Method and apparatus for phase evaluation of pattern images used in optical measurement

The invention relates to measuring a phase-modulated signal 5. The signal is measured along at least five different steps (P1-P5) corresponding to preselected phase angles of the carrier wave 4. From the associated sets of measured values, at least three sets of measured values are formulated in a manner that from each of the sets a phase value [.phi..sub.i =arctan (Z.sub.i /N.sub.i) where i is equal to or greater than 3] can be calculated. The same correct phase value is computed based upon these three sets for a signal with the frequency of the carrier wave. The essence of the invention is finding that linear combinations of a.sub.i Z.sub.i and a.sub.i N.sub.i can be used for the computation of an accurate phase measurement where the factors a.sub.i are selected so that the phase error, as a function of the preselected phase steps, has at least three zero positions among the measured phase steps (P1-P5). As a result, the systemic errors that normally accompany phase measuring are significantly reduced. The invention is particularly suitable for the evaluation of bar pattern images and multiple-bar pattern images.

TECHNICAL FIELD 
The invention relates to measuring the phase of a signal which is modulated 
on a carrier wave by phase modulation, the intensity of the modulated 
signal being recorded and measured at different phase angles of the 
carrier wave. The measured intensities are used to determine values 
representative of the sine (Z) and cosine (N) of the phase value, which is 
then calculated from the ratio Z/N. 
BACKGROUND 
This type of measurement is used in many fields of measuring technology and 
is a particularly useful technique for optical measuring. The actual 
measuring signal is superimposed on a carrier wave so that the measurement 
is expressed by a modulation of the frequency or phase of the combined 
signal. In phase-measuring technology, the signal is measured along a 
plurality of different phase steps defined by preselected phase angles of 
the carrier wave. For instance, the simplest known formula for such 
evaluation results when three measured values, at phase angles .pi./4, 
3.pi./4, and 5.pi./4 relative to the carrier wave, are recorded. 
Due to the effect of the phase modulation, the combined signal is actually 
measured at slightly different phase angles, that is, the phase angles 
relative to the combined signal do not exactly correspond to the desired 
phase angles relative to the carrier wave. As a result, erroneous phase 
values of the modulation signal are measured systematically. Therefore, 
the results achieved with this type of method are adequately accurate only 
when the frequency of the modulation signal is considerably lower than the 
frequency of the carrier wave. 
Phase-measuring techniques are often used in optical-measuring technology 
for the evaluation of bar pattern images. For example, U.S. Pat. No. 
4,744,659 to Kitabayashi discloses an interferometer where the reference 
and measuring beams interfere at predetermined angles of inclination 
relative to a detector surface. As a result of this inclination of the two 
beams, a bar pattern representing a spatial carrier wave is generated on 
the detector surface. The frequency of this carrier wave is determined by 
the angle of inclination. Deviations of the surface profile of the 
measured surface from the surface profile of the reference mirror result 
in a spatial modulation of the bar image, that is, the phase angle of the 
bar image deviates locally from the phase angle of the carrier wave by an 
amount which is determined by the angle of inclination. The intensity 
distribution of the bar pattern is measured and, as a result of two 
Fourier transformations of the intensity distribution, the deviation of 
the phase angle of the bar pattern from the carrier wave is computed. By 
means of a window function, a sideband of the spatial frequency spectrum 
is filtered out. 
However, the two Fourier transformations require such a significant amount 
of computation time that an evaluation of the interferograms in video real 
time is not possible. In addition, filtering out the sideband has the 
effect of a low-pass filter, thereby changing the measured value. 
An alternative method for evaluating multiple-bar interferograms by Fourier 
transformation has been disclosed in Optical Engineering, Vol. 23, No. 4, 
page 391 (1984), where the measured intensity distribution of the bar 
pattern is first multiplied by a function of the frequency of the carrier 
wave, and then a convolution of the product is performed with a window 
function. This window function is selected in such a way that--for 
calculating the phase value in one point of the interferogram--the 
interferogram intensities of a spatial region covering several periods of 
the carrier wave are used. However, also with this method, the convolution 
of the measured intensity values over several periods of the carrier wave 
has the effect of a low-pass filter, resulting in a reduction of spatial 
resolution. Further, this prior art method does not provide an analysis of 
errors occurring in the computation of phase values, particularly when the 
bar frequency deviates from the carrier frequency. 
