Surface shape measurement apparatus

A surface shape measurement apparatus of non-contact type is based upon acousto-optical scanning means for producing a pair of polarized light beams differing in frequency by a fixed amount, which are scanned twice in succession across the surface to be measured in a succession of steps, with an electrical signal being produced by heterodyne interference of the resultant reflected light. Variations in phase of this signal, produced during the first scan, are processed to derive a set of data values representing measurement errors produced during the successive steps, which are utilized to correct a set of phase measurement values obtained during the second scan. The corrected phase measurement values thus obtained are integrated to produced surface shape data, with an accuracy of approximately 0.01 .mu.m or better being attainable.

BACKGROUND OF THE INVENTION 
In recent years, the accuracy of machining which is possible using 
precision machine tools has progressed substantially, and as a result 
there is an urgent requirement for apparatus to measure the shape and 
roughness of a machined surface to a substantially higher level of 
accuracy than has been possible in the prior art. 
The principal method of precision surface shape measurement which has been 
used on a practical engineering bases until now is based upon homodyne 
interferency of light. With this method, two light beams of identical 
frequency are directed onto the surface under measurement, and 
interference (i.e. homodyne interference) resulting from this is sensed 
and measured. Such a method provides a maximum level of measurement 
accuracy of the order 0.1 .mu.m, which is insufficient for many 
present-day applications. 
A modification of the homodyne interference method has been proposed, 
whereby data representing the phase relationships and amplitude of 
interference fringes produced by homodyne interference is processed, to 
thereby derive optical path differences and hence measure surface height 
variations to a high degree of precision. However such a method requires 
complex and hence expensive data-processing circuits, which has prevented 
its practical implementation. 
A further disadvantage of prior art types of surface shape measurement 
apparatus is that in order to scan light beams over the surface, to derive 
surface shape information, the light beams are generally held in a fixed 
orientation while body having the surface to be measured is moved with 
respect to the light beams. The resultant errors which result, due to this 
physical movement of the surface, set a limitation to the accuracy which 
can be obtained by practical types of apparatus. 
SUMMARY OF THE INVENTION 
It is an objective of the present invention to overcome the disadvantages 
of the prior art described above, and to provide a surface shape 
measurement apparatus which achieves a very high degree of measurement 
accuracy by utilizing heterodyne interference of light to measure minutes 
variations in surface height as differences in path length of a pair of 
light beams incident on the surface under measurement, which are scanned 
across the surface in a succession of uniform steps. In order to maintain 
the very high level of accuracy attainable with the heterodyne 
interference of light method, the light beams used for measurement are 
successively shifted, by means of an acousto-optical light deflector, 
while the body having the surface under measurement is held stationary. 
Due to this shifting of the direction of the scanning light beams, the beam 
positions will deviate substantially from the optical axis of the optical 
system used to direct and focus the light beams onto the surface, which 
inevitably results in measurement errors. With the very high level of 
measurement accuracy involved (e.g. to 0.01 .mu.m or better), it is 
impossible to apply any form of fixed compensation for such errors, which 
can vary in an unpredictable manner for a variety of caused, for example 
as a result of small changes in the operating temperature of the optical 
system, etc. For this reason a surface shape measurement apparatus 
according to the present invention performs each surface measurement 
operation, along a line section of the surface under measurement, as two 
successive scans over the surface. During the first scan, a pair of light 
beams which differ in frequency by a fixed amount and are spaced apart by 
a fixed distance light beams are moved in a succession of steps of uniform 
amplitude along the surface. The level of interference produced between 
the resultant light beams reflected from the surface is converted into an 
electrical signal, whose deviation from a reference phase value during 
each step is measured. The resultant phase difference values are then 
processed to derive a set of values each representing an amount of 
measurement error produced during a scanning step. A second scan is then 
performed, with the amplitude of the steps in this case being identical to 
that for the first scan, and with the distance separating the light being 
made identical to the scanning step amplitude, so that the trailing light 
beam successively overlaps positions previously illuminated by the leading 
light beam. As a result, a set of phase difference values are derived for 
the scanning steps, each of which results from a combination of a 
difference between the heights of the positions of incidence of the light 
beams on the surface under measurement during the corresponding step and a 
certain amount of measurement error, the latter resulting from the causes 
described above. Processing is then performed on the latter measured phase 
difference values, to subtract the previously derived measurement error 
values from the measured phase difference values. In this way, a set of 
corrected phase difference values are produced. These are integrated, to 
provide the required data representing the shape of the surface under 
measurement. 
