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RELATED APPLICATIONS
This application claims the priorities of Japanese Patent Application No.
2000-277444 filed on Sep. 13, 2000 and Japanese Patent Application No.
2001-022633 filed on Jan. 31, 2001, which are incorporated herein by
reference.
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to a fringe analysis error detection method
and fringe analysis error correction method using Fourier transform method
when analyzing fringe images by using phase shift methods; and, in
particular, to a fringe analysis error detection method and fringe
analysis error correction method in which PZTs (piezoelectric elements)
are used for shifting the phase, and Fourier transform method is utilized
when analyzing thus obtained image data with fringe patterns such as
interference fringes, whereby the analyzed value can be made more
accurate.
2. Description of the Prior Art
While light wave interferometry, for example, has conventionally been knows
as an important technique concerning precise measurement of a wavefront of
an object, there have recently been urgently demanded for developing an
interferometry technique (sub-fringe interferometry) which can read out
information from a fraction of a single interference fringe (one fringe)
or less due to the necessity of measuring a surface or wavefront
aberration at an accuracy of 1/10 wavelength or higher.
Known as a typical technique widely used in practice as such a sub-fringe
interferometry technique is the phase shift fringe analyzing 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, one or more phase shift element such as PZTs
(piezoelectric element), for example, are used for phase-shifting the
relative displacement between an object to be observed and the reference,
interference fringe images are captured each time when a predetermined
phase amount is shifted, the interference fringe intensity at each point
on the surface to be inspected is measured, and the phase of each point on
the surface is determined by using the result of measurement.
For example, when carrying out a four-step phase shift method, respective
interference fringe intensities I.sub.1, I.sub.2, I.sub.3, I.sub.4 at the
individual phase shift steps are expressed as follows:
##EQU1##
where
x and y are coordinates;
.phi.(x, y) is the 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 and
expressed as:
##EQU2##
Though the phase shift methods enable measurement with a very high accuracy
if the predetermined step amount can be shifted correctly, it may be
problematic in that errors in measurement occur due to errors in the step
amount and in that it is likely to be influenced by the disturbance during
measurement since it necessitates a plurality of interference fringe image
data items.
For sub-fringe interferometry other than the phase shift method, attention
has been paid to techniques using the Fourier transform method as
described in "Basics of Sub-fringe Interferometry," Kogaku, Vol. 13, No. 1
(February, 1984), pp. 55 to 65, for example.
The Fourier transform fringe analysis method is a technique in which a
carrier frequency (caused by a relative inclination between an object
surface to be observed and a reference surface) is introduced, so as to
make it possible to determine the phase of the object with a high accuracy
from a single fringe image. When the carrier frequency is introduced,
without consideration of the initial phase of the object, the interference
fringe intensity i(x, y) is represented by the following expression (3):
i(x,y)=a(x,y)+b(x,y)cos[2.pi.f.sub.x x+2.pi.f.sub.y y+.PHI.(x,y)] (3)
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 to be observed; and
f.sub.x and f.sub.y are carrier frequencies in the x and y directions
respectively expressed by:
##EQU3##
where .lambda. is the wavelength of light, and .theta..sub.x and
.theta..sub.y are the respective inclinations of the object in the x and y
directions.
The above-mentioned expression (3) can be rewritten as the following
expression (4):
i(x,y)=a(x,y)+c(x,y)exp[i(2.pi.f.sub.x x+2.pi.f.sub.y
y)]+c*(x,y)exp[i(2.pi.f.sub.x x+2.pi.f.sub.y y)] (4)
where c(x, y) is the complex amplitude of the interference fringes, and
c*(x, y) is the complex conjugate of c(x, y).
Here, c(x, y) is represented as the following expression (5):
##EQU4##
The Fourier transform of expression (4) gives:
I(.eta.,.zeta.)=A(.eta.,.zeta.)+C(.eta.-f.sub.x,.zeta.-f.sub.y)+C*(.eta.-f.
sub.x,.zeta.-f.sub.y) (6)
where A(.eta., .zeta.) is the Fourier transform of a(x, y), and
C(.eta.-f.sub.x, .zeta.-f.sub.y) and C*(.eta.-f.sub.x, .zeta.-f.sub.y) are
the Fourier transforms of c(x, y) and c*(x, y), respectively.
