WO2012022955A1

WO2012022955A1 - Apparatus and method for measuring distance

Info

Publication number: WO2012022955A1
Application number: PCT/GB2011/051397
Authority: WO
Inventors: Matthew Warden; David Urner
Original assignee: Isis Innovation Limited
Priority date: 2010-08-19
Filing date: 2011-07-22
Publication date: 2012-02-23
Also published as: GB201013894D0

Abstract

A method of measuring an unknown distance, comprises providing waves simultaneously to: a) a first, reference, interferometer (30'') including first and second arms of lengths differing by a first path difference, b) a second interferometer(30') including first and second arms of lengths differing by an initially unknown second path difference, and c) at least one further interferometer(30), each further interferometer including first and second arms of lengths differing by an initially unknown respective path difference. The waves are scanned by tuning them over a tuning range. During a scan, a plurality of interference measurements are generated from the waves in each interferometer (30, 30', 30'') dependent upon the path difference of that interferometer (30, 30', 30''). The unknown path difference of at least one of the further interferometers (30) is determined from the interference measurements for that scan.

Description

Apparatus and method for measuring distance

Field of the Invention

The present invention concerns an apparatus and method for measuring distance. More particularly, the invention is in the field of interferometric measurement of distance.

Background to the Invention

An interferometer is a well-known instrument for measuring properties of light or other waves. In its simplest form, an interferometer comprises a source of at least partially coherent waves, a splitter/combiner for splitting the waves into two portions and subsequently recombining them, two "arms" which the two split portions of waves propagate along and back prior to recombination, and a detector for detecting variations in the intensity of the recombined waves. Many variations exist, however; for example the splitting and combining may be carried out by separate elements, the arms may Include waveguides or simply involve free-space propagation to a reflector of some kind, and there may in some cases be more than two arms. The significant property of an interferometer is that the intensity of the detected light waves varies with changes in the relative lengths of the arms or the wavelength of the light, due to interference between the portions of waves that have propagated along each arm.

As is well-known, waves will interfere constructively (the instantaneous amplitudes will reinforce each other) when two or more identical waves are superimposed on each other in phase (such that peaks and troughs in each wave line up with each other), and will interfere destructively (the instantaneous amplitudes will cancel each other out) when such waves are superimposed on each other in anti-phase (such that peaks and troughs in one wave line up with troughs and peaks, respectively, in another wave). Phase differences that fall between fully in-phase and fully anti-phase interference result in reduced amounts of constructive or destructive interference, relative to those extreme cases. In an interferometer, as the length of an arm is increased, the wave that has traversed that arm will move in and out of phase with respect to an identical wave that has traversed an arm of fixed length. Therefore, the degree of constructive and destructive interference between the waves will vary cyclically and so therefore will the intensity detected at the detector. Thus the intensity at the detector is an indication of the relative lengths of the arms.

An interferometric distance measurement method known as frequency- scanning interferometry (FSI) uses a source of waves having a frequency that can be varied (for an introduction to FSI, see Zheng J. Optical frequency- modulated continuous-wave interferometers", Applied Optics, 2006, 45, pp2723- 2730). In most FSI schemes, the waves are fed simultaneously into two interferometers: a reference interferometer having arms of known length and another, measurement, interferometer including a measurement arm, of unknown length that is to be determined. Providing the lengths of all arms remain constant, when the frequency of the waves is varied, the resulting change in the phase of the detected intensity in each interferometer is proportional to the optical path difference between the arms of that interferometer; equivalently, the ratio of optical path differences between the interferometers is equal to the ratio of the change in the phases in the detected signal. So if tuning over a fixed frequency interval results in the detected intensity in the reference interferometer passing through, say, 3 intensity maxima, and in the measurement

interferometer, say, 6 intensity maxima, then the path difference between the arms of the measurement interferometer is twice as large as that between the arms of the reference interferometer. If the lengths of all interferometer arms except the measurement arm are known, the unknown length is determined. FSI has the advantage over other interferometric measurement methods that it can measure distances without any physical movements required. Regrettably, FSI tends to be less accurate and precise than some of those other methods of interferometry, particularly for long unknown distances. There are two main reasons for that. Firstly, most FSI systems measure length relative to the optical path difference of a 'reference' interferometer, whereas other methods of interferometry measure length relative to a fixed laser wavelength. Limits in the accuracy of knowledge of the length of the reference-interferometer optical path difference, or of the fixed laser wavelength will degrade the accuracy of the final measurement. Typically, it is easier to construct a laser with an accurately known fixed wavelength than it is to construct a useful reference interferometer with a length that is accurately known. Therefore FSI measurements are typically less accurate than measurements with the simple method of interferometry described earlier. Secondly, the nature of an FSI measurement is such that it is more sensitive to errors in determining the phase of the interferometers than is differential interferometry, i.e., a given error in phase measurement will lead to a larger error in distance measurement in FSI than in differential interferometry.

FSI also suffers from an error commonly known as ^'drift error' whereby if the measured length changes during a measurement, this creates an error in the distance measurement that is typically much larger than the drift itself, being magnified by the ratio of the scan average frequency to the change in frequency over the scan.

Drift can often be limited at least to some extent by controlling the environment of the interferometers. Where that is not possible or not sufficient, various other methods of ameliorating the effects of drift exist. Kinder et al., in "Progress in absolute distance interferometry based on a variable synthetic wavelength", Messtechnik, (1) describe an FSI system utilising superheterodyne interferometry, in which the FSI phase measurements are made at beat frequencies generated between a first, fixed-frequency, laser and a second, tuneable, laser. In this case, the change of beat frequency over the scan is twice the scan average frequency, and so the magnification of the drift is 0.5 (i.e. the error in the measurement of the unknown length is smaller than the relative interferometer drift). However, the Kinder system suffered from slow recording of multiple measurements, and an improved system was developed by Bechstein and Fuchs, "Absolute interferometric distance measurements applying a variable synthetic wavelength", Journal of Optics 29, 1998, p179. This system utilised specialised optics, based on a Kosters prism, to generate four phase quadrature signals (one for each of two orthogonally polarised waves from two tuneable lasers). Use of the quadrature signals enabled simultaneous measurement of the interferometer phase for each laser. The lasers were tuned in opposite directions, to reduce the effects of interferometer drift.