To ensure accuracy of phase measurement, a pair of light beams which emerge 
from the acousto-optical light deflector, and which are successively 
deflected in uniform angular increments to provide the scanning steps 
described above, are split into two pairs of light beams. These constitute 
a probe light beam pair, which are directed onto the surface under 
measurement, as described above, and a reference light beams pair which 
are used to produce a phase reference signal, used for comparison with the 
signal produced by heterodyne interference of the reflected light. Use of 
such a reference light beam pair serves to eliminate many sources of 
measurement inaccuracy, such as the effects of temperature upon the 
characteristics of the optical system, etc, and ensures practicability of 
use of such an apparatus in a normal industrial environment.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Before describing in detail an embodiment of a surface shape measurement 
apparatus according to the present invention, a brief summary will be 
given of the basic principles of an acousto-optical light deflector 
suitable for use with the present invention. Referring to FIG. 1A, an 
acousto-optical light deflector 10 comprises a medium 9 which is 
transparent to light and suited to propogation of ultrasonic acoustic 
waves, with a piezo-electric transducer 15 acoustically coupled to the 
medium. Numeral 17 denotes an acoustic absorption member. Numeral 14 
denotes a combination of a double-balanced modulator circuit and a power 
amplifier circuit for amplifying the modulator output. A carrier signal 
source 12 inputs a high-frequency carrier signal at frequency f.sub.2 to 
the modulator circuit, while a modulation signal source applies a 
sinusoidal waveform modulation signal of lower frequency, f.sub.m, which 
is applied as a modulation signal to the modulator circuit. As a result, 
an output signal is applied to piezo-electric transducer 15 having the 
waveform illustrated in FIG. 1B, i.e. which comprises the sideband 
frequency components (f.sub.a +f.sub.m) and (f.sub.a -f.sub.m). Ultrasonic 
travelling acoustic waves are thereby generated in medium 9, as indicated 
by numeral 11, which produce successive regions of relatively increased 
and decreased coefficient of refraction in medium 9. As a result, medium 9 
serves to diffract and to frequency modulate light which passes 
therethrough. Numeral 18 denotes a beam of polarized light which is 
emitted from a source such as a laser and enters medium 9 at an angle 
.theta..sub.B with respect to the transverse direction of the transverse 
direction of the travelling acoustic waves, and is thereby split into a 
non-diffracted component 24 and two diffracted component light beam 20 and 
22. This diffraction phenomenon has been described in detail in, for 
example, a paper by Baronian presented at the IEEE 1974 Region Six 
Conference, entitled "Acousto-optic Bragg Diffraction Devices and their 
Applications", and is basically analagous to the diffraction of X-rays in 
crystals. The optimum value of the input angle of incidence .theta..sub.B, 
with regard to maximizing the diffracted light component, is therefore 
referred to as the Bragg angle. 
Designating the frequency of the light waves in the incident light beam 18 
as f.sub.0, frequency modulation of the incident light beam is performed 
within medium 9, whereby the diffracted component light beams 20 and 22 
have the respective frequencies (f.sub.0 +f.sub.a +f.sub.m) and (f.sub.0 
+f.sub.a -f.sub.m). 
The angle of divergence between the two diffracted light beams 20 and 22 is 
determined by the frequency difference between them, and hence can be 
varied by variation of the modulation frequency f.sub.m. 
The degree of diffraction produced, and hence the direction at which the 
pair of light beams 20 and 22 emerge from the acousto-optical light 
diflector, can be varied by varying the carrier frequency f.sub.a. However 
to produce major changes in this direction, it is necessary to ensure that 
the direction of incidence of the incoming light beam is held close to the 
Bragg angle. In a surface shape measurement apparatus according to the 
present invention, scanning of a pair of light beams across a surface 
under measurement is performed by producing the light beams from an 
acousto-optical light deflector as illustrated in FIG. 1A, and sweeping 
the carrier frequency f.sub.a in a succession of decrements or increments 
of uniform value, to thereby successively deflect the light beams 20, 22 
by successive angular amounts. The value of the Bragg angle varies in 
accordance with the frequency of the acoustic waves generated within 
medium 11, i.e. in accordance with the carrier frequency f.sub.a. Thus if 
only a single piezoelectric transducer 15 is utilized, as shown in FIG. 1A 
it will not be possible to vary the beam deflection over a substantial 
angle. However this problem can be overcome by using a "phased beam 
array", i.e. a set of piezoelectric transducers which are successively 
selected as the carrier frequency changes, and are respectively positioned 
such as to maintain the angle between the wavefront of the travelling 
acoustic waves and the incident laser light beam close to the Bragg angle 
throughout the carrier frequency sweep. 