Subsequently, C(.eta.-f.sub.x, .zeta.-f.sub.y) is taken out by filtering,
the peak of the spectrum positioned at coordinates (f.sub.x, f.sub.y) is
transferred to the origin of a Fourier frequency coordinate system (also
referred to as Fourier spectra plane coordinate system; see FIG. 6), and
the carrier frequencies are eliminated. Then, inverse Fourier transform is
carried out, so as to determine c(x, y), and the wrapped measured phase
.phi.(x, y) can be obtained by the following expression (7):
##EQU5##
where Im[c(x, y)] is the imaginary part of c(x, y), whereas Re[c(x, y)] is
the real part of c(x, y).
Finally, unwrapping processing is carried out, so as to determine the phase
.PHI.(x, y) of the object to be measured.
In the Fourier transform fringe analyzing method explained in the
foregoing, the fringe image data modulated by carrier frequencies is
subjected to a Fourier transform method as mentioned above.
As mentioned above, the phase shift method captures and analyzes the
brightness of images while applying a phase difference between the object
light of an interferometer and the reference light by a phase angle
obtained when 2.pi. is divided by an integer in general, and thus can
theoretically realize highly accurate phase analysis.
For securing highly accurate phase analysis, however, it is necessary to
shift the relative displacement between the sample and the reference with
a high accuracy by predetermined phase amounts. When carrying out the
phase shift method by physically moving the reference surface or the
similar by using phase shift elements, e.g., PZTs (piezoelectric
elements), it is necessary to control the amount of displacement of PZTs
(piezoelectric elements) with a high accuracy. However, errors in
displacement of the phase shift elements or errors in inclination of the
reference surface or sample surface are hard to eliminate completely.
Controlling the amount of phase shift or amount of inclination is actually
a difficult operation. Therefore, in order to obtain favorable results, it
is important to detect the above-mentioned errors resulting from the phase
shift elements, and correct them according to thus detected values when
carrying out the fringe analysis.
SUMMARY OF THE INVENTION
In view of the circumstances mentioned above, it is an object of the
present invention to provide a fringe analysis error detection method
utilizing a Fourier transform fringe analyzing method which can favorably
detect, without complicating the apparatus configuration when analyzing
fringe image data obtained by use of the phase shift method, influences of
errors in the amount of displacement of phase shift and/or in the amount
of relative inclination between the object to be observed and the
reference.
It is another object of the present invention to provide a fringe analysis
error correction method utilizing a Fourier transform fringe analyzing
method which can favorably correct, without complicating the apparatus
configuration when analyzing fringe image data obtained by use of the
phase shift method, influences of errors in the amount of displacement of
phase shift and/or in the amount of relative inclination between the
object to be observed and the reference.
The present invention provides a fringe analysis error detection method in
which phase shift elements are used for relatively phase-shifting an
object to be observed and a reference with respect to each other, and a
wavefront of the object is determined by fringe analysis;
the method comprising the steps of Fourier-transforming two pieces of
carrier fringe image data respectively carrying wavefront information
items of the object before and after the phase shift; and carrying out a
calculation according to a result of the transform so as to detect an
amount of error of the phase shift.
Here, the "amount of error of the phase shift" includes at least "amount of
relative inclination between the object to be observed and the reference"
and "amount of translational displacement of the phase shift."
In the fringe analysis error detection method in accordance with the
present invention, the fringe image data may be carrier fringe image data
on which carrier fringes are superposed.
In the fringe analysis error detection method in accordance with the
present invention, the two pieces of carrier fringe image data may be
Fourier-transformed so as to determine carrier frequencies, and a position
of a spectrum may be calculated according to two of the carrier
frequencies so as to detect an amount of relative inclination between the
object and reference generated by the phase shift.
When determining the carrier frequencies of carrier fringes in this case, a
positional coordinate of a predetermined peak among peaks on a frequency
coordinate system obtained by the Fourier transform may be determined, and
an arithmetic operation for calculating the carrier frequencies may be
carried out according to the positional coordinate.
In the fringe analysis error detection method in accordance with the
present invention, the two pieces of carrier fringe image data may be
Fourier-transformed so as to determine phase information of the object,
and
thus obtained phase information of the object may be subjected to a
predetermined arithmetic operation so as to detect an inclination of the
object.
When determining the phase information of the object in this case, a
predetermined spectrum distribution among spectrum distributions on a
frequency coordinate system obtained by the Fourier transform may be
determined, and an arithmetic operation for calculating the phase
information according to the spectrum distribution may be carried out.
The predetermined arithmetic operation may be an arithmetic operation for
determining a least square of the phase information of the object.
