WO1997023766A1

WO1997023766A1 - Fibre-optic sensor in the form of a fabry-perot interferometer having one or both reflecting elements made as a bragg grating

Info

Publication number: WO1997023766A1
Application number: PCT/SE1996/001731
Authority: WO
Inventors: Anders Henriksson
Original assignee: Flygtekniska Försöksanstalten
Priority date: 1995-12-22
Filing date: 1996-12-20
Publication date: 1997-07-03
Also published as: SE9504601D0; SE9504601L; AU1219897A; SE504772C2

Abstract

The present invention relates to a fibre-optic sensor in the form of a Fabry-Pérot interferometer having one or both beam-reflecting elements designed as a Bragg grating (BG). What especially distinguishes the invention is that the chirp function of at least one Bragg grating is balanced against one non-desired measurand, such that the shift of the centre (xr) of the reflection which is a result of the variation of the non-desired measurand exactly compensates for the optical path difference which this measurand generates in the interferometer. This can be carried out by the chirp function having the form Μ0(x) = μ0/2n0 (1+Ax/BL0-Cx), wherein A, B and C are constants which are determined analytically or experimentally for each different case of desired invariance.

Description

Fibre-optic sensor in the form of a Fabry-Pέrot interferometer having one or both reflecting elements made as a Bragg grating

The present invention relates to a fibre-optic sensor in the form of a Fabry-Perot interferometer with one or both beam-reflecting elements designed as a Bragg grating.

Fibre-optic sensors have been gaining ground for many years and have frequently been recommended for their many good properties, such as smallness, sensitivity, great dynamic range, flexibility, non-interfering, immunity to electromagnetic inter¬ ference, corrosion immunity, signal range etc. In spite of all these good properties, the fibre-optic sensors have had difficulty in keeping up with conventional electric sensors. Above all this probably depends on the fact that their advantages have not been great enough in relation to drawbacks such as higher expenses and greater complexity.

One of the drawbacks that many of these types of sensor have suffered from and which have contributed to the complexity of the measurements is the difficulty in distinguishing different measurands, usually strain and temperature. This weakness is more inconvenient than one first realises since many quantities, chemical, electrical and mechanical, are often measured by coating the fibre with a material that reacts to the specific quantity by straining the fibre. It is then uncertain whether the sensor reacts to the measurand or to a variation in temperature, and therefore also the temperature must be measured although it is per se irrelevant. For a long time, efforts have therefore been made to find different ways of separating above all strain and temperature from the measured signal. Most prior-art methods aim at measuring by means of double sensors of slightly varying sensitivity to the different measurands, and then separating the signals in post processing.

As background of the following specification of the present invention, it should be mentioned that a fibre-optic Bragg grating is a periodic modulation of the refractive index in the core of an optical fibre. When the grating pitch of the Bragg grating coincides with half the wavelength of the light, the reflection conditions of the first order are satisfied and part of the light will be reflected and still be conducted in the core of the fibre. The reflectance and bandwidth of the grating are decided by the length and modulation depth of the grating. Longer gratings and greater modulation depths result in higher reflectance, while essentially the length of the grating decides the bandwidth thereof. This means that the gratings can be tailor-made to a great extent. By the grating having a extension in the direction of propagation of the light, the reflection will occur successively and increase the longer the grating is. In the grating, an "efficient reflective surface" is usually defined, which corresponds to the centre of the grating lines that have contributed to the reflection. This is not only a definition but also a physical consequence of the reflected light simultaneously satisfying the reflection condition.