A suitable drive circuit to perform the functions of circuit blocks 12 and 
14 in the example of FIG. 1A, is commercially available from IntraAction 
Corp. of Bensenville, Ill., USA, under the designation Model DE-70M VCO 
Deflector Drive. This incorporates a voltage-controlled oscillator (VCO) 
serving as carrier signal source 12, whose frequency can be varied to 
provide the sweep function described above by application of a suitable 
control signal waveform. An acousto-optical light deflector for use with 
this drive circuit is marketed by the same company under the designation 
Model AOD-70 Acousto-Optic Light Deflector. 
With the heterodyne interference of light method of surface shape 
measurement, a pair of light beams of mutually different frequencies, 
spaced a fixed distance apart, directed onto the surface under measurement 
to be reflected therefrom, and the reflected beams are made to interfere 
while incident on a photo-electric transducer. A beat-frequency signal is 
thereby produced by the transducer, varying in accordance with the 
interference, whose frequency is equal to the frequency difference between 
the two light beams. 
For example, if light of frequency fl is designated as E1, and light of 
frequency f2 is designated as E2, then these can be expressed as time 
functions as follows: 
EQU E.sub.1 (t)=A.sub.1 (t) cos (2.pi.f.sub.1 t+.phi..sub.1 (t)) 
EQU E.sub.2 (t)=A.sub.2 (t) cos (2.pi.f.sub.2 t+.phi..sub.2 (t)) 
Here, A1 and A2 denote amplitudes, and o1, o2 denote phase. 
If these light waves are allowed to interfere, then the amplitude I(t) of 
the interference is given as: 
EQU I(t)=.vertline.E1(t)+E2(t).vertline..sup.2 
If this is converted to an electric current i(t) by a photo-sensor, then 
the following electrical signal can be obtained: 
EQU i(t).alpha.A.sub.1.sup.2 +A.sub.2.sup.2 2.A.sub.1 A.sub.2 cos 
(2.alpha..DELTA.ft +.DELTA..phi.) 
Here, .DELTA.f=f1-f2,.DELTA..phi.=.phi.1-.phi.2 
Changes in phase of this signal can easily be measured if f is in the range 
10.sup.5 to 10.sup.6 Hz, approximately, and this phase will vary in direct 
accordance with changes in phase difference between the two reflected 
light beams. Thus, if such a pair of light beams is reflected from a 
surface, to then fall upon a photo-electric transducer, then any 
difference in surface height between the points on which the beams fall 
will result in a change in the respective path lengths of the reflected 
beams, which can be regarded as change in phase difference between the 
beams. Thus, such a difference in surface height can be measured as a 
shift in phase of the output signal from the photo-electric transducer 
sensing ther light interference. 
This process is illustrated in FIG. 2, in which a pair of light beams 20b, 
22b, are incident on a surface 34 at positions A and B respectively, with 
there being a difference in height z between these positions (i.e. a 
height difference measured along the direction of the incident light 
beams). Such a height difference will produce a corresponding difference 
in path length of the reflected light beams 20c, 22c, when these reach an 
photo-electric transducer, which can be expressed as a phase shift of the 
photo-electric transducer output signal as described above. 
FIG. 3 is a block diagram of an embodiment of a surface shape measurement 
apparatus according to the present invention. A single light beam 18, at a 
frequency f.sub.0, is emitted from a laser light source 28, and is 
incident upon an acousto-optical element 10. Ultrasonic travelling waves 
are generated within acousto-optical light deflector 10 by a drive signal 
applied from a double-balanced modulator circuit 14, operating from a 
carrier signal of frequency f.sub.a produced from a carrier signal source 
12 which comprises a voltage-controlled oscillator circuit (VCO), and a 
modulation signal of frequency f.sub.m applied from a modulation signal 
source 16, as described hereinabove referring to FIG. 1. A pair of light 
beams 20, 22 are thereby output from acousto-optical light deflector 10, 
with an angle of divergence and frequency difference between them whose 
values are determined by the value of modulation frequency f.sub.m and 
which are deflected with respect to the incident laser light beam 18 by an 
angle determined by the carrier frequency f.sub.a. Light beams 20 and 22 
contain the frequency components (f.sub.0 +f.sub.a -f.sub.m) and (f.sub.0 
+f.sub.a +f.sub.m) respectively, where f.sub.0 is the frequency of laser 
light beam 18. 