In the fringe analysis error detection method in accordance with the
present invention, the two pieces of carrier fringe image data may be
Fourier-transformed so as to determine complex amplitudes of carrier
fringes, and a position may be calculated according to two of the complex
amplitudes so as to detect the amount of translational displacement of the
phase shift.
In the fringe analysis error detection method in accordance with the
present invention, the two pieces of carrier fringe image data may be
Fourier-transformed so as to determine carrier frequencies and complex
amplitudes, and a position may be calculated according to two of the
carrier frequencies and two of the complex amplitudes so as to detect the
amount of relative inclination between the object and reference and the
amount of tilt displacement of the phase shift which are generated by the
phase shift.
In the fringe analysis error detection method in accordance with the
present invention, the fringe images may be an interference fringe images.
Also, the present invention provides a fringe analysis error correction
method in which, after the detection is carried out in the fringe analysis
error detection method in accordance with the present invention, a
correction calculation for compensating for the difference of the detected
amount of inclination generated by the phase shift from a target amount of
inclination and/or the difference of a predetermined amount of
translational displacement of phase shift from a target predetermined
amount of displacement of phase shift is carried out in the fringe
analysis of the fringe image data.
The above-mentioned methods in accordance with the present invention are
applicable to fringe image analyzing techniques using the Fourier
transform method in general, such as analysis of interference fringes and
moire fringes, three-dimensional projectors based on fringe projection, or
the like, for example.
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 for realizing the method shown in FIG. 1;
FIG. 3 is a block diagram showing a part of FIG. 2 in detail;
FIGS. 4A and 4B are conceptual views specifically showing a part of FIG. 2;
FIG. 5 is a flowchart for explaining the method in accordance with another
embodiment of the present invention;
FIG. 6 is a conceptual view for explaining a part of the method shown in
FIG. 5;
FIG. 7 is a block diagram for carrying out the method shown in FIG. 5;
FIG. 8 is a block diagram specifically showing a part of FIG. 7; and
FIG. 9 is a flowchart showing a partly modified example of the flowchart
shown in FIG. 5.
DESCRIPTION OF THE PREFERRED EMBODIMENTS
In the following, the fringe analysis error detection method and fringe
analysis error correction method in accordance with embodiments of the
present invention will be explained with reference to the drawings.
These methods are techniques for determining a wavefront of an object to be
observed by using the phase shift method, in which fringe image data
carrying wavefront information of the object obtained according to a
relative form between the object surface and a reference surface is
subjected to the Fourier transform method so as to determine carrier
frequencies and complex amplitudes occurring due to the deviation between
the wavefront from the object and the wavefront from the reference, and
the amount of relative inclination between the object surface and
reference surface and the amount of displacement of phase shift are
detected according to the carrier frequencies and complex amplitudes.
Thereafter, a correcting calculation for compensating for the detected
amount of inclination and amount of displacement is carried out in the
fringe analysis of the fringe image data according to the phase shift
method.
In the following, the detection and correction of the amount of
displacement of phase shift and the detection and correction of the amount
of relative inclination between the object to be observed and the
reference in the phase shift will be explained separately from each other.
Detection and Correction of Translational Displacement of Phase Shift
The outline of the phase shift error amount detecting and correction method
in accordance with an embodiment will now be explained with reference to
the flowchart of FIG. 1.
First, an interference fringe image carrying phase information of an object
to be observed, on which spatial carrier fringes are superposed, is
obtained by a CCD imaging camera (S1). Subsequently, thus obtained
interference fringe image data is subjected to Fourier transform (S2), a
spatial carrier frequency (f.sub.x, f.sub.y) is extracted (S3), and a
Fourier transform fringe analysis is carried out according to this carrier
frequency, so as to determine a complex amplitude c(x, y) (S4) which will
be explained later. Then, the amount of displacement of the reference
surface is determined (S5), whereby the amount of displacement of phase
shift can be determined (S6). Further, when analyzing the fringe image
according to the phase shift amount, the amount of displacement determined
at S6 is corrected so as to determine the phase of the object (S7).
In general, the Fourier transform fringe analyzing method can determine the
phase with 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 (8):
i(x,y)=a(x,y)+b(x,y)cos[2.pi.f.sub.x x+2.pi.f.sub.y y+.phi.(x,y)+.xi.] (8)
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 to be observed;
.xi. is the initial phase (2.pi.x/.lambda.) of the object to be observed;
and
f.sub.x and f.sub.y are carrier frequencies.
