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This invention relates to displacement-measuring interferometers, more
especially to Michelson interferometers arranged to provide an output
suitable for use with an automatic reversible fringe counting system.
Use of a reversible fringe counting system allows automatic correction for
any vibration or retraced motion. Conventional systems require two
electrical input signals which vary sinusoidally with path difference in
the interferometer, which are in phase quadrature, and which ideally have
fixed amplitudes.
Usually the signals are provided by photodetectors each arranged to sense
an interferogram, that is, a part of an interference fringe pattern which
changes as the optical path difference in the interferometer changes.
In many prior art arrangements, interferograms are generated which each
comprise a regularly alternating component, which will be referred to as
an a.c. component and which is dependent on path difference between its
two constituent light beams, and a component which will be referred to as
a d.c. component which does not alternate regularly and does not depend on
the path difference.
The magnitude of the d.c. component depends on variations in the alignment
and relative sizes of the interfering beams, attenuation in one or both
interferometer arms, losses due to limiting apertures, and on intensity
fluctuations of the light source, so that this component cannot be removed
by simple subtraction. If the d.c. component increases to a certain level,
operation of the fringe counting system may not be possible.
In U.S. Pat. No. 3,771,875, Russo, an arrangement is disclosed in which
three interferogram signals are provided which can be combined by sum and
difference to give two purely a.c. components with no d.c. contribution.
However, in the Michelson interferometer, in the measuring arm the
radiation is at one direction of polarisation, and in the reference arm it
is at the orthogonal direction of polarisation. The disadvantage of this
is that the interferometer beam splitter must be constructed so that an
area of its surface is polarising and the remaining area is
non-polarising; further the non-polarising area must have values of
reflectance and transmittance which are the same for each state of
polarisation; this is a difficult condition to achieve.
It is an object of the invention to provide an interferometer system
capable of supplying two signals which vary sinusoidally with fixed
amplitude as a function of path difference, which are in phase quadrature,
in which any signal level changes not related to path difference are
substantially eliminated, and in which the interferometer beam splitter is
relatively easy to make.
According to the invention, an improved interferometer system comprises:
a Michelson interferometer comprising an interferometer beam splitter and
two reflecting means arranged to receive radiation from opposite sides of
the beam splitter and to return radiation to those sides whereby two exit
beams of radiation are provided, one from each side of the beam splitter;
a beam splitting and polarising means arranged to receive one exit beam and
to separate the received radiation into radiation at two orthogonal
directions of polarisation whereby two optical interferograms are
provided;
means arranged to receive the other exit beam and to separate from it
radiation at one of said orthogonal directions of polarisation whereby a
third optical interferogram is provided;
the third interferogram being in phase quadrature with one of the first and
the second interferograms, and being in antiphase with the other of the
first and the second interferograms;
characterised in that the interferometer beam splitter has for radiation at
each orthogonal direction of polarisation reflection coefficients on its
two faces which are equal, and has for radiation at one orthogonal
direction of polarisation a transmission coefficient which is equal to its
reflection coefficient.
The effect of this equality of the coefficients is that, as will be
explained in detail below, three interferograms can be provided having the
same ratio of the a.c. to the d.c. component.
It has been stated that the three interferograms must be in a phase
quadrature or antiphase relationship with one another. While the system
operates most effectively, so far as the signal-to-noise ratio is
concerned, when the interferograms differ in phase by, ideally, 90.degree.
and 180.degree., the system can operate when the phases differ
considerably from the ideal situation. Depending on the accuracy required,
the quadrature relationship may be from 89.degree. to 91.degree. with no
significant decrease in accuracy, may be from 75.degree. to 150.degree. if
lower accuracy is acceptable, and the system may still be operable when
the phase difference is 50.degree. or 130.degree..
Further, it has been stated that three of the reflection and transmission
coefficients of the interferometer beam splitter must be equal. While the
tolerance in this equality is not as wide as the tolerance on phase
relationships, it is not essential for operation of the system to have
precise equality, although this will obviously be the preferred condition.
In one arrangement of the interferometer the two reflecting means each
comprise a plane reflector, and the means to receive the other exit beam
comprises a further beam splitter arranged to reflect that beam to a
second beam splitting and polarising means. Four interferograms are
available with this arrangement, and three are selected, depending on
which coefficients of the interferometer beam splitter have been
equalised. The reflectors will usually be plane mirrors.
In another arrangement of the interferometer the two reflecting means each
comprise a retro reflector such as a corner cube, and the means to receive
said other exit beam comprises a polariser arranged to allow passage of
light in one of the two orthogonal directions. In practice the use of
corner cubes is preferred because the system is insensitive to tilt of the
reflectors and light is not reflected back to the source. The corner cubes
can be coated so that the reflections do not introduce unwanted
polarisation effects.
