amounts of displacement and tilt of phase shift
are detected according to the carrier frequency and complex amplitude, so
as to correct results determined by the phase shift method, whereby
influences caused by errors in the amount of tilt/displacement of the
phase shift amount are eliminated.
The above-mentioned technique proposed by the assignee is a quite effective
technique in that it can easily alleviate influences of errors caused by
the phase shift device without using an expensive phase shift device.
However, since calculations for correcting the amount of error from a
predetermined step amount (e.g., 90 degrees in the case of 4-bucket (step)
method), the time required for fringe analysis becomes longer. Also, even
in the case where 3-bucket method is theoretically sufficient when no
shift error exists in the phase shift device, it is necessary to use
4-bucket method or 5-bucket method from the viewpoint of accuracy, which
may also hinder the time required for fringe analysis from becoming
shorter.
In view of the foregoing circumstances, it is an object of the present
invention to provide a phase shift fringe image analysis method which can
eliminate influences caused by errors in the amount of displacement of
phase shift and/or relative tilt amounts between the object and the
reference without making the apparatus configuration complicated and
expensive when analyzing fringe image data obtained by using a phase shift
method, whereby the fringe analysis can be carried out rapidly and
favorably; and an apparatus using the same.
The present invention provides a phase shift fringe image analysis method
comprising the steps of shifting an object to be observed and a reference
relative to each other by using a phase shift device, obtaining fringe
image data at a plurality of phase shift positions, and determining a
phase of the object by analyzing thus obtained plurality of fringe image
data items;
wherein the plurality of phase shift positions are at least three phase
positions having a given phase gap therebetween; and
wherein positional data of the above-mentioned at least three phase
positions are specified, and the whole or part of the fringe image data on
which carrier fringes at these phase positions are superposed is subjected
to a predetermined arithmetic operation so as to carry out a phase
analysis and determine the phase of the object.
The positional data of the above-mentioned at least three phase positions
may be determined by a Fourier transform fringe analysis method.
The predetermined arithmetic operation may be carried out in view of data
concerning relative tilt between the object and the reference at the
above-mentioned at least three phase positions.
The data concerning relative tilt between the object and the reference may
be determined from a difference in frequency of the carrier fringes.
The data concerning relative tilt between the object and the reference may
be determined from a difference in phase of the object.
The number of phase shift positions for determining the fringe image data
may be 3, and the phase of the object may be represented by the following
conditional expression (1):
##EQU5##
where
a(x, y) is the background of interference fringes;
b(x, y) is the visibility of fringes;
.phi. (x, y) is the phase of the object; and
.delta..sub.m is the phase shift amount of the phase shift device expressed
by:
.delta..sub.m =2.pi..function..sub.xm x+2.pi..function..sub.ym y+.xi..sub.m
where
.xi..sub.m is the phase of the phase shift device (not including the part
involved with the tilt of the phase shift device); and
f.sub.xm and f.sub.ym are the carrier frequencies (including the part of
the error in inclination of the phase shift device) after the m-th phase
shift expressed by:
##EQU6##
where
.lambda. is the wavelength of light;
.theta..sub.xm and .theta..sub.ym are respective inclinations of the object
surface upon the m-th phase shift in x and y directions; and
z.sub.m is the amount of displacement of the phase shift device at the m-th
shift position (not including the part involved with the tilt of the phase
shift device).
The phase shift fringe analysis method in accordance with the present
invention may comprise the steps of determining a complex amplitude of a
fringe image by the Fourier transform fringe analysis method, and
obtaining the above-mentioned at least three phase positions according to
thus determined complex amplitude.
The phase shift fringe analysis method in accordance with the present
invention may comprise the steps of selecting a plurality of sets of at
least three local fringe image data items corresponding to each other from
fringe image data at the above-mentioned at least three phase shift
positions, obtaining positional data of the above-mentioned at least three
phase positions concerning each set according to the fringe image data of
respective set, and averaging positional data of phase positions by a
number corresponding to the number of the sets, so as to determine final
positional data of the above-mentioned at least three phase positions.
The fringe image may be an interference fringe image.