As mentioned above, .xi. can be expressed by .xi.=2.pi.x/.lambda., where
.lambda. is the wavelength of light, and x is the phase shift amount of
the phase shift element, whereby the above-mentioned expression (8) can be
modified as the following expression (9):
i(x,y)=a(x,y)+c(x,y)exp[i(2.pi.f.sub.x x+2.pi.f.sub.y
y)]+c*(x,y)exp[i(2.pi.f.sub.x x+2.pi.f.sub.y y)] (9)
where c*(x, y) is the complex conjugate of c(x, y).
Here, c(x, y) is represented by the following expression (10):
##EQU6##
When the above-mentioned expression (9) is Fourier-transformed, the
following expression (11) can be obtained:
I(.eta.,.zeta.)=A(.eta.,.zeta.)+C(.eta.-f.sub.x,.zeta.-f.sub.y)+C*(.eta.-f.
sub.x,.zeta.f.sub.y) (11)
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); and
C*(.eta.-f.sub.x, .zeta.-f.sub.y) is the Fourier transform of c*(x, y).
In general, in the Fourier transform method, C(.eta.-f.sub.x,
.zeta.-f.sub.y) is determined by filtering and then is subjected to
inverse Fourier transform, so as to yield c(x, y). Here, from the
interference fringe image data at the start position, the following
expression (12) is obtained:
##EQU7##
where .xi..sub.0 is the initial phase of phase shift, and .xi..sub.1 is the
phase after the object to be observed is moved.
Subsequently, from the interference fringe data after the object to be
observed is moved, the following expression (13) is obtained:
##EQU8##
As a consequence, the following expression (14) is obtained:
##EQU9##
Therefore, the phase difference between before and after moving the object
to be observed is represented by the following expression (15):
##EQU10##
Hence, the amount of displacement of the object to be observed is
represented by the following expression (16):
##EQU11##
When the average of the respective amounts of displacement determined for
the individual interference fringe images obtained by a predetermined
phase shift is determined, the displacement can be detected with a higher
accuracy. When detecting the displacement of the object by the Fourier
transform fringe analyzing method, it is not always necessary to use the
whole fringe image data, whereas even a part of the fringe image data
alone makes it possible to detect the displacement with a sufficiently
high accuracy.
A method of correcting thus detected amount of displacement of phase shift
will now be explained.
First, the principle of correction will be explained with reference to a
general expression of the phase shift method.
In an n-bucket phase shift system which moves the reference surface by
using PZTs or the like, the intensity distribution of interference fringes
on the object to be observed in which the reference surface is shifted for
j times (j=1, 2, . . . , n) is represented by the following expression
(17):
i.sub.j (x,y)=a(x,y)+b(x,y)cos[2.pi.f.sub.xj x+2.pi.f.sub.yj
y+.phi.(x,y)+.xi..sub.j ] (17)
In general, in a highly accurate actuator such as PZT, the j-th phase shift
amount .xi..sub.j (j=1, 2, . . . , n) is given as represented by the
following expression (18):
##EQU12##
where .xi..sub.j is the j-th phase shift amount (j=1, 2, . . . , n).
The above-mentioned expression (17) may be expanded, whereby the phase
.phi.(x, y) to be observed can be determined by utilizing the
orthogonality of trigonometric functions. Thus determined expression (19)
is as follows:
##EQU13##
where
s.sub.j is a constant; and
c.sub.j is a constant.
Meanwhile, an actuator with a high accuracy is expensive. If a method for
detecting the displacement of the actuator is used, a highly accurate
phase shift can be obtained without using the expensive actuator.
Namely, assuming the j-th (j=1, 2, . . . , n) phase shift amount to be
.xi..sub.j (whose specific value is unknown), the following expression
(20) is obtained:
.xi..sub.j =(j-1).multidot.a+.delta..sub.j (20)
where .delta..sub.j is the phase shift error of the actuator (.delta..sub.j
<<.pi./2).
Since .delta..sub.j can be determined by the above-mentioned phase shift
error amount detection method, the phase analysis error can be corrected
with a high accuracy by using the following expressions (21) and (22):
##EQU14##
In the following, a 5-bucket method, which is often used, will be explained
by way of example. The 5-bucket method is represented by the following
expression (23):
##EQU15##
When .sup.a.sub.j =.pi.(.sup.j -.sup.1)/.sup.2 (j=1, 2, . . . , 5) is
utilized here, the following expression (24) is obtained:
##EQU16##
Since .epsilon..sub.j.sup.2 is small in general, expression (24) is
represented by the following expression (25):
##EQU17##
Therefore, the following expression (26) is obtained:
##EQU18##
Hence, the phase having corrected the shift error of the actuator can be
determined.