A technique that has frequently been described to increase the bandwidth of the grating is to change the grating pitch over the length of the grating and then obtain a so-called chirped grating. Light of a wider spectrum could then be reflected in the same grating, which would then of course obtain a greater bandwidth in the charac¬ teristic. The different wavelengths of the light, however, would not be reflected from the same area in the grating, which is utilised by Kersey and Davis, 1994, to measure strain by fibre-optic Michelson interferometers. The Michelson interfero- meter is a construction where one measures the differences in optical path length between two arms which are independent of one another, one measuring arm and one reference arm. The light is conducted in the two arms, is reflected so as then to interfere when meeting again. To utilise the function of a chirped Bragg grating in the interferometer, it is used as a reflector in the measuring arm, and the used light is monochromatic and coherent. When the measuring arm and the grating are strained, not only the grating in its entirety is moved, but also the centre of the reflection inside the grating, which results in an additionally increased optical path difference. In other words, the chirp in the grating functions as a mechanical signal amplification of the measurand.

It may be discussed how far it is from the mechanically signal-amplified Michelson interferometer to a mechanically signal-amplified Fabry-Perot interferometer. Fig. 1 illustrates a Fabry-Perot interferometer having reflectors of chirped Bragg gratings BG. The position of the centre of the reflection within a chirped Bragg grating is called x_r. The Fabry-Perot interferometer is an interferometer where both the modulated light beam and the reference beam take the same way or the same fibre. The beams are divided by a semitransparent mirror, where the reflected light constitutes the reference beam and the transmitted light is allowed to pass the measuring range L so as to be then reflected in a second mirror and back through the semitransparent mirror, whereupon the two beams can interfere. What such a sensor senses is changes of the optical path length over the measuring range between the mirrors, such as displacement ε of the mirrors in relation to each other, which for a sensor can be caused by, for instance, applied strain. Instead, the two mirrors can of course consist of Bragg gratings. In a mechanically signal-amplified Fabry-Perot interferometer, the two mirrors would consist of chirped gratings, where the chirps have a varying or opposite inclination. Applied strain would then not only cause a displacement of the mirrors in relation to each other, but also displace the centre of the reflection in the gratings and thus obtain an increased deflection.

The present invention does not concern a signal-amplified Fabry-Perot interfero¬ meter per se, since it may be discussed if such an interferometer should be consid¬ ered obvious in view of prior art, but a sensor comprising such an interferometer, which is made invariant to some non-desired measurands by containing at least one grating having a specially designed chirp. This occurs by the invention having the design as defined in the independent claim. Suitable embodiments of the inven¬ tion are stated in the dependent claims.

The invention will now be described in more detail with reference to the accompany¬ ing drawings, in which

Fig. 1 shows a Fabry-Perot interferometer with reflectors of chirped Bragg gratings, Fig. 2 shows the phase shift as a function of strain and temperature in an ordinary Fabry-Perot interferometer with straight Bragg gratings as reflectors, Fig. 3 shows an example of the chirp function Λ₀(x) in the Bragg grating, Fig. 4 shows the phase shift as a function of strain and temperature for a temperature-compensated Fabry-Perot interferometer,

Fig. 5 shows the phase shift as a function of strain and temperature for an embedded Fabry-Perot interferometer with straight gratings as reflec¬ tors, and Fig. 6 shows the phase shift as a function of strain and temperature for an embedded Fabry-Perot interferometer with gratings having a tempe¬ rature-compensating chirp.

The signal obtained from an interferometer represents an optical phase difference. In an interferometer, this phase difference is determined by the difference in optical path length between two reflections. This optical path difference is then affected by several factors such as: 1) temperature

2) mechanical strain in different directions

3) changes in the refractive index of the glass caused by 1) and 2)

4) change of the position of the centre of the area in a chirped grating where the reflection condition is satisfied, also caused by 1) and 2).

The principle accomplished according to the invention is controlling 4) by controlling the chirp of the Bragg grating such that the combined change of the optical path difference, caused by one arbitrary measurand of 1) or 2) and 3) and 4) will be zero under influence.

This is possible only because the changes in 4) caused by temperature and mechanical strain are not uniform compared with the way in which these meas¬ urands directly affect the optical path difference in the interferometer outside the grating.

