Methods and apparatus for determining shape parameter(s) using a sensing fiber having a single core with multiple light propagating modes

0001This application claims the priority and benefit of <patcit id="pcit0001" dnum="US62440035"><text>United States Provisional Patent Application 62/440,035, filed December 29, 2016</text></patcit>, entitled "METHODS AND APPARATUS FOR DETERMINING SHAPE PARAMETER(S) USING A SENSING FIBER HAVING A SINGLE CORE WITH MULTIPLE LIGHT PROPAGATING MODES," which is incorporated herein by reference in its entirety.

TECHNICAL FIELD

0002The technology described in this application relates to optical interrogation system measurements used for fiber optic shape and other sensing applications.

INTRODUCTION

0003Optical strain sensing is a technology useful for measuring physical deformation of a waveguide caused by, for example, the change in tension, compression, or temperature of an optical fiber. Measuring the shape of an optical fiber using multiple single mode cores to determine parameters including pitch, yaw, twist, and strain has been demonstrated. Shape is the position or orientation of the fiber in three dimensions. A continuous measure of strain along the length of a core can be derived by interpreting the optical response of the core using swept wavelength inteferometery typically in the form of Optical Frequency Domain Reflectometry (OFDR) measurements. With knowledge of the relative positions of the cores along the length of the fiber, these independent strain signals may be combined to gain a measure of the strain profile applied to the multi-core optical fiber. The strain profile of the fiber refers to the measure of applied bend strain, twist strain, and/or axial strain along the length of the fiber at a high (e.g., less than 50 micrometers) sample resolution. A non-limiting example application is robotic arms used in surgical or other environments. Previous patents have described OFDR-based shape sensing with multiple single mode cores (e.g., see <patcit id="pcit0002" dnum="US7781724B"><text>U.S. patents 7,781,724</text></patcit> and <patcit id="pcit0003" dnum="US8773650B"><text>8,773,650</text></patcit> incorporated by reference).

0004Fibers with multiple single mode cores are difficult and costly to manufacture. Cost is further increased when multi-core fibers must be spun or helically-twisted during manufacture, which is the case for prior multi-core shape sensing fiber. If a single core, unspun fiber could be used to sense shape, then the cost of the fiber sensor could be significantly reduced. For example, mass produced, standard telecom fiber that has single cores and is unspun sells for pennies per meter. But there are several technological challenges (described below) that must be addressed and overcome in order for a single core, multiple mode fiber to sense shape.

SUMMARY

0005Example embodiments include an optical interrogation system and method that includes a sensing fiber having a single core, where the single core has multiple light propagating modes. The system includes interferometric apparatus that probes the single core over a range of predetermined wavelengths and detects measurement interferometric data associated with the multiple light propagating modes of the single core for each predetermined wavelength in the range. Data processing circuitry processes the measurement interferometric data associated with the multiple light propagating modes of the single core to determine a shape-sensing parameter of the sensing fiber. In an example application, the data processing circuitry determines a shape of the sensing fiber based on the shape-sensing parameter.

0006Examples of a shape-sensing parameter include a bend parameter and/or a twist parameter. In one example embodiment, the data processing circuitry processes the measurement interferometric data associated with the multiple light propagating modes of the single core to determine a plurality of shape-sensing parameters including the shape-sensing parameter, and wherein the plurality of shape-sensing parameters includes strain, bend, and twist parameters.

0007The single core may have a shape that limits a number of the multiple light propagating modes below a predetermined number while providing a predetermined sensitivity to the twist parameter. One example is where the single core is ring-shaped in cross-section. An even more specific ring-shaped core example is a ring-shaped cored that has a radius that permits fewer than 40 modes of light to propagate along the single core. Another example is where the single core is a solid core. A radius of the solid core permits less than six modes of light to propagate along the single core.

0008In example embodiments, the interferometric apparatus may include multiple interferometers with multiple reference branches and multiple measurement branches having an array of corresponding multiple single core, single mode fibers. A tunable laser generates light over the range of predetermined wavelengths. Each fiber of the array has a different optical delay. A collimator collimates light from the single core and de-collimate light to the single core, and a microlens array receives and focuses collimated light from the collimator onto the array of corresponding multiple single core, single mode fibers. The different optical delays produce multiple coupling coefficients that appear on different ones of the array of corresponding multiple single core, single mode fibers.

0009In an example implementation, the single core includes an overlapping grating pattern, where each overlapping grating in the overlapping grating pattern is tilted with respect to a longitudinal axis of the single core. The overlapping grating pattern varies with bend, strain, and twist applied to the sensing fiber. The overlapping grating pattern is associated with (i) a cross-sectional index perturbation for the sensing fiber as a function of distance along the sensing fiber and (ii) coupling coefficients between back-scattered light propagating modes for the sensing fiber.

0010In an example application, the interferometric apparatus measures a phase and an amplitude of the coupling coefficients, and the data processing circuitry determines a difference between the measured coupling coefficient phase and amplitude and a predetermined baseline coupling coefficient phase and amplitude for the sensing fiber.

BRIEF DESCRIPTION OF THE FIGURES

0011<ul id="ul0001" list-style="none" compact="compact"><li><figref idref="f0001">Figures 1A and 1B</figref> show a fiber with two single-mode cores displaced from the central axis and a Bragg grating in an unbent and bent shape.</li><li><figref idref="f0001">Figure 2</figref> illustrates first and second order mode fields in a two dimensional waveguide.</li><li><figref idref="f0001">Figure 3</figref> shows an example of a sum and a difference of first and second order modes in a two dimensional.</li><li><figref idref="f0002">Figures 4A and 4B</figref> show straight and bent conditions of a two dimensional waveguide and how it affects the spacing in a Bragg grating in the waveguide.</li><li><figref idref="f0002">Figure 5</figref> illustrates coupling of a forward propagating mode to a backward propagating mode using a Bragg grating.</li><li><figref idref="f0003">Figures 6A-6D</figref> show all of the forward to backward coupling modes in the example from <figref idref="f0002">Figure 5</figref>.</li><li><figref idref="f0004">Figure 7A</figref> shows a grating in a multi-core fiber; <figref idref="f0004">Figure 7B</figref> shows the same fiber when bent; and <figref idref="f0004">Figure 7C</figref> shows a large core fiber with the same grating pattern bent in the same way.</li><li><figref idref="f0005">Figure 8</figref> is a reference diagram of a single core fiber with axis and angle labels.</li><li><figref idref="f0005">Figure 9</figref> shows an example tilted Bragg grating.</li><li><figref idref="f0006">Figure 10</figref> shows an example intensity plot of the index modulation of a tilted Bragg grating.</li><li><figref idref="f0006">Figure 11</figref> shows an example phase mask to write a Bragg grating.</li><li><figref idref="f0006">Figure 12</figref> illustrates an example tipped phase mask to produce a tilted Bragg grating.</li><li><figref idref="f0007">Figure 13</figref> shows an example intensity plot of the index change created by a tilted grating.</li><li><figref idref="f0007">Figure 14</figref> shows an example intensity plot of a pattern created when two tilted gratings are written overtop of one another.</li><li><figref idref="f0008">Figure 15</figref> shows an example amplitude envelope of overlapped tilted gratings.</li><li><figref idref="f0008">Figure 16</figref> is an example intensity plot of a germanium doping concentration in a fiber core.</li><li><figref idref="f0009">Figure 17</figref> illustrates intra-core features created by writing tilted gratings over top of one another within the optical fiber core.</li><li><figref idref="f0010">Figure 18</figref> is an example intensity plot of an end view of the fiber with features created by writing tilted gratings over top of one another in the core.</li><li><figref idref="f0010">Figure 19</figref> shows scalar field plots of three lowest order LP modes.</li><li><figref idref="f0010">Figure 20</figref> shows decomposition functions for decomposing index perturbation.</li><li><figref idref="f0011">Figure 21</figref> shows simulated compound tilted grating index profiles and the profiles reconstructed from the overlap function decomposition.</li><li><figref idref="f0011">Figure 22</figref> shows an example of a bulk-optic design for coupling laser light from a fiber array into a single core fiber.</li><li><figref idref="f0012">Figure 23</figref> shows an example system for interrogating a single core, multiple mode sensing fiber.</li><li><figref idref="f0013">Figure 24</figref> illustrates delay locations of time-domain measurements of the coupling coefficients.</li><li><figref idref="f0013">Figure 25</figref> shows an example model of an optical connection system as a matrix.</li><li><figref idref="f0014">Figure 26</figref> is a graph of an example impulse response of a few-mode fiber with a cleaved end.</li><li><figref idref="f0014">Figure 27</figref> is a graph of an example impulse response drawing with coupling coefficients labeled.</li><li><figref idref="f0015">Figure 28</figref> graphs example fiber mode responses with exaggerated time-length differences.</li><li><figref idref="f0015">Figure 29</figref> graphs example resampling and alignment of coupling coefficients.</li><li><figref idref="f0015">Figure 30</figref> graphs example phase accumulations for different coupling coefficients as a function of delay.</li><li><figref idref="f0016">Figure 31</figref> is a flowchart showing example procedures for using a single core, multiple mode fiber for sensing shape in accordance with example embodiments.</li><li><figref idref="f0017">Figure 32</figref> is a flowchart showing example procedures for using a single core, multiple mode fiber for sensing shape in accordance with example embodiments.</li><li><figref idref="f0018">Figure 33</figref> is a flowchart showing example procedures for calibrating and then using a single core, multiple mode fiber for sensing shape in accordance with example embodiments.</li><li><figref idref="f0019">Figures 34A and 34B</figref> show a skew ray in a fiber core and an axial ray in a fiber core, respectively.</li><li><figref idref="f0019">Figure 35</figref> shows an example single core fiber with three modes.</li><li><figref idref="f0019">Figure 36</figref> is a graph showing modal propagation values for Linear Polarization Modes vs. V-number.</li><li><figref idref="f0020">Figure 37</figref> is a graph showing normalized propagation constant B versus normalized frequency V for TE and TM modes.</li><li><figref idref="f0021">Figure 38</figref> shows an example single large core fiber with 200 modes.</li><li><figref idref="f0021">Figure 39</figref> shows an example single core fiber where the core is annular and supports 30 modes.</li></ul>

DETAILED DESCRIPTION

0012The following description sets forth specific details, such as particular embodiments for purposes of explanation and not limitation. But it will be appreciated by one skilled in the art that other embodiments may be employed apart from these specific details. In some instances, detailed descriptions of well-known methods, interfaces, circuits, components, and devices are omitted so as not to obscure the description with unnecessary detail. It will be appreciated by those skilled in the art that diagrams herein can represent conceptual views of illustrative circuitry, components, or other functional units.