Numeral 30 denotes an optical isolator, which is made up of a polarizing 
beam splitter and a 1/4 wavelength plate, the latter being disposed on the 
opposite side of the polarizing beams splitter to acousto-optical 
deflector 10. This optical isolator 30 is positioned between 
acousto-optical element 10 and the surface under measurement 34. The two 
light beams 20 and 22 are each split by optical isolator 30 into light 
beams travelling in two different directions. As a result, reference light 
beams 20a, 22a are produced, which do not impinge upon the surface under 
measurement 34, while the light which emerges from optical isolator 30 in 
the other direction, and will be referred to as a probe light beam pair, 
passes through a condenser lens 32 to be thereby focussed onto the surface 
under measurement as two extremely small-diameter spots separated by a 
fixed spacing. The value of this spacing can be varied by altering 
modulation frequency f.sub.m, to thereby alter the angle of divergence 
between light beams 20, 22. 
The resultant light beam pair reflected from surface 34, designated by 
numerals 38, 40, then passes through condenser lens 32 and is reflected by 
optical isolator 30 onto a photo-electric transducer section 56 which 
includes a photo-receptor 55 to perform photo-electric conversion of 
interference between the reflected light beams 38, 40. Numeral 54 denotes 
a photo-electric transducer section which includes a photo-receptor 53 
positioned to perform photo-electric conversion of the interference 
between reference light beams 20a, 22a. Each of these photo-electric 
transducer sections can comprise for example a PIN photo-diode used as a 
photo-receptor, and a current-voltage converter, with the beat frequency 
signal current produced by the PIN photo-diode being converted to a 
voltage signal. 
As described hereinabove, there will be a shift in phase between the 
reflected light beams 38 and 40, whose magnitude will be determined by the 
difference in surface height between the points on surface 34 at which the 
light beams from condensor lens 32 are respectively incident, and with the 
direction of the phase shift being determined by the direction of that 
height difference. As a result, corresponding shifts will occur in the 
phase of the beat frequency signal generated by interference between light 
beams, from photo-electric transducer section 56. If the DC component is 
removed from each of the output signals from photo-electric transducer 
sections 54 and 56, then the resultant AC voltage signal which is output 
on line 43 will be fixed in phase, and will be referred to in the 
following as the phase reference signal, while a phase measurement output 
signal from photo-electric transducer 56 appearing on line 44 will vary in 
phase in direct accordance with any phase difference between reflected 
light beams 38, 40, i.e. in direct accordance with any difference between 
the heights of the positions on surface 34 from which light beams 38, 40 
are reflected. The frequency of both the phase reference signal and the 
phase measurement signal is the difference between the frequencies of 
light beams 22, 20, i.e. 2f.sub.m. In the following, it will be assumed 
that the phase reference signal represents zero phase. That is to say, if 
the probe light beam pair 20b, 22b fall upon positions on surface 34 which 
are perfectly coplanar in a plane aligned perfectly perpendicular to the 
direction of the probe light beam pair, then the phase measurement signal 
will be exactly in phase with the phase reference signal. Any deviation 
from this condition will result in a change in phase of the phase 
measurement signal whose polarity and amplitude are respectively 
determined by the direction and the magnitude of the difference in height 
between the positions of incidence of the probe light beam pair (i.e. 
quantity .DELTA.z illustrated in FIG. 2). The value of the phase of the 
phase measurement signal appearing on line 44 is measured by means of a 
phase comparator circuit 46, which compares the phase measurement signal 
with the phase reference signal phase. 
The value of surface height difference .DELTA.z can be expressed as: 
EQU .DELTA.z=.lambda...DELTA..phi./(4.pi.) 
In the above, .lambda. is the wavelength of the light output from the 
laser, and .phi. denotes the phase of the phase measurement signal, 
defined as described above assuming the phase reference signal phase as 
zero. If a He-Ne laser is used, then .lambda.=0.6328 .mu.m, so that 
.DELTA.z has a value of 8.8 angstroms per degree of change in phase of the 
phase measurement signal. The maximum value of surface height difference 
which can be measured in this way is equal to .+-..lambda./4. 
Numeral 48 denotes a central processing unit, comprising for example 
analog-digital converters, data-processing circuits and memory circuits, 
for processing the phase values produced by phase comparator 46 as 
described hereinafter, and which can be based on a personal computer. 