An apparatus for carrying out the above-mentioned embodiment of the present
invention will now be explained with reference to FIGS. 2 and 3.
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 formed by respective reflected
luminous fluxes from an object surface 2 to be observed and a reference
surface 3 are captured at an imaging surface of a CCD 5 of an imaging
camera 4, and are fed into a computer 7 equipped with a CPU and an image
processing memory by way of an image input board 6. Thus fed interference
fringe image data is subjected to various arithmetic operations, and the
results of operations are displayed on a monitor screen 7A. Here, the
interference fringe image data output from the imaging camera 4 is
temporarily stored into the memory upon an operation of the CPU.
As shown in FIG. 3, the computer 7 comprises, in terms of software, an
FFT-operated complex amplitude calculating means 11, a phase shift
displacement amount detecting means 12, and a phase shift displacement
amount correcting means 13. The FFT-operated complex amplitude calculating
means 11 carries out the operation of step 3 (S3) for subjecting the
obtained interference fringe image to a Fourier transform method and
extracting the FFT-operated complex amplitude as mentioned above. The
phase shift displacement amount detecting means 12 carries out the
operations corresponding to the above-mentioned step 4 (S4) to the
above-mentioned step 6 (S6) according to the FFT-operated complex
amplitude calculated in the FFT-operated complex amplitude arithmetic
means 11. According to the amount of displacement detected by the phase
shift displacement amount detecting means 12, the phase shift displacement
amount correcting means 13 compensates for the amount of displacement, and
determines the phase of the object having corrected the error (S7).
As a consequence, even in the case where a PZT (piezoelectric element)
actuator 10 shifts by a predetermined amount, so that an error occurs in
the amount of shift of the PZT (piezoelectric element) actuator 10, it is
adjusted such that the finally determined phase of the object is not
affected by the error. In FIG. 2, since the surface 2 of the object is
fixed, the relative displacement in shift amount between the object
surface 2 and the reference surface 3 depends on only the amount of
displacement of the shift amount of the reference surface 3 shifted by the
PZT (piezoelectric element) actuator 10.
Therefore, the system constructed by the constituents shown in FIGS. 2 and
3 as mentioned above can calculate, according to the determined
FFT-operated complex amplitude, the phase of the object in a state where
the phase shift displacement amount of the reference surface 3 is
eliminated.
FIGS. 4A and 4B show two modes of the PZT (piezoelectric element) actuator
10, respectively.
As shown in FIG. 4A, the first mode comprises three piezoelectric elements
121, 122, 123 for supporting 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. 4B, the second mode is
constructed such that the center part of the back side 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.
Detection and Correction of Inclination Error in Phase Shift
In general, a plurality of phase shift elements, or a phase shift element
and a support guide are necessary in the case where the object to be
observed or the reference surface has a large size. In this case, the
straightness of phase shift is hard to secure, whereby an inclination may
occur in the object or the reference surface. Therefore, in the method in
accordance with the embodiment explained in the following, the amount of
inclination of the phase shift element (the amount of relative inclination
between the object surface and the reference surface) is detected and
corrected by a technique substantially similar to the method of the
above-mentioned embodiment.
The method in accordance with the second embodiment of the present
invention for detecting and correcting the amount of inclination of the
phase shift element will now be explained.
FIG. 5 is a flowchart conceptually showing the method of the second
embodiment.
First, an interference fringe image carrying surface form information of an
object to be observed, on which spatial carrier fringes are superposed, is
obtained by a CCD imaging camera (S11). Subsequently, thus obtained
interference fringe image data is subjected to a Fourier transform method
(S12), a spatial carrier frequency (f.sub.x, f.sub.y) is extracted (S13),
and a Fourier transform fringe analysis is carried out according to this
carrier frequency, so as to determine c(x, y) which will be explained
later, whereby the amount of inclination of the reference surface is
determined (S14). The amount of inclination is compensated for (S15), and
the phase of the object after the correcting calculation is determined
(S16).
In the following, the method of the second embodiment will be explained
with reference to expressions.
