Optical Sensing System with Separable Spectrally Overlapping Sensor Responses

Abstract

An optical sensing system including an optical interrogator is operative with an array of reflective sensors, each sensor providing a separable reflected spectral response parameter such as a unique Gaussian standard deviation or reflected response compared to other sensors in the same operating wavelength range. The optical interrogator provides narrowband swept or broadband continuous optical power source to the array of FBG sensors, and an optical interrogator generates a g(x) representation of power vs wavelength of the reflected optical power and decomposes the representation into the wavelength of the individual sensors, thereby allowing operation of two or more FBG sensors in the same operating wavelength range.

Claims

1. A measurement system comprising a plurality of fiber Bragg grating (FBG) sensors arranged on a single optical fiber, the FBGs operating in a common range of wavelengths and receiving optical power from an optical interrogator; each of the plurality of gratings having a unique full width half max (FWHM) bandwidth, each of the plurality of gratings reflecting less than 10% of incoming optical power; the optical interrogator receiving reflected optical power, the optical interrogator having at least one optical detector operative to measure reflected optical power at a plurality of wavelengths; the optical interrogator operative to generate a representation of response power vs wavelength; the optical interrogator operative to identify for a peak response, at least one of: a standard deviation, a Full Width Half Maximum (FWHM) value, or a bandwidth; the optical interrogator subtracting a response corresponding to a closest match response of the representation and associating a wavelength to the subtracted response.

2. The measurement system of claim 1 where the optical interrogator continues to subtract a response corresponding to a closest match and associating a wavelength of the subtracted response until a residual error term is less than 10% of a previous peak.

3. The measurement system of claim 1 where the reflected optical power comprises a plurality of Fiber Bragg Grating (FBG) Gaussian reflection responses.

4. The measurement system of claim 1 where the at least one optical detector comprises a plurality of wavelength-specific optical detectors, each detector providing a reflected power for each detector wavelength.

5. The measurement system of claim 1 where the optical power from the optical interrogator is broadband optical source.

6. The measurement system of claim 1 where optical power from the optical interrogator is a tunable laser.

7. The measurement system of claim 1 where the at least one optical detector comprises a plurality of optical detectors operating concurrently and the optical power from the optical interrogator is a broadband optical source.

8. The measurement system of claim 1 where subtracting a response is operative on a Fourier transform of discrete amplitude values from the plurality of optical detectors.

9. The measurement system of claim 1 subtracting a response is operative on a continuous representation of reflected amplitude response measured by the at least one optical detector in response to a tunable laser tuned over a range and generating optical power from the optical interrogator.

10. The measurement system of claim 1 where continuing to subtract a response is performed for each measurement sensor until all measurement sensors are subtracted.

11. A sensor system comprising: a plurality n of Fiber Bragg Gratings (FBG) arranged on a single optical fiber, each FBG having a unique standard deviation σ; a tunable laser coupling optical energy into the plurality of FBGs; a detector operative to measure reflected optical power from the plurality of FBGs; a controller operative to generate a response plot g(x) of power vs wavelength, the power as measured by the detector and the wavelength as provided by the tunable laser; the controller operative to iteratively identify a peak response and subtract a proposed FBG response from the response plot to generate a residue; the controller continuing to iteratively identify a peak response and subtract a proposed FBG response until the residue is below a threshold, or a number n of iterations have been performed.

12. The sensor system of claim 11 where the response plot is a Fourier transform of amplitude vs wavelength multiplied by $e^{\frac{{\gamma }^2{\omega }^2}{2}}.$

13. The sensor system of claim 11 where the peak response identifies a wavelength associated with a sensor response.

14. The sensor system of claim 11 where the tunable laser has a bandwidth which is narrower than a narrowest bandwidth of a sensor FBG by a factor of at least 4.

15. The sensor system of claim 11 where at one plurality of sensors is in a first range of the tunable laser and a second plurality of sensors is in a second range of the tunable laser.

16. A process operative on a controller for identifying FBG reflections from a plot of responses g(x) for a plurality n of sensors, each sensor having a known x and σ, and an unknown μ, the process comprising: a first step of taking Fourier Transform (FT) of the observed composite function g(x); a second step of multiplying the FT of g(x) by $e^{\frac{{\gamma }^2{\omega }^2}{2}}$ a third step of taking the inverse FT of a result of the second step to generate ƒ*(x); extracting a mean (u.sub.1) of the result of the third step using the smallest σ.sub.min=σ.sub.1 if other mean values u.sub.i are not easily identifiable, repeating the procedure with new updated ψ for ψ.sub.1(x)=ƒ(x)−α.sub.1ƒ.sub.1(x.sub.1,μ.sub.1,σ.sub.1) for each of the n sensors.