Numeral 47 denotes a control signal generating circuit, for producing a 
signal of analog type on output line 60 to control the value of carrier 
frequency f.sub.a produced by VCO 12 to thereby control the angle of 
deflection of light beams 20, 22, and a control signal on line 58 for 
controlling the frequency f.sub.m of the modulation signal from modulation 
signal source 16 to thereby control the angle of divergence between light 
beams 20, 22. 
The change .DELTA..theta.d in the angle of deflection of light beams 20, 22 
produced by a change .DELTA.f.sub.a of the carrier frequency is given as: 
EQU .DELTA..theta.d=.lambda..DELTA.f.sub.a /V.sub.a 
Where V.sub.a is the acoustic velocity within acousto-optical light 
deflector 10, and .lambda. is the wavelength of laser light beam 18. 
The angle of divergence .theta..sub.B between deflected light beams 20, 22 
is given as: 
EQU .theta..sub.b =.lambda..2fm/V.sub.a 
Referring now to FIG. 4, the method of scanning over surface 34 is 
illustated. Initially, the probe light beam pair 20b, 22b, spaced apart by 
the fixed separation S.sub.0, are respectively incident on positions r1, 
r2 of the surface, and the height difference Z between them is measured as 
a phase difference as described above. A first scanning step is then 
performed, whereby light beams 20b, 22b are shifted by an amount equal to 
the separation S.sub.0 between them, so that now they are incident on 
positions r2, r3 respectively, that is to say, the trailing beam 20b is 
now incident on the previous position of leading beam 22b. A second step 
is then performed, of equal amplitude to the first, then a third, and so 
on. The intervals between each step, during which the phase measurement is 
performed, are of uniform duration. 
FIG. 5 illustrates a waveform for the analog control signal applied over 
line 60 in FIG. 3 to VCO 12, whereby a succession of uniform increments of 
the angle of deflection of the light beam pair 20, 22 from acousto-optical 
light deflector 10 is produced, to hereby produce a succession of steps of 
the form shown in FIG. 4. Each control voltage step is of uniform 
amplitude, V.sub.i, and of uniform duration, t. 
FIG. 6A shows an example of beam scanning across a surface having a very 
simple shape, for ease of explanation. Initially, the probe light beam 
pair are positioned at positions r1, r2 respectively, while measurement of 
the corresponding value of phase is performed at time t1. The beams are 
then stepped into positions r2, r3 respectively, and phase measurement 
performed at time t2, and so on, until time t7. The resultant set of 
measured phase values is represented graphically in FIG. 6B, on the 
assumption that the overall system does not introduce measurement errors. 
At time t1, both light beams are incident on a perfectly plane portion of 
surface 34 which is aligned perpendicular to the beam direction, so that 
the measured phase value is zero as described hereinabove. At time t2 and 
at time t3, the beams are both incident on a slope of fixed inclination, 
so that the height difference between the positions of incidence is equal 
in each case, and hence the measured phase values are equal. In a similar 
manner, phase values corresponding to the surface shape are measured at 
times t4 to t7, with the phase polarity being reversed when the slope 
inclination angle is reversed. However as the angle of deflection of beams 
20, 22 shown in FIG. 3 deviates from the direction of the optical axis of 
the system which directs and focusses probe light beam pair 20b, 22b, 
errors of increasing magnitude are introduced into the measured phase 
values. As a result, the set of measured phase values obtained for the 
case of FIG. 6A might appear as shown in FIG. 6C. Each of this set of 
values can be regarded as the sum of an error value (whose magnitude 
varies with the degree of beam deflection, i.e. with the scanning 
position) and a true phase value. This set of error values can be regarded 
as varying in accordance with a continuously varying function, which will 
be referred to as the phase error function. The phase error function 
varies as the modulation frequency f.sub.m is varied, but basically 
approximates to a third-order curve having a value of zero when the angle 
of deflection of light beam pair 20, 22 is zero. This is illustrated in 
the graphs of FIG. 7, in which phase error functions .phi.(s)1 and 
.phi.(s)2 are plotted as variations of measured phase o with respect to 
angle of beam deflection, and result for modulation frequencies f.sub.m 1 
and f.sub.m 2 respectively, with phase error function .phi.(s)2 being 
equal to phase error function .phi.(s)1 multiplied by the factor f.sub.m 
2/f.sub.m 1. 