Therefore, known phase-measuring technology is burdened by the 
above-mentioned systemic errors, and the methods for evaluation of 
multiple-bar interferograms provide correct phase values principally only 
when the bar frequency of the bar pattern corresponds to the bar frequency 
determined by the angle of inclination, that is, when the profile of the 
measured surface corresponds fairly closely to the profile of the 
reference surface. Further, since this method measures the deviations of 
both profiles, the values relating to the sample being measured exhibit 
this systemic error. 
Another known phase-measuring technique, sometimes referred to as 
phase-shift interferometry, has been described in Applied Optics, Vol. 22, 
No. 21, page 3421 (1983). According to this method, several interferograms 
are recorded at time intervals without a spatial carrier wave. Instead, a 
time carrier wave is generated in that, between the recording of each 
interferogram, the reference mirror is shifted parallel to the optical 
axis (n-1) times by the same distance .lambda./2n, where .lambda. is the 
wavelength of the light in the interferometer. This results in a phase 
shift of 2.pi./n, where n is the number of interferograms. By using at 
least four interferograms, identical points on each of the interferograms 
can be used to calculate a phase value .phi.=arc tangent (Z/N), where Z 
and N (relating, respectively, to the sine and cosine functions of the 
phase value) are computed from the intensities of the respective 
interferograms. 
The accuracy attainable with this just-described phase-shift method is 
essentially a function of the accuracy with which the reference mirror is 
shifted relative to the intended position. Therefore, high-quality 
expensive piezo translators are used for shifting. 
An analysis of the error in phase value, as a function of the phase shift 
which has in fact occurred, shows that, in the case of the intended phase 
shift, the error has a value of zero; and in the case of any deviation 
from the intended phase shift, the error increases quantitatively in a 
linear direction. The last-cited reference suggests that the measuring 
procedure be carried out twice in sequence. Between the two passes, the 
phase is shifted again by .pi./2. If, after the first pass, the phase 
value is computed based on the equation tan(.phi..sub.1)=Z.sub.1 /N.sub.1 
and, after the second pass, based on the equation tan(.phi.2) =Z.sub.2 
/N.sub.2, an improved phase value tan (.phi.)=(Z.sub.1 +Z.sub.2)/(N.sub.1 
+N.sub.2) is obtained. 
The present invention is a method of the above-described type in which 
systemic measuring errors are significantly reduced. 
SUMMARY OF THE INVENTION 
At least three sets of measured values are recorded so that, from each of 
said three sets of measured values, a phase value .phi..sub.1 =arctan 
(Z.sub.i /N.sub.i) can be computed (i is the i-th measured value set, i=1, 
2, . . . , m; m.gtoreq.3). For a signal having the frequency of the 
carrier wave, the same correct phase value .phi.=.phi..sub.i for all i is 
computed correctly from all three sets of measured values. The significant 
feature of the inventive method is that now the linear combination of 
Z.sub.i and N.sub.i can be used to compute a correct phase value 
##EQU1## 
The a.sub.i are selected in such a manner that the phase error, as a 
function of the phase shift, has at least three zero positions. It is also 
possible to select the a.sub.i so that two or more of the zero positions 
may change to a single zero position, so that the error function, as well 
as the first and the second derivation of the error function, has a value 
of zero at the zero position. Defining equations for the a.sub.i may be 
obtained by applying the same values to the expressions 
##EQU2## 
for several different phase steps. 
As has been disclosed in Applied Optics, Vol. 22, No. 21, Page 3421 (1983), 
the phase error response, as a function of the actual phase steps, is 
proportional to sin 2.phi. with one amplitude. Quantitative maxima of the 
phase error occur in the immediate range of phase values 
.phi.=-135.degree., .phi.=-45.degree., .phi.=45.degree., 
.phi.=135.degree.. The error function of the phase value is identified as 
the maximum amplitude that results when the actually measured phase steps 
deviate from the nominal phase steps. 