FSI has also been carried out without using a reference interferometer. Barwood G, Gill P and Rowley W describe in "High-accuracy length metrology using multiple-stage swept-frequency interferometry with laser diodes",

Measurement Science and Technology, 1998, 9(7), pp 1036-1041 an FSI-like technique in which, rather than using a reference interferometer, uses the beat frequencies between, on the one hand, a laser locked successively to two different spectral absorption features of rubidium and, on the other hand, a laser scanned in frequency between start and end points that are locked to etalon peaks. Barwood's method builds up accuracy by using sequentially larger frequency sweep ranges.

Another approach to FSI, using fast, coarse tuning and fine-tuned subscans, was proposed by Fox-Murphy etal. in "Frequency scanned

interferometry (FSI): the basis of a survey system for ATLAS using fast automated remote interferometry", Nuclear Instruments and Methods in Physics, 1996,383(1), pp229-237. Another, using linked subscans has been proposed by Coe PA, Howell DF, and Nickerson RB, "Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment", Measurement Science and Technology, 2004,15(1 1), pp 2175-2187.

Cabral A, RebordSo J and Abreu M describe in "Dual frequency sweeping interferometry with range invariant accuracy" Proc. SPIE, Vol. 7063, 70630T (2008) an FSI measurement system that uses a pair of reference interferometers, one longer than the other. FSI is used to measure repeatedly the length of a longer reference interferometer as a multiple of the length of a shorter interferometer. The repeated measurements reduce the random error on the length measurement of the longer reference interferometer. The longer reference interferometer can then be used as a reference interferometer when measuring still longer measurement interferometers.

Although making repeated measurements will reduce random error, it will not reduce the various systematic errors that arise in an FSI measurement, so this technique cannot continuously increase the measurement accuracy of the longer reference interferometer length.

There remains a need for a method and apparatus capable of making precise and unambiguous measurements of lengths that are longer than can be achieved by prior-art systems.

Summary of the Invention

The present invention provides, according to a first aspect, a method of measuring an unknown distance, comprising:

(i) providing waves simultaneously to:

a. a first, reference, interferometer including first and second arms of lengths differing by a first path difference,

b. a second interferometer including first and second arms of lengths differing by an initially unknown second path difference, c. at least one further interferometer, each further interferometer including first and second arms of lengths differing by an initially unknown respective path difference;

(ii) scanning the waves by tuning them over a tuning range;

(Hi) during a scan, generating from the waves in each interferometer a plurality of interference measurements dependent upon the path difference of that interferometer; and (iv) determining from the interference measurements for that scan the unknown path difference of at least one of the further interferometers.

The invention thus provides determination of the unknown path difference in a single shot (i.e. a single scan) measurement.

Preferably, the invention enables large distances to be measured to a better relative precision than was possible using comparable prior-art techniques. Advantageously, that precision may be achieved whilst using lasers that have a smaller tuning range than those used by prior-art techniques, and that are therefore less expensive.

One may label the interferometers with a label, for example P. The first, reference, interferometer may then be labelled P=1, the second interferometer P=2, the first further interferometer P=3, and so on. It may be that the i^th interference measurement made with the P^th interferometer has an instantaneous fringe number F^F consisting of an integer part Νζ and an unwrapped fractional fringe number u . The determination of the interference measurements may involve calculating:

a. the integer part N₀ ^J from the interference measurements from the first, reference, interferometer;

b. from that calculated integer part Ν_ϋ ^ι , and from the interference measurements from the first, reference, interferometer and from the second interferometer, the integer part Ν ;

c. from that calculated integer part Nl, and from the interference measurements from the second interferometer and from the first of the further interferometers, the integer part Ν ; cf. optionally, in cases in which there is more than one of the

further interferometers, from the calculated integer part N ~¹ , and from the interference measurements from the (P-1)^th further interferometer and from the P^lh further interferometer, the

integer part N£ ; and

e. thereby, from at least two of the calculated integer parts, the unknown path difference of at least one of the further

interferometers.

It may be that the path difference of one or more of the interferometers, for example the first interferometer, is a known path difference, in which case step (iv) may involve determining the unknown path difference of at least one of the further interferometers relative to the known path difference.

Alternatively or additionally, step (iv) may involve determining the unknown path difference of at least one of the further interferometers relative to a reference length, which may for example be a reference wavelength. The method may include the step of identifying an interference measurement for which the waves provided to the interferometers are of the same wavelength as the reference wavelength.

The determination of the unknown path difference may involve calculating the frequency and phase of interference patterns comprised in the interference measurements.

Interferometer systems exist which measure distances in terms of a "synthetic wavelength" formed form two real wavelengths. Thus, the waves may be waves of a synthetic wavelength formed from real waves of two different frequencies.

It is possible to construct a "synthetic" optical path difference from the difference between the optical path differences of a pair of interferometers. It may be that the method includes the step of constructing one or more "synthetic" interferometer, with a corresponding synthetic optical path difference. Any of the first, second or further interferometers may be a synthetic interferometer. Optionally, the method is used to determine the wavelength of the waves. The method may be used to determine the ratios of all distances in the interferometer array, including the wavelengths of the waves.

The method may be a method of, for example, precision manufacturing, machine tool calibration, or optical fibre network testing.

The invention also provides, according to a second aspect, an

interferometer array comprising:

(i) a source of waves of a frequency tuneable over a tuning range;

(ii) a first, reference, interferometer including first and second arms of lengths differing by a first path difference and arranged to receive the waves and to generate therefrom a plurality of interference measurements dependent upon the first path difference,

(iii) a second interferometer including first and second arms of lengths differing by an initially unknown second path difference and being arranged to receive the waves and to generate therefrom a plurality of interference measurements dependent upon the second path difference,

(iv) at least one further interferometer, each further interferometer including first and second arms of lengths differing by an initially unknown respective path difference and being arranged to receive the waves and to generate therefrom a plurality of interference measurements dependent upon the path difference of that further interferometer,

wherein the interference measurements from all of the interferometers are generated during a scan of the source of waves over the tuning range, the array further comprising:

(v) a signal processor arranged to receive the interference

measurements from each interferometer, and to calculate therefrom the unknown path difference of at least one of the further interferometers. It may be that the first path difference is a known path difference, in which case the signal processor may be configured to calculate the unknown path difference of at least one of the further interferometers from the known path difference and the interference measurements received from each interferometer. It may be that the array further comprises a device configured to provide an alternative or additional reference length, which may for example be a reference wavelength, in which case the signal processor may be configured to calculate the unknown path difference of at least one of the further interferometers from the reference length. The reference wavelength may be provided by a laser of a fixed wavelength. The reference wavelength may be provided by a spectral feature of a material, for example a gas, which may for example be in a gas cell. A laser of a fixed wavelength may be locked to the spectral feature.