As mentioned above, the Fourier transform fringe analyzing method can
determine the phase by a single sheet of 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
(27):
i(x,y)=a(x,y)+b(x,y)cos[2.pi.f.sub.x x+2.pi.f.sub.y y+.phi.(x,y)+.xi.]
(27)
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 to be observed;
.xi. is the phase shift amount (2.pi.x/.lambda.); and
f.sub.x and f.sub.y are carrier frequencies.
The carrier frequencies f.sub.x and f.sub.y are represented by the
following expression (27a):
##EQU19##
As mentioned above, assuming .lambda. to be the wavelength of light,
.theta..sub.x and .theta..sub.y to be the respective inclinations
(postures) of the object surface in X and Y directions, and x to be the
phase shift amount of the phase shift element, the above-mentioned
expression (27) can be deformed as the following expression (28):
i(x,y)=a(x,y)+c(x,y)exp[i(2.pi.f.sub.x x+2.pi.f.sub.y
y)]+c*(x,y)exp[i(2.pi.f.sub.x x+2.pi.f.sub.y y)] (28)
where c*(x, y) is the complex conjugate of c(x, y).
Here, c(x, y) is represented by the following expression (29):
##EQU20##
When the above-mentioned expression (28) is Fourier-transformed, the
following expression (30) can be obtained:
I(.eta.,.zeta.)=A(.eta.,.zeta.)+C(.eta.-f.sub.x,.zeta.-f.sub.y)+C*(.eta.-f.
sub.x,.zeta.-f.sub.y) (30)
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); and
C*(.eta.-f.sub.x, .zeta.-f.sub.y) is the Fourier transform of c*(x, y).
In the Fourier transform method, in general, C(.eta.-f.sub.x,
.zeta.-f.sub.y) is determined by filtering, the peak of the spectrum
located at the position (f.sub.x, f.sub.y) on a frequency coordinate
system is transferred to the origin of coordinates as shown in FIG. 6, so
as to eliminate the carrier frequencies. Then, inverse Fourier transform
is carried out, so as to determine c(x, y), whereby a wrapped phase is
obtained. Subsequently, the phase .PHI.(x, y) of the object is determined
by unwrapping. While (f.sub.x, f.sub.y) is a carrier frequency, respective
values of (f.sub.x, f.sub.y) are determined in view of the fact that a
predetermined angular relationship (relative posture), specifically the
relationship of expression (27a), exists between the object surface and
the reference surface. According to these values, the angular relationship
between the object surface and the reference surface is determined.
The respective values of (f.sub.x, f.sub.y) are obtained when a sub-peak
position other than the maximum peak located at the origin of coordinates,
i.e., the position of C(.eta.-f.sub.x, .zeta.-f.sub.y), is determined. As
a consequence, .theta..sub.x and .theta..sub.y, which are the respective
inclinations (postures) of the object surface in X and Y directions, can
be determined.
.theta..sub.x and .theta..sub.y, which are the respective inclinations
(postures) of the object surface in X and Y directions, can also be
determined by the following technique (second technique) in place of the
above-mentioned technique (first technique).
The outline of this inclination detection method is represented by
individual steps (S21 to S27) in the flowchart of FIG. 9 in place of the
steps 11 to 14 (S11 to S14) in FIG. 5.
First, interference fringe images carrying form information of the object
to be observed, on which spatial carrier fringes are superposed, are
captured by a CCD camera (S21). Subsequently, thus obtained interference
fringe image data is subjected to a Fourier transform method (S22), and
C(n-f.sub.x, .zeta.-f.sub.y), which is a spectrum distribution (side lobe)
of carrier frequencies is extracted therefrom by filtering (S23). Then,
this distribution C(n-f.sub.x, .zeta.-f.sub.y) is subjected to inverse
Fourier transform, so as to yield c(x, y), whereby a wrapped phase is
obtained (S24). Thereafter, upon unwrapping, a phase p(x, y) of the object
is determined according to the form information of the object (S25).
Subsequently, the method of least squares is used for determining the
least-square plane of the phase p(x, y) (S26). Finally, the inclination of
the object is determined according to differential coefficients of the
least-square plane (S27).
In the Fourier fringe analyzing method in the first technique, after
C(n-f.sub.x, .zeta.-f.sub.y), which is a spectrum distribution (side lobe)
of carrier frequencies on a frequency coordinate system, is extracted by
filtering, for example, a peak thereof is moved from its position
(f.sub.x, f.sub.y) to the origin of coordinates so as to eliminate the
carrier frequencies, and then an inverse Fourier transform method is
carried out so as to determine the phase (form) of the object.