The invention is based on the finding that the number of zero positions of 
the error function may be greater when more sets of measured values in the 
linear combination are used. Preferably, for each of said sets, the signal 
is measured at three different phase angles. The number of measured values 
in each set can be minimized. Further, a few of the same measured values 
can belong to several different sets of measured values. If the signal is 
measured overall at m different phase angles, the evaluation may be 
expressed by a formula with an error function of (m-2) zero positions. 
Preferably, all measured values should also be recorded within one period 
of the carrier wave. This results in a high measuring resolution, and 
there is little low-pass filtering. 
The inventive method is especially suitable for the evaluation of bar 
images, particularly multiple-bar images. Multiple-bar images may be 
generated by interferometry or by recording a bar pattern projected on a 
sample. The inventive method permits a highly accurate separate analysis 
of individual multiple-bar images. However, even when the inventive method 
is used in multiple-bar interferometry, a high measuring accuracy is still 
assured. 
The hardware for carrying out the method of the invention comprises at 
least five detectors that measure the signal at five different phase 
angles. The output signals of each detector are fed to two different 
amplification units where said output signals are amplified by fixed 
factors and accumulated in two different addition units. Using the 
amplified and accumulated detector signals, an analyzing unit computes a 
corresponding phase value by forming the arc tangent of the quotient of 
the output signals of both addition units. The just-described operation 
may be carried out rapidly with modern digital electronics, the 
computation of the phase .phi. from the two values Z and N being 
accomplished by accessing a look-up table. The amplification factors of 
the two amplification units for each detector are selected so that the 
phase error, as a function of the phase steps, has at least three zero 
positions. When the detectors are opto-electronic sensors, the phase value 
for each point of an optical-bar pattern can be calculated very 
accurately. 
In the preferred embodiment, opto-electronic sensors are arranged in a 
matrix in which the numbers of lines and columns are equal, and spatial 
resolution is improved by running the bars of the bar pattern diagonal to 
the lines and columns of the sensors. Preferably, the camera comprises a 
CCD sensor, and the bar pattern is run diagonal to the columns and lines 
of the camera sensor. 
The invention achieves high spatial resolution and accurate analysis of an 
individual multiple-bar image by providing two convolution components 
that--by convolving the brightness values of an image recorded by the 
camera--compute the sine (Z) and cosine (N) of the phase angle of the 
modulated signal from which a phase value belonging to each image point is 
computed. The weighted convolution values (with which the brightness 
values are individually multiplied) are selected in such a manner that the 
phase error, as a function of the phase shift, has at least three zero 
positions. 
The invention permits analysis of multiple-bar image patterns in video real 
time with an arrangement that is insensitive to external interferences, 
for example, vibrations.

DETAILED DESCRIPTION 
FIG. 1a shows a phase-modulated signal 5 expressed as I(P)=A+Bcos(.phi.+P), 
where the variable P may define either location or time. Signal 5 results 
from the phase modulation of the periodic carrier wave 4 which is 
expressed as T(P)=A+Bcos(P). The problem to be solved by phase-measuring 
technology is to measure the phase value .phi. as a function of the 
variable P. The first option, in which the variable P identifies location, 
is particularly applicable to the analysis of multiple-bar images. The 
second option, in which the variable P identifies time, has particular 
applicability to phase-shift interferometry where several temporally 
recorded interferograms are analyzed. 
The phase-modulated signal I(P) is measured at five different phase angles, 
that is, at five different values (P1 to P5) of the variable P. The 
differences between these phase angles represent preselected "phase 
steps". The related phase-modulated signals I1 through I5 are given by the 
following 5 equations: 
EQU I1=A+cos[P1]*B*cos[.phi.]-sin[P1]*B*sin[.phi.] 
EQU I2=A+cos[P2]*B*cos[.phi.]-sin[P2]*B*sin[.phi.] 
EQU I3=A+cos[P3]*B*cos[.phi.]-sin[P3]*B*sin[.phi.] 
EQU I4=A+cos[P4]*B*cos[.phi.]-sin[P4]*B*sin[.phi.] 