It may be that the first path difference is shorter than the second path difference and the path difference of each of the further interferometers.

Again, one may label the interferometers with a label, for example P. The first, reference interferometer may then be labelled P=1 , the second

interferometer P=2, the first further interferometer P=3, and so on. It may be that:

(i) the i^th interference measurement made with the P^th interferometer has an instantaneous fringe number F^p consisting of an integer part

Νζ and an unwrapped fractional fringe number u ; and

(ii) the signal processor is arranged to calculate:

a. the integer part N from the measurements from the first,

reference, interferometer;

b. from that calculated integer part N₀ ^l , and from the interference measurements from the first, reference, interferometer and from the second interferometer, the integer part N₀ ² ; c. from that calculated integer part N₀ ² , and from the interference measurements from the second interferometer and from the first of the further interferometers, the integer part N₀ ³ ;

d. optionally, in cases in which there is more than one of the

further interferometers, from the calculated integer part N< ^-1 , and from the interference measurements from the (P-1)^th further interferometer and from the P^th further interferometer, the integer part N£ ; and

interferometers.

It may be that the path difference of the P^th interferometer is shorter than the path difference of the (P+1)^th interferometer, for all of the interferometers.

The interferometer array may further comprise a source of waves of a fixed frequency.

It may be that at least one arm of at least one interferometer is formed in substantially free space, a bulk dielectric material (e.g. glass) or an optical fibre.

It may be that each arm of each interferometer is substantially free space, a bulk dielectric material (e.g. glass) or an optical fibre.

The interferometer array may comprise a second source of waves of a tuneable frequency.

It may be that at least two interferometers, optionally all of the

interferometers, in the array share components. It may be that all of the components of the interferometers are shared except for a reflector at the end of an arm of each interferometer. Thus, it may be that the path differences of the arms of each interferometer are provided by providing a plurality of reflectors at different locations.

The waves may be in a beam that is split to form a plurality of angularly separated beams, with the plurality of reflectors each located to receive a different one of the beams. The beam may be split, for example, by mirrors, a diffraction grating, or a holographic element.

Alternatively, the plurality of reflectors may be located along a line and may have reflectivities that allow waves to be reflected by each reflector. Such an arrangement will result in multiple interferences in a "single" interferometer, and hence effectively a plurality of interferometers. Thus, for example, the reflectors may be grating lines of a fibre Bragg grating.

The interferometer array may comprise a gas cell including a gas having a spectral feature for stabilising the laser.

The invention provides, according to a third aspect, an instrument comprising an interferometer array according to the second aspect of the invention. The instrument may be, for example, a surveying tool (e.g. a laser tracker/scanner), an instrument for machine tool calibration, an instrument for optical frequency domain reflectometry, an instrument for optical fibre network testing, a monitoring component on a satellite for formation flying with other satellites, a monitoring component on a telescope, a monitoring component on a particle accelerator, or an optical fibre sensor.

The present invention provides, according to a fourth aspect, an interferometer array comprising:

(i) a source of waves of a fixed frequency;

(ii) a source of waves of a frequency tuneable over a tuning range;

(iii) a first interferometer including first and second arms of lengths differing by a known path difference and arranged to receive the waves of the fixed frequency and the waves of the tuneable frequency and to generate an interference signal dependent upon the known path difference;

(iv) a second interferometer, including first and second arms of lengths differing by an initially unknown path difference and arranged to receive the waves of the fixed frequency and the waves of the tuneable frequency, and to generate an interference signal dependent upon the unknown path difference; and

(v) a signal processor arranged to receive from each interferometer the interference signals indicative of the path differences, and to calculate, on the one hand, using features of the signal from the second interferometer and resulting from the fixed-frequency waves, an ambiguous but more precise measurement of the unknown path difference and, on the other hand, using features of the signals from both interferometers and resulting from the tuneable-frequency waves, together with the known path difference of the first interferometer, an absolute but less precise measurement of the unknown path difference, the signal processor also being arranged to combine the ambiguous but more precise measurement of the unknown path difference with the absolute but less precise measurement of the unknown path difference to provide an absolute and more precise measurement of the unknown path difference.

Preferably, the interferometer array comprises a series of at least three interferometers, the series comprising the first and second interferometers and at least one additional interferometer,

each additional interferometer including first and second arms of lengths differing by an initially unknown path difference and arranged to receive the waves of the fixed frequency and the waves of the tuneable frequency, and to generate an interference signal indicative of that path difference,

wherein the unknown path difference of the second and each subsequent interferometer in the series is longer than the path difference of the interferometer preceding that interferometer in the series,

and wherein the signal processor is arranged to receive from each additional interferometer the interference signal indicative of the path difference, and to calculate: on the one hand, using features of the signal from each respective additional interferometer and resulting from the fixed-frequency waves, an ambiguous but more precise measurement of the unknown path difference of the arms of each additional interferometer

and, on the other hand, using features of the signals from each respective additional interferometer in the series and from the interferometer preceding that additional interferometer in the series and resulting from the tuneable-frequency waves, together with the absolute and more precise measurement of the path difference of the interferometer preceding that additional interferometer in the series, an absolute but less precise measurement of the unknown path difference of each additional interferometer,

the signal processor also being arranged to combine the ambiguous but more precise measurement of the unknown path difference with the absolute but less precise measurement of the unknown path difference to provide an absolute and more precise measurement of the unknown path difference of each additional interferometer.