With a surface shape measurement apparatus according to the present 
invention, each measurement operation comprises scanning the probe light 
beam pair over the surface in steps of uniform amplitude. A set of phase 
error values are derived by CPU 48 from the resultant output signals 
produced by phase comparator 46, i.e. values which vary according to an 
error characteristic of the form shown in FIG. 7. These values are then 
subtracted from a set of measured phase values derived by a scanning 
operation the form described hereinabove with reference to FIG. 4, i.e. 
with the probe light beam pair being stepped in a successively overlapping 
manner, a set of phase values being obtained thereby which include the 
measurement errors described above, and will be referred to as the 
uncorrected phase values. The phase error value corresponding to each 
scanning step is then subtracted from the uncorrected phase value obtained 
for that step, to thereby derive a set of corrected phase values, from 
which the effects of measurement system errors have been eliminated. These 
values are then integrated, to produce a set of data values representing 
the surface shape. By repeating a plurality of such double scan operations 
over successively adjacent strips of the surface under measurement, the 
overall surface shape can be obtained. 
A first method of deriving the phase error function will now be described, 
in which each measurement operation comprise a pair of immediately 
consecutive scans over the same surface portion. During the first scan, a 
control signal applied over line 58 from control signal generating circuit 
47 to modulation signal source 16 sets the modulation frequency f.sub.m to 
a value such that angle of divergence of deflected light beams 20, 22 is 
reduced to such a degree that the spacing between probe light beams 20b, 
22b becomes less than the minimum for which the beams can be resolved, 
i.e. less than the distance set by the Rayleigh criterion for resolution 
of adjacent light beams of circular cross-section. The probe light beam 
pair 20b, 22b thereby substantially mutually overlap, so that the phase of 
the output signal produced by photo-electric transducer 56, resulting from 
heterodyne interference between the reflected light beams 38, 40, will be 
independent of variations in height of surface 34. Thus, the set of phase 
values obtained during this first scan will vary in accordance with the 
phase error function, e.g. if the modulation frequency f.sub.m during the 
first scan is assumed to be f.sub.m 1 shown in FIG. 7, then the set of 
values thus derived will vary in accordance with phase error function 
.phi.(s)1. Assuming that the modulation frequency value to be used during 
actual surface measurement is f.sub.m 2 shown in FIG. 7, CPU 48 then acts 
to multiply each of the phase values obtained from the first scan by the 
factor f.sub.m 2/f.sub.m 1. In this way, a set of phase error value is 
obtained, which vary in accordance with phase error function .phi.(s)2. A 
second scan is then performed, as described hereinabove with reference to 
FIG. 4, with control signal generating circuit producing a signal causing 
modulation signal source to set the modulation frequency f.sub.m to the 
value f.sub.m 2, to thereby set the spacing between probe light beam pair 
20b, 22b to an amount equal to the amplitude of each scanning step, i.e. 
spacing S.sub.0 shown in FIG. 4. A set of uncorrected phase values are 
thereby produced from phase comparator 46, each representing the sum of a 
true phase value and an error value. The corresponding phase error values 
are then subtracted from these uncorrected phase values, to derive the 
corrected phase values. These are then integrated, to derive data values 
representing the shape of the surface strip measured during that pair of 
scans. 
The set of true (i.e. corrected) phase values can be expressed as: 
EQU (.phi..sub.p 1, .phi..sub.p 2, . . . , .phi..sub.p n) 
while the set of phase measurement error values can be expressed as: 
EQU (.phi..sub.s 1, .phi..sub.s 2, . . . , .phi..sub.s n). 
The set of uncorrected phase values generated as described above by the 
second scan, can therefore be expressed as: 
EQU .phi..sub.(s+p) 1, .phi..sub.(s+p) 2, . . . , .phi..sub.(s+p) n 
Thus the set of corrected phase values .phi..sub.p i can be derived from 
the set of uncorrcted phase values .phi..sub.(s+p) i by the operation: 
EQU .phi..sub.p i=.phi..sub.(s+p) i-.phi..sub.s i 
The surface shape as measured along the line of scan is thereby obtained by 
integrating the corrected phase values, e.g. is given as the integral: 
##EQU1## 
If CPU 48 comprises a general-purpose type of microcomputer or 
minicomputer, then the above operations can readily be accomplished by 
suitable programming. 