17. The process of claim 16 where, after completing an nth step, the residual value ψ.sub.1(x) which remains is less than 10% of a previous peak residue value.

18. A measurement sensor comprising a plurality of fiber Bragg gratings (FBGs) operative arranged in sequence on a single optical fiber, the FBGs operating in a common range of wavelengths and receiving optical power from an optical interrogator; each of the plurality of gratings having a unique full width half max (FWHM) bandwidth, each of the plurality of gratings reflecting less than 10% of incoming optical power.

19. The measurement sensor of claim 18 where a first plurality of FBGs is operative over a first range of wavelengths, each FBG having a unique FWHM bandwidth from other FBG sensors in the first range of wavelengths.

20. The measurement sensor of claim 18 where a first plurality of FBGs is operative over a first range of wavelengths, each FBG having a unique FWHM bandwidth from other FBG sensors in the first range of wavelengths and a second plurality of FBGs is operative over a second range of wavelengths, each FBG having a unique FWHM bandwidth from other FBG sensors in the second range of wavelengths; and where at least one FBG of the first plurality of FBGs has a FWHM which is substantially the same as at a FWHM of least one FBG of the second plurality of FBGs.

21. (canceled)

22. (canceled)

23. (canceled)

24. (canceled)

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] FIG. 1 shows a block diagram for an FBG sensing system and optical interrogator, each FBG sensor operating in a spectral wavelength range.

[0014] FIG. 1A shows a plot of FBG reflected wavelength profiles for the system of FIG. 1.

[0015] FIG. 2 shows a block diagram for a separable wavelength sensor system operating with an array of FBG sensors generating unique optical power vs wavelength shapes.

[0016] FIG. 2A shows a plot of FBG reflected optical power vs wavelength profiles for the sensors of FIG. 2, where the FBG sensors have Gaussian reflected power vs wavelength responses with unique standard deviations or Gaussian sigma (σ) values (Gaussian FWHM≈2.355 σ).

[0017] FIG. 3 shows a block diagram of a tunable laser source-based optical interrogator and array of FBGs.

[0018] FIG. 4 shows a plot of the optical detector 304 of FIG. 3 over a range of wavelengths for an FBG array with separable sensor characteristics.

[0019] FIG. 5A shows an equation form for a superposition of Gaussian responses.

[0020] FIGS. 5B, 5C, 5D, 5E, and 5F are example plots of the operation of the FBG interrogator system of FIG. 2 or 3.

[0021] FIGS. 6A, 6B, and 6C show plots for FBGs with unique response shapes.

[0022] FIG. 7 shows an optical interrogator with cross correlator for use with the grating responses of FIG. 6A.

DETAILED DESCRIPTION OF THE INVENTION

[0023] FIG. 1 shows an optical power vs wavelength interrogator 106 coupled to an array of narrowband FBG sensors 107. In an example of the invention, each FBG sensor 108, 110, 112, 114 through 116 receives broadband optical source emission power from the optical power vs wavelength interrogator 106, and each FBG sensor is operative to reflect a fraction of the incoming optical power at a respective range of wavelengths 122, 126, and 134. Each FBG sensor operates to substantially pass or transmit optical power outside of its particular reflective narrow spectral band of operation to a subsequent sensor and is operative to generate a narrowband (typically with a bandwidth chosen between a fraction of nanometer up to several nanometers) reflection centered at its Bragg wavelength 120, 124, 128, corresponding to its own FBG sensor reflective center wavelength, shown for FBG sensors 108, 110, and 112 as example wavelengths λ1, λ2, and λ3, respectively.