In this manner, a sensor which selectively is invariant to certain measurands would be obtained. The most important application is assessed to be a temperature- invariant strain sensor, but also other types of sensors can be based on the same concept. Here follow some examples:

• Strain sensor invariant to transverse strain

• Strain sensor invariant to longitudinal strain

• Temperature sensor invariant to longitudinal strain • Sensor for acoustic waves invariant to global strain

In the following theoretical part, it is proved that the above is possible under given conditions, which in turn are obtained from the literature and from experimental work. Moreover, the theoretical part also describes temperature-invariant strain sensors for two cases, one in which the sensor is free, and one in which the sensor is embedded in a cross laminate of carbon fibre/epoxy composite. There is also a description of strain sensors which are invariant to both longitudinal and transverse strain. All these descriptions include how the chirp should be designed for the desired effect. The case involving the embedded sensor illustrates an example of how the form of the solution is adapted to amended marginal conditions. In a practi¬ cal application of the sensor, the chirp algorithm is, however, expected to need adjustment in order to compensate for various deviations from the theoretical case. The form of the chirp algorithm, however, must be the one that is given in the solution.

The analysis proves that the reason for the function of the concept is the fortunate circumstance that the refractive index in the fibre core is affected by strain to a higher degree inside than outside the grating, and by temperature to a lower degree inside the grating than outside the same. This difference could be due to a struc¬ tural orthotropy in the structure of the glass in the grating. New findings regarding the physics behind Bragg gratings in optical fibres argue that exposure of the fibre to UV light, among other things, contracts the atomic structure, so-called compact¬ ing, which consequently increases the density and refractive index locally in the glass. When Bragg gratings are written or made of optical fibres, the core of the fibre is exposed to an interference pattern of UV laser light such that the refractive index and density will vary periodically along the fibre for some fraction of a milii- metre up to thirty or forty millimetres. The period is typically about 500 nm and the radius of the core is typically about 5 μm. This then induces a local structural orthotropy in the manufacture but also internal residual stress. These two results can explain to some extent the difference noticed in strain and temperature dependence of the refractive index inside and outside the grating.

Theoretical Part

In the further description, the following designations are used.

α: Thermal expansion coefficient Λ: Grating pitch

Λ₀(x): Grating chirp function in non-loaded state

Δ: Small variation ε: Strain ^εxt Strain in x direction caused by thermal expansion ^εyt Strain in y direction caused by thermal expansion ^εzt Tension in z direction caused by thermal expansion λ: Optical wavelength

2π 360 k: Wave number for radians and degrees: — ; λ λ p: Density p₀: Density in non-loaded state σ: Stress n: Refractive index n₀: Refractive index in non-loaded state ng: Average refractive index within the grating v: Lateral contraction constant, Poisson's ratio x_r: Position of the centre of the reflection within a chirped grating relative to a nominal centre of the grating

ΔC_X: Change in sensor gauge length intended to compensate for the influ¬ ence of the non-desired measurand, i.e. the combined shift of the position of the centres of the reflections within the two gratings of the sensor, ΔC_X = x_r-| - X_f

T: Temperature

L₀: Sensor gauge length of the Fabry-Perot interferometer in non-loaded state

Φ: Optical phase f_g: Strain correction factor for variation of refractive index within the grating tg: Temperature correction factor for variation of refractive index within the grating.

First follows a general description of the mode of action for deriving the specific grating chirp which compensates for an optional non-desired measurand. Then follows a more accurate study of the calculations for some examples.

The optical phase difference of the interferometer must be expressed as a function of all measurands that influence the signal, such as strain, temperature and the combined shift of the positions of the centres of mass of the reflections in the two gratings of the sensor and can be expressed as ΔΦ = ΔΦ , AT, ΔC ). To solve the relation between the non-desired measurand and the shift of position of the centre of the reflection in the grating that gives compensation, all the other parameters and finally also the phase difference are set to be zero. Then remains but an expression containing the non-desired measurand and the shift of position of the centre of the reflection in the grating, and the relation can easily be solved. This relation will be valid in the zero point for the remaining parameters, but since the relations are rela¬ tively linear, the errors will be small also for moderate variations.