Regarding the requirement that one interferogram is respectively in phase
quadrature and in antiphase with the other two interferograms, this may be
achieved by:
(a) arranging that the interferometer beam splitter provides the required
phase difference;
(b) the combination of an interferometer beam splitter that would normally
produce a 180 or zero degree phase difference between reflected and
transmitted orthogonal polarisations, with an eigth-wave plate in the
optical path between the interferometer beam splitter and one of the plane
reflectors, radiation passing through the plate twice;
(c) the combination of an interferometer beam splitter providing a 180 or
zero degree change with a quarter-wave plate in the optical path between
the interferometer beam splitter and one of the corner cube reflectors,
radiation passing through the plate only once; or
(d) the combination of an interferometer beam splitter which produces a
phase difference between the orthogonal polarisations which differs from
180.degree. or 0.degree., with a complementary phase plate of thickness
suitable for either one or two passages of the radiation so that the sum
of the phase changes gives phase quadrature.
Although three interferograms have now been provided in which the ratios of
the a.c. component to the d.c. component are equal, it is a preferred but
not essential further condition that the overall magnitudes of the three
interferogram signals are equal. The interferograms are usually sensed by
respective photosensitive detectors, and it is preferable if these
detectors can be coupled to amplifiers having equal gains so that any
electronic drift will tend to be the same for each signal. This condition
can be achieved by the use of a variable optical attenuator between the
interferometer outputs and each photocell; in this case the light beam
illuminating the interferometer must comprise components at the said two
orthogonal directions of polarisation of approximately equal magnitude.
Alternatively, when corner cubes are used, the relative intensities of
light in the two orthogonal directions of polarisation can be adjusted to
provide three interferogram signals of equal magnitude.
The invention will now be described by way of example with reference to the
accompanying drawings in which:
FIG. 1(a) illustrates an interferogram signal having optimum fringe
contrast;
FIG. 1(b) illustrates an interferogram having reduced fringe contrast;
FIG. 2 illustrates an interferometer according to the invention;
FIGS. 3(a), 3(b) and 3(c) illustrate the effect of subtracting two pairs of
signals;
FIG. 4 shows a full optical system incorporating an interferometer
according to the invention;
FIG. 5 illustrates a full optical system using corner cubes in place of
plane reflectors in the interferometer; and
FIG. 6 shows an automatic gain control system,
FIG. 1(a) illustrates the output of a photodetector arranged to sense an
interferogram. As the path difference between the two interfering beams
changes, the output varies between zero, corresponding to complete absence
of light, and a maximum on an arbitrary scale. This represents a perfect
situation.
In practice, the output is typically as shown in FIG. 1(b) on the same
arbitrary scale; the output signal of the photodetector varies between a
maximum, which is less than in the perfect arrangement, and a minimum
which is well above the zero level and is due to background illumination.
The signal can be regarded as an a.c. component of constant maximum
amplitude superimposed on a constant d.c. level. (Normally, neither
component will have a constant value).
The problems of using such output signals in conjunction with automatic
fringe counting systems have been explained above.
FIG. 2 illustrates schematically an interferometer according to the
invention comprising a dielectric inteferometer beam splitter 10, and two
plane reflectors 12, 14, arranged as a conventional Michelson
interferometer. The apparatus further comprises an eighth-wave plate 16 in
one of the optical paths, a further beam splitter 18, and two polarising
beam splitters 20, 22.
An illuminating light beam 24, for example from a frequency-stabilised
laser (not shown) passes through the further beam splitter 18 into the
Michelson interferometer, and the further beam splitter also receives one
exit beam from the Michelson interferometer and reflects it to the first
polarising beam splitter 20. The second polarising beam splitter 22
receives the other exit beam directly from the Michelson interferometer.
The entire interferometer according to the invention is indicated by
reference numeral 8.
The polarising beam splitters 20, 22, separate the light incident on them
into two orthogonally polarised components, P and S, the letters referring
to polarisations respectively parallel and perpendicular to the plane of
incidence at the beam splitter 10. Each component contains contributions
from beams which have travelled through the Michelson interferometer by
different routes, and therefore interfere to form interferograms, which
may be numbered from 1 to 4 as shown.
The combination of the dielectric beam splitter 10 and the eighth-wave
plate 16, through which light passes twice, ensures that the
interferograms formed by orthogonal polarisations at each output at
90.degree. out of phase, while the interferograms for the same
polarisations at the different outputs are 180.degree. out of phase. For
example, interferograms 1 and 4 are 90.degree. out of phase, and
interferograms 1 and 2 are 180.degree. out of phase.