[0024] Optical power vs wavelength interrogator 106 includes optical source 102 for generating broadband optical emission power which spans the operating wavelength ranges 122, 126, and 134 of FIG. 1A. The broadband optical power is directed from optical source 102 to circulator 105, which directs the source optical power to sensor array 107 with low to negligible attenuation, and the reflected optical power from each FBG sensor, shown only as 108, 110, 112, and 116, is directed back to circulator 105, which directs the reflected optical power from various FBG sensors positioned on the fiber to wavelength selector 104 and wavelength-selective optical detector array 105. The wavelength selective detector array 105 may use any of a wide variety of different spectral wavelength partitioning and detection methods, one example method is to utilize an Arrayed Waveguide Grating (AWG) filter, which provides a series of wavelength-selective optical outputs which can be applied to individual optical detectors (or plurality of optical detectors occupying a spectral band), and the detector responses will indicate optical power levels at various wavelengths which can be compared in adjacent spectral channels to resolve the FBG reflected optical power to great precision. A pair of spectrally adjacent AWG channels is used to generate a differential signal for improved wavelength discrimination, as described in U.S. Pat. No. 8,983,250, assigned to the applicant in the present patent application, and is incorporated by reference.

[0025] The method of FIGS. 1 and 1A requires that individual FBGs operate in unique wavelength ranges, such as 122, 126, and 134 of FIG. 1A for FBG reflected responses 120, 124, and 128. A limitation of this method is shown when a fourth FBG response 130 is added into the same wavelength range 134, which results in the spectrally overlapping FBG responses 128 and 130, which superimpose to form the overlapping response 132. It is no longer possible to determine which FBG sensor is which if one sensor moves on the opposite side of the other sensor's reflection wavelength, and it is not possible to provide accurate measurement wavelengths for shape 132 after divergence of the responses.

[0026] FIG. 2 shows an example separable shape detector 202 of the present invention. An array of FBG sensors 221 comprises individual FBGs 220a, 220b, 220c, 220d, 220e, 220f, etc., and each FBG returns a unique spectral shape (with unique Gaussian Full Width Half Maximum (FWHM) or near-Gaussian FWHM) rather than a fixed Gaussian response as was shown in FIG. 1A. In one example of the invention, each grating 220a, 220b, 220c, 220d, 220e, 220f is a Gaussian response grating but each with a different FWHM or bandwidth, or alternatively a Gaussian response standard deviation σ (wherein Gaussian FWHM≈2.355σ). Each grating 220a, 220b, 220c, 220d, 220e, 220f may have a different FWHM, or range of wavelengths accommodating each FWHM grating may be used, so that the FWHM profile is unique in each wavelength range, but may be repeated in a different wavelength range. For example, FBGs 220a and 220d may have the same first FWHM, 220b and 220e may have the same second FWHM, and 220c and 220f may have the same third FWHM, but 220a, 220b, and 220c are operative in a first wavelength range as a group 280 which is distinct from a second wavelength range containing a group 282 having FBGs 220d, 220e and 220f. The number of groups of the present invention may be 1 or an integer greater than 1. The optical power vs wavelength interrogator 202 may operate with a broadband optical source 204, circulator 206 for directing optical power to the FBG array 221, and directing reflected optical power to AWG detector 208, and individual detectors, at least one for each wavelength range, to shape detection processor 210.

[0027] FIG. 2A shows plots for the operation of separable shape sensor 202 of FIG. 2. An example set of four FBGs, each with a unique FWHM 231 for Gaussian profile 234, FWHM 240 for Gaussian profile 241, FWHM 250 for Gaussian profile 251, and FWHM 260 for Gaussian profile 261. Each FWHM for a particular FBG response is unique and/or distinguishable from the FWHM of a different FBG response. The optical interrogator 202 of FIG. 2 is operative with individual discrete detectors operating at specific wavelengths, such as provided by an AWG filter (demultiplexer) having a plurality of optical separation channels, typically one for each detector, and so the detector response comprises discrete points for each detector shown as detector samples 233 for each detector. The optical interrogator of FIG. 2 has the advantage of simultaneous interrogation operation, since it can sample all FBGs on the same optical fiber concurrently during the same time interval, but also has the disadvantage of being unable to provide a continuous distribution or plot of the reflected wavelength over a range of wavelengths.

[0028] The tunable laser optical interrogator of FIG. 3 addresses this problem with narrowband tunable laser source 302, which directs a swept range of wavelengths through circulator 306, which is reflected by the FBG array 307, as before. The FBG array responses are directed to a broadband optical detector 304, which, in combination with the laser wavelength control signal 303, provides for a temporal construction of the continuous plot shown in FIG. 4 of optical power 418 vs wavelength 416 limited only by the granularity of power quantization.