When the combined shift of position of the centres of mass of the reflections in the grating is given by ΔC = x_ή - Xa, wherein x, = x_r (εg, AT) for each grating, ΔC can of course be obtained through an optional number of combinations, and the selected combination is determined by boundary conditions and technical restric¬ tions in the manufacture. For stronger reflections and a better signal, it may as a rule be assumed that a lower chirp is to be preferred, which also results in x_f being considerably greater than ΔC . For the sake of simplicity, the chirp of the grating which yields ΔC_X in the shift of position of a grating is solved below. The chirp in the function can then easily be divided by an optional scaling factor, whereupon the shift of position of the centre of the reflection will be multiplied by the same scaling factor.

The grating period of a chirped grating depends on the position, all strains and the temperature and can be expressed as Λ = Λ(x, εy, AT). The basic reflection condi- λ tion of a Bragg grating is Λ₀ = — — . The length of the physical period as well as

2n₀ the refractive index, however, depend on strain and temperature, so that the techni¬ cally decisive matter is to determine the appearance of the chirp function of the grating.

Apart from refractive index variation with strain and temperature within the grating, n_g = n , AT), also the physical length of each period in the grating will change with strain and temperature as Λ(^, AT) = Λ„^*(1 + ε* + ^8₂- ^_{1 3} +a^AT), all according to Hooke's law. At the first glance, the chirp function will then be a func¬ tion of merely x, Λ(x), which of course is shifted in strain and temperature changes and may then be written as: Λ(x, ε,_j-, AT) - Λ₀(x)^*(1 + ε_\ + V2*ε₂ - v^ι-?₃ +αιΔT).

With a view to determining the chirp function, the reflection condition with n_g = n(εt_j, AT) is set for x =ΔC_X, in which case the relation between ΔC_X and the non-desired measurand is already given, and the remaining (desired) parameters

Λ A are set to be zero. To start with, use is made of Λ(x) = Λo + — x . With three equa- dx

Λ A tions and four unknown, a relation between — and x and thus also Λ(x) and Λ(x, ε dx i_j, AT) can be determined.

Merely thanks to the variation of the correction terms for refractive index with strain and temperature within the grating, this concept can be applied without the com¬ pensation eliminating the dependence of all measurands at once. All relations are optionally accurate, analytical or empirical and can be adapted to prevailing bound- ary conditions, such as fibre coatings, temperature-induced strain, stress fields, direction dependence and properties of the surrounding material, etc.

Refractive Index n = n(ε, T)

To be able to calculate phase shifts as a function of strain and temperature changes, a relation for refractive index variation with strain and temperature changes must, of course, be determined. It is frequently assumed that the refractive index is proportional to the density, but since the strain dependence is different in the direction of polarisation of the light and the remaining directions, use is gener¬ ally made of the photoelastic constants P** and P₁₂ for the strain dependence and dn

— for the temperature dependence. For moderate variations, the refractive index can be assumed to have the following behaviour:

Experimentally Determined Correction Factors

The above expression of how the refractive index varies with strain and temperature cannot fully explain the response of the Bragg grating to these measurands. The wavelength of the reflected light from a Bragg grating does in fact not correspond to what is anticipated. The reason for this irregularity is probably a local orthotropy in the structure of the glass and a local residual stress state caused by the manufac¬ turing procedure for Bragg gratings. This could explain the deviations. For handling of the problem, e.g. correction factors are introduced into the conventional expres¬ sions such that the variation of the refractive index with strain and temperature in the grating can be calculated. In the expression of the variation of the refractive index within the grating with strain and temperature, use is thus made of a correc¬ tion factor for each measurand:

n_g(ε,AT) (2)

wherein f_g and t_g are correction factors for strain and temperature, respectively. To explain the strain and temperature dependency of the Bragg grating as reported in the literature, the values f _g = 1.19 and t_g = 0.47 are to be used. Phase Shift ΔΦ - ΔΦ(ε, ΔT, ΔC_X)

The phase shift caused by strain and temperature in a Fabry-Perot interferometer with common straight Bragg grating reflectors has been simulated over the range +1 % and -50° - +100°C in Fig. 2, the phase being expressed in degrees.