A second method of deriving the phase measurement error function will also 
be described, which is applicable to measurement of a surface shape of 
basically planar configuration, e.g. comprising a number of flat portions 
which may be situated in different mutually parallel planes as illustrated 
in cross-section in the example of FIG. 8A. Assuming that probe light beam 
pair 20b, 22b are scanned across the surface in a succession of steps as 
described above, in direction r, the resultant phase value .phi. produced 
by phase comparator 46 will vary as shown in FIG. 8B. Prior to position r1 
the phase will be zero (assuming a perfectly smooth planar surface), while 
at position r1 the phase will attain a positive value .phi.1 representing 
the magnitude of surface height difference Z1, and will thereafter return 
to zero, and so on, with phase values -.phi.1, .phi.2, -.phi.3 
respectively representing changes in surface height a positions r2, r3 and 
r4 along the line of scan. With this second method of deriving the phase 
measurement error function, each measurement operation consists of a 
single scan along a line portion of the surface, with the spacing between 
probe light beam pair 20b, 22b, being held fixed at the value used for 
surface height measurement (i.e. spacing S.sub.0 which is equal to the 
scanning step amplitude, as shown in FIG. 4). During this scan, a set of 
uncorrected phase values are produced by phase comparator 46 as described 
hereinabove, i.e. with each value representing the sum of a quantity 
corresponding to a surface height difference and a quantity corresponding 
to measurement error, and these are stored by memory means contained in 
CPU 48. In addition, CPU 48 uses these phase values to derive a set of 
measurement error values corresponding to the phase measurement error 
function, which are then used to correct the stored uncorrected phase 
values. Such a set of uncorrected phase values, derived from the surface 
example of FIG. 8A, is illustrated graphically as curve 72 in FIG. 8C. As 
shown, this comprises a smoothly varying characteristic (corresponding to 
successive phase measurements of the planar portions of the surface) 
connected by jumps in phase value of large amplitude, e.g. 74, 75, which 
correspond to the surface height changes of magnitude Z1, Z2, . . . . As 
these phase values are produced by phase comparator 46 during the scan and 
are successively memorized, CPU 48 acts to detect any phase value which 
differs in magnitude from the phase value of the immediately preceding 
scanning step by more than a predetermined maximum value, and to ignore 
that large-magnitude phase value in the subsequent processing. In this 
way, a set of phase values are produced which can be represented as shown 
by characteristic 76 in FIG. 8D. 
The surface measured will not in practice be ideally smooth, however the 
variations in measured phase value resulting from surface roughness are 
essentially random in distribution, e.g. as illustrated by expanded 
portion 78 of characteristic 76 shown in FIG. 9A. This set of phase values 
is therefore subjected to statistical averaging, by CPU 48, to thereby 
derive a set of phase values which follow a smoothly changing average 
characteristic as illustrated by numeral 80 in FIG. 9A. This average 
characteristic is an accurate approximation to the phase measurement error 
function. 
The set of uncorrected phase values, stored as described above, is 
illustrated graphically in FIG. 7C. These uncorrected phase values are 
then read out of storage, and the corresponding phase measurement error 
values for the successive steps, derived as described above, are 
subtracted from the uncorrected phase values by CPU 48. A set of corrected 
phase values are thereby derived, e.g. as illustrated graphically in FIG. 
9B, which include both the small-magintude phase variations resulting from 
surface roughness and the large-magnitude phase variations which result 
from large changes in surface height, e.g. changes brought about by 
machining. This set of corrected phase values is then integrated by CPU 48 
to derive data values representing the surface shape, as measured along 
the line of scan. 
FIG. 10 illustrates an example of an optical system for the heterodyne 
interference of light surface measurement apparatus shown in FIG. 3. 
Numerals 82 and 90 denote cylindrical lenses each of which has a focal 
length of L.sub.1. 84 and 86 are plano-convex lenses, each having a focal 
length of L.sub.2. Numeral 88 denotes a polarizing beam-splitter, and 92 
is a 1/4 wave plate. Numeral 32 denotes a laser condenser lens having a 
focal length of L.sub.0. 
In order to attain aximum resolution for the diffraction produced by 
acousto-optical deflector 10, the light beam produced by laser light 
source 28 should be converted to have an elongated, i.e. elliptical 
cross-sectional shape, with the elongated axis being directed parallel to 
the paper as viewed in FIG. 10. For this purpose, a broad-width beam is 
produced by the combination of cylindrical lens 82 and plano-convex lens 
84. In general, the beat frequency signals which are output from 
photo-electric transducer sections 54 and 56 will differ in amplitude from 
one another, but it is desirable to apply input signals to phase 
comparator 46 which are as uniform in amplitude as possible. The 
difference between the output signal amplitudes from photo-electric 
transducer sections 54 and 56 is basically due to the fact that the 
intensity of light reflected back from the surface under measurement will 
vary in accordance with the reflectance of that surface. To compensate for 
this, the relative amplitudes of signals 43 and 44 can be adjusted, if 
laser light source 28 produces a linearly polarized beam, by rotating the 
laser tube in light source 28 about the optical axis, so as to adjust the 
axis of linear polarization of the output light beam. This will result in 
a corresponding change in the relative proportions of light which is 
transmitted through beam splitter 88 to the surface under measurement and 
the light which is reflected onto photo-receptor 56. 