If .alpha..sub.1 and .alpha..sub.2 are the interfering amplitudes in each
interferogram and .DELTA. is their phase difference, then at any part of
the overlapping light distributions the resultant intensity is given by:
I=.alpha..sub.1.sup.2 +.alpha..sub.2.sup.2 +2.alpha..sub.1 .alpha..sub.2
cos .DELTA. (1)
The intensity of the background illumination is given by
.alpha..sub.1.sup.2 +.alpha..sub.2.sup.2 which determines the d.c.
component; the maximum intensity change due to variations of .DELTA. is
4.alpha..sub.1 .alpha..sub.2, and the maximum a.c. component therefore has
a peak-to-peak value of 4.alpha..sub.1 .alpha..sub.2. Thus in each
interferogram, the ratio of the a.c. to the d.c. component is determined
by the root mean squares of the interfering signals divided by the sum of
these signals. The d.c. components due to non-overlapping areas of the
beams are, for the moment, neglected.
[NB. In optics, signals are referred to on the basis of their intensity,
which equals the square of the amplitude. In electronics, signals are
referred to by the value of their amplitude.]
The values of .alpha. depend on the reflection and transmission
coefficients of the interferometer beam splitter, depending on the path
travelled by the interfering beam, and on beam expansion, alignment, etc.
In the interferometer beam splitter 10, let the face nearer the further
beam splitter 18 be the "air side" and the face nearer the reflector 14 be
the "substrate side", and let the reflectance and transmission
coefficients of the interferometer beam splitter be represented by R and
T. The six coefficients of the interferometer beam splitter at the two
orthogonal directions of polarisations are set out in Table I below FIG.
2. Each value of .alpha. is dependent on two coefficients, either on two
different coefficients, or twice dependent on the same coefficient. For
example a beam having polarisation direction S which is reflected by the
interferometer beam splitter 10, reflected by reflector 12, reflected
again by beam splitter 10 and passes to the polarising beam splitter 20,
the amplitude .alpha..sub.1 is given by:
.alpha..sub.1 =k.sub.1 AR.sub.s (2)
where A is the initial amplitude of the S component and k.sub.1 is a term
dependent on beam expansion, alignment, attenuation etc. For a P component
beam transmitted by beam splitter 10, reflected by reflector 14 and
reflected by beam splitter 10, the amplitude .alpha..sub.2 is given by:
##EQU1##
The amplitudes .alpha..sub.1 and .alpha..sub.2 for the interfering beams
in all four interferograms are given in Table II below FIG. 2, where
k.sub.2 is a term similar to k.sub.1 and the initial amplitude of the
P-component is also A. The values of k.sub.1 and k.sub.2 will be the same
only if the beams are of the same size and perfectly aligned, and if there
is no relative attenuation. The effect of the further beam splitter 18 is
ignored; this does not affect the reasoning. The polarising beam splitters
are assumed to be perfect.
It can be seen from Table II that the values .alpha..sub.1.sup.2
+.alpha..sub.2.sup.2 for the background illumination and the ratio of a.c.
to d.c. components are different for each interferogram. If all values of
.alpha..sub.1.sup.2 +.alpha..sub.2.sup.2 and the a.c. components were
equal, the d.c. component could be removed by subtraction of pairs of
signals. The sinusoidal components would not be removed because they are
not in phase.
Inspection at Table II shows that three interferogram signals all having
equal ratios of a.c. to d.c. components can be provided by selecting the
values of some of the 6 coefficients of the interferometer beam splitter
10. Hence it is easy to obtain three signals having equal d.c. components
and also equal a.c. components, for example by suitable orientation of the
plane of polarisation of the input beam. When this condition is achieved,
it can be shown that the d.c. components of the three signals due to
non-overlapping areas of the interfering beams are also equal. It is
believed that this has not previously been realised, and this selection is
the basis for the present invention.
Two alternative possibilities exist:
(a) for equality in interferograms 1, 2 and 3,
R.sub.s =R.sub.s.sup.1 =T.sub.s and R.sub.p =R.sub.p.sup.1 (4)
(b) for equality in interferograms 2, 3 and 4,
R.sub.p =R.sub.p.sup.1 =T.sub.p and R.sub.s =R.sub.s.sup.1 (5)
The dielectric beam splitter 10 is selected to meet one of the two
conditions (a) or (b), and from the three output signals from the
photodetectors sensing the appropriate interferograms, two pairs of
difference signals are provided, and supplied to a conventional fringe
counting system.