[0029] A special consideration of systems which have multiple FBGs reflecting optical power at a particular wavelength is to use care to avoid creating unintentional Fabry Perot interferometric cavities in the optical fiber separating the FBGs when the FBGs are operating at the same wavelength. This may be addressed by the geometrical spacing between FBGs, which spacing acts to reduce or eliminate the coherence and increase the randomness of the reflected optical power between the gratings, or to reduce the reflectivity of the FBG. It is believed that high reflectivity FBGs may need to be separated by separation distances at least on the order of at least a few centimeters (cm), and low reflectivity FBGs may need to be spaced by separation distances at least on the order of only a few millimeters (mm).

[0030] FIG. 5A shows a linear combination of Gaussian functions, each component of the well-known Gaussian function form

[00001]$\gamma =\propto e^{-\frac{{\left(x-x{\mu }_1\right)}^2}{2{\sigma }^2}}$

[0031] where ∝ is a peak amplitude (or the peak function value), μ.sub.1 is a peak offset in x, and σ is the standard deviation (derivable from bandwidth and FWHM).

[0032] FIG. 5A shows an example of a linear combination or superposition of three terms, all with ∝=5, and with corresponding offsets 502 μ.sub.1=0 and σ.sub.1=0.17; μ.sub.2=2 and σ.sub.2 nearly doubled to 0.35; and μ.sub.3=5 and σ.sub.3 nearly doubled again to 0.71. These terms correspond to the peaks 510, 512, and 514, respectively of FIG. 5B. As can be seen, each peak remains discernable. Using the same respective σ.sub.1, σ.sub.2, and σ.sub.3 but leaving μ.sub.0 at 0 and shifting μ.sub.1 to 0.5 and μ.sub.2 to 2 results in the superposition of responses 520, 522, and 524, respectively, shown in plot of FIG. 5C, and leaving μ.sub.0 unchanged and shifting μ.sub.1 to 1 and μ.sub.2 to 2 results in the superposition of responses shown in the respective responses 530, 532, and 534 shown in FIG. 5D.

[0033] FIG. 5E shows the plot of FIG. 5D with a Gaussian of the term contributing to superposition profile response 532 subtracted out, and FIG. 5F shows the additional subtraction of the Gaussian term corresponding to superposition profile at 538 of FIG. 5E, leaving only plot 540 of FIG. 5F.

[0034] In an example of the invention using a quantity of n FBGs with Gaussian spectral reflections, each grating reflecting a unique center (or Gaussian mean) wavelength, width of reflection −3 dB points (FWHM), and magnitude (reflected peak optical power), the plurality of reflected responses forms the series:

ƒ(x)=Σ.sub.i=1.sup.n∝.sub.iƒ.sub.i(x.sub.i,μ.sub.i,σ.sub.i)  (Eq 1)

representing a linear combination of Gaussian functions of different bandwidths or sigma values, Gaussian peaks (means), and magnitudes. Each FBG has an optical reflection bandwidth corresponding to the FWHM of the FBG. In the present invention, ∝.sub.iσ.sub.i, and n are known a-priori. Further,

[00002]$\begin{array}{cc}{f}_i\left(x_i,{\mu }_i,{\sigma }_i\right)=\frac{1}{{\sigma }_i\sqrt{2\pi }}{e}^{-\frac{{\left(x-x{\mu }_i\right)}^2}{2{\sigma }_i^2}}\mathrm{for}i=1.Math.n& \left(\mathrm{Equation}2\right)\end{array}$

[0035] ƒ(x.sub.i,μ.sub.i,σ.sub.i) will be the model for the observed composite signal g(x) comprising the superposition of reflected FBG responses from the array of FBG sensors.

[0036] In an example of the invention, a search for u.sub.i will be conducted such that the difference between the observed signal g(x) and the theoretical value ƒ(x.sub.i,μ.sub.i,σ.sub.i) can be considered random at a chosen level of significance. The objective is to find the mean μ.sub.i of each Gaussian component through localizing and separating the components x.sub.i,μ.sub.i,σ.sub.i, which may be iteratively performed using automated methods, preferably using an on-board computer integrated with the optical interrogator.

[0037] In an example iterative localization method, a first step of decreasing deviations is found, selecting γ>0 which satisfies the condition:

γ<Min(σ.sub.i)  (Eq 3)

[0038] where Min(σ.sub.i) is the smallest standard deviation value of all FBGs with Gaussian profiles.