The invention provides, according to a fifth aspect, a method of measuring an unknown distance, comprising:

(i) providing waves of a fixed frequency and waves of a varying

frequency to a first, reference interferometer having a known optical path difference and to a second, measurement interferometer that includes an arm of unknown length;

(ii) using a variation, resulting from the fixed-frequency waves, in the intensity detected in the measurement interferometer to calculate an ambiguous but more precise measurement of the unknown length;

(iii) tuning the waves of the varying frequency, and using a variation, resulting from the tuning of the varying-frequency waves, in the intensity detected in the reference and the measurement interferometers to calculate an unambiguous but less precise measurement of the unknown length;

(iv) combining the absolute measurement and the ambiguous

measurement to resolve the ambiguity in the ambiguous measurement, thus providing a more precise absolute measurement of the unknown length.

The formerly unknown length of the arm of the second interferometer thus becomes known, and so the method may further comprise:

(v) repeating steps (i) to (iv) with the second interferometer taking the role of the reference interferometer, and a third interferometer, which includes an arm of unknown length, taking the role of the measurement interferometer, wherein the unknown path difference of the third interferometer is longer than the path difference of the second interferometer.

Again, the formerly unknown length of the arm of the third interferometer thus becomes known, and so the method may further comprise:

(vi) repeating steps (i) to (iv) with the third interferometer taking the role of the reference interferometer, and a fourth interferometer, which includes an arm of unknown length, taking the role of the measurement interferometer, wherein the unknown path difference of the fourth interferometer is longer than the path difference of the third interferometer.

As will be understood, the repeating of steps (i) to (iv) may be repeated a plurality of times, each time with an additional interferometer, each additional interferometer including an arm of initially unknown length, longer than that of the interferometers preceding it in the array, that is measured in step (iv). Thus the method may comprise repeating a plurality of times steps (i) to (iv), with the (n- 1)th interferometer taking the role of the reference interferometer, and an nth interferometer, which includes an arm of unknown length, longer than the initially unknown arm length of the (n-1)th interferometer, taking the role of the measurement interferometer, and where n is 3 on a first of the plurality of times and increases by one with each successive one of the plurality of times.

Thus, the method may be a method of measuring a wavelength, in which the more precise absolute measurement measured in at least one iteration of step (iv) is used to calculate an unknown wavelength, which may be from a further wavelength source.

It will of course be appreciated that features described in relation to one aspect of the present invention may be incorporated into other aspects of the present invention. For example, the method of the invention may incorporate a feature described with reference to the apparatus of the invention and vice versa.

Description of the Drawings

Embodiments of the present invention will now be described by way of example only with reference to the accompanying schematic drawings of which:

Figure 1 is (a) an interferometer array used in the prior-art method of frequency-scanning interferometry, and (b) and (c) show interferometers within the array in more detail;

Figure 2 is a diagram illustrating graphically a data analysis step in the prior-art method of frequency scanning interferometry;

Figure 3 is a diagram illustrating graphically a data analysis step in a method according to an example embodiment of the invention;

Figure 4 is a diagram showing the variation of measurement uncertainty with measurement distance, in the array of Fig. 1 ;

Figure 5 is an interferometer array according to an example embodiment of the invention;

Figure 6 is (a) an interferometer array according to another example embodiment of the invention, and (b) and (c) show details of two alternative implementations of this embodiment; and Figure 7 is a flow chart showing steps in a method according to an example embodiment of the invention.

Detailed Description

An example, standard, prior-art interferometer system 10 (Fig. 1(a)) comprises a laser 20, with vacuum wavelength λ and two interferometers 30 and 30', referred to as a ^"measurement interferometer' 30 and a ^"reference interferometer' 30'. Light is sent from the laser 20 into both interferometers 30 and 30'

simultaneously.

Interferometers 30 and 30' have the construction shown in Fig. 1(b) and Fig. 1(c), respectively. Considering interferometer 30, light with intensity enters the interferometer and is split by a splitter 31. A portion travels a round- trip over a fixed distance to reference mirror 32 and back to splitter 31. The remainder of the light travels a round-trip to measurement mirror 33 and back to the splitter 31. At splitter 31 , the reflected light arriving from both mirrors 32 and 33 is recombined and exits splitter 31 as a combined beam. The intensity, I^meas , of this beam is measured by a photo-detector 34 and is given by:

where D^meas is a quantity known as the optical path difference, which is the effective difference in the round trip distances travelled between the splitter 31 and each of the two mirrors 32 and 33, taking into account the refractive index, n , of the medium through which the light travels. D^meas is thus related to the physical distance, L^{me s} , (i..e. the physical, single-pass path difference, as shown in Fig. 1(b) and (c)) by the following:

D^mea* = 2L^measn

Interferometer 30' has the same construction as interferometer 30, as shown in Fig. 3(c) (in which elements are labelled with a primed reference number corresponding to the reference numeral used on the equivalent feature in Fig. 3(b)).

The mirrors 32 and 33 in measurement interferometer 30 are arranged such that the distance L^meas corresponds to the distance to be measured. The corresponding mirrors 32' and 33' in reference interferometer 30' are held in a fixed in position and the fixed distance L^ref in this interferometer is pre-calibrated by another method.

To perform a distance measurement with this prior-art system, the laser wavelength, λ , is varied smoothly and monotonically from a starting value through a range ΑΛ whilst the varying intensity I^meas is recorded on the photo- detector 34 and the intensity I^ref is recorded on photo-detector 34'. That process is referred to as a wavelength scan.

The unknown distance L ^eas is calculated from an analysis of intensity values recorded during a wavelength scan, given knowledge of L^ref . This analysis will be described in the following paragraphs. The wavelength scan and its analysis together constitute a frequency scanning interferometry (FSI) measurement.