The present invention provides a phase shift fringe analysis apparatus for
shifting an object to be observed and a reference relative to each other
by using a phase shift device, obtaining fringe image data at a plurality
of phase shift positions, and analyzing thus obtained plurality of fringe
image data items so as to determine a phase of the object;
wherein the plurality of phase shift positions are at least three phase
positions having a given phase gap therebetween; and
wherein the apparatus comprises data acquiring means for obtaining
positional data of the at least three phase positions, and phase analysis
means for carrying out a phase analysis by subjecting the whole or part of
the fringe image data on which carrier fringes at these phase positions
are superposed to a predetermined arithmetic operation.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1 is a flowchart for explaining the method in accordance with an
embodiment of the present invention;
FIG. 2 is a block diagram of an apparatus for carrying out the method shown
in FIG. 1;
FIG. 3 is a graph for explaining characteristic features of the method in
accordance with the present invention;
FIG. 4 is a graph corresponding to FIG. 3 in a conventional technique;
FIGS. 5A and 5B are views specifically showing a part of FIG. 2;
FIG. 6 is a view showing a fringe image employed for verifying the method
shown in FIG. 1;
FIG. 7 is a view showing the results having verified the method shown in
FIG. 1; and
FIG. 8 is a view showing a frequency coordinate system employed in the
method shown in FIG. 1.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
In the following, the phase shift fringe analysis method in accordance with
an embodiment of the present invention will be explained with reference to
the drawings.
This method is one comprising the steps of shifting an object to be
observed and a reference relative to each other by using a phase shift
device, obtaining fringe image data at a plurality of phase shift
positions, and determining a phase of the object by analyzing thus
obtained plurality of fringe image data items. The plurality of phase
shift positions are at least three phase positions having a given phase
gap therebetween, positional data of the above-mentioned at least three
phase positions are specified, and the whole or part of the fringe image
data on which carrier fringes at these phase positions are superposed is
subjected to a predetermined arithmetic operation so as to carry out a
phase analysis and determine the phase of the object. This embodiment
relates to a case where it is applied to an interference fringe analysis
which is a typical example of fringe image analysis.
Though the phase position is not required to be a specific value in this
phase shift fringe analysis method, it is necessary that, after the fringe
image data is obtained, positional data of each phase position where the
fringe image data was obtained be specified.
Namely, in the conventional phase shift fringe analysis method, respective
fringe intensity values are plotted at predetermined phase shift positions
(0.degree., 90.degree., 180.degree., 270.degree., and 360.degree. here)
arranged at equally-spaced intervals (at intervals of a .alpha.=90.degree.
here) as shown in FIG. 4, and a sine wave function f(x) passing near these
points (a, b, c, d, e) is determined. According to such a conventional
technique, the phase shift positions are required to be set at
predetermined positions with a high accuracy, whereby processing for
correcting them to values at these predetermined positions and the like
are necessary in practice.
In the phase shift fringe analysis method in accordance with this
embodiment, by contrast, respective fringe intensity values of at least
three phase shift positions having a given phase gap
(.alpha..sub.1,.alpha..sub.2) are plotted as shown in FIG. 3, and a sine
wave function f(x) passing near points (p, q, r) is obtained by using a
computing equation which will be explained later. Here, assuming that
.function.(x)=A cos(x+.phi.)+B,
the number of unknowns is 3, i.e., A, .phi., and B. Therefore, the function
f(x) can be specified when positional data in at least three phase shift
positions and their corresponding fringe intensities are known.
An outline of this embodiment will now be explained with reference to the
flowchart of FIG. 1, before explaining it separately in terms of
specifying positional data at individual phase positions and calculating
the phase of the object to be observed by using thus specified positional
data.
In this embodiment, as shown in FIG. 1, interference fringe images carrying
form information of an object to be observed on which spatial carrier
fringes are superposed by imparting a predetermined relative inclination
between a reference surface and an object surface is captured by a CCD
image pickup camera at a plurality of phase shift positions while driving
a phase shift device (S1). Subsequently, each of thus obtained
interference fringe image data is subjected to Fourier transform (S2), and
a spatial carrier frequency (f.sub.x, f.sub.y) is found, whereby
C(.eta.-f.sub.x, .zeta.-f.sub.y) is determined (S3). The spatial carrier
frequency (f.sub.x, f.sub.y) at that time corresponds to the total tilt
formed by the predetermined inclination imparted beforehand between the
reference surface and the object surface and the error in inclination
generated upon driving the phase shift device.