EQU I5=A+cos[P5]*B*cos[.phi.]-sin[P5]*B*sin[.phi.] (1) 
With these five equations, three different determinations can be made of 
the phase values at point P3, namely, by using (I1, I2, I3); (I2, I3, I4); 
and (I3, I4, I5); each of which should produce the same phase value 
.phi.=.phi.1=.phi.2=.phi.3: 
EQU .phi..sub.1 =arctan [(I1*(cos[P2]-cos[P3])+I2*(cos[P39 
-cos[P1])+I3*(cos[P1]-cos[P2]))/(I1*(sin[P2]-sin[P3])+I2*(sin[P3]-sin[P1]) 
+I3*(sin[P1]-sin[P2]))] 
EQU .phi..sub.2 =arctan [(I2*(cos[P3]-cos[P4])+I3*(cos[P49 
-cos[P2])+I4*(cos[P2]-cos[P3]))/(I2*(sin[P3]-sin[P4])+I3*(sin[P4]-sin[P2]) 
+I4*(sin[P2]-sin[P3]))] 
EQU .phi..sub.3 =arctan [(I3*(cos[P4]-cos[P5])+I4*(cos[P59 -cos[P39 
)+I5*(cos[P3]-cos[P4]))/(I3*(sin[P4]-sin[P5])+I4*(sin[P5]-sin[P3])+I5*(sin 
[P3]-sin[P4]))] 
However, in accordance with the invention herein, the three phase values 
.phi.1, .phi.2, .phi.3 are not computed separately as just set forth. 
Instead, an average phase value .phi. is determined as follows: 
EQU .phi.=arctan Z/N=arctan [(a.sub.1 Z.sub.1 +a.sub.2 Z.sub.2 +a.sub.3 
Z.sub.3)/(a.sub.1 N.sub.1 +a.sub.2 N.sub.2 +a.sub.3 N.sub.3)](2) 
The expressions Z and N represent, respectively, sine and cosine functions 
of the phase angle of the modulated wave and, therefore, their ratio Z/N 
represents the tangent of the phase angle. The individual expressions 
Z.sub.1, Z.sub.2, Z.sub.3 and N.sub.1, N.sub.2, N.sub.3 are calculated 
from the measured intensities of the signal, while the factors a.sub.1, 
a.sub.2, a.sub.3 are selected such that the phase error, as a function of 
the phase steps, has at least three zero positions. These various 
expressions will now be discussed in greater detail. 
If the phase-modulated signal is measured generally at k values of the 
variable P, instead of at just five values of the variable P, the linear 
combinations of a total of (k-2) expressions Z.sub.1, N.sub.1 with i=1 
through (m=k-2) can be used to compute the phase value: 
##EQU3## 
In this general case, the a.sub.i are selected so that the phase error, as 
a function of the phase steps, has (m) zero positions. A few of these zero 
positions may also have changed to a higher order. In order to compute 
these a.sub.i, additional defining equations are formulated by using the 
intensity values for each of these further positions and then making their 
respective equations 
##EQU4## 
equal to each other. 
In the specific example shown in FIG. 1a, phase-modulated signal 5 is 
measured at values (P1-P5) of the variables P which are shifted relative 
to carrier wave 4 by 90.degree. in each case. The nominal phase steps are 
all identical in this case and are 90.degree.. 
FIG. 1b shows examples of phase errors, plotted as a function of the phase 
steps, when the measured phase step deviates from the nominal step. As 
disclosed in Applied Optics, Vol. 22, 21, page 3421 (1983), phase error as 
a function of the phase step indicates a response between the measured 
points which is proportional to sin (2.phi.) with an amplitude E. 
Quantitative maxima occur in this case in the immediate vicinity of the 
phase values .phi.=-135.degree., .phi.--45.degree., .phi.=45.degree., 
.phi.=135.degree.. The error function of the phase values is the maximum 
amplitude E of the resulting deviation between the measured and nominal 
phase steps. 
In FIG. 1b, the phase error curve (1) results with the use of only three 
measured points for computation of the phase value .phi.. This error 
function shows that the correct phase value can be computed only when the 
measured distance between adjacent phase-step points (P1-P5) is exactly 
equal to the nominal phase step, .pi./2. If the phase steps deviate from 
this nominal phase step, the phase error increases quantitatively linearly 
for each successive deviation from the nominal phase step. 