It is useful to introduce a quantity that we shall call the instantaneous fringe number, F , which is the number of sinusoidal intensity cycles one would observe on the photo-detector of an interferometer if the optical path difference, D , were smoothly and monotonically increased from zero to a value, D , whilst the laser wavelength is held constant at a value λ . F is therefore a function of D and λ and is given by:

There are several well known methods for calculating F from measured intensity values in wavelength scan data. One such method is described in Suematsu M, Takeda M, "Wavelength-shift interferometry for distance measurements using the Fourier transform technique for fringe analysis", Applied Optics, 1991,30(28). Because the intensity in a wavelength scan is cyclical, repeating itself every time F is changed by one, these methods can only determine the fractional part of F , leaving the integer part unknown. Therefore, it is helpful to consider F as the sum of two components; the measured fractional part, / , and the unmeasured integer part, N :

F = N + f (2)

It is convenient to express the laser vacuum wavelength, λ , in terms of the corresponding electromagnetic wave frequency, υ . The two are related by the speed of light in vacuum, c , as follows: λ = -° (3) v

Substituting equations (2) and (3) into equation (1), we can therefore express the fringe number as follows:

N + f = ^ (4) c

Equation (4) neatly summarises the information obtained from measuring / and it will be used to explain how the prior-art measurement method works.

In a wavelength scan, several measurements of the fractional fringe number, /, , are made at several different laser frequencies, υ, . These multiple measurements are represented in the following equation:

. N, + (5) c

where from now on we assume that the wavelength scan is rapid enough that we may approximate D to be constant for its duration and thus for all values of i . The smooth and monotonic change in υ during the measurement creates a smooth and monotonic change in the fringe number, F . As this happens we observe a smooth change in the measured variable, / , punctuated by discontinuous jumps of / that occur when there is a change in the integer part of the fringe number, N . By counting up, or 'unwrapping', these discontinuous jumps one may track the variation of fringe number over more than one fringe. Note that, in order to do that, we must make at least two measurements for every change of N , in order to count up the changes. This process is commonly known as 'phase unwrapping'. We can now re-write equation (5), taking into account the knowledge we gain from this phase-unwrapping process:

_{No + Uj =} ^L (6)

c

Here, the f s have been replaced with their unwrapped counterparts, u, . As a consequence of the phase-unwrapping process there is now only a single unknown integer part of the fringe number, N₀ , for the entire wavelength scan.

We shall call this the ^'integer fringe number'.

Equation (6) applies to both interferometers 30 and 30'. We may therefore write out one equation for each of these interferometers:

neaa

N₀ ^meoi — - (⁷)

where the ^'meas' superscripts indicate a quantity is associated with the measurement interferometer 30 and 'ref superscripts indicate a quantity is associated with the reference interferometer 30'. These equations may now be combined and solved to calculate D^meas . In an example solution with only two measurement points (corresponding to i = \ and i = 2 ), we may calculate D^meas as follows:

meas _ ..meas

jyneus _{= D}ref ^M2 ^U\ ,

.ref _ ref '

U 2

To further understand this measurement and the improved version of it, it will be helpful to consider equation (9) graphically. Fig. 2 shows a graph of the fringe number in the reference interferometer, F^ref , on the horizontal axis, versus the fringe number in the measurement interferometer, F^meas , on the vertical axis. On this graph are two points corresponding to the two measurements in equation (9). These are [u? ,u?^eas ), marked 201 on the figure, and (u^r ,u™^as), marked 202 on the figure. Note that according to equation (9) these points lie along a straight line of gradient D^meos j D^*1 . Thus the final part of an FSI measurement may be considered as fitting a line to points (u^ ,u™^as) in order to determine this gradient and thus the unknown distance, D^meas , from the known distance D^ref .

This concludes the description of the prior-art measurement system. We now move on to discuss the improved measurement system, with the key advantage that it offers higher measurement precision. Setting aside the precision to which D^re/ is known, the precision to which we may make a distance measurement using the prior-art measurement system depends upon the precision to which we may determine the gradient of the line in Fig. 2. The precision to which we may determine the gradient increases with the separation of the points along the line, so for a high precision distance measurement we wish to have points spread as far as possible.

Note that, according to equation (4), if we could measure the fringe number of an interferometer corresponding to a laser frequency of zero, v = 0 , the result of this measurement would be F = 0 for any distance, D . That fact can be used to effectively give us an additional measurement point, and greatly increase the precision to which we can determine the gradient of the line in Fig. 2. (Note that we cannot actually measure the fringe number corresponding to v = 0 , but we do not need to.) Fig. 3 is a reproduction of Fig. 2, with an extra point, labelled 203, at the origin, corresponding to the zero optical frequency. We cannot use this extra point straight away. In the prior-art measurement system, we needed to perform the phase-unwrapping procedure before we could fit a line to the measured fringe numbers. If we wish to use this extra point, then we must consistently include it into the unwrapping procedure. However, we cannot perform the standard phase-unwrapping procedure here: the standard phase- unwrapping procedure counts up the integer jumps in N , and thus requires at least two points for each of those jumps. The difference in N between the extra point 203 and the nearest 'real' data point 201 will be too large for the standard phase unwrapping procedure to work; if we wish to use this additional point in the straight line fit, we must determine the constants N"^f and N™^as from equations (7) and (8) by other means.

We start by discussing how to determine N₀ ^re with a method that we shall refer to as ^¾N-bootstrapping'. A coarse estimate of N₀ ^re/ may be obtained by rearranging equation (4), using knowledge of the reference interferometer OPD, D^ref , rough knowledge of the laser frequency, v , and the measured fractional fringe number f^ref :

- r D^refv

N =— ~-f^reI

c

where N (with a hat) is an estimate of the true value, N₀ ^ref , that is subject to errors arising from errors in determining D^K/ , v and f^ef . We may make use of the fact that N^r is an integer quantity and round our estimate, N₀ ^ref , to the nearest integer in order to calculate the exactly correct value of N_Q ^rei :

N = round(N?) (10)

This requires that we are able to reliably determine N by the above method with an error smaller than ±0.5 .

In order to reliably calculate N exactly we must ensure that the uncertainty on N"^f is less than ±0.5. An estimate of the uncertainty on N₀ ^ref is given by: σ^ « ^β^ θν (11) where σ_Μ is the uncertainty on N , σ_υ is the uncertainty in our knowledge of laser frequency, υ , and a_D„_f is the uncertainty in our knowledge of reference interferometer optical path difference, D^ref . The Θ symbol represents a quadrature sum. The uncertainty on measurements of f^ref may easily be made significantly smaller than ±0.5 and therefore their contribution is small enough to be neglected and is therefore not mentioned in equation (11).