[0039] In a second step of the iterative localization method, the equation shown below in evaluated:

[00003]$\begin{array}{cc}{f}^{*}\left(n\right)=\underset{i=1}{\overset{N}{.Math.}}\frac{{\alpha }_1}{\sqrt{{\sigma }_i^2-{\gamma }^2}\sqrt{2\pi }}{e}^{-\frac{{\left(x-{\mu }_i\right)}^2}{2\left({\sigma }_i^2-{\gamma }^2\right)}}& \left(\mathrm{Eq}4\right)\end{array}$

[0040] Equation 4 above is a linear combination of Gaussian functions, each component of the sum of Gaussian functions having a standard deviation √{square root over (σ.sub.i.sup.2−γ.sup.2)} instead of σ.sub.i, where 0<√{square root over (σ.sub.1.sup.2−γ.sup.2)}<σ.sub.i for every i=1, 2, . . . , n and the other constants α.sub.i,μ.sub.i,n remain unchanged.

[0041] In the graphical plots of both functions ƒ(x) and ƒ*(x), ƒ*(x) has a more pronounced local maximum, and each peak is narrower and of greater value than those of ƒ(x). If the value of γ is sufficiently close to the smallest standard deviation, it may be assumed that the mean component with the smallest standard deviation is exactly the value in which the function has an absolute maximum value.

[0042] This maximum value for each component can be found in the manner in which not only the mean of one component, but from the other local maxima, the means of further components with similar standard deviations by iterating with a new value of γ which is slightly larger than the previous one.

[0043] The next step is to assume success in finding the mean of at least one component of the mixture of responses, and subtract the previous one. In this subsequent step, the difference is formed:

ψ.sub.1(x)=ƒ(x)−α.sub.1ƒ.sub.1(x,μ.sub.1,σ.sub.1)  (Equation 5)

[0044] and repeat the procedure with a larger value of γ. Thus, we get a sequence of functions {ψ.sub.i(x)} until for some integer i, ψ.sub.i(x)<ε for all x. In this case, i=n+1, where we have found one of the Gaussian components in each of the steps.

[0045] In the specific case where we are using FBGs with known characteristics in a string of FBG sensors returning the superposition of FBG reflection responses over a range of wavelengths, we know the different standard deviation σ.sub.1 (which can be derived from FWHM), number of sensors n and associated magnitudes (FBG reflectivity) α.sub.i.

[0046] The process becomes much simpler in choosing γ.

[0047] Localizing of each mean (wavelength at the peak response) is fully described if we specify how to form the function ƒ*(x) from a given ƒ(x). The preferred FBG spectral partitioning method is the use of the Fourier transform.

[0048] The Fourier transform (F{.}) of the Gaussian function may be expressed as:

[00004]$\begin{array}{cc}F\left\{{\alpha }_i{f}_i\left(x_i,{\mu }_i,{\sigma }_i\right)\right\}={\alpha }_i{e}^{-i{\mu }_i\omega }{e}^{-\frac{{\sigma }_i^2{\omega }^2}{2}}& \left(\mathrm{Eq}6\right)\end{array}$

and similarly:

[00005]$\begin{array}{cc}F\left\{{\alpha }_i{f}_i^{*}(x_i,{\mu }_i,\sqrt{{\sigma }_i^2-{\gamma }^2}\right\}={\alpha }_i{e}^{-i{\mu }_i\omega }{e}^{-\frac{\left({\sigma }_i^2-{\gamma }^2\right){\omega }^2}{2}}=e^{\frac{{\gamma }^2{\omega }^2}{2}}F\left\{f\left(x_i,{\mu }_i,{\sigma }_i\right)\right\}& \left(\mathrm{Eq}7\right)\end{array}$

by linearity of the Fourier transform:

[00006]$\begin{array}{cc}F\left\{f^{*}\left(x\right)\right\}=e^{\frac{{\gamma }^2{\omega }^2}{2}}F\left\{f\left(x\right)\right\}& \left(\mathrm{Eq}8\right)\end{array}$

in this manner we may get ƒ*(x) from a given ƒ(x)

[0049] These methods may be used to perform numerical calculations by selecting an inverse Fourier transform, such as by using a Fast Fourier Transform (FFT) or Discrete Fourier Transform (DFT), and it becomes possible to also estimate the accuracy of the resulting calculation.