TWO DIMENSIONAL WAVEGUIDE EXAMPLE

0013Before considering a single core fiber in three dimensions, consider a fiber with two single-mode cores displaced from the central axis and a Bragg grating present in the structure from two dimensions. <figref idref="f0001">Figures 1A and 1B</figref> show (two-dimensionally) a fiber 8 with two single-mode cores 12, 14 displaced from the central axis and a Bragg grating 16 in an unbent and bent shape, respectively. The grating forms a set of vertical planes in the core in <figref idref="f0001">Figure 1A</figref> and tilted planes in <figref idref="f0001">Figure 1B</figref>. With the fiber unbent, the two cores reflect the same wavelength. When the fiber is bent, the top core is stretched and the bottom core is compressed, which changes the periodicity of the Bragg grating 16 seen by each core. As a result, a different wavelength is reflected at the grating in each of the two single-mode cores.

0014If there is only a single core in the fiber, but one that supports two field propagation modes, then the two modes will have electric field envelopes that look like those in <figref idref="f0001">Figure 2</figref> which illustrates a first lower order mode field <b>E</b>₀ and a second higher order mode field <b>E</b>₁ in a two dimensional fiber waveguide. If both modes are launched, they propagate down the fiber with a linearly varying phase between them. When the two modes are in phase, their fields will sum and the total [<b>E</b>₀ + <b>E</b>₁] will be weighted toward the top of the waveguide, and when they are out of phase, their fields will subtract and the combination [<b>E</b>₀ - <b>E</b>₁] will be weighted toward the bottom of the waveguide as illustrated by the "peaks" shown in <figref idref="f0001">Figure 3</figref>.

0015By using the sum and difference of the modes, reflections preferentially weighted in the top or the bottom of the core can be used to detect bending of the waveguide, i.e., to detect strain which is one of the shape determination parameters. <figref idref="f0002">Figures 4A and 4B</figref> show straight and bent conditions of a two dimensional fiber waveguide and how it affects the spacing in a Bragg grating in the waveguide. Bending the waveguide causes the top and bottom of the waveguide to expand or compress. If there is a grating physically in the core, then reflections from this core grating can be detected in both the top and the bottom of the waveguide. In <figref idref="f0002">Figure 4B</figref>, the sum of the fields will measure the grating in the top of the core, and the difference of the fields will measure the grating in the bottom of the core.

0016Rather than directly measuring the propagating modes, measuring the cross coupling of those modes in the core grating can provide a measurement of the deformation of the grating which can be used to determine shape parameters. Cross coupling coefficients, <i>κ</i>, represent the scattering of light from the forward propagating modes to the backward propagating modes in the core grating. <figref idref="f0002">Figure 5</figref> illustrates an electric field incident on and interacting with the grating resulting in some of the incident field E⁺ being transmitted (a forward propagating mode) and some of the incident field E^- being reflected (a backward propagating mode). Another way of describing this interaction is "cross coupling" (or simply "coupling") the forward propagating mode and the backward propagating mode in the grating.

0017For a two mode, two dimensional wave guide, four types of backward coupling can occur including: coupling from the forward propagating first order mode to the backward propagating first order mode (and having a coupling coefficient <i>κ</i>₀₀), coupling from the forward propagating first order mode to the backward propagating second order mode (and having a coupling coefficient <i>κ</i>₀₁), coupling from the forward propagating second order mode to the backward propagating second order mode (and having a coupling coefficient <i>κ</i>₁₁), and coupling from the forward propagating second order mode to the backward propagating first order mode (and having a coupling coefficient <i>κ</i>₁₀). <figref idref="f0003">Figure 6</figref> shows all of the forward to backward coupling modes in this example.

THREE DIMENSIONAL WAVEGUIDE EXAMPLE

0018Modeling waveguide systems as ideal lossless waveguide systems, coupling coefficients <maths id="math0001"><math display="inline"><msub><mi>κ</mi><mn>01</mn></msub><mo>=</mo><msubsup><mi>κ</mi><mn>10</mn><mo>∗</mo></msubsup></math><img file="EP4249971A2_D0001.tif" /></maths>, and so there are three independent coupling coefficients available to measure: <i>κ</i>₀₀, <i>κ</i>₀₁, and <i>κ</i>₁₁. The local pattern of the Bragg grating determines the magnitude of the cross coupling or the coupling coefficients. As a result, distributed measurements of these three coupling coefficients can be used to calculate the local frequency of the grating (which determined by the tilt of the grating, which in turn determined by the amount of bending of the grating) at every point along the fiber wave guide. The manner in which these coupling coefficients are measured, and the corrections used to calculate the local frequency of a Bragg grating in the fiber core are described in detail below for a non-limiting and example case of a three dimensional waveguide.

0019Previous fiber-optic shape sensing systems use a multi-core fiber or multiple single core fibers, where each core is a single mode fiber and typically has gratings written along the length of the core. Bend, twist, and/or strain cause local stretching or compressing of the grating pattern. These grating changes relate directly to the applied bend, twist, or strain and can be quantified by measuring the phase change vs. distance along the fiber with respect to a reference state. A matrix can be calculated via knowledge of the fiber geometry and/or calibration, and used to convert phase change in four cores to bend, twist, and strain.

0020In this example case of a single core fiber with multiple modes, the variations in the core grating pattern are measured and processed with distance along the fiber and with cross-sectional location in the fiber core. For example, if the fiber is bent in one direction, the grating pattern is compressed on the side of the core on the inside of the bend, stretched on the side of the core on the outside of the bend, and unchanged along the neutral axis. <figref idref="f0004">Figure 7A</figref> shows a grating in an unbent multi-core single mode fiber, and <figref idref="f0004">Figure 7B</figref> shows the same multi-core when bent. <figref idref="f0004">Figure 7C</figref> shows a single core multiple mode fiber with the same grating pattern bent in the same way. The periodicity of the grating pattern varies across the cross-sectional area of the core.

0021<figref idref="f0005">Figure 8</figref> is a reference diagram of a single core fiber with x, y, z axes, a vector/radius r, and an angle θ. These labels are used in the some of the following figures and in the description below.

0022Although a single grating in the core may be used to determine strain, it is not sufficient by itself to determine another important shape parameter-twist. Twist may be measured using a helixed fiber that uses multiple single mode cores disposed in a helical manner as described in the patents referenced in the introduction. But with a single core, multiple mode sensing fiber, a helixed fiber is no longer available for measuring twist.

0023In various embodiments, a grating pattern can be used to detect twist in the fiber. Construction of an example grating pattern is now described. <figref idref="f0005">Figure 9</figref> shows an example of a tilted Bragg grating written in a single core. <figref idref="f0006">Figure 10</figref> shows an example intensity plot of the index modulation of an example tilted Bragg grating. An index modulation is another way to represent the grating pattern. Note the x axis for the core is plotted horizontally, and z axis for the core is plotted vertically.

0024This type of grating can be written in a fiber core using a phase mask that is tilted with respect to the fiber. <figref idref="f0006">Figure 11</figref> shows an example phase mask to write a Bragg grating in a typical orientation with respect to the fiber when writing gratings. <figref idref="f0006">Figure 12</figref> shows how the mask can be tilted to write a tilted grating. Alternatively, the mask can be made with the pattern tilted in order to write the desired tilted grating.

0025Then, a second tilted grating is written on the core as shown in the example intensity plot of the index modulation or index change of <figref idref="f0007">Figure 13</figref>, but tilted in an opposing direction in the same location. The result is constructive and destructive interference. <figref idref="f0007">Figure 14</figref> shows an example intensity plot of a grating pattern created when the two tilted gratings are written overtop of one another in the core. The dim/blurry areas along the vertical z axis at about 20, 60, 140, and 180 units of distance on the hroizontal x axis represent areas where destructive interference has "washed out" the grating amplitude.

0026<figref idref="f0008">Figure 15</figref> shows an example amplitude envelope of the overlapped tilted gratings. Lighter areas represent stronger grating amplitude, and darker areas represent weaker grating amplitude where the grating amplitude is washed out. This plot represents the grating amplitude along the length of the fiber (axis z) vs. one cross sectional axis of the fiber, x.

0027<figref idref="f0008">Figure 16</figref> is an example intensity plot of a germanium doping concentration in an example fiber core looking at a cross section of the core in the x-y plane. This example fiber is only photosensitive in the core, and the grating only exists in the core.