Subsequently, with respect to thus determined C(.eta.-f.sub.x,
.zeta.-f.sub.y), a spectrum peak located at coordinates (f.sub.x, f.sub.y)
is moved to the origin of the frequency coordinate system (see FIG. 8),
and is subjected to inverse Fourier transform (S4) after eliminating the
carrier frequency (including the part of the error in inclination). Then,
a complex amplitude c(x, y) is obtained, and the relative displacement
amount between the reference surface and the object surface is determined
(S5). This relative displacement amount does not include the part
generated by the error in inclination of the phase shift device. Namely,
the relative displacement amount represents the pure displacement amount
of the phase shift device toward the optical axis.
Subsequently, without carrying out processing for moving the spectrum peak
located at coordinates (f.sub.x, f.sub.y) to the origin of the frequency
coordinate system, the above-mentioned C(.eta.-f.sub.x, .zeta.-f.sub.y) is
subjected to inverse Fourier transform in a state where the carrier
frequency (including the part of the error in inclination of the phase
shift device) is not eliminated, so as to determine the relative tilt
amount between the reference surface and the object surface (S6). Thus
determined relative tilt amount includes a predetermined inclination
imparted to the phase shift device in order to introduce the spatial
carrier frequency, and the part of the error in inclination of the phase
shift device generated upon driving the phase shift device.
From thus determined relative displacement amount and relative tilt amount
between the reference surface and the object surface, each phase shift
position (phase shift amount) of the phase shift device including the
phase displacement amount concerning the part of the error in inclination
of the phase shift device can be determined (S7). Then, thus determined
plurality of (at least three) phase shift positions (phase shift amounts)
are inputted into a predetermined computing equation which will be
explained later, so as to determine a wrapped phase of the object (S8).
Thereafter, the unwrapped phase of the object is determined by a known
unwrapping process (S9).
Step of Specifying Positional Data of Each Phase Position
This step corresponds to the processing of S1 to S5 in the flowchart of
FIG. 1.
In general, the Fourier transform fringe analysis method can determine a
phase from a single fringe image alone by introducing a carrier frequency
(relative inclination between the object surface and the reference
surface). When the carrier frequency is introduced, the interference
fringe intensity is represented by the following expression (9):
i(x, y)=a(x, y)+b(x, y)cos[2.pi..function..sub.x x+2.pi..function..sub.y
y+.phi.(x, y)+.xi.] (9)
where
a(x, y) is the background of interference fringes;
b(x, y) is the visibility of fringes;
.phi. (x, y) is the phase of the object;
.xi. is the initial phase of the object; and
f.sub.x and f.sub.y are carrier frequencies.
As mentioned above, .xi. is expressed by .xi.=2.pi.z/.lambda., where
.lambda. is the wavelength of light, and z is the displace amount of the
phase shift device (not including the part involved with the inclination
of the phase shift device). Therefore, the above-mentioned expression (9)
can be deformed as the following expression (10):
i(x, y)=a(x, y)+c(x, y)exp[i(2.pi..function..sub.x
+2.pi..function..sub.y)]+c.sup.* (x, y)exp[-i(2.pi..function..sub.x
+2.pi..function..sub.y)] (10)
where c(x, y) is the complex amplitude of interference fringes, and c.sup.*
(x, y) is the complex conjugate of c(x, y):
##EQU7##
When the above-mentioned expression (10) is subjected to Fourier transform,
the following expression (12) can be obtained:
I(.eta.,.zeta.)=A(.eta.,.zeta.)+C(.eta.-.function..sub.x,.zeta.-.function..
sub.y)+C.sup.* (.eta.+.function..sub.x,.zeta.+.function..sub.y) (12)
where
A(.eta., .zeta.) is the Fourier transform of a(x, y);
C(.eta.-f.sub.x, .zeta.-f.sub.y) is the Fourier transform of
c(x,y)exp[i(2.pi..function..sub.x +2.pi..function..sub.y)]; and
C.sup.* (.eta.+f.sub.x, .zeta.+f.sub.y) is the Fourier transform of c.sup.*
(x, y)exp[(-i(2.pi..function..sub.x +2.pi..function..sub.y)].