The phase shift in an optional interferometer with at least one Bragg grating as reflector as a function of the strain, ε, temperature change, ΔT, and the combined shifting of the centres of the reflections in the gratings, ΔC_X , can be expressed as:

AΦ(ε, AT,AC_x) o (3)

If the terms of the higher order are omitted and ε and ΔΦ are set to be zero, the relation between _4Tand ΔC_X which results in a temperature-invariant sensor can be determined:

The relative error caused by omitting the terms of the higher order is less than 1 l_∞ in the maximum temperature range.

In the case of using two Bragg gratings in e.g. a Fabry-Perot interferometer, of course ΔC_X - x_rι - XΛ , i e- the difference in shifting the centre of the reflection between the two gratings. In order to accomplish small changes ΔC_X, the differential effect must be used since the chirp of the grating would otherwise have to be too steep and then give very weak reflections. With a view to obtaining correct chirp functions of the two gratings, equation (6) is to be solved for the shifts x_r1 and Xr2 , respectively, inserted instead of C_x . Chirp Function of the Grating; Λ = Λ(x, ε, ΔT)

The grating pitch in a Bragg grating which is to reflect the wavelength λ₀ is given by λ the condition Λ₀ = — — . The refractive index of course varies with strain and tem-

2»o perature and so does the grating pitch. In a chiφed grating, the grating pitch how¬ ever also varies with the x coordinate within the grating. If the function for the refractive index within the grating is used together with the expression of the exten¬ sion of the grating pitch caused by strain and thermal expansion, a reflection crite¬ rion for x = x_r can be set up:

A(ε, AT,x_r) = Λ₀ (x_/. )(l + £^,+ α Δ7^") = (5)

2nJε,AT)

If ε = 0 is inserted in equation 5 together with the desired shift of the centre of the reflection, ΔC_X, the chiφ function Λ₀ = Λ₀(x) can be derived. See also Fig. 3.

dA

Λ₀(x) = Λ₀(ΔC_X) =

2n_n d 2n_fl(f = 0,Δ7)(l + α Δ7)

The terms of the higher order have been omitted, which results in a relative error of 1.4 ^• 10^*6 at a full temperature range, +75°C.

Position of the Centre of the Reflection x_r = x_r [ε, ΔT)

The desired shift of the centre of the reflection, ΔC_X , which compensates for temperature variations, has been derived, but the shift also depends on the strain. For calculating the effect on the strain signal of the grating chiφ, complete knowl¬ edge of the shifting function x, = x_r (ε, ΔT) is required, wherein x_r is the x coordi¬ nate for the centre of the reflection from the nominal centre of the grating. If x is solved from equation 6 and by inserting equation 5, the following is obtained (7)

Also in this case, the terms of the higher order have been omitted, which results in a relative error of 0.6%o at a full strain and temperature range, +1% and +75°C.

The Resulting Phase Shift

The combined results from equations 3-7 cause a phase shift ΔΦ= ΔΦ (ε, AT, ΔC_X ) in a Fabry-Perot interferometer with a compensating chiφ of the Bragg grating reflectors. The phase shift caused by strain and temperature has been simulated over the range +1% and -50° - +100°C in Fig. 4, where the phase is also expressed in degrees. Obviously, the strain signal has been completed inverted compared with the previous non-compensated case; a positive strain results in a negative phase response. The strain signal is obviously overcompensated, and consequently it would also be possible to make such a sensor invariant to longitudinal or transverse strain with other chirps.