0028<figref idref="f0009">Figure 17</figref> shows the grating amplitude features created by writing tilted gratings over top of one another within the optical fiber core. <figref idref="f0010">Figure 18</figref> is an example intensity plot of an end view of the fiber in the x-y plane created by writing tilted gratings over top of one another.

0029A grating pattern (like the pattern described above and an amplitude like that shown in <figref idref="f0008">Figure 15</figref>) written in a single core varies with bend, strain, and twist. Thus, various embodiments determine the cross-sectional variation in the grating pattern-which can also be described as the cross-sectional index perturbation-as a function of distance along the fiber. If the cross-sectional variation in the grating pattern can be determined, then the bend, strain, and twist as function of distance along the fiber can be determined.

0030Applying coupled-mode theory, the index perturbation in a fiber core determines the cross-coupling between forward traveling and backward traveling modes. In this application, the index perturbation is the core grating. The core is probed with light having forward traveling modes, and the backward traveling modes created by the grating are detected. Because the input (forward traveling) modes may be determined in advance, and because the output (backward traveling) modes can be measured and the coupling coefficients determined, mode coupling is used the example embodiments to determine the index perturbation (the state of the grating such as pulled, compressed, bent, and/or twisted) that was present.

0031The following assumes that the single core guides three modes for a detailed example. However, a single core that guides more than three modes may also be used.

0032A representation of the index perturbation (the core grating) across the cross section of the example fiber at any given location may be determined by measuring the coupling coefficients between the back scattered modes. Assume that the single core fiber supports three modes E₀ (circular), E_H (horizontal), and Ev (vertical), two of them degenerate. The three modes correspond to three linear polarizations LP₀₁ (circular), LP_11x (horizontal), LP_11y (vertical) as set forth here: <maths id="math0002"><math display="block"><msub><mi>E</mi><mn>0</mn></msub><mo>=</mo><msub><mi mathvariant="italic">LP</mi><mn>01</mn></msub><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mfrac><msup><mi>r</mi><mn>2</mn></msup><msub><mi>w</mi><mn>0</mn></msub></mfrac></mrow></msup><msup><mi>e</mi><mrow><msub><mi mathvariant="italic">iβ</mi><mn>0</mn></msub><mi>z</mi></mrow></msup></math><img file="EP4249971A2_D0002.tif" /></maths><maths id="math0003"><math display="block"><msub><mi>E</mi><mi>H</mi></msub><mo>=</mo><msub><mi mathvariant="italic">LP</mi><mrow><mn>11</mn><mi>x</mi></mrow></msub><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mfrac><msup><mi>r</mi><mn>2</mn></msup><msub><mi>w</mi><mn>1</mn></msub></mfrac></mrow></msup><mi>r</mi><mspace width="1ex" /><mi>sin</mi><mspace width="1ex" /><msup><mi mathvariant="italic">θe</mi><mrow><msub><mi mathvariant="italic">iβ</mi><mn>1</mn></msub><mi>z</mi></mrow></msup></math><img file="EP4249971A2_D0003.tif" /></maths><maths id="math0004"><math display="block"><msub><mi>E</mi><mi>V</mi></msub><mo>=</mo><msub><mi mathvariant="italic">LP</mi><mrow><mn>11</mn><mi>y</mi></mrow></msub><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mfrac><msup><mi>r</mi><mn>2</mn></msup><msub><mi>w</mi><mn>1</mn></msub></mfrac></mrow></msup><mi>r</mi><mi>cos</mi><msup><mi mathvariant="italic">θe</mi><mrow><msub><mi mathvariant="italic">iβ</mi><mn>1</mn></msub><mi>z</mi></mrow></msup></math><img file="EP4249971A2_D0004.tif" /></maths> The variables r and θ and the z axis are shown in <figref idref="f0010">Figure 18</figref>. The variable w is the Gaussian width of the mode and β is the propagation constant of the mode.

0033<figref idref="f0010">Figure 19</figref> shows scalar field plots of three lowest order LP modes E₀ (circular), E_H (horizontal), and E_V (vertical).

0034Based on perturbation theory, (see, e.g., <nplcit id="ncit0001" npl-type="b"><text>Fundamentals of Optical Waveguides, Katsunari Okamoto, Elsevier 2006</text></nplcit>), the coupling factor, <i>κ_mn</i> between any forward propagating mode, <i>E_m</i> and any backward propagating mode <i>E_n</i> is given by the equation below. <maths id="math0005"><math display="block"><msub><mi>κ</mi><mi mathvariant="italic">mn</mi></msub><mo>=</mo><mstyle displaystyle="true"><mrow><mo>∬</mo><mrow><msub><mi>E</mi><mi>m</mi></msub><mfenced><mi>x</mi><mi>y</mi></mfenced><mi>η</mi><mfenced><mi>x</mi><mi>y</mi></mfenced><msubsup><mi>E</mi><mi>n</mi><mo>*</mo></msubsup><mfenced><mi>x</mi><mi>y</mi></mfenced><msup><mi>e</mi><mrow><mi>i</mi><mfenced><msub><mi>β</mi><mi>m</mi></msub><mo>+</mo><msub><mi>β</mi><mi>n</mi></msub><mo>−</mo><mi>k</mi></mfenced><mi>z</mi></mrow></msup><mi mathvariant="italic">dx</mi><mspace width="1ex" /><mi mathvariant="italic">dy</mi></mrow></mrow></mstyle></math><img file="EP4249971A2_D0005.tif" /></maths> Here <i>η</i>(<i>x,y</i>)<i>e</i>^-<i>ikz</i> is the periodic index perturbation (the grating in the core) within the optical fiber, <i>β_m</i> and <i>β_n</i> are the propagation constants of the two non-degenerate modes, x and y are the cross sectional axes, and z is the dimension along the axis of the fiber.

0035Expressing this equation in polar coordinates: <maths id="math0006"><math display="block"><msub><mi>κ</mi><mi mathvariant="italic">mn</mi></msub><mo>=</mo><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mi>a</mi></munderover><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></munderover><mrow><msub><mi>E</mi><mi>m</mi></msub><mfenced><mi>r</mi><mi>θ</mi></mfenced><mi>η</mi><mfenced><mi>r</mi><mi>θ</mi></mfenced><msubsup><mi>E</mi><mi>n</mi><mo>*</mo></msubsup><mfenced><mi>r</mi><mi>θ</mi></mfenced><msup><mi>e</mi><mrow><mi>i</mi><mfenced><msub><mi>β</mi><mi>m</mi></msub><mo>+</mo><msub><mi>β</mi><mi>n</mi></msub><mo>−</mo><mi>k</mi></mfenced><mi>z</mi></mrow></msup><mi mathvariant="italic">rdrdθ</mi><mo>.</mo></mrow></mrow></mstyle></mrow></mstyle></math><img file="EP4249971A2_D0006.tif" /></maths>

0036Six different decomposition functions, <i>ξ_mn</i>(<i>r,θ</i>) are available for determining <i>η</i>(<i>r,θ</i>): <maths id="math0007"><math display="block"><msub><mi>ξ</mi><mi mathvariant="italic">mn</mi></msub><mfenced><mi>r</mi><mi>θ</mi></mfenced><mo>=</mo><msub><mi>E</mi><mi>m</mi></msub><mfenced><mi>r</mi><mi>θ</mi></mfenced><msubsup><mi>E</mi><mi>n</mi><mo>*</mo></msubsup><mfenced><mi>r</mi><mi>θ</mi></mfenced></math><img file="EP4249971A2_D0007.tif" /></maths>

0037<figref idref="f0010">Figure 20</figref> illustrates the six decomposition functions of the six cross coupling possibilities for this example.

0038<figref idref="f0011">Figure 21</figref> shows decomposition functions for decomposing index perturbation for this example.

0039By measuring the phase and amplitude of all six coupling coefficients, <i>κ_mn</i>, the grating modulation (e.g., changes in the grating caused by strain, stress, and/or twist), <i>η</i>(<i>r,θ</i>)<i>e</i>^-<i>^ikz,</i> may be reconstructed using these coefficients, <i>κ_mn</i>, as the weight factor on summation of the decomposition function <i>ξ_mn</i>(<i>r,θ</i>). <maths id="math0008"><math display="block"><mi>η</mi><mfenced><mover accent="true"><mi>p</mi><mo>→</mo></mover></mfenced><mo>≈</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><msub><mi>κ</mi><mi mathvariant="italic">mn</mi></msub><msub><mi>ξ</mi><mi mathvariant="italic">mn</mi></msub><mfenced><mi>r</mi><mi>θ</mi></mfenced><mo>=</mo><mstyle displaystyle="true"><mo>∑</mo><mrow><msub><mi>κ</mi><mi mathvariant="italic">mn</mi></msub><msub><mi>E</mi><mi>m</mi></msub></mrow></mstyle></mrow></mstyle><mfenced><mi>r</mi><mi>θ</mi></mfenced><msubsup><mi>E</mi><mi>n</mi><mo>*</mo></msubsup><mfenced><mi>r</mi><mi>θ</mi></mfenced><mo>.</mo></math><img file="EP4249971A2_D0008.tif" /></maths>

0040This decomposition method of determining the distribution of a Bragg grating across the cross section of a single multimode core was simulated for two tilted and overlapped Bragg gratings, such as described above. The plots in <figref idref="f0011">Figure 21</figref> represent the grating amplitude across the core cross-section and show simulated, compound, tilted grating index profiles on the left and the grating profiles successfully reconstructed from the overlap function decomposition on the right. The top two images are without fiber twist, the bottom two with twist. The simulation results show that the overlap integrals provide significant reconstruction capability and the lower right reconstruction shows that twist of the grating structure can be determined.