[0050] In a first example, two FBGs with Gaussian reflectivity spectral responses and unique bandwidths (expressed as σ.sub.1 and σ.sub.2) produce the reflected response:

[00007]$\begin{array}{cc}f\left(x\right)=\frac{{\alpha }_1}{{\sigma }_1\sqrt{2\pi }}{e}^{\frac{{\left(x-{\mu }_1\right)}^2}{2{\sigma }_1^2}}+\frac{{\alpha }_2}{{\sigma }_2\sqrt{2\pi }}{e}^{-\frac{{\left(x-{\mu }_2\right)}^2}{2{\sigma }_2^2}}& \left(\mathrm{Eq}9\right)\end{array}$

[0051] where:

[00008]$$\begin{gathered} {\alpha }_1<{\alpha }_2\mathrm{and}\gamma <\mathrm{Min}\left\{{\sigma }_1,{\sigma }_2\right\} \\ {f}^{*}\left(x\right)=\frac{{\alpha }_1}{\sqrt{{\sigma }_1^2-{\gamma }^2}\sqrt{2\pi }}{e}^{-\frac{{\left(x-{\mu }_1\right)}^2}{2\left({\sigma }_1^2-{\gamma }^2\right)}}+\frac{{\alpha }_2}{\sqrt{{\sigma }_2^2-{\gamma }^2}\sqrt{2\pi }}{e}^{-\frac{{\left(x-{\mu }_2\right)}^2}{2\left({\sigma }_2^2-{\gamma }^2\right)}} \\ F\left\{f\left(x\right)\right\}={\alpha }_1{e}^{-i{\mu }_1\omega }{e}^{-\frac{{\sigma }_1^2{\omega }^2}{2}}+{\alpha }_2{e}^{-i{\mu }_2\omega }{e}^{-\frac{{\sigma }_2^2{\omega }^2}{2}} \\ F\left\{f^{*}\left(x\right)\right\}=e^{\frac{{\gamma }^2{\omega }^2}{2}}F\left\{f\left(x\right)\right\}=e^{\frac{{\gamma }^2{\omega }^2}{2}}\left[{\alpha }_1{e}^{i{\mu }_1\omega }{e}^{-\frac{{\sigma }_1^2{\omega }^2}{2}}+{\alpha }_2{e}^{i{\mu }_2\omega }{e}^{-\frac{{\sigma }_2^2{\omega }^2}{2}}\right] \end{gathered}$$

[0052] Where the procedure steps are:

[0053] 1) Take the Fourier Transform (FT) of the observed composite function g(x)

[0054] 2) Multiply the FT of g(x) by

[00009]$e^{\frac{{\gamma }^2{\omega }^2}{2}}$

[0055] 3) Take the inverse FT to find ƒ*(x)

[0056] 4) Extract the mean (u.sub.1) of the Gaussian with the smallest σ.sub.min=σ.sub.1, which is the response associated with the closest match in wavelength.

[0057] 5) If other mean values u.sub.i are not easily identifiable, repeat the procedure with new updated ψ for ψ.sub.1(x)=ƒ(x)−α.sub.1ƒ.sub.1(x.sub.1,μ.sub.1,σ.sub.1) for each of the n sensors. At the n+1 step, ψ.sub.1(x)≈ϵ is a very small residual value, typically less than 1/100th (1%) of the smallest peak value, or alternatively less than 1/10th (10%) of the smallest peak value.

[0058] If ψ.sub.1(x) is sufficiently small, in one example of the invention ψ.sub.1(x) is less than 10% of the smallest peak value, or in another example of the invention, less than 1% of the smallest peak value, the set of matches are considered approximately Gaussian shaped. In another example of the invention, an approximately Gaussian shaped response is one which results in a difference between ψ.sub.1(x) and the a response associated with a closest match which is less than 10%. In another example of the invention, a series of approximately shaped Gaussian responses results in a residual error ψ.sub.1(x) of less than 10% on the final n+1 step. In another example of the invention, an approximately Gaussian shaped response is one with a FWHM which is within 20% of an envelope of a Gaussian response associated with α.sub.1ƒ.sub.1(x.sub.1,μ.sub.1,σ.sub.1). In another example of the invention, an approximately Gaussian shaped response is one with a FWHM which is within 10% of a FWHM of a Gaussian response associated with α.sub.1ƒ.sub.1(x.sub.1,μ.sub.1,σ.sub.1). In another example of the invention, an approximately Gaussian shaped response is one where the FWHM of the approximately Gaussian shaped response is within 10% of a true Gaussian shaped response.