As in the procedure carried out in a typical Fourier transform method,
C(.eta.-f.sub.x, .zeta.-f.sub.y) is determined by filtering. After a
spectrum peak located at coordinates (f.sub.x, f.sub.y) is moved to the
origin of the frequency coordinate system so as to eliminate the carrier
frequency (including the part of the error in inclination of the phase
shift device), C(.eta.-f.sub.x, .zeta.-f.sub.y) is subjected to inverse
Fourier transform (S4), whereby c(x, y) is obtained. Here, assuming that
.xi..sub.0 is the initial phase of the phase shift device, and .xi..sub.1
is the phase after the movement, the following expression (13):
##EQU8##
is obtained from the interference fringe image data at the start position.
Then, from the interference fringe image data after moving the phase shift
device, the following expression (14):
##EQU9##
is obtained.
As a consequence, the following expression (15):
##EQU10##
is obtained.
Therefore, the phase difference between before and after moving the phase
shift device is represented by the following expression (16):
##EQU11##
Therefore, the displacement amount of the phase shift device is represented
by the following expression (17):
##EQU12##
When the respective displacement amounts determined for individual
interference fringe images obtained by a predetermined phase shift are
averaged, displacement can be detected at a high accuracy. When detected
the displacement of the object by the Fourier transform fringe analysis
method, it is not always necessary to use the whole fringe image data.
Displacement can also be detected with a sufficiently high accuracy from a
part of fringe image data alone.
Step of Calculating Object Phase
In the following, a technique for determining the phase of the object
according to individual phase position data of the phase shift device
obtained as mentioned above will be explained.
In summary, this technique is carried out such that, as shown in the
flowchart of FIG. 1, the above-mentioned C(.eta.-f.sub.x, .zeta.-f.sub.y)
is subjected to inverse Fourier transform in a state not eliminating the
carrier frequency (including the part of the error in inclination of the
phase shift device), so as to determine the relative tilt amount
(including the part of the error in inclination of the phase shift device)
between the reference surface and the object surface (S6), each phase
shift position (phase shift amount) of the phase shift device including
the phase displacement amount involved with the inclination of the phase
shift device is determined according to the relative tilt amount and the
displacement amount of the phase shift device determined as mentioned
above (S7), the wrapped phase of the object is obtained according to a
predetermined computing equation which will be explained later (S8), and
the unwrapped phase of the object is determined by unwrapping (S9).
Though the technique for determining the phase in this embodiment can
correspond to any number of buckets (3, 4, 5, 7, 11, and N buckets, etc.),
only 3-bucket, 4-bucket, and 5-bucket methods will be explained here for
the convenience of explanation.
In the 3-bucket phase shift method based on a given shift amount, the
interference fringe intensity at the m-th shift position introducing the
carrier frequency is represented by the following expression (18):
i.sub.m (x, y, .xi..sub.m)=a(x, y)+b(x, y)cos[2.pi..function..sub.xm
x+2.pi..function..sub.ym y+.phi.(x, y)+.xi..sub.m ]=a(x, y)+b(x,
y)cos[.phi.(x, y)+.delta..sub.m ] (18)
where
a(x, y) is the background of interference fringes;
b(x, y) is the visibility of fringes;
.phi. (x, y) is the phase of the object; and
.delta..sub.m is the phase shift amount of the phase shift device expressed
by:
.delta..sub.m =2.pi..function..sub.xm x+2.pi..function..sub.ym y+.xi..sub.m
(18a)
where
.xi..sub.m is the phase of the phase shift device (not including the part
involved with the tilt of the phase shift device); and
f.sub.xm and f.sub.ym are the carrier frequencies (including the part of
the error in inclination of the phase shift device) after the m-th phase
shift expressed by:
##EQU13##
where
.lambda. is the wavelength of light;
.theta..sub.xm and .theta..sub.ym are respective relative inclinations
between the reference surface and the object surface upon the m-th phase
shift (including the part of the error in inclination of the phase shift
device) in X and Y directions; and
z.sub.m is the amount of displacement of the phase shift device at the m-th
shift position (not including the part involved with the tilt of the phase
shift device).