0041Before measurements are taken with a single multimode core, a "baseline" or reference measurement with the fiber in a known orientation, in this case straight and untwisted, as performed. Thereafter, normal measurements may be made with fiber moved, e.g., bent and/or twisted. A phase "deformation" δφ is then calculated based upon the phase change between these baseline and normal measurements. <maths id="math0009"><math display="block"><mi mathvariant="italic">δφ</mi><mfenced><mi>r</mi><mi>θ</mi></mfenced><mo>=</mo><mo>∠</mo><mfenced open="{" close="}"><msub><mi>η</mi><mi mathvariant="italic">move</mi></msub><mfenced><mi>r</mi><mi>θ</mi></mfenced><mo>⋅</mo><msubsup><mi>η</mi><mi mathvariant="italic">base</mi><mo>*</mo></msubsup><mfenced><mi>r</mi><mi>θ</mi></mfenced></mfenced></math><img file="EP4249971A2_D0009.tif" /></maths> Or equivalently in the x, y coordinate system: <maths id="math0010"><math display="block"><mi mathvariant="italic">δφ</mi><mfenced><mi>x</mi><mi>y</mi></mfenced><mo>=</mo><mo>∠</mo><mfenced open="{" close="}"><msub><mi>η</mi><mi mathvariant="italic">move</mi></msub><mfenced><mi>x</mi><mi>y</mi></mfenced><mo>⋅</mo><msubsup><mi>η</mi><mi mathvariant="italic">base</mi><mo>*</mo></msubsup><mfenced><mi>x</mi><mi>y</mi></mfenced></mfenced></math><img file="EP4249971A2_D0010.tif" /></maths>

0042Between the base and the moved measurement for this example, there can be a horizontal bend, b_x, that will be a linear phase change as a function of x, a vertical bend, b_y, that will be a linear phase change as a function of y, a stretch, ε, that will be constant change over the surface, or a twist, τ, that will be a rotation of the entire complex (phase and amplitude) scattering cross section.

0043Estimates can be determined for each of these shape parameters b_x, b_y, ε, and τ. These estimates can essentially function as a shape conversion matrix to convert from phase measurements to pitch, yaw, twist and strain. <maths id="math0011"><math display="block"><mi>ε</mi><mo>=</mo><mo>∠</mo><mstyle displaystyle="true"><mrow><mo>∬</mo><mrow><msub><mi>η</mi><mi mathvariant="italic">move</mi></msub><mfenced><mi>x</mi><mi>y</mi></mfenced><mo>⋅</mo><msubsup><mi>η</mi><mi mathvariant="italic">base</mi><mo>*</mo></msubsup><mfenced><mi>x</mi><mi>y</mi></mfenced><mi mathvariant="italic">dA</mi></mrow></mrow></mstyle></math><img file="EP4249971A2_D0011.tif" /></maths><maths id="math0012"><math display="block"><msub><mi>b</mi><mi>x</mi></msub><mo>=</mo><mi>max</mi><mstyle displaystyle="true"><mrow><mo>∫</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><msub><mi>η</mi><mi mathvariant="italic">move</mi></msub><mfenced><mi>x</mi><mi>y</mi></mfenced><mo>⋅</mo><msubsup><mi>η</mi><mi mathvariant="italic">base</mi><mo>*</mo></msubsup><mfenced><mi>x</mi><mi>y</mi></mfenced><msup><mi>e</mi><msub><mi mathvariant="italic">ib</mi><mi>x</mi></msub></msup><mi mathvariant="italic">dA</mi></mrow></mrow></mstyle></mrow></mstyle></math><img file="EP4249971A2_D0012.tif" /></maths><maths id="math0013"><math display="block"><msub><mi>b</mi><mi>y</mi></msub><mo>=</mo><mi>max</mi><mstyle displaystyle="true"><mrow><mo>∫</mo><mstyle displaystyle="true"><mrow><mo>∫</mo><mrow><msub><mi>η</mi><mi mathvariant="italic">move</mi></msub><mfenced><mi>x</mi><mi>y</mi></mfenced><mo>⋅</mo><msubsup><mi>η</mi><mi mathvariant="italic">base</mi><mo>*</mo></msubsup><mfenced><mi>x</mi><mi>y</mi></mfenced><msup><mi>e</mi><msub><mi mathvariant="italic">ib</mi><mi>y</mi></msub></msup><mi mathvariant="italic">dA</mi></mrow></mrow></mstyle></mrow></mstyle></math><img file="EP4249971A2_D0013.tif" /></maths><maths id="math0014"><math display="block"><mi>τ</mi><mo>=</mo><mi>max</mi><mstyle displaystyle="true"><mrow><munderover><mo>∫</mo><mn>0</mn><mrow><mn>2</mn><mi>π</mi></mrow></munderover><mrow><msub><mi>η</mi><mi mathvariant="italic">move</mi></msub><mfenced><mi>r</mi><mi>θ</mi></mfenced><mo>⋅</mo><msubsup><mi>η</mi><mi mathvariant="italic">base</mi><mo>*</mo></msubsup><mfenced><mi>r</mi><mo>,</mo><mi>θ</mi><mo>+</mo><mi>τ</mi></mfenced><mi mathvariant="italic">dθ</mi></mrow></mrow></mstyle></math><img file="EP4249971A2_D0014.tif" /></maths>

0044With the bend, twist, and strain terms b_x, b_y, ε, and τ, the shape of the fiber may be determined using three dimensional rotations and projections described in <patcit id="pcit0004" dnum="US7781724B"><text>U.S. patents 7,781,724</text></patcit> and <patcit id="pcit0005" dnum="US8773650B"><text>8,773,650</text></patcit> identified in the introduction.

MEASUREMENT OF COUPLING COEFFICIENTS

0045In addition to using the measured coupling coefficients (<i>κ_mn</i>) to determine the index perturbation in a single multimode core, some embodiments also measure the coupling coefficients. Some embodiments get the energy from single mode fibers into and out of each of the different modes in a way that is power efficient (the detecting fibers are typically coupled to photodiode detectors in an Optical Frequency Domain Reflectometry (OFDR) system). As one option, a fused tapered coupler, tapering multiple single mode fibers and fusing them to different locations on the multimode core, may be an option if the tapering can be done as the mode coupling was monitored in a 3x3 coupler. Another option abuts a bundle of single mode cores up against the larger multimode core. Some embodiments use this option when the cores are surrounded by little or no cladding. Another option is to image the multimode core directly onto a bundle of single mode fibers with a large magnification. Some embodiments use this option when the system can function with the inefficiencies resulting from light that may be captured by the cladding surrounding each of the single mode fibers.

0046Some embodiments including this last option of directly imaging the multimode core reduce the inefficiencies by using a micro lens array with a sufficiently large fill-factor and a numerical aperture (NA) that matches the optical fiber with the multimode core more closely (e.g., between 0.1 and 0.25 in some cases). An example of a bulk-optic design for getting energy from each of the different modes into a corresponding single-mode detecting fiber in a way that is power efficient is shown in <figref idref="f0011">Figure 22</figref>. The design may use a commercially-available microlens array 22 and includes a fiber array 20 of three single core, single mode detecting fibers for directing light from each of the three modes of light provided from a single multimode core 13 in a single core multimode sensing fiber 10 into three single core, single mode detecting fibers in the fiber array 20. Multimode light from multimode core 13 is imaged by lens 24 to the microlens array 22.

0047<figref idref="f0011">Figure 22</figref> shows specific values for dimensions and other parameters that are useful for specific embodiments, and are only examples and are not limiting.

0048The optical coupling mechanism shown in <figref idref="f0011">Figure 22</figref> is included in <figref idref="f0013">Figure 24. Figure 24</figref> shows an example system for interrogating a single core multimode sensing fiber 10 with multiple modes. The interrogation system is controlled by an OFDR controller 30 that includes one or more computers 32 coupled to (i) a laser controller 34 that controls a tunable laser 36 through a range of frequencies or wavelengths, (ii) a display, and (iii) acquisition circuitry including photodiodes, analog to digital conversion circuitry, sampling circuitry. Because there are three modes, there are three reference branches and three measurement branches of different lengths coupled to three single mode fibers that are coupled to three polarization beams splitters (PBSs) that convert the interfered light for each mode into s and p polarizations for detection by respective s and p photodiodes. The laser controller 34 controls the tunable laser 36 to probe the single core multimode sensing fiber 10 with three separate single core fibers in the three measurement branches, each with a unique delay corresponding to a different length of fiber L, L1, and L2. The laser light from each of the three input fibers is coupled via the fiber array 20 through the microlens array 22 and lens 24 into the single core 13, and the reflected light from each of the three modes from the single core grating is delivered by the microlens array 22 to its corresponding single core single mode detecting fiber in the fiber array 20 for detection and processing in the OFDR controller 30.

0049If the delays of the three input fibers are larger than the total delay associated with the single core multimode sensing fiber 10, then all of the cross coupled terms between input and output fibers of the measurement interferometer will appear at different delays and on unique detectors. If the fiber lengths for each output fiber are the same, then the overall delay is determined by the input fiber in this example.

0050<figref idref="f0013">Figure 24</figref> are graphs that illustrate 9 coupling terms m11-m33. The three graphs of amplitude vs. delay represent the amplitude of light detected on each of the three detectors at the acquisition circuitry 38 corresponding to the three single mode fibers in the fiber array 20. The first subscript on the coupling terms, m, identifies the input fiber and the second subscript identifies the output fiber (the detector the signal is detected on). So m31 represents the amount of light sent in on fiber 3 and detected on detector fiber 1.