We may use the laser manufacturer's specifications to give us a coarse estimate of the laser frequency, v . That may not be very accurate, but we may make N relatively insensitive to errors in our knowledge of v by making D^nf small. A small D^ref will also make it easier to ensure D^ref is stable and well known enough so that the uncertainty contribution from a_D„_f is small. With a sufficiently small D^ref , it will be possible to ensure that the uncertainty on N₀ ^re/ is smaller than ±0.5 , and thereby ensure that we can reliably determine N

exactly.

Our improved measurement method requires that we determine N and Νζ^εα* . The above sections described how to determine N"^J exactly, given that we know the reference interferometer's optical path difference, D^ref . We will now describe how to use this knowledge of N_Q ^ref to determine N"^eas , the absolute fringe number of the measurement interferometer, which has unknown optical path difference, D^meas . We shall refer to this as the ^'N-propagation' method.

The information gained from a two wavelength scan with two fringe- number measurements may be summarised by the following equations:

D^mea3v,

D^refv,

N_n ^re/ + _U;^ef =

D^refv.

These may be combined and rearran ed to solve for N™^a - as follows:

where N™^as (with a hat) is an estimate of the true value of N™^as , which is subject to errors arising from errors in determining the quantities on the right hand side of the equation. N , is the value of N as determined by the method described in the 'N-bootstrapping' section. Note that no estimate of the approximate length j meas jg _req_Uj_reci j_n order to determine N™^eOT .

We can determine the exact value of N^*e<" by rounding that estimate to the nearest integer: ™^as = round(N™^as) (¹³) In order to reliably calculate N™^eas exactly we must ensure that the uncertainty on

N™^as is less than ± 0.5 . The inventors have investigated the errors in N™^as and have found that the dominant contributions come from errors in measurements of vT" and u^reJ . The error on N™^as is iven by:

where each quantity written in the form σ_χ represent the uncertainty in determining a corresponding quantity X . The quantities A and B are constants that depend upon the size of random errors in determining f "^eas and f,^ref , respectively, and the Θ symbol represents a quadrature sum.

The variation of σ.„„„ is plotted against D^meas in Fig. 4. The two thin lines

Ϊ and Ίϊ' represent the variation of σ_Ληιί„ with D^meas for two different values of

D^ref : the line Ί' corresponds to a smaller D^ref whereas the line Ίϊ' corresponds to a larger D^ref . The dashed line Tip is the line of = 0.5 ; we want σ_Λ,„_ιαι to stay below this value. As the measurement uncertainty increases with distance, there will be a distance above which σ increases above 0.5. The two thick solid lines Ίν' and V indicate when this occurs for the two different values of D^ref : the line Ίν' corresponds to the smaller D^ref whereas the line V corresponds to the larger D^ref . As can be seen, a larger D^re/ means that we may measure a larger D^ma* and still keep a .„,_m below ± 0.5 . There is effectively a maximum ratio,

R_mm , of D^meas to D^ref at which this N-propagation method can work. Thus, in the absence of another technique, there is an incentive to choose a large D^ref in order to maximise this largest range at which we can use this technique to obtain an exact value for N™^as .

However, the above discussion presents conflicting requirements for the choice of reference-interferometer optical path difference, D^ref : we must ensure that we determine N"^f with an error smaller than + 0.5 , which requires D^ref to be small; yet the maximum distance reliably measurable with this technique is proportional to D^ref , so to measure large distances we require D^ref to be large.

That conflict can be resolved with an interferometer system containing more than two interferometers. Given any pair of interferometers in this system, we may choose which has the role of reference and which has the role of measurement interferometer. We may therefore use the N-propagation method on any pair of interferometers to determine the N™^as of one, given that we know the N of the other and that their ratio of OPDs does not exceed the limit . We may also use the N-bootstrapping method to determine N₀ ^re of any suitably short interferometer. We arrange the OPDs of a series of interferometers according to a geometric sequence where the shortest OPD is short enough that we can use the N-bootstrapping method. We then use the N-propagation method on successive pairs of interferometers to calculate the JV₀ ^meo1 of each interferometer in the series.

Fig. 5 shows an example of such a system. Light from a laser 20 is fed into interferometers 30" and 30' and 30. The laser frequency is varied whilst the intensity output from all three interferometers 30", 30' and 30 is recorded.

Interferometer 30" has a short optical path difference, short enough to determine its N using the N-bootstrapping method. We then use the N-propagation method on the pair of interferometers 30", acting as the reference interferometer, and 30', acting as the measurement interferometer, which determines N™^as for interferometer 30'. We then use the N-propagation method on the pair of interferometers 30', acting as the reference interferometer, and 30, acting as the measurement interferometer, which determines N™^as for interferometer 30. By choosing appropriate OPDs for this trio of interferometers 30", 30' and 30 we can determine N™"⁵ for interferometer 30 even though it is not possible to directly use the N-propagation method on the pair of interferometers 30" and 30 due to the ratio of their OPDs being above the maximum limit.

It is important to note a point about the errors in this method. The error at each stage is discretised, so there is a finite probability of determining the N™^as of an interferometer exactly. It is possible, with care, to make this probability large at each stage of the cascading process, in which case, there will be a high probability of determining N™^as exactly for each and every interferometer. When this happens there is no build up of errors as one would expect from a system based upon repeated measurements with a continuous error distribution.

It should also be noted that only one laser wavelength scan is required for this process. It is not necessary to take another scan for every step in the iterative process of determining the N₀ ^we∞ values, as we may simply reuse data recorded from a single wavelength scan.

The final step in our method is to calculate the actual distance we wish to measure. So far, all we have done is to calculate N™^as of the interferometer we wish to measure. Using that, we may calculate D^meas by comparing it to a length standard. Two options for length standards in this measurement system are: (i) incorporating into the series of interferometers an interferometer with a previously calculated D (this interferometer therefore acts as a physical length standard), and (ii) comparing D^meas to a wavelength or frequency standard.