Error curves (2) and (3) result when five measured points (P1-P5) are used, 
i.e., where k=5 in equation (3). Curve (2) identifies the error when the 
following values are selected: a.sub.1 =1, a.sub.2 =.sqroot.3, and a.sub.3 
=1; namely, the error is zero at phase steps of 60.degree., 90.degree., 
and 120.degree.. Using different values, curve (3) identifies the error 
when a.sub.1 =1, a.sub.2 =2, and a.sub.3 =1; error function (3) has a 
triple zero position at the phase step of 90.degree.. It can be seen that 
curves (2) and (3) are substantially flatter than curve (1), and that as a 
result of the values selected for a.sub.i in each of these cases, even if 
the distances between measured points (P1-P5) deviate from the nominal 
phase steps, the error of the calculated phase values remains low. When 
the phase value .phi. is calculated using equation (3) with selected 
values for a.sub.i which result in zero error at least at three of the 
phase-step angles, the error remains low. 
A camera 11 records a two-dimensional bar pattern, and its output signals 
are digitized in an analog-to-digital converter and fed to an image 
storage 13. The image storage acts only as a buffer. It may be omitted if 
the subsequent analytical circuit is sufficiently fast. The digitized 
camera signals are then fed to two convolution blocks 14a, 14b. 
Convolution block 14a computes a first expression (Z) by a two-dimensional 
convolution operation, and convolution block 14b computes a second 
expression (N), also by a convolution operation. 
These two-dimensional convolution operations in convolution blocks 14a, 14b 
can best be explained by reference to FIG. 2b, which shows a 
10.times.10-pixel section of camera sensor 11a in which each image point 
is identified as a square. A portion of the recorded bar pattern is 
indicated by the dotted lines 22. This bar pattern extends diagonally to 
the lines and columns of camera sensor 11a. 
For each image point of camera sensor 11a, a related phase value is 
computed that considers the measured intensity values of its eight 
adjacent image points. Therefore, to determine the intensity value for 
each point, the measured intensity values within a convolution window of 
3.times.3 points are convolved; and, within each such convolution window, 
each point (P.sub.ij)(i=4,5,6; j=2,3,4) is associated with two weighted 
convolution values (Z.sub.ij) and (N.sub.ij), where (Z.sub.ij) is 
implemented in convolution block 14a and (Nij) is implemented in 
convolution block 14b. 
Below is a list of weighted convolution values (Zij) and (Nij) associated 
with the points (Pij) of the convolution window: 
______________________________________ 
P.sub.ij Z.sub.ij 
N.sub.ij 
______________________________________ 
P42 -2 -2 
P52 3 -1 
P62 0 2 
P43 3 -1 
P53 0 4 
P63 -3 -1 
P44 0 2 
P54 -3 -1 
P64 2 -2 
______________________________________ 
Using these weighted convolution values (Z.sub.ij, N.sub.ij), the measured 
intensity values of the camera sensor at the respective points (P.sub.ij) 
are multiplied in convolution block 14a, and then these nine convolution 
products are added to the expression 
##EQU5## 
Analogously, the weighted convolution values (N.sub.ij) are multiplied in 
convolution block 14b with the measured intensity values of the camera 
sensor at the point (P.sub.ij) and then added to the expression 
##EQU6## 
These summed values for expressions Z and N are then associated with that 
image point which is in the center of each convolution window, that is, 
with the point marked P53 in the window illustrated in FIG. 2b. 
The outputs of both convolution blocks 14a, 14b are fed to an arctan block 
15 that computes the arc tangent from the ratio Z/N of the two calculated 
expressions Z and N. For this purpose, the function values of the arc 
tangent are stored in a look-up table. The phase values .phi., which are 
computed in this manner for each image point, are then stored as a phase 
image in an image storage 17b. 
Of course, to determine phase values over the entire camera image, all 
image points of the camera sensor are scanned by the convolution window. 
This is indicated by the two arrows Pf1, Pf2 in FIG. 2b. 
In a subsequent subtraction unit 18, reference values stored in another 
image storage 17a are deducted from the phase values in image storage 17b. 
These reference phase values are mathematically computed phase values 
corresponding to a nominal measured value, as well as phase values 
obtained with a calibration measurement. The difference between the 
measured and reference phase values is stored temporarily in another image 
storage 19, then converted to analog in a digital-to-analog converter 20, 
and finally displayed in graph form on a monitor 21 used for data output. 