EXTRACTION OF FIBER MODAL COUPLING FROM INSTRUMENTATION COEFFICIENTS

0051As demonstrated above, the index perturbation (grating) may be determined by measuring the cross coupling coefficients detected at the multiple (three in the example) single mode fibers in the fiber array 20 that correspond to but are not identical to the modes of the single core 13 in the single core multimode sensing fiber 10. <figref idref="f0013">Figure 24</figref> described above illustrates the coupling coefficients between the three measured components coupled to the single mode fibers in the fiber array 20. These cross-coupling coefficients for these single mode fibers can be measured directly by the OFDR controller 30, but they need to be connected or converted to the actual modal cross-coupling coefficients for the three modes of the single core 13 in the single core multimode sensing fiber 10.

0052Some embodiments make this connection or conversion using a matrix, <maths id="math0015"><math display="inline"><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0015.tif" /></maths>, that describes how the light from the three input fibers in the fiber array 20 couples into the three modes supported in the single multimode core 13 of the single core multimode sensing fiber 10. <figref idref="f0013">Figure 25</figref> shows an example model of an optical connection system as two matrices connected by the alpha matrix <maths id="math0016"><math display="inline"><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0016.tif" /></maths>. <figref idref="f0013">Figure 25</figref> is represented using the mathematical expression below: <maths id="math0017"><math display="block"><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>E</mi><mi>V</mi></msub></mtd></mtr><mtr><mtd><msub><mi>E</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><msub><mi>E</mi><mi>H</mi></msub></mtd></mtr></mtable></mfenced><mo>=</mo><msub><mover accent="true"><mi>A</mi><mo>⇀</mo></mover><mi mathvariant="italic">out</mi></msub><mo>=</mo><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover><mover accent="true"><mi>E</mi><mo>⇀</mo></mover></math><img file="EP4249971A2_D0017.tif" /></maths> where <maths id="math0018"><math display="inline"><msub><mover accent="true"><mi>A</mi><mo>⇀</mo></mover><mi mathvariant="italic">out</mi></msub></math><img file="EP4249971A2_D0018.tif" /></maths> is the vector formed by the fields in each of the single-mode input/probe fibers in the fiber array 20, <maths id="math0019"><math display="inline"><mover accent="true"><mi>E</mi><mo>⇀</mo></mover></math><img file="EP4249971A2_D0019.tif" /></maths> is the vector composed of the field in each of the three fiber modes in the single core 13, and <maths id="math0020"><math display="inline"><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0020.tif" /></maths> is the connecting matrix that couples the multimode fiber modes <maths id="math0021"><math display="inline"><mover accent="true"><mi>E</mi><mo>⇀</mo></mover></math><img file="EP4249971A2_D0021.tif" /></maths> to the single mode fiber fields <maths id="math0022"><math display="inline"><msub><mover accent="true"><mi>A</mi><mo>⇀</mo></mover><mi mathvariant="italic">out</mi></msub></math><img file="EP4249971A2_D0022.tif" /></maths>. <i>E</i>₀ is the circularly symmetric LP01 mode, <i>E_V</i> is the vertically oriented LP11 mode, and <i>E_H</i> is the horizontally oriented LP 11 mode. Light can travel in both directions, and the conversion from light in the individual single mode cores in the fiber array 20 to the light in the modes of the single core multimode sensing fiber 10 are given by, <maths id="math0023"><math display="block"><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>E</mi><mi>V</mi></msub></mtd></mtr><mtr><mtd><msub><mi>E</mi><mn>0</mn></msub></mtd></mtr><mtr><mtd><msub><mi>E</mi><mi>H</mi></msub></mtd></mtr></mtable></mfenced><mo>=</mo><msup><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mover accent="true"><mi>A</mi><mo>⇀</mo></mover><mi mathvariant="italic">out</mi></msub><mo>=</mo><mover accent="true"><mi>E</mi><mo>⇀</mo></mover></math><img file="EP4249971A2_D0023.tif" /></maths>

0053The expression is now written for the light travelling from the single-mode fibers in the fiber array 20, through the optics (microlens array 22 and lens 24) into the single core multimode sensing fiber 10, then being coupled into the backward travelling mode, and then back through the optics (microlens array 22 and lens 24) and into the single-mode fibers in the fiber array 20 as follows: <maths id="math0024"><math display="block"><msub><mover accent="true"><mi>A</mi><mo>⇀</mo></mover><mi mathvariant="italic">out</mi></msub><mo>=</mo><msup><mover accent="true"><mover accent="true"><mrow><mi>α</mi><mi mathvariant="normal">K</mi><mi>α</mi></mrow><mo>‾</mo></mover><mo>‾</mo></mover><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mover accent="true"><mi>A</mi><mo>⇀</mo></mover><mi mathvariant="italic">in</mi></msub><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>κ</mi><mn>11</mn></msub></mtd><mtd><msub><mi>κ</mi><mn>21</mn></msub></mtd><mtd><msub><mi>κ</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>κ</mi><mn>21</mn></msub></mtd><mtd><msub><mi>κ</mi><mn>22</mn></msub></mtd><mtd><msub><mi>κ</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>κ</mi><mn>31</mn></msub></mtd><mtd><msub><mi>κ</mi><mn>32</mn></msub></mtd><mtd><msub><mi>κ</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced></math><img file="EP4249971A2_D0024.tif" /></maths>

0054The combination of the above three matrices, <maths id="math0025"><math display="inline"><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover><mover accent="true"><mover accent="true"><mi mathvariant="normal">K</mi><mo>‾</mo></mover><mo>‾</mo></mover><msup><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover><mrow><mo>−</mo><mn>1</mn></mrow></msup></math><img file="EP4249971A2_D0025.tif" /></maths>, forms the overall coupling coefficients between input fibers and output fibers shown in <figref idref="f0013">Figure 24</figref> and is referred to hereafter as the <maths id="math0026"><math display="inline"><mover accent="true"><mover accent="true"><mi>M</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0026.tif" /></maths> matrix. <maths id="math0027"><math display="block"><msub><mover accent="true"><mi>A</mi><mo>⇀</mo></mover><mi mathvariant="italic">out</mi></msub><mo>=</mo><msup><mover accent="true"><mover accent="true"><mrow><mi>α</mi><mi mathvariant="normal">K</mi><mi>α</mi></mrow><mo>‾</mo></mover><mo>‾</mo></mover><mrow><mo>−</mo><mn>1</mn></mrow></msup><msub><mover accent="true"><mi>A</mi><mo>⇀</mo></mover><mi mathvariant="italic">in</mi></msub><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>m</mi><mn>11</mn></msub></mtd><mtd><msub><mi>m</mi><mn>21</mn></msub></mtd><mtd><msub><mi>m</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>m</mi><mn>21</mn></msub></mtd><mtd><msub><mi>m</mi><mn>22</mn></msub></mtd><mtd><msub><mi>m</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>m</mi><mn>31</mn></msub></mtd><mtd><msub><mi>m</mi><mn>32</mn></msub></mtd><mtd><msub><mi>m</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>A</mi><mn>1</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>2</mn></msub></mtd></mtr><mtr><mtd><msub><mi>A</mi><mn>3</mn></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mover accent="true"><mover accent="true"><mi>M</mi><mo>‾</mo></mover><mo>‾</mo></mover><msub><mover accent="true"><mi>A</mi><mo>⇀</mo></mover><mi mathvariant="italic">in</mi></msub></math><img file="EP4249971A2_D0027.tif" /></maths>

0055In this example, <maths id="math0028"><math display="inline"><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover><mover accent="true"><mover accent="true"><mi mathvariant="normal">K</mi><mo>‾</mo></mover><mo>‾</mo></mover><msup><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover><mrow><mo>−</mo><mn>1</mn></mrow></msup></math><img file="EP4249971A2_D0028.tif" /></maths> is what is measured with the time-delayed interrogation network shown in <figref idref="f0012">Figure 23</figref>. Although the objective is to determine the modal cross-coupling coefficients represented by the cross-coupling coefficients matrix <maths id="math0029"><math display="inline"><mover accent="true"><mover accent="true"><mi mathvariant="normal">K</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0029.tif" /></maths> corresponding to the state of the core grating, what can be actually measured by the OFDR interrogation system is the <maths id="math0030"><math display="inline"><mover accent="true"><mover accent="true"><mi>M</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0030.tif" /></maths> matrix.

0056Embodiments determine the elements that form the matrix <maths id="math0031"><math display="inline"><mover accent="true"><mover accent="true"><mi>M</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0031.tif" /></maths> in the following manner. An OFDR scan of a single core multimode fiber with a cleaved end is used to separate out the <i>E</i>₀(LP01) mode based upon propagation time to the end of the cleave and also to separate the cross coupling terms. <figref idref="f0014">Figure 26</figref> is a graph of an example impulse response or propagation delay of a single core multimode fiber with a cleaved end. The reflection from the cleaved end is shown and labeled as "fiber endface." The different propagation modes propagate along the fiber at different speeds, which means they can be separated and identified. For example, because the <i>E</i>₀(LP01) mode has a longer propagation time, its reflection will arrive later at the OFDR detector than the <i>E_V</i> and <i>E_H</i>(LP11) mode reflections. The light that travels down the fiber in the <i>E</i>₀(LP01) mode and couples to the <i>E_V</i> and <i>E_H</i>(LP11) modes at the cleave will arrive with a delay exactly in between the two modes.