Once we have determined N₀ for all the interferometers, we may calculate the distance ratios of any pair of interferometers by combining equations

N"^eas- + u™^as =—— ^ and N + u = =— as follows:

c c

r^meas \rtne s , _^ meas

U _ No + ^ui

_Dref - _Nre/ _{+ U}ref

(Note that we may now actually calculate the distance for each and every sample, /, which allows this method to measure a changing distance. However this measurement can not tolerate too rapid a distance change as this would cause an error in determining N₀ , due to a drift error.)

In the first option for a length standard, we choose any one of the

Interferometers in the system which has a pre-calibrated length to act as the reference interferometer. The system then measures length relative to the length of that interferometer. It is often preferable to measure optical path differences using the second option, relative to a fixed reference wavelength or frequency, as these are often more stable than maaoscopic length reference artefacts, and therefore offer greater measurement accuracy. This may be done by comparing the swept laser wavelength to a known wavelength and identifying a data point i = p where the swept wavelength is identical to the reference wavelength.

^reference

A reference wavelength may for example be defined by an atomic absorption feature which can be probed in an atomic vapour cell (provided it lies within the laser wavelength range).

Examples of a system using a frequency reference are shown in Fig. 6. Light from a variable frequency laser 20 is fed into interferometers 30" and 30' and 30 and frequency reference component 60. The laser frequency is varied whilst the intensity output from all three interferometers 30", 30' and 30 is recorded, and frequency reference component 60 determines the swept laser frequency for at least one data point in the scan.

Two possible implementations for the frequency reference component are shown in Fig. 6 (b) and Fig. 6 (c) respectively.

In the example of Fig. 6(b), frequency reference component 60 includes a fixed frequency laser 120 and a photodetector 50. Light coming from the variable frequency laser 20 is combined with light from the fixed frequency laser 120 and sent onto the photodetector 50. When the varying frequency of the light from the variable frequency laser 20 passes through the value of the light frequency from the fixed frequency laser 120, an optical beat signal will be observed on photodetector 50. That allows the measurement system to determine at which point in time the variable frequency laser had the same frequency as the fixed frequency laser.

In the example of Fig. 6(c), frequency reference component 60 includes a gas cell 70 and a photodetector 50. Light from the variable frequency laser passes 20 through the gas cell 70 and is detected on photodetector 50. When the light frequency from the variable frequency laser 20 passes through an atomic transition of the gas in the gas cell, the intensity on the photodetector 50 will vary. This allows the measurement system to determine at which point in time the variable frequency laser had the same frequency as various atomic transition frequencies.

Higher accuracy implementations may combine the two approaches by locking the fixed frequency laser to the atomic absorption feature and using method (a) to compare its frequency to that of the variable frequency laser.

The flow diagram of Fig. 7 describes, by way of example only, a summary of an analysis method that can be used after a set of interferometer data has been taken corresponding to a laser wavelength scan. The system contains multiple interferometers, labelled 1 to F. Interferometer 1 is made such that we may determine its absolute integer fringe number using the N-bootstrapping method. Interferometers 2 to (F-1) are made such that their optical path differences are in a series of increasing length. (Note that we do not require a precise a-priori knowledge of these lengths). The optical path difference of interferometer F embodies the distance we wish to measure. Therefore its length is unknown.

in the preceding discussion, the symbol N₀ has been used to refer to the absolute integer fringe number of an interferometer. The superscripts 'ref and 'meas' have been used with this symbol to identify the interferometer to which it refers. The system now under consideration contains multiple interferometers labelled as described in the preceding paragraph. So here, instead of 'ref and 'meas' superscripts, numbered superscripts are used to identify interferometers, for example, N¹o is the absolute integer fringe number of the 1^st interferometer, and N²o is the absolute integer fringe number of the 2^nd interferometer, and so on.

The steps of the example method are as follows:

Step 300: Determine N¹o using the N-bootstrapping method.

Step 310: Set a counter, m = 1. Step 320: Determine N^m+ ₀ using knowledge of N^m ₀ and the N- propagation method.

Step 330: Have we determined N₀ for all interferometers (i.e. does m = F-1)? If yes, go to step 350; if no, go to Step 340.

Step 340: Increment m (i.e. set m = m+1) and go to Step 320.

Step 350: Determine distance ratio relative to reference length.

Step 360: If using a physical reference length embodied by

interferometer q.

Step 370: Calculate the measurement interferometer's optical path difference as follows:

Alternatively, or in addition, Steps 360 and 370 can be replaced, or supplemented, by the following steps:

Step 360': If using a reference length embodied by a known wavelength

Vreference.

Step 365': Identify a data point i=p where the swept wavelength is identical to the reference wavelength

Step 370': Calculate the measurement interferometer's optical path difference as follows: reference

The above explanations have been given in terms of the gradient and offset of phases calculated from sinusoidially time-varying interferometric intensity signals. Those skilled in the art will note that an equivalent explanation, and therefore an equivalent analysis method, may be given in terms of the fringe frequency and phase of the above sinusoidal signals. A general discussion of this equivalence is given by Tretter S. in "Estimating the frequency of a noisy sinusoid by linear regression", IEEE Transactions on Information Theory.

1985;31(6):832-835.

Another equivalent interpretation of this measurement process is that the determination of N₀ is done by an imprecise, but absolute prior art FSI measurement, and that imprecise measurement is then improved with a more precise (but ambiguous) fixed-frequency interferometry measurement.

All previous explanations so far have been given for an interferometer which has fringe number zero at zero OPD. A simple modification to equation (4) may be made in order to accommodate an arbitrary constant shift, s , of the fringe number:

+ — _{+ s} (15) c

One may then define M as an ^'offset' version of N :

M = N -s (16) substituting equation (16) into equation (15), we find:

This has the same form as equation (4), and therefore the above discussions apply to this equation, except that the rounding process in equations (10) and (13) must be done differe ly because of the shift, s , in fringe number:

Some methods for determining the fractional fringe number / from wavelength scan data, including the previously cited method by Suematsu and Takeda ("Wavelength-shift interferometry for distance measurements using the Fourier transform technique for fringe analysis", Applied Optics. 1991 , vol. 30, issue 28) work best when dealing with longer optical path differences (OPDs). As the N-bootstrapping method requires a short OPD it may be advantageous to instead use a short ^'synthetic' OPD defined as the difference in OPDs between two, longer, interferometers. The fringe number corresponding to this synthetic length may be obtained by taking the difference in the fringe numbers of the two longer interferometers; once this is done, these fringe numbers may be treated as if coming from a real interferometer with OPD equal to the length difference of the two longer interferometers.