The phase shift amount .delta..sub.m of the phase shift device in the
above-mentioned expression (18) represents the relative phase difference
between the reference surface and the object surface including the phase
displacement amount involved with the pure displacement of the phase shift
device toward the optical axis and the phase displacement amount involved
with the inclination of the phase shift device, and is determined by the
following procedure.
First, the above-mentioned C(.eta.-f.sub.x, .zeta.-f.sub.y) is subjected to
inverse Fourier transform (S6) in a state not eliminating the carrier
frequency (including the part of the error in inclination of the phase
shift device), so as to determine the form of the object surface in a
state where the relative tilt amount (including the part of the error in
inclination of the phase shift device) between the reference surface and
the object surface is superposed thereon. Subsequently, the method of
least squares is used for determining a plane fitting the above-mentioned
surface form, and the above-mentioned relative tilt amounts .theta..sub.xm
and .theta..sub.ym are determined according to the differential
coefficient of the least-square plane.
Thereafter, the relative tilt amounts .theta..sub.xm and .theta..sub.ym are
inputted into their corresponding expressions mentioned above in
connection with expression (18), so as to determine the carrier frequency
(f.sub.xm, f.sub.ym) (including the part of the error in inclination of
the phase shift device), whereas the displacement amount z.sub.m of the
phase shift device determined by the procedure mentioned above is inputted
into its corresponding expression set forth in connection with the
above-mentioned expression (18) so as to determine the phase .xi..sub.m of
the phase shift device (not including the part involved with the
inclination of the phase shift device). Further, thus determined carrier
frequency (f.sub.xm, f.sub.ym) and phase .xi..sub.m of the phase shift
device are inputted into the above-mentioned expression (18a), so as to
determine the phase shift amount .delta..sub.m of the phase shift device.
For determining the phase .phi.(x, y) of the object, the following
expression (19):
##EQU14##
will now be studied.
From this expression, the following expression (20):
##EQU15##
is obtained.
The above-mentioned procedure can determine the phase shift amount
.delta..sub.m representing the relative phase difference between the
reference surface and the object surface. Therefore, the wrapped phase of
the object form can be determined by using the above-mentioned expression
(20). Further, a known unwrapping method can be used for determining a
continuous object phase .PHI..
In the 4-bucket phase shift method with a given shift amount, the following
expressions (21) and (22) are used in place of the above-mentioned
expressions (19) and (20):
##EQU16##
In the 5-bucket phase shift method with a given shift amount, the following
expressions (23) and (24) are used in place of the above-mentioned
expressions (19) and (20):
##EQU17##
An apparatus for carrying out the above-mentioned method will now be
explained with reference to FIG. 2.
This apparatus is used for carrying out the method in accordance with the
above-mentioned embodiment. As shown in FIG. 2, in a Michelson type
interferometer 1, interference fringes are formed on the imaging surface
of CCD 5 in an image pickup camera 4 by respective reflected luminous
fluxes from an object surface 2 and a reference surface 3, and are
inputted into a computer 7 equipped with a CPU and an image processing
memory by way of an image input board 6. Thus inputted interference fringe
image data is subjected to various arithmetic operations, and the results
of operations are displayed onto a monitor screen 7A. The interference
fringe image data outputted from the image pickup camera 4 is temporarily
stored into the memory due to the processing carried out by the CPU. The
piezoelectric driving signal outputted from the computer 7 is fed into a
piezoelectric driver 9, whereas a PZT (piezoelectric device) actuator 10
is driven in response thereto.
The computer 7 subjects thus obtained interference fringe image data to
Fourier transform, so as to determine the (relative) phase shift amount of
the phase shift device (between the object surface and the reference
surface), and an arithmetic operation of the computing equation in which
the phase shift amount is inputted is carried out, so as to calculate the
phase of the object surface.