0057<figref idref="f0014">Figure 27</figref> is a graph of the example impulse response delays in <figref idref="f0014">Figure 26</figref> relabeled to identify the modal assignments (coupling coefficients). From <figref idref="f0014">Figure 27</figref>, the three groups of coupling coefficients can be separated and filtered around each of the different delays (the distinct and separate impulse response amplitude lines). The terms are isolated by first windowing the complex data around the reflection peak, and then performing a Fourier transform to compute the frequency domain complex spectrum of the peak. First, the reflected coupling coefficient κ₂₂ for mode <i>E</i>₀(LP01) is isolated using the windowing and transformation to mathematically separate the peak on all channels and at all offset delays to give a matrix of filtered values. <maths id="math0032"><math display="block"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>0</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>22</mn></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></math><img file="EP4249971A2_D0032.tif" /></maths> To implement this isolation in this example, the OFDR controller applies a time domain window filter (like a bandpass filter) around the <i>E</i>₀(LP01) delay peak.

0058Next, the light that travels exclusively in the <i>E_V</i> and <i>E_H</i> (LP11) modes and corresponding to the following is isolated: <maths id="math0033"><math display="block"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>1</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>κ</mi><mn>11</mn></msub></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>κ</mi><mn>13</mn></msub></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></math><img file="EP4249971A2_D0033.tif" /></maths>

0059And finally the light that crosses between <i>E</i>₀ (LP01) and the <i>E_V</i> and <i>E_H</i> (LP11) modes at the cleave is isolated as follows: <maths id="math0034"><math display="block"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>2</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>21</mn></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><msub><mi>κ</mi><mn>12</mn></msub></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>23</mn></msub></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></math><img file="EP4249971A2_D0034.tif" /></maths>

0060Next, assuming a flat, 0 degree cleave at the end of the fiber, the cross coupling terms (<i>κ</i>₁₃ and <i>κ</i>₃₁) are driven to zero because the perturbations function <i>η</i>(<i>r,θ</i>) for a zero degree cleave is a constant, giving a simplified matrix: <maths id="math0035"><math display="block"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>1</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>κ</mi><mn>11</mn></msub></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></math><img file="EP4249971A2_D0035.tif" /></maths>

0061Adding <maths id="math0036"><math display="inline"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>0</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0036.tif" /></maths> and <maths id="math0037"><math display="inline"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>1</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0037.tif" /></maths> together gives the following diagonal matrix between the input and output coupling matrices. <maths id="math0038"><math display="block"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>0</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>+</mo><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>1</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>κ</mi><mn>11</mn></msub></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>22</mn></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><msup><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mrow><mo>−</mo><mn>1</mn></mrow></msup></math><img file="EP4249971A2_D0038.tif" /></maths> which can be rewritten as by moving the inverse matrix to the left side of the equality: <maths id="math0039"><math display="block"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>0</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>+</mo><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>1</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>κ</mi><mn>11</mn></msub></mtd><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>22</mn></msub></mtd><mtd><mn>0</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>0</mn></mtd><mtd><msub><mi>κ</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced></math><img file="EP4249971A2_D0039.tif" /></maths>

0062This last equation takes the form of an eigenvalue problem: <maths id="math0040"><math display="block"><mover accent="true"><mover accent="true"><mi>A</mi><mo>‾</mo></mover><mo>‾</mo></mover><mover accent="true"><mover accent="true"><mi>S</mi><mo>‾</mo></mover><mo>‾</mo></mover><mo>=</mo><mover accent="true"><mover accent="true"><mi>S</mi><mo>‾</mo></mover><mo>‾</mo></mover><mover accent="true"><mover accent="true"><mi mathvariant="normal">Λ</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0040.tif" /></maths> where S is a matrix whose columns are the eigenvectors of A and A is a diagonal matrix whose elements are the eigenvalues. Since matrix A can be measured by the OFDR interrogation system, the coupling coefficients (<i>κ</i>₁₁, <i>κ</i>₂₂, and <i>κ</i>₃₃) can be determined.

0063Finding the matrices that convert the measurements, <maths id="math0041"><math display="inline"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>0</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>+</mo><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>1</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0041.tif" /></maths>, to the mode coupling coefficients (<i>κ</i>₁₁, <i>κ</i>₂₂, and <i>κ</i>₃₃) is now an eigenvalue problem where the coupling coefficients (<i>κ</i>₁₁, <i>κ</i>₂₂, and <i>κ</i>₃₃) are the eigenvalues of the matrix, <maths id="math0042"><math display="inline"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>0</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>+</mo><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>1</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0042.tif" /></maths>.

0064The connector matrix: <maths id="math0043"><math display="block"><mfenced open="[" close="]"><mtable><mtr><mtd><msub><mi>α</mi><mn>11</mn></msub></mtd><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>21</mn></msub></mtd><mtd><msub><mi>α</mi><mn>22</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd></mtr><mtr><mtd><msub><mi>α</mi><mn>31</mn></msub></mtd><mtd><msub><mi>α</mi><mn>32</mn></msub></mtd><mtd><msub><mi>α</mi><mn>33</mn></msub></mtd></mtr></mtable></mfenced></math><img file="EP4249971A2_D0043.tif" /></maths> can be constructed from the eigenvectors of <maths id="math0044"><math display="inline"><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>0</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover><mo>+</mo><mover accent="true"><mover accent="true"><msub><mi>M</mi><mn>1</mn></msub><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0044.tif" /></maths> using linear algebra.

0065Once the connector <maths id="math0045"><math display="inline"><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0045.tif" /></maths> matrix, which defines coupling between the single mode fibers and the single core multimode sensing fiber, is determined, it remains constant for a given fiber connection. This connector <maths id="math0046"><math display="inline"><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0046.tif" /></maths> matrix and its inverse define how light couples from the OFDR interrogation system to the modes of the single core multimode sensing fiber and back. Knowing the values for this connector <maths id="math0047"><math display="inline"><mover accent="true"><mover accent="true"><mi>α</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0047.tif" /></maths> matrix and the input light, the output light is measured by the OFDR interrogation system, and from these three things, the OFDR interrogation system calculates the values of the current modal cross coupling coefficients, or the <maths id="math0048"><math display="inline"><mover accent="true"><mover accent="true"><mi mathvariant="normal">K</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0048.tif" /></maths> matrix. Based on these modal cross coupling coefficient values, the phase perturbations in the single core multimode sensing fiber 10 that produced the measured coupling (<i>κ</i>) coefficients and which represent the state of the grating in the single core multimode sensing fiber 10 are calculated. From these determined phase perturbations corresponding to the state of the core grating, the bend and twist of the single core multimode sensing fiber 10 can be calculated at every point along the single core multimode fiber 10. These bend and twist values can then be used to calculate the shape of the multimode sensing fiber 10.

ACCOUNTING FOR GROUP DELAY DIFFERENCES BETWEEN MODES

0066The approach described above provides measurements of the coupling between modes along the length of the single core multimode sensing fiber 10 as a function of time delay. In some embodiments, further corrections are made to address differing group velocities between modes. For example, to address the differing group velocities causing the mode coupling coefficients ( <maths id="math0049"><math display="inline"><mover accent="true"><mover accent="true"><mi>κ</mi><mo>‾</mo></mover><mo>‾</mo></mover></math><img file="EP4249971A2_D0049.tif" /></maths>) to be spread over different delay ranges depending on which modes the light propagated in and for how long.

0067<figref idref="f0015">Figure 28</figref> graphs example fiber mode responses with exaggerated time-length differences to illustrate how the effective length of the fiber appears to be different for each of the modes due to differences in group delay between the modes. In <figref idref="f0013">Figure 24</figref>, the time delay amounts are the same, while in <figref idref="f0015">Figure 28</figref>, the time delay amounts are different due to group delay effects. These group delay effects can be accounted for by resampling the data when the coupling coefficients are identified. In some embodiments, the coupling coefficients identification includes mapping the physical location in the fiber where the coupling took place to the same index for each mode. Here, the term index is the location along the length of the fiber before it has been scaled to engineering lengths. The data is lined up to represent the same location at the same index. This mapping is performed for each of the 6 independent coupling coefficients in this example (although there are 9 coupling coefficients, several have the same effective group index such as κ₁₃ and κ₃₁).

0068<figref idref="f0015">Figure 29</figref> is a graph showing an example of resampling and alignment of coupling coefficients κ₁₁, κ₂₁, and κ₃₁ showing conversion of them from the time domain to the spatial domain (the horizontal axis in <figref idref="f0015">Figure 29</figref> aligns with the z axis of the single core multimode sensing fiber 10. This conversion is accomplished by multiplying the time label of each of the coupling coefficients κ₁₁, κ₂₁, and κ₃₁ in the time domain by the speed of light for that coupling coefficient.

0069In addition to propagating with different group delays, each mode also propagates with a different wavenumber, or effective refractive index, meaning that each mode accumulates phase at a different rate as it propagates down the single core multimode sensing fiber 10. <figref idref="f0015">Figure 30</figref> graphs example phase accumulations for different coupling coefficients as a function of delay.

0070The LP11 and LP01 self-coupling terms are both weighted integrals over the single core multimode sensing fiber 10 in this example. As a result, even for random Rayleigh scatter, the LP11 and LP01 self-coupling terms are substantially similar. If the sensing fiber is straight, this common term in the coupling terms allows precise measurements of the difference between the phase propagation terms for the LP 11 and LP01 modes along the fiber length. Further, this difference in effective refractive index can be used to predict the phase change in the cross term phases, such as by assuming the cross term phase accumulations are exactly halfway between the pure LP 11 and LP0 1 phase delays. These calculated phase changes due to the different effective refractive indexes of the modes are then applied to the measured and resampled coupling coefficients before the scattering cross section is calculated.