Usually, the properties of dielectric beam splitters are such that the
reflection coefficients on opposite sides of the beam splitter are equal
for the same plane of polarisation. Thus the interferograms 2 and 3 will
in any case comprise signals having the same ratio of the a.c. to the d.c.
component. It follows that it is usually necessary to deliberately select
an interferometer beam splitter in which the coefficients of transmission
and reflection are equal for one of the two orthogonal polarisation
states; this determines whether interferogram 1 or interferogram 4 is
used.
It is possible for all six coefficients to be equal but beam splitters with
this property are not readily available or easy to manufacture whereas it
is fairly easy to equalise R and T for one polarisation.
The effect of subtracting two pairs of signals, each having a constant
ratio of the a.c. to the d.c. component and with the same magnitude is
illustrated in FIG. 3. The upper part, FIG. 3(a), shows by the
chain-dotted, dotted and full lines respectively the outputs from
photodetectors arranged to sense the interferograms 1, 2 and 3. The phase
difference between interferograms 1 and 3 and between 3 and 2 is
90.degree., because interferogram 3 is produced by the polarisation
orthogonal to that producing 1 and 2; this condition is imposed by the use
of a dielectric interferometer beam splitter (which gives a 180.degree.
phase difference between interferograms 1 and 2) plus an eighth wave plate
in one interferometer arm.
It is essential to subtract signal pairs which differ in phase by
90.degree., so that the resultants of the subtracted signals also differ
in phase by 90.degree., and can therefore be presented to the fringe
counting system. This is illustrated in the vector diagrams in FIG.
3(b)--subtraction of the pair of signals 3 and 1, and the pair of signals
3 and 2 provides resultant signals (3-1) and (3-2) which differ in phase
by 90.degree.. This can also be seen in FIG. 3(c) which represents by the
chain-dotted and full lines the respective outputs of two subtractor units
each arranged to subtract one pair of signals. The subtractors provide a
pair of subtraction output signals which vary about a zero level, i.e.
which have no d.c. component, and which are in quadrature; this pair of
signals fulfils the conditions required by an automatic fringe counting
system. With variation about a zero, a suitable constant triggering level
is easily chosen.
A full optical system, incorporating an interferometer according to the
invention and an automatic fringe counting system is shown in FIG. 4 in
which a beam of light 24 from a helium-neon frequency stabilised laser 26
is incident on an interferometer 8 according to the invention through a
quarter wave plate 28 and a plane polariser 30; this combination provides
an input beam having a plane of polarisation which can be adjusted with
respect to the interferometer 8 so that it contains approximately equal
components in the P and S directions, which is independent of the
polarisation state of the light emerging from the laser 26.
The four interferogram outputs 1 to 4 of the interferometer 8 are each
sensed by a photodetector 31, 32, 33, 34 respectively. The outputs of
photodetectors 31 and 33 and the outputs of photodetectors 32 and 33 are
connected to respective subtractor units 36, 38 which each provide an
output which is connected to an automatic fringe counting system 40 of
conventional type, which can count fringes corresponding to movement of
one of the interferometer reflectors 12, 14. The count is not affected by
vibration, etc., and reversal of the movement is possible.
For reasons explained above, it is convenient if the photodetectors receive
signals of equal magnitude. One method of providing such signals is to
arrange variable attenuators 61, 62, 63, 64 (shown dotted) between the
outputs of interferometer 8 and the photocells 31, 32, 33, 34. The
settings of the attenuators can be adjusted until three equal signals are
supplied to the subtractors 36 and 38.
FIG. 4 illustrates an optical system operating on condition (a), given by
equation (4); therefore the output of photodetector 34 is not used. If
condition (b), given by equation (5) was used, the appropriate pairs of
photodetectors would be connected to the subtractor units.
An alternative interferometer according to the invention is illustrated in
FIG. 5; comparison with FIG. 2 will show that the plane mirrors 12, 14
have been replaced by corner cubes 42, 44, and the polarising beam
splitter 20 has been replaced by a plane polariser 46. Otherwise, like
components have been give the same reference numerals. Selection of
interferogram 1 or interferogram 4 is accomplished by changing the
orientation of the polariser 46.
It is an advantage of this arrangement that, instead of use of attenuators
as in FIG. 4, the interferogram signals may be equalised by altering the
plane of polarisation of the polariser 30, so that the two orthogonal
components in the incident beam are unequal.