When the bar pattern is oriented diagonal to the lines and columns of the 
camera sensor, a 3.times.3 convolution block of image points forms, in 
effect, a detector arrangement which measures the bar pattern along five 
equally-spaced steps. Referring to the 3.times.3 block of image points 
outlined in FIG. 2b, the five step detectors are formed by the following 
image points: (1) P64; (2) the average of P54 and P63; (3) the average of 
P44, P53, and P62; (4) the average of P43 and P52; and (5) P42. 
This diagonal orientation of the bar pattern is advantageous for two 
reasons: First, since a 3.times.3 block of image points measures five 
phase steps, the error function can have three zero positions 
perpendicular to the direction of the bars so that, even with a change of 
the bar frequency, the phase values can be determined with great accuracy. 
Second, the distance between the image points measured perpendicular to 
the direction of the bars, is 1/.sqroot.2 smaller than the distance of 
pixels in adjacent lines and columns of camera sensor 11a, thereby 
enhancing spatial resolution. 
For simplification of the explanation, FIG. 2b shows the window as applied 
to a 3.times.3-pixel section of the camera sensor. However, phase value 
errors may be reduced even further if the intensity values measured over 
an area of 5.times.5 image points are used for the convolution operation. 
With this larger section, the bar pattern can be measured over nine 
equally-spaced phase steps; and phase errors can be definitely reduced, 
since the weighted convolution values can be selected so that the error 
function (perpendicular to the direction of the bars, that is, diagonal to 
the columns and lines of the camera sensor) has seven zero positions, and 
three zero positions are possible in the direction of the lines and 
columns, respectively. 
Another embodiment of the invention is illustrated in FIG. 3, which shows a 
measuring arrangement comprising nine photosensors 30a-30i arranged in a 
square. The output signals of each photosensor 30a-i are fed to two 
amplification groups 31a-i, 32a-i, respectively, which amplify the output 
signals of the photosensors 30a-i by fixed factors. 
The output signals of the first amplification group 31a-i are accumulated 
in a first addition unit 33 to calculate an expression Z, and the output 
signals of the second amplification group 32a-i are added in a second 
addition unit to calculate a second expression N. From the ratio of both 
calculated expressions (Z/N), the arc tangent is computed in an arc 
tangent block 35 and fed to data output as indicated by arrow 37. For the 
determination of the arc tangent, the function values of the arc tangent 
are stored in a look-up table 36. 
The amplification factors of the two amplification groups 31a-i32a-i are 
selected analogous to the weighted convolution values of a 3.times.3 
convolution window, namely, so that the error function of the phase value 
in a direction diagonal to the square of the photosensors 30a-i has at 
least three zero positions. Such a permanently wired sensor arrangement is 
particularly advantageous for distance measurement with an interferometer 
such as that represented schematically in FIG. 4. 
The interferometer shown in FIG. 4 comprises a polarizing beam splitter 41 
which deflects an incident laser beam 40 into a measuring beam 42 and a 
reference beam 43. After passing through a .lambda./4--plate 44, reference 
beam 43 is reflected back into itself by a reference mirror 45. 
After passing through a second .lambda./4--plate 46, measuring beam 42 is 
reflected by a reflecting surface 49 which is movable in the direction of 
the two arrows 47 and 48. Reflected measuring beam 42 and reflected 
reference beam 43 are deflected through polarizing beam splitter 41 to a 
Wollaston prism 50 which causes the two partial beams to be inclined 
relative to each other. 
A polarizer 51, positioned behind the Wollaston prism, allows the beams to 
interfere with each other. As a result of the inclination, when reflective 
surface 49 is positioned perpendicular to measuring beam 42, a bar pattern 
is generated which has a carrier frequency. This bar pattern is 
appropriately detected by a detector 52 such as that indicated by FIG. 3. 
A shift of reflective surface 49 in the direction of the arrow causes a 
change of the phase (.phi.) detected by detector 52. 
When reflective surface 49 is tilted, the carrier frequency changes. The 
phase value computed by detector 52, however, is largely independent of 
this carrier frequency, so that the phase value is also largely 
independent of a tilting of reflective surface 49.