0071<figref idref="f0016">Figure 31</figref> is a flowchart showing example procedures for using a single core, multiple mode fiber for sensing shape in accordance with example embodiments. In step S1, light is input into three (or more) modes supported by the single multimode core fiber. The grating of the core causes forward traveling modes to be reflected into backward traveling modes as shown in step S2. In step S3, reflected light is detected, converted into electrical signals, converted from analog format into digital format, processed in the OFDR controller to calculate the cross coupling coefficients. In step S4, decomposition functions are used to determine the index perturbation, or the current state of the grating, when compared to a measurement in a known state (such as the fiber positioned in a straight line). At step S5, the pitch, yaw, twist, and strain from the change in the index perturbation is determined. Then, in step S6, the fiber shape is determined from the pitch, yaw, twist, and strain determined in step S5.

0072<figref idref="f0017">Figure 32</figref> is a flowchart showing example procedures for using a single core, multiple mode fiber for sensing shape in accordance with example embodiments. In step S10, light from the tunable laser is split between measurement and reference paths using for example the OFDR system in <figref idref="f0012">Figure 23</figref>. In step S11, the measurement light is split into three input fibers, each input fiber having a different delay. In step S12, light from the three input fibers is coupled into the single core multimode sensing fiber using a lens array. In step S13, forward propagating life from the three modes is coupled into backward traveling light in the three modes via the core grating. In step S14, the light in the backward traveling modes is coupled into the three single mode fibers (now output fibers) via the lens array. In step S15, this backward traveling light is combined with the reference light and detected on three detectors. In step S16, data from the three detectors is processed to determine the M matrix which couples the coupling coefficients between the input light and the output light. In step S17, using the alpha matrix, which is the matrix describing the coupling between the single mode input fibers and the modes of the single core multimode fiber, cross coupling coefficients between the modes of the single core multimode fiber are determined. These cross coupling coefficients form the K matrix. In step S18, these cross coupling coefficients are then used to calculate the current index perturbation, which is the core grating, and from there, the shape of the single core multimode sensing fiber is calculated as in steps S4-S6 in <figref idref="f0016">Figure 31</figref>.

0073<figref idref="f0018">Figure 33</figref> is a flowchart showing example procedures for calibrating and then using a single core, multiple mode fiber for sensing shape in accordance with example embodiments. Two different process stages are included and coupled loosely with a dashed line. The first process stage is performed before the fiber is used for shape measurements. The second process stage is used when the sensing fiber is to perform shape measurement.

0074In step S20, the single core multimode fiber is connected to a multimode interrogation system such as the OFDR system shown in <figref idref="f0012">Figure 23</figref>. In step S21, the shape sensing fiber is positioned in a straight line or in another known orientation. The cleaved end response of the fiber is measured in step S22. Then, the decoupling matrix is calculated to convert the measurement modes to fiber modes in step S23. The decoupling matrix is referred to above as the alpha matrix and is determined following steps S20-S23. The next steps S24 through S26 relate to determining reference or baseline measurements of the shape sensing fiber in a known state. In step S24, the cleaved end is terminated, and the coupling coefficients are measured for the shape sensing fiber. In step S25, the coupling coefficients are resampled into the same physical frame. Then, in step S26, corrections are determined for the wavenumber differences.

0075Moving over to the next process stage of the flowchart, step S30 describes changing the shape of the fiber to a measurement orientation. In step S31, the fiber response is measured using the interrogation system. Then, the decoupling or alpha matrix is applied to the measured response in step S32. The coupling coefficients are resampled in step S33, and the wavenumber corrections are applied in step S34. The index modulation profile change is calculated at each point along the sensing fiber in step S35. In step S36, the bend, twist, and strain are calculated each point along the fiber, and from these values, the shape of the sensing fiber is calculated in step S37.

EXAMPLE MULTIMODE SINGLE CORE FIBER SENSOR TWIST SENSITIVITY

0076Having determined how to make and use a single core multimode fiber to determine shape using an OFDR interrogation method and overlapped tilted gratings, the inventors tested different fiber core sizes and estimated their sensitivity to detecting twist.

0077An example fiber that admits only three modes was designed and then checked to verify that this fiber provides reasonable twist sensitivity. The number of modes in a step-index multimode fiber is described by a normalized frequency parameter known as the V-number. A V-number of around 3.5 was selected in order to guide the LP11 modes tightly, while excluding the LP21 and LP02 modes.

0078<figref idref="f0020">Figure 37</figref> is a graph showing modal propagation values for Linear Polarization Modes vs. V-number. The V-number is given by the equation below. <maths id="math0050"><math display="block"><mi>V</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="italic">πr</mi></mrow><mi>λ</mi></mfrac><msqrt><mrow><msubsup><mi>n</mi><mn>1</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>n</mi><mn>2</mn><mn>2</mn></msubsup></mrow></msqrt><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="italic">πr</mi></mrow><mi>λ</mi></mfrac><mi mathvariant="italic">NA</mi></math><img file="EP4249971A2_D0050.tif" /></maths> where <i>n</i>₁ is the index of refraction of the core, <i>n</i>₂ is the index of the cladding, <i>λ</i> is the wavelength of the light, <i>r</i> is the core radius, and <i>NA</i> is the Numerical Aperture. Using V=3.5, the selected operating wavelength, and a selected Numerical Aperture permits calculation of the core radius as follows: <maths id="math0051"><math display="block"><mn>3.5</mn><mo>=</mo><mfrac><mrow><mn>2</mn><mi mathvariant="italic">πr</mi></mrow><mrow><mn>1540</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>9</mn></mrow></msup></mrow></mfrac><mn>0.2</mn></math><img file="EP4249971A2_D0051.tif" /></maths> Solving for r yields: <maths id="math0052"><math display="block"><mfrac><mrow><mn>3.5</mn><mfenced><mn>1540</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>9</mn></mrow></msup></mfenced></mrow><mrow><mn>0.2</mn><mo>⋅</mo><mn>2</mn><mi>π</mi></mrow></mfrac><mo>=</mo><mi>r</mi><mo>=</mo><mn>4.3</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup><mo>.</mo></math><img file="EP4249971A2_D0052.tif" /></maths>

0079The rotational period of the skew rays can be calculated from the difference in the wavenumbers between the modes present in the fiber. This difference may be characterized by the parameter b , which can be used to calculate the wavenumbers for each mode. <figref idref="f0021">Figure 38</figref> is a graph showing a normalized propagation constant B versus normalized frequency V for TE and TM modes. This graph is used to identify values for <i>b</i> at the selected V number of 3.5 as <i>b</i>₀ = 0.34 and <i>b</i>₁ = 0.75. The definition of the b number is <maths id="math0053"><math display="inline"><mi>b</mi><mo>=</mo><mfrac><mrow><msup><mi>β</mi><mn>2</mn></msup><mo>−</mo><msubsup><mi>k</mi><mn>2</mn><mn>2</mn></msubsup></mrow><mrow><msubsup><mi>k</mi><mn>1</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>k</mi><mn>2</mn><mn>2</mn></msubsup></mrow></mfrac></math><img file="EP4249971A2_D0053.tif" /></maths>, where <i>β</i> is the propagation constant and <i>k</i> is the wavenumber. The definition of the wave number is <maths id="math0054"><math display="inline"><msub><mi>k</mi><mn>1</mn></msub><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>λ</mi></mfrac><msub><mi>n</mi><mn>1</mn></msub></math><img file="EP4249971A2_D0054.tif" /></maths>.

0080The propagation constant β as a function of <i>b</i> is solved in accordance with: <maths id="math0055"><math display="block"><msqrt><mrow><mi>b</mi><mfenced><msubsup><mi>k</mi><mn>1</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>k</mi><mn>2</mn><mn>2</mn></msubsup></mfenced><mo>+</mo><msubsup><mi>k</mi><mn>2</mn><mn>2</mn></msubsup></mrow></msqrt><mo>=</mo><mi>β</mi></math><img file="EP4249971A2_D0055.tif" /></maths> Factoring out the vacuum wavenumber, <maths id="math0056"><math display="inline"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>λ</mi></mfrac></math><img file="EP4249971A2_D0056.tif" /></maths>, collecting terms, and applying the definition of Numerical Aperture (NA) in an optical fiber, <maths id="math0057"><math display="inline"><mi mathvariant="italic">NA</mi><mo>=</mo><msqrt><mrow><msup><msub><mi>n</mi><mn>1</mn></msub><mn>2</mn></msup><mo>−</mo><msup><msub><mi>n</mi><mn>2</mn></msub><mn>2</mn></msup></mrow></msqrt></math><img file="EP4249971A2_D0057.tif" /></maths>, this equation may be expressed as: <maths id="math0058"><math display="block"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>λ</mi></mfrac><mi mathvariant="italic">NA</mi><msqrt><mrow><mi>b</mi><mo>+</mo><mfrac><msubsup><mi>n</mi><mn>2</mn><mn>2</mn></msubsup><msup><mi mathvariant="italic">NA</mi><mn>2</mn></msup></mfrac></mrow></msqrt><mo>=</mo><mi>β</mi><mo>.</mo></math><img file="EP4249971A2_D0058.tif" /></maths>