With either the FIG. 2 or the FIG. 5 arrangement of the interferometer, it
may be a requirement that the fringes are sub-divided, and sub-divisions
of a fringe counted. This can be achieved by setting the trigger levels at
values other than zero, but for satisfactory performance the two final
signals having 90.degree. phase difference must have constant and equal
maximum amplitudes as well as a zero d.c. component. In order to maintain
the equal amplitude condition, an automatic gain control system can be
used. A suitable system is illustrated in FIG. 6.
The photodetectors 31, 32, 33, are connected to respective amplifiers 41,
42, 43; amplifiers 41 and 43 are connected to difference amplifier 44 and
amplifiers 42 and 43 are connected to difference amplifier 46. The
difference amplifiers are connected through respective squarers 48, 50 to
a summing amplifier 52, the output of which is connected through a
square-rooter 54 to one input of each of the two ratio amplifiers 56, 58,
the other inputs being supplied directly from the difference amplifiers
44, 46.
Suppose a is the amplitude of the a.c. signal and b is the amplitude of the
d.c. signal; then the output of each amplifier 41, 42, 43 will comprise
both a and b components. When the output signals of the difference
amplifiers 44, 46, are squared by squarers 48, 50, and summed by summing
amplifier 52, the sum comprises a signal proportional to the square of the
amplitude because the phase difference of 90.degree. between each pair of
difference signals reduces the sum to the addition (sine.sup.2
+cosine.sup.2).times.amplitude.sup.2. Taking the square root of this sum
provides a signal proportional to the peak amplitude of the quadrature
input signals and which can be used with the respective ratio amplifiers
56, 58 to provide two output signals in phase quadrature and of constant
magnitude. These signals can be supplied to the automatic fringe counting
unit 40.
It will be appreciated that the function of the subtractor units 36, 38 in
FIG. 4 have been incorporated into the automatic gain control circuit of
FIG. 6.
It is to be understood that the apparatus described with reference to the
drawings may be modified in several ways.
For example, in place of one of the plane mirrors 12, 14 in FIG. 2, a
reflecting surface of a workpiece may be used to derive one or two of the
three required interferograms; alternatively, a tilted plane workpiece
surface plus a lens may replace one of the corner cubes 42, 44, in FIG. 5.
In these arrangements, it is particularly advantageous that the
interferometer system can operate when the a.c. signal has fallen from its
maximum value by a considerable factor; the factor may be as great as 9.
The invention has been described with reference to the use of a stabilised
laser as a light source. This is not essential; an unstabilised laser or a
line source such as a cadmium lamp may be used as a source, provided the
working distance of the Michelson interferometer is within the coherence
length of the chosen source.
Several other changes are possible. Instead of using polarising beam
splitters, references 20 and 22 in FIG. 2, a non-polarising beam splitter
and two polaroids may be used in each position. By the incorporation of an
additional plane reflection in the Michelson interferometer to give
parallel working and reference beams, a tilt-measuring system can be
provided. Instead of a .lambda./8 plate, which is usually extremely
fragile, a 3.lambda./8 plate may be provided. When the additional plane
reflector described above is used, it may be designed to provide the
required phase conditions.
It would also be possible to arrange the Michelson interferometer so that
the beams between the interferometer beam splitter and the reflecting
means are at an angle other than 90.degree. without affecting the
operation of the invention.
The output from the light source may be circularly polarised by attaching a
circular polariser to the source or the interferometer with the advantage
that rotation of the source and/or the interferometer system about the
axis of the input light beam or axes parallel to it have, in practice,
little effect on the instrument performance. In this arrangement the
positions of the polariser 30 and the .lambda./4 plate 28 in FIG. 5 are
interchanged. If the source 26 is already polarised as in the case of most
frequency stabilised lasers and lasers with Brewster windows, the
polariser 30 may be dispensed with. The .lambda./4 plate is then suitably
orientated with respect to the plane of vibration of the source. However,
optical attenuators as shown in FIG. 4 or different light detector
amplifier gains are required.
The two antiphase signals may also be subtracted to provide a d.c.-free
signal. This and the two other subtractor signals already available and
free of d.c. components can be added or subtracted in all the possible
combinations to provide signals or intermediate phases suitable for fringe
sub-division. In turn this may be applied indefinitely by similarly adding
and subtracting the intermediate phase signals but in practice will be
limited by the signal to noise ratios. A method of producing intermediate
phase signals has been described in the specification of U.K. Pat. No.
1,345,204 but this has the disadvantage of requiring a number of
photodetectors equal to the number of signals and in addition the signals
are not free of d.c. components.
The interferometer will usually be used to measure distance, but by
measuring the frequency of the fringes in the interferograms, a velocity
measurement can be made.
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