[0059] In a physical example with FBGs having a Gaussian reflection response, the measured FWHM is approximately equal to 2.355σ. For the following examples,

[0060] FWHM.sub.1 (specified as frequency band)=25 GHz, which is ˜0.2 nm in the wavelength band (typical of an FBG with ˜2 mm grating extent and operating in a 1550 nm wavelength range)

[0061] FWHM.sub.2 (specified as frequency band)=50 GHz, which is ˜0.4 nm in the wavelength band (typical of an FBG with ˜1 mm grating extent and operating in a 1550 nm wavelength range)

σ.sub.1=0.2 nm λ corresponding to 10.6 GHz bandwidth at 1550 nm

σ.sub.2=0.4 nm λ corresponding to 21.2 GHz bandwidth (σ.sub.2=2σ.sub.1)

[0062] The two FBG reflection spectra may overlap and the wavelength of each discernable in a system converting center wavelength to a strain, vibration, temperature, or other proxy for grating period measured by the optical power vs wavelength optical interrogator receiving the superimposed reflected Gaussian responses.

[00010]$$\begin{gathered} f\left(x\right)=\frac{{\alpha }_1}{{\sigma }_1\sqrt{2\pi }}{e}^{-\frac{{\left(x-{\mu }_1\right)}^2}{2{\sigma }_1^2}}+\frac{{\alpha }_2}{{\sigma }_2\sqrt{2\pi }}{e}^{-\frac{{\left(x-{\mu }_2\right)}^2}{2{\sigma }_2^2}} \\ {f}^{*}\left(x\right)=\frac{{\alpha }_1}{\sqrt{1-{\left(\frac{\gamma }{{\sigma }_1}\right)}^2}{\sigma }_1\sqrt{2\pi }}{e}^{-\frac{{\left(x-{\mu }_1\right)}^2}{2{{\sigma }_1^2\left(1-\frac{\gamma }{{\sigma }_1}\right)}^2}}+\frac{{\alpha }_2}{\sqrt{1-{\left(\frac{\gamma }{{\sigma }_2}\right)}^2}{\sigma }_2\sqrt{2\pi }}{e}^{-\frac{{\left(x-{\mu }_2\right)}^2}{2{{\sigma }_2^2\left(1-\frac{\gamma }{{\sigma }_2}\right)}^2}} \end{gathered}$$

[0063] if γ=0.95σ.sub.1=√{square root over (1−(0.95).sup.2)}=0.31 then the magnitude of the first component increases by a factor of

[00011]$$\begin{gathered} \frac{1}{0.31}\approx 3. \\ F\left\{f^{*}\left(x\right)\right\}=e^{\frac{{\gamma }^2{\omega }^2}{2}}F\left\{f\left(x\right)\right\} \end{gathered}$$

[0064] Another example of the invention may use chirped gratings 720a and 720b of FIG. 7 having response profiles 622a and 622b of FIG. 6A mixed with a Gaussian 622c and flat-top response 622d also shown in FIG. 6A to provide separable reflected wavelength characteristics. For example, the chirped gratings 720a and 720b with amplitude vs wavelength responses of 622a and 622b, respectively, may overlap and form the superimposition waveform 640 of FIG. 6B. By cross correlation of the kernel shapes of 622a, 622b, 622c, and 622d, the overlapping chirped gratings of 640 and 642 may be resolved into 646 and 648, and 650 and 652, respectively, each of which may be further resolved to a particular FBG grating and wavelength, and converted to a corresponding sensor measurement such as strain or temperature.

[0065] Spectral response 644 shows an example where the superposition of gratings 720a, 720b, and 720e having respective spectral shapes 622a, 622b, and 622d, may be decomposed or partitioned into the individual responses 654, 656, and 658, respectively. Each of the individual responses 654, 656, and 658 may then be resolved to individual wavelengths and associated temperatures or strains.

[0066] In practice, a plurality of FBG sensors with unique Gaussian or near-Gaussian shaped reflection spectra FWHM (or sigma) values may be produced by a number of methods including adjusting the lengths of the FBG sensors, with shorter FBGs having fewer gratings providing larger FWHM values compared to longer FBGs with more gratings. For instance, this can be accomplished by fabricating FBGs with lengths in the range of about 1 mm up to over 10 mm.

[0067] The present examples are provided for illustrative purposes only, and are not intended to limit the invention to only the embodiments shown.