It is preferable to arrange that the light beams which fall upon 
photo-receptors 52 and 54 are of elongated, i.e. elliptical cross-section. 
One advantage of this is that the intensity of light falling on each 
photo-receptor is thereby increased, by comparison with a circular 
cross-section beam pattern. Another important advantage is that, as the 
position of a photo-receptor is moved along the axis of elongation of such 
an incident beam pattern comprising two interfering beams, the phase of 
the output beat-frequency signal produced by that photo-receptor will 
vary. This allows the relative phases of the output signals from 
photo-electric transducer sections 54, 56 to be adjusted so that the 
output signal from section 54 serve as a "zero phase" reference signal, 
for phase comparison by phase comparator 46. As a result of the shaping of 
the incident beam entering acousto-optical deflector 10, the beams 
emerging from deflector 10 are also of elliptical cross-section, and so 
therefore will the reference beams falling on photo-receptor 54. The pair 
of beams which pass through cylindrical lens 90 are converted to have a 
ciurcular cross-section thereby, so as to minimize the spot size incident 
on the surface under measurement. However the resultant reflected beams 
pass back through sylindrical lens 90, and are thereby converted to 
elliptical cross-section before being reflected onto photo-receptor 58. 
In the embodiment of FIG. 10, the two light beams which are split by 
acousto-optical deflector 10 are indicated as a single beam, for clarity 
of description. In addition, non-diffracted light is omitted from the 
drawing. The actual shapes of the light beams within the optical system 
are as illustrated in the partical detailed cross-sectional view of FIG. 
11, which shows how the probe light beam pair are directed and focussed 
onto surface 34, and the reflected light beams 38, 40 are formed to have 
elongated cross-sections (i.e. elongated in a direction parallel to the 
plane of the paper). The cross-hatched portion in which reflected beams 38 
and 40 overlap constitutes the region in which heterodyne interference 
between these beams is produced, and this portion is directed onto 
photo-receptor 55 to thereby produce a corresponding beat-frequency 
signal. 
Designating the modulation frequency which determines the angle of 
divergence of the two light beams after leaving deflector 10 as f.sub.m, 
the resultant distance S.sub.0 separating the two light beams when they 
become incident upon the body surface being measured is given as follows: 
EQU S.sub.0 =2.multidot.L.sub.2 .multidot.L.sub.0 .multidot..lambda.fm/(L.sub.1 
.multidot.V.sub.0) 
Here, V.sub.0 is the velocity of travel of sound waves within 
acousto-optical deflector 10. If for example V.sub.0 =3.8 km/sec, L.sub.1 
=15 mm, L.sub.2 =500 mm, and L.sub.0 =7 mm, then if f.sub.m =100 kHz, the 
value of separation distance S.sub.0 will be 7 micronmeters. 
The amount of lateral displacement .DELTA.S.sub.d applied to probe beam 
pair 20b, 22b as a result of a change f.sub.a in the carrier frequency of 
the drive signal applied to acousto-optical deflector 10, is given as: 
EQU .DELTA.S.sub.d =L.sub.2 .multidot.L.sub.0 .multidot..lambda.f.sub.a 
/(L.sub.1 .multidot.V.sub.a) 
From the above, it will be understood that the present invention enables 
surface shape to be measured to a very high degree of accuracy, which has 
hitherto been attainable only with extremely complex and expensive 
equipment and moreover with equipment which is generally suitable for use 
only under carefully controlled environmental conditions, rather than in a 
normal industrial environment in which various disturbing factors such as 
oprating temperature fluctuations, vibration, etc are ineviatable, and 
that an apparatus acording to the present invention can be of simple 
configuration yet will provide satisfactory operation in spite of such 
external disturbing factors, due to the use of a dual-scan technique 
whereby the magnitude of errors due to such factors or other causes are 
accurately measured immediately prior to each measurement operation, to 
thereby provide error-free measurement results. 
Although the present invention has been described in the above with 
reference to specific embodiments, it should be noted that various changes 
and modifications to the embodiments may be envisaged, which fall within 
the scope claimed for the invention as set out in the appended claims. The 
above specification shoud therefore be interpreted in a descriptive and 
not in a limiting sense.