FIGS. 5A and 5B show two modes of the PZT (piezoelectric device) actuator
10, respectively.
As shown in FIG. 5A, the first mode comprises three piezoelectric elements
121, 122, 123 for supporting the backside of the reference surface
(reference mirror) 3, whereas two lines L.sub.x, L.sub.y respectively
connecting the piezoelectric element 121, also acting as a fulcrum member,
to the other piezoelectric elements 122, 123 on the reference mirror
having the reference surface 3 are arranged orthogonal to each other. A
phase shift is effected when the three piezoelectric elements 121, 122,
123 expand/contract by the same amount. When the piezoelectric element 122
expands/contracts alone, the reference surface 3 of the reference mirror
inclines in the x-axis direction so as to rotate about the y axis. When
the piezoelectric element 123 expands/contracts alone, the reference
surface 3 of the reference mirror inclines in the y-axis direction so as
to rotate about the x axis. As shown in FIG. 5B, the second mode is
constructed such that the center part of the backside of the reference
surface (reference mirror) 3 is supported by a cylindrical piezoelectric
tube 124. A phase shift is effected by an unbiased expansion/contraction
of the piezoelectric tube 124. On the other hand, a biased
expansion/contraction freely tilts the reference surface 3 of the
reference mirror in x- and y-axis directions.
For verifying the accuracy of the method in accordance with the
above-mentioned embodiment, a phase analysis was carried out by the
above-mentioned 3-bucket method while using three fringe images whose
initial phases were 0.degree., 76.degree., and 123.degree., respectively,
in interference fringes (with 256 gradations) shown in FIG. 6 obtained
when an ideal plane is observed. Thus determined phase shift amounts were
75.99860.degree. and 122.9973.degree.. FIG. 7 shows the error between the
results of phase analysis and the ideal plane. The maximum error is about
.+-.0.01 rad (7.9.times.10.sup.-4 .lambda.), which is on a negligible
order. Therefore, it has been verified that highly accurate phase analysis
can be carried out by the 3-bucket method as well.
Without being restricted to the above-mentioned embodiment, the method of
the present invention can be modified in various manners. When detected
the inclination of the object in the method in accordance with the
above-mentioned embodiment, fringe image data carrying the phase
information from the object is acquired, the whole or part of each fringe
image data is subsequently subjected to Fourier transform, so as to
determine the phase information of the object in the fringe image data,
and the inclination of the object is detected according to thus determined
phase information. However, for example, the frequency of carrier fringes
in the fringe image data may be determined in place of the phase
information in the fringe image data, and the inclination of the object
may be detected according to thus determined frequency of carrier fringes.
A technique for detecting the inclination of the object by determining the
frequency of carrier fringes in fringe image data as such is disclosed in
detail in the specification of commonly-assigned U.S. patent application
Ser. No. 10/021,014.
The phase shift element is not limited to the above-mentioned PZT, but may
be those which can achieve the phase shift method by physically moving the
reference surface or object surface or changing the optical path length by
use of an AO element or EO element, or a transmission type element which
can change the refractive index or the like so as to alter the optical
path length when inserted into the reference optical path and/or
observation optical path.
Though three PZT elements are exactly positioned at the respective vertices
of a rectangular triangle as a mode for arranging them in the
above-mentioned embodiment, the aimed effect can be obtained if the three
members are arranged so as to form the respective vertices of a given
triangle on the reference mirror.
Though the above-mentioned embodiment is explained while using spatial
carrier frequencies as carrier frequencies, temporal carrier frequencies
or temporal spatial carrier frequencies can also be used as the carrier
frequencies of the present invention.
Though the interference fringe image data is captured with a Michelson type
interferometer in the above-mentioned embodiment, the present invention is
similarly applicable to interference fringe image data obtained by other
types of interferometers such as those of Fizeau type as a matter of
course.
Further, the present invention is similarly applicable to various kinds of
fringe images such as moirefringes and speckle fringes in addition to
interference fringes.
In the phase shift fringe analysis method in accordance with the present
invention and the apparatus using the same, a phase shift device is
employed for phase-shifting an object to be observed and a reference
relative to each other, fringe image dat | | |