In the second technique, by contrast, the inclination of the object is
assumed to be a part of form thereof and, without moving peaks of
C(n-f.sub.x, .zeta.-f.sub.y), which is a spectrum distribution (side lobe)
of carrier frequencies in the above-mentioned expression (30), i.e.,
without eliminating the carrier frequencies, the spectrum distribution
C(n-f.sub.x, .zeta.-f.sub.y) is subjected to inverse Fourier transform. As
a consequence, the finally obtained phase p(x, y) of the object includes
an inclination component.
Namely, the p(x, y) is represented by the following expression (31):
##EQU21##
where
a is the differential coefficient of the least-square plane in the x
direction; and
b is the differential coefficient of the least-square plane in the y
direction.
Therefore, according to the second technique, the method of least squares
is used for determining the least-square plane of the form of the object
determined without eliminating carrier frequencies (the plane obtained by
fitting the form with the method of least squares), the respective
differential coefficients of the least-square plane in the x and y
directions are determined, and the above-mentioned expression (31) is used
for determining inclinations q.sub.x and q.sub.y of the object, whereby
the inclination of the object can be determined easily.
When determining a plane indicative of the form of the object in the
above-mentioned technique, a desirable plane to which a curved surface is
fitted can also be obtained when other fitting techniques are employed in
place of the method of least squares.
Thus, the posture (inclination) of the object to be observed can be
detected when the Fourier transform fringe analyzing method is used. When
determining the posture (inclination) of the object by using the Fourier
transform fringe analyzing method, it is not necessary to use the whole
region of fringe image, whereas sufficiently effective data can be
obtained even when a part of the fringe image region is analyzed.
A method of correcting thus detected amount of inclination will now be
explained.
First, the principle of correction will be explained with reference to a
general expression of the phase shift method.
In an n-bucket phase shift system which moves the reference surface by
using a PZT, the intensity distribution of interference fringes on the
object to be observed in which the reference surface is shifted for j
times (j=1, 2, . . . , n) is represented by the following expression (32):
i(x,y)=a(x,y)+b(x,y)cos[2.pi.f.sub.xj x+2.pi.f.sub.yj
y+.phi.(x,y)+.xi..sub.j ] (32)
In general, the j-th phase shift amount .xi..sub.j (j=1, 2, . . . , n) is
represented by the following expression (33) in a highly accurate actuator
such as PZT:
##EQU22##
where .xi..sub.j is the j-th phase shift amount (j=1, 2, . . . , n).
The above-mentioned expression (33) may be expanded, whereby the sample
phase .phi.(x, y) can be determined by utilizing the orthogonality of
trigonometric functions. Thus determined expression (34) is as follows:
##EQU23##
where
s.sub.j is a constant; and
c.sub.j is a constant.
When the posture of the object relative to the reference surface is
changed, the j-th (j=1, 2, . . . , n) phase .xi..sub.j (whose specific
value is unknown) is represented by the following expression (35):
##EQU24##
where .theta..sub.xj and .theta..sub.yj are the inclinations of the
actuator (.epsilon..sub.j <<.pi./2).
Since .theta..sub.xj and .theta..sub.yj can be determined by the
above-mentioned method, the phase analysis error can be corrected with a
high accuracy by using the following expressions (36) and (37):
##EQU25##
In the following, a 5-bucket method, which is often used, will be explained
by way of example. The 5-bucket method is represented by the following
expression (38):
##EQU26##
When .sup.a.sub.j =.pi.(.sup.j -.sup.1)/.sup.2 (j=1, 2, . . . , 5) is
utilized here, the following expression (39) is obtained:
##EQU27##
Since .epsilon..sub.j.sup.2 is small in general, expression (39) is
represented by the following expression (40):
##EQU28##
Therefore, the following expression (41) is obtained:
##EQU29##
Hence, the phase having corrected the shift error of the actuator can be
determined.
An apparatus for carrying out the above-mentioned embodiment of the present
invention will now be explained with reference to FIGS. 7 and 8.
This apparatus is used for carrying out the method in accordance with the
above-mentioned embodiment. (As for the inclination detection, the first
technique is illustrated.) As shown in FIG. 7, in a Michelson type
interferometer 1, interference fringes formed by respective reflected
luminous fluxes from an object surface 2 to be observed and a reference
surface 3 are captured at an imaging surface of a CCD 5 of an imaging
camera 4, and are fed into a computer 7 equipped with a CPU and an image
processing memory by way of an image input board 6. Thus fed interference
fringe image data is subjected to various arithmetic operations, and the
results of operations are displayed on a monitor screen 7A. Here, the
interference fringe image data output from the imaging camera 4 is
temporarily stored into the memory upon an operation of the CPU.