Formulae for an Embedded Sensor

For an embedded sensor, a number of other loads arise which can be difficult to anticipate. Above all, the difference in thermal expansion between the surrounding material and the sensor material will manifest itself as mechanical strain, and if the material is anisotropic, the influence will vary with the direction.

The variation of the refractive index with strain and temperature changes occurs in a manner similar to that of the free sensor, except that there will be one more temperature dependence. With a view to making exact calculations for an orthotropic material, polarisation-maintaining fibres must be used which also have a controlled orientation in the laminate. An approximation for single-mode fibres, where the orientation of the fibre in the laminate does not make any difference, can be made if one disregards the effect on the refractive index of temperature-generat¬ ing strain and uses an average value of the lateral contraction. The error that arises is below 0.1 %o for the full strain and temperature range. The resulting approximative expression will be:

n_r δ n _n(ε,AT) = n₀ -^ ^■[P_.i - _.(P.. +P ]*+ L T (8) δ T

This change of the refractive index results in all the previous expressions of ΔC (A T), x_r(ε, ΔT) and ΔΦ (ε, ΔT, ΔC_X) having new forms. The phase shift will be:

A (ε,AT,ACx)

(9)

( ₀(l + f+( + ^)Δ7) ₊ ΔCx) n₀ -^_[P₁₂ - _βrø (P_{11 +} P₁₂ )μ ₊ τΔ7 n_n

2 Θ T oLH)

resulting in the centre of the reflection for the embedded sensor with temperature compensation being:

(10

Assumptions, simplifications and the omission of the terms of the higher order have been applied as in previous cases.

Analysis of an Example of an Embedded Sensor

To evaluate the effect of the surrounding material on an embedded fibre-optic sen¬ sor and a possible temperature compensation, an analysis was made of the results from an FEM model of such a fibre-optic sensor embedded in a cross laminate with respect to strain caused by load and temperature. It is obvious that the three- dimensional strain field and temperature-induced strain must be taken into consid- eration. The results are shown in Figs 5 and 6.

Invariance to other measurands

By using the relations above, the conditions for invariance can be satisfied also for other measurands. Some of these are: Chirp-generated optical path difference for longitudinal strain invariance of a free sensor:

Chirp function for longitudinal strain invariance of a free sensor

Chirp-generated optical path difference for temperature invariance of an embedded sensor (approximation):

Chirp function for temperature invariance of an embedded sensor:

Chiφ-generated optical path difference for longitudinal strain invariance of an embedded sensor:

ΔC_{X *} -Jl [P₁₂ - v_avβ(P + P₁₂)] (15)

Chiφ function for longitudinal strain invariance of an embedded sensor:

Chirp-generated optical path difference for transverse strain invariance of an embedded sensor:

Chirp function for transverse strain invariance of an embedded sensor:

Summary

The invention concerns the technique of utilising the internal residual tension state in a fibre-optic Bragg grating for separating temperature signals and different strain signals. This is achieved by controlling the chirp of a Bragg grating, such that a fibre-optic interferometer, in which at least one reflector consists of a chirped Bragg grating, is made invariant to one arbitrary measurand of either temperature change or mechanical strain in any direction. This is achieved by the chiφ function of the grating being balanced against one non-desired measurand, such that the shifting of the centre of the reflection which is a result of the variation of the non-desired measurand exactly compensates for the optical path difference which this meas¬ urand generates in the interferometer. Consequently, the combined change of the optical path difference in an interferometer will be zero, when influence is effected by one arbitrary measurand of either temperature changes or mechanical strain in any direction, changes of the refractive index of the glass caused by this arbitrary measurand and the change of the position of the centre of the reflective area in a chiφed grating caused by the arbitrary measurand.

Specifically, this can take place by the grating pitch of the grating approximatively following a chirp function of the form

Λ,(x) = ι+

2n_n BL ₀ - CxJ wherein A, B and C are constants which can be determined analytically or experi¬ mentally. The constants are determined for each different case of desired invari¬ ance.