In further alternative example embodiments of the invention, the role of multiple interferometers may be taken up by multiple round trips within a multi- path interferometer such as a Fabry-Perot or Fizeau interferometer.

Where in the foregoing description, integers or elements are mentioned which have known, obvious or foreseeable equivalents, then such equivalents are herein incorporated as if individually set forth. Reference should be made to the claims for determining the true scope of the present invention, which should be construed so as to encompass any such equivalents. It will also be appreciated by the reader that integers or features of the invention that are described as preferable, advantageous, convenient or the like are optional and do not limit the scope of the independent claims. Moreover, it is to be understood that such optional integers or features, whilst of possible benefit in some embodiments of the invention, may not be desirable, and may therefore be absent, in other embodiments.

Claims

1. A method of measuring an unknown distance, comprising:

(i) providing waves simultaneously to:

b. a second interferometer including first and second arms of

lengths differing by an initially unknown second path difference, c. at least one further interferometer, each further interferometer including first and second arms of lengths differing by an initially unknown respective path difference;

(ii) scanning the waves by tuning them over a tuning range;

(iii) during a scan, generating from the waves in each interferometer a plurality of interference measurements dependent upon the path difference of that interferometer; and

(iv) determining from the interference measurements for that scan the unknown path difference of at least one of the further interferometers.

2. A method as claimed in claim 1 , in which the i^th interference measurement made with the P^th interferometer has an instantaneous fringe number

F^F consisting of an integer part Νζ and an unwrapped fractional fringe number u , and the determination of the interference measurements involves calculating:

a. the integer part N_Q ^l from the interference measurements from the first, reference, interferometer;

b. from that calculated integer part , and from the interference measurements from the first, reference, interferometer and from the second interferometer, the integer part N₀ ² ; c. from that calculated integer part N₀ ², and from the interference measurements from the second interferometer and from the first of the further interferometers, the integer part N₀ ³ ;

d. optionally, in cases in which there is more than one of the

further interferometers, from the calculated integer part N^^~l, and from the interference measurements from the (P-1)^th further interferometer and from the further interferometer, the integer part Nf and

interferometers.

3. A method as claimed in claim 1 or claim 2, in which the path difference of one or more of the interferometers is a known path difference, and step (iv) involves determining the unknown path difference of at least one of the further interferometers relative to the known path difference.

4. A method as claimed in any preceding claim, in which step (iv) involves determining the unknown path difference of at least one of the further

interferometers relative to a reference length.

5. A method as claimed in claim 4, in which the reference length is a reference wavelength.

6. A method as claimed in claim 5, further including the step of identifying an interference measurement for which the waves provided to the interferometers are of the same wavelength as the reference wavelength.

7. A method as claimed in claim 1 , in which the determination of the unknown path difference involves calculating the frequency and phase of interference patterns comprised in the interference measurements.

8. An interferometer array comprising:

(i) a source of waves of a frequency tuneable over a tuning range; (ii) a first, reference, interferometer including first and second arms of lengths differing by a first path difference and arranged to receive the waves and to generate therefrom a plurality of interference measurements dependent upon the first path difference,

(v) a signal processor arranged to receive the interference

measurements from each interferometer, and to calculate therefrom the unknown path difference of at least one of the further interferometers.

9. An interferometer array as claimed in claim 8, in which the first path difference is shorter than the second path difference and the path difference of each of the further interferometers.

10. An interferometer array as claimed in claim 8 or claim 9, further comprising a source of waves of a fixed frequency.

11. An interferometer array as claimed in any of claims 8 to 10, in which at least one arm of at least one interferometer is formed in substantially free space, a bulk dielectric material (e.g. glass) or an optical fibre.

12. An interferometer array as claimed in any of claims 8 to 11 , further comprising a second source of waves of a tuneable frequency.

13. An interferometer array as claimed in any of claims 8 to 12, wherein at least two interferometers in the array share components.

14. An interferometer array as claimed in claim 13, in which the waves are in a beam that is split to form a plurality of angularly separated beams, with a plurality of reflectors each located to receive a different one of the beams.

5. An interferometer array as claimed in any of claims 8 to 14, in which a plurality of reflectors are located along a line and have reflectivities that allow the waves to be reflected by each reflector.

16. An interferometer array as claimed in any of claims 8 to 15, further comprising a gas cell including a gas having a spectral feature for stabilising one or more of the lasers.

17. An instrument comprising an interferometer array according to any preceding claim.

18. An interferometer array comprising:

(i) a source of waves of a fixed frequency;

(ii) a source of waves of a frequency tuneable over a tuning range; (Hi) a first interferometer including first and second arms of lengths differing by a known path difference and arranged to receive the waves of the fixed frequency and the waves of the tuneable frequency and to generate an interference signal dependent upon the known path difference;

(v) a signal processor arranged to receive from each interferometer the interference signals indicative of the path differences, and to calculate, on the one hand, using features of the signal from the second interferometer and resulting from the fixed-frequency waves, an ambiguous but more precise measurement of the unknown path difference and, on the other hand, using features of the signals from both interferometers and resulting from the tuneable-frequency waves, together with the known path difference of the first interferometer, an absolute but less precise

measurement of the unknown path difference, the signal processor also being arranged to combine the ambiguous but more precise measurement of the unknown path difference with the absolute but less precise measurement of the unknown path difference to provide an absolute and more precise measurement of the unknown path difference.

A method of measuring an unknown distance, comprising:

(i) providing waves of a fixed frequency and waves of a varying

(iii) tuning the waves of the varying frequency, and using a variation, resulting from the tuning of the varying-frequency waves, in the intensity detected in the reference and the measurement

interferometers to calculate an unambiguous but less precise measurement of the unknown length;

(iv) combining the absolute measurement and the ambiguous