As shown in FIG. 8, the computer 7 comprises, in terms of software, an
FFT-operated complex amplitude calculating means 21, an inclination amount
detecting means 22, and an inclination amount correcting means 23. The
FFT-operated complex amplitude calculating means 21 carries out the
operation of step 13 (S13) for subjecting the obtained interference fringe
image to a Fourier transform method and extracting the FFT-operated
complex amplitude as mentioned above. The inclination amount detecting
means 22 carries out the operation corresponding to the above-mentioned
step 14 (S14) according to the FFT-operated complex amplitude calculated
in the FFT-operated complex amplitude arithmetic means 21. The inclination
amount correcting means 23 carries out the operations corresponding to
steps 15 and 16 (S15 and S16) in which, according to the amount of
inclination of the reference surface detected by the inclination amount
detecting means 22, the amount of inclination is compensated for, and the
phase of the object is determined.
Both of the two methods (the method of detecting and correcting the amount
of displacement of the phase shift element, and the method of detecting
and correcting the amount of inclination) in accordance with the
above-mentioned embodiments may be carried out in a single inspection step
or correction step. In this case, the accuracy in analyzing fringe images
can be raised more efficiently.
In general, it is necessary for a phase shift interferometer to control the
determined phase shift amount (e.g., 45 degrees, 90 degrees, and 120
degrees) precisely, whereby an expensive actuator with a high accuracy is
used therein. In the methods of the present invention, a carrier frequency
is determined from the measured fringe image, and the amount of
translational displacement of the phase shift and the amount of
inclination of the reference surface and/or object based on the phase
shift are determined according to the carrier frequency, whereby the
displacement and inclination of the actuator can be detected. Also, a
correction for compensating for the amount of displacement or amount of
inclination is possible in the analysis of fringe image. Therefore, an
expensive actuator having a high accuracy is not always necessary in
accordance with the present invention. Also, when employing an N-bucket
method, it is not always necessary to accurately adjust 1 step (360/N
(degrees)).
In the following, the case where the methods of the present invention are
applied to an arbitrary bucket phase shift interferometer will be
explained. The arbitrary bucket phase shift interferometer is a phase
shift interferometer in which the amount of phase shift can take any
values (individual shift amounts may differ from each other) without being
restricted to specific values (e.g., 45 degrees, 90 degrees, 120 degrees,
and so forth). Though only 3-, 4-, and 5-bucket methods will be mentioned
in the following explanation for convenience of explanation, the number of
buckets is not limited to them as a matter of course.
First, in the arbitrary shift amount 4-bucket phase shift method, the
interference fringe intensity at the m-th introduction of carrier
frequencies is represented by the following expression (42):
##EQU30##
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 to be observed;
.xi..sub.m the phase shift amount of the phase shift element; and
f.sub.xm and f.sub.ym are carrier frequencies after the m-th phase shift:
##EQU31##
where .lambda. is the wavelength of light, .theta..sub.xm and
.theta..sub.ym are the respective inclinations (amounts of inclination) of
the object in x and y directions after the m-th phase shift, and x.sub.m
is the m-th phase shift amount.
Subsequently, in order to determine the object phase .theta.(x, y), the
following expression (43) is considered:
##EQU32##
Hence, the following expression (44) is obtained:
##EQU33##
Since the amount of phase shift displacement and amount of inclination of
the reference surface or object generated upon driving the phase shift
element PZT can be detected, .delta..sub.m in expression (42a) can be
determined. Therefore, the wrapped phase .phi. of the object form can be
determined by using the above-mentioned expression (44). Furthermore, the
continuous object phase .PHI. can be determined by using a known
unwrapping method.
In the arbitrary shift amount 4-bucket phase shift method, the following
expressions (45) and (46):
##EQU34##
are used in place of the above-mentioned expressions (43) and (44).
In the arbitrary shift amount 5-bucket phase shift method, the following
expressions (47) and (48):
##EQU35##
are used in place of the above-mentioned expressions (43) and (44).
Without being restricted to the above-mentioned embodiments, the methods of
the present invention can be modified in various manners. For example, 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 embodiments, 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 embodiments are 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 | | |