The cases which are first of all taken into consideration when it is a matter of creat¬ ing invariance to non-desired influence involve invariance to temperature changes for a free fibre-optic interferometer, where the formula used is equation (6) above, invariance to longitudinal strain for a free fibre-optic interferometer, where the for¬ mula used is equation (12), invariance to temperature changes for an embedded fibre-optic interferometer, where the formula used is equation (14), invariance to longitudinal strain for an embedded fibre-optic interferometer, where the formula used is equation (16), and invariance to transverse strain for an embedded fibre- optic interferometer, where the formula used is equation (18).

In all the Examples above, it is understood that a plurality of material parameters which as a rule are determined experimentally, may vary from case to case, with the result that the required chirp algorithm may be changed so as to obtain the desired effect for varying cases. The constants of the algorithm can be determined experi¬ mentally, and this can be regarded as a calibration of the chirp. The main behaviour of the algorithm, however, will be the same. Moreover, the terms of the higher order are omitted from the analytical expressions.

It is also possible to derive a similar algorithm by starting from other expressions of the variation of the refractive index with strain and temperature. However, this must be considered to be a special case of the same algorithm.

Claims

Claims:

1. A fibre-optic sensor in the form of a Fabry-Perot interferometer with one or both beam-reflecting elements designed as a Bragg grating (BG), c h a r a c t e r - i s e d in that at least one Bragg grating is a chiφed Bragg grating, whose chirp function is balanced against one non-desired measurand, such that the combined shift of the centre (x_r1, x^) of the reflection which is a result of the variation of the non-desired measurand, compensates for the optical path difference generated by this measurand in the interferometer.

2. The fibre-optic sensor as claimed in claim 1 , c h a r a c t e r i s e d in that the chiφ function Λ is calculated, based on the possibility of expressing the chirp function as a function of the position in the grating, strain and temperature Λ = Λ(x, ε,_j-, AT), which according to Hooke's law can be expressed as Λ₀(x)*(1 + ε\ + V21S2 ^■ 3i£3 ⁺αιΔ7), wherein Λ₀(x) is the reflection condition λ Λ₀ = — — , and that the optical phase difference of the interferometer can be

2"₀ expressed as a function of strain, temperature and the change of position of the centre of the reflection in the grating, ΔΦ = ΔΦ tø, AT, ΔC ), that the latter expression is set to be zero, when all parameters are zero except the specific measurand which is to be compensated for, and the change of position of the centre of the reflection in the grating, which yields a relation between ΔC_X and said measurand, that ΔC_X is replaced by x in the expression of ΔΦ, since ΔC_X = x is the condition for the desired compensation, which yields a relation between x and said measurand, and that this relation is inserted in the expression of Λ, which thus becomes a function of x.

3. A fibre-optic sensor as claimed in claim 2, c h a r a c t e r i s e d in that the chirp function has the form Λ . (x) = — *- 1+ _Dt ^nΛ _r 1 , wherein A, B and C are

2A7₀ DL ₀-LXJ constants which are determined analytically or experimentally for each different case of desired invariance.

4. A fibre-optic sensor as claimed in claim 3, c h a r a c t e r i s e d in that it is a free fibre-optic interferometer which is invariant to variations in temperature, the chirp function having the form

5. A fibre-optic sensor as claimed in claim 3, c h a r a c t e r i s e d in that it is a free fibre-optic interferometer which is invariant to longitudinal strain, the chirp function having the form

6. A fibre-optic sensor as claimed in claim 3, c h a r a c t e r i s e d in that it is an embedded fibre-optic interferometer which is invariant to variations in tempe-

7. A fibre-optic sensor as claimed in claim 3, c h a r a c t e r i s e d in that it is an embedded fibre-optic interferometer which is invariant to longitudinal strain, the chirp function having the form

8. A fibre-optic sensor as claimed in claim 3, c h a r a c t e r i s e d in that it is an embedded fibre-optic interferometer which is invariant to transverse strain, the chirp function having the form