0081An expression for the beat length between the two non-degenerate modes is: <maths id="math0059"><math display="block"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>λ</mi></mfrac><mi mathvariant="italic">NA</mi><mfenced open="[" close="]"><msqrt><mrow><msub><mi>b</mi><mn>1</mn></msub><mo>+</mo><mfrac><msubsup><mi>n</mi><mn>2</mn><mn>2</mn></msubsup><msup><mi mathvariant="italic">NA</mi><mn>2</mn></msup></mfrac></mrow></msqrt><mo>−</mo><msqrt><mrow><msub><mi>b</mi><mn>0</mn></msub><mo>+</mo><mfrac><msubsup><mi>n</mi><mn>2</mn><mn>2</mn></msubsup><msup><mi mathvariant="italic">NA</mi><mn>2</mn></msup></mfrac></mrow></msqrt></mfenced><mo>=</mo><msub><mi>β</mi><mn>1</mn></msub><mo>−</mo><msub><mi>β</mi><mn>0</mn></msub><mo>.</mo></math><img file="EP4249971A2_D0059.tif" /></maths>

0082Pulling the common terms out from under the radicals and factoring out the common terms results in: <maths id="math0060"><math display="block"><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi>λ</mi></mfrac><msub><mi>n</mi><mn>2</mn></msub><mfenced open="[" close="]"><msqrt><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>b</mi><mn>1</mn></msub><msubsup><mi>n</mi><mn>2</mn><mn>2</mn></msubsup></mfrac><msup><mi mathvariant="italic">NA</mi><mn>2</mn></msup></mrow></msqrt><mo>−</mo><msqrt><mrow><mn>1</mn><mo>+</mo><mfrac><msub><mi>b</mi><mn>0</mn></msub><msubsup><mi>n</mi><mn>2</mn><mn>2</mn></msubsup></mfrac><msup><mi mathvariant="italic">NA</mi><mn>2</mn></msup></mrow></msqrt></mfenced><mo>=</mo><msub><mi>β</mi><mn>1</mn></msub><mo>−</mo><msub><mi>β</mi><mn>0</mn></msub><mo>.</mo></math><img file="EP4249971A2_D0060.tif" /></maths>

0083A binomial approximation is applied to the terms under the radicals, and terms are canceled and common factors removed to yield: <maths id="math0061"><math display="block"><mfrac><mi>π</mi><mi>λ</mi></mfrac><mfrac><msup><mi mathvariant="italic">NA</mi><mn>2</mn></msup><msub><mi>n</mi><mn>2</mn></msub></mfrac><mfenced open="[" close="]"><msub><mi>b</mi><mn>1</mn></msub><mo>−</mo><msub><mi>b</mi><mn>0</mn></msub></mfenced><mo>=</mo><msub><mi>β</mi><mn>1</mn></msub><mo>−</mo><msub><mi>β</mi><mn>0</mn></msub><mo>=</mo><mi mathvariant="italic">Δβ</mi><mo>.</mo></math><img file="EP4249971A2_D0061.tif" /></maths>

0084Substituting numerical values gives: <maths id="math0062"><math display="block"><mfrac><mi>π</mi><mrow><mn>1540</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>9</mn></mrow></msup></mrow></mfrac><mfrac><msup><mn>0.2</mn><mn>2</mn></msup><mn>1.47</mn></mfrac><mfenced open="[" close="]"><mn>0.75</mn><mo>−</mo><mn>0.34</mn></mfenced><mo>=</mo><mfenced><mn>55,510</mn></mfenced><mfenced><mn>0.41</mn></mfenced><mo>=</mo><mn>23</mn><mo>×</mo><msup><mn>10</mn><mn>3</mn></msup><mo>=</mo><mi mathvariant="italic">Δβ</mi><mo>.</mo></math><img file="EP4249971A2_D0062.tif" /></maths> Using <maths id="math0063"><math display="inline"><mi>L</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>π</mi></mrow><mi mathvariant="italic">Δβ</mi></mfrac></math><img file="EP4249971A2_D0063.tif" /></maths> gives a beat length of 270 microns.

0085Estimating twist sensitivity using an approximate radius for the higher order modes of <i>r</i> = <i>0.002mm</i> produces: <maths id="math0064"><math display="block"><mfrac><msup><mi>r</mi><mn>2</mn></msup><mi>L</mi></mfrac><mo>=</mo><mfrac><msup><mfenced><mn>0.002</mn><mi mathvariant="italic">mm</mi></mfenced><mn>2</mn></msup><mn>0.270</mn></mfrac><mo>=</mo><mn>14</mn><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>6</mn></mrow></msup><mi mathvariant="italic">mm</mi><mo>.</mo></math><img file="EP4249971A2_D0064.tif" /></maths>

0086This twist sensitivity may be doubled to a value that is about half of the sensitivity in a multi-core shape sensing fiber. This demonstrates that a simple step-index, three-mode single core fiber has sufficient twist sensitivity to function as an effective shape sensor.

0087Different shape cores with different numbers of multimodes may be used. For example, a single large core shown in <figref idref="f0021">Figure 38</figref> supports many (e.g., 200) modes in addition to the axial ray and skew rays described above. Further, all of these modes can couple together when the fiber is perturbed, leading to a multitude of interactions that are measured in order to get accurate results. This large number of measurements may or may not be worthwhile depending on the application.

0088<figref idref="f0021">Figure 39</figref> shows an example single core multimode fiber where the core is annular and supports 30 modes. Additional sensitivity afforded by the larger core shown in <figref idref="f0021">Figure 38</figref> may be achieved using an annular core and eliminate the interior modes. Recognizing that axial rays can actually propagate parallel to the axis, but not on the axis, the annular core limits the number of modes present and the complexity of the measurement.

0089The technology described provides a single core multimode fiber that can be used to accurately sense shape and can be manufactured quite simply and cost effectively as compared to multicore shape sensing fiber.

0090The technology described above also has wide and diverse applications. One non-limiting example application is to a fiber optic shape sensing system for a robotic surgical arm in which one or more of the various technical features and/or embodiments described above may be used.

0091The techniques described herein can be implemented using a control system including at least one memory and at least one processor, and often a plurality of processors. The control system also includes programmed instructions (e.g., a computer-readable medium storing the instructions) to implement some or all of the methods described in accordance with aspects disclosed herein. The control system may include two or more data processing circuits with one portion of the processing optionally being performed on or adjacent the tool, and another portion of the processing being performed at a station (e.g. an operator input system or central processing system or the like) remote from the tool. Any of a wide variety of centralized or distributed data processing architectures may be employed. Similarly, the programmed instructions may be implemented as a number of separate programs or subroutines, or they may be integrated into a number of other aspects of the teleoperational systems described herein. In one embodiment, the control system supports wireless communication protocols such as Bluetooth, IrDA, HomeRF, IEEE 802.11, DECT, and Wireless Telemetry.

0092Although various embodiments have been shown and described in detail, the claims are not limited to any particular embodiment or example. None of the above description should be read as implying that any particular element, step, range, or function is essential such that it must be included in the claims scope. The scope of patented subject matter is defined only by the claims. The extent of legal protection is defined by the words recited in the allowed claims and their equivalents. All structural and functional equivalents to the elements of the above-described preferred embodiment that are known to those of ordinary skill in the art are expressly incorporated herein by reference and are intended to be encompassed by the present claims. Moreover, it is not necessary for a device or method to address each and every problem sought to be solved by the technology described, for it to be encompassed by the present claims. No claim is intended to invoke 35 USC § 112(f) unless the words "means for" or "step for" are used. Furthermore, no embodiment, feature, component, or step in this specification is intended to be dedicated to the public regardless of whether the embodiment, feature, component, or step is recited in the claims.

0093The present application also includes the following numbered clauses: <ol id="ol0001"><li>1. An optical interrogation system for a sensing fiber, the system comprising: <ul id="ul0002" list-style="none" compact="compact"><li>interferometric apparatus to probe a single core of the sensing fiber over a range of predetermined wavelengths, the single core having multiple light propagating modes, and to detect measurement interferometric data associated with the multiple light propagating modes of the single core for each predetermined wavelength in the range; and</li><li>data processing circuitry to process the measurement interferometric data associated with the multiple light propagating modes of the single core to determine a shape-sensing parameter of the sensing fiber.</li></ul></li><li>2. The optical interrogation system in clause 1, wherein the data processing circuitry is configured to determine a shape of the sensing fiber based on the shape-sensing parameter.</li><li>3. The optical interrogation system in clause 1, wherein the data processing circuitry is configured to process the measurement interferometric data associated with the multiple light propagating modes of the single core to determine a plurality of shape-sensing parameters including the shape-sensing parameter, and wherein the plurality of shape-sensing parameters includes strain, bend, and twist parameters.</li><li>4. The optical interrogation system in clause 1, wherein the shape-sensing parameter includes a bend parameter.</li><li>5. The optical interrogation system in clause 1, wherein the shape-sensing parameter includes a twist parameter.</li><li>6. The optical interrogation system in clause 5, wherein the single core comprises a shape that limits a number of the multiple light propagating modes below a predetermined number while providing a predetermined sensitivity to the twist parameter.</li><li>7. The optical interrogation system in clause 6, or of any of clauses 1 to 5, wherein the single core is ring-shaped in cross-section.</li><li>8. The optical interrogation system in clause 7, wherein a radius of the single core permits fewer than 40 modes of light to propagate along the single core.</li><li>9. The optical interrogation system in clause 6, or of any of clauses 1 to 5, wherein the single core is a solid core.</li><li>10. The optical interrogation system in clause 9, wherein a radius of the solid core permits less than six modes of light to propagate along the single core.</li><li>11. The optical interrogation system in clause 1, or of any of clauses 1 to 4, wherein the interferometric apparatus includes: <ul id="ul0003" list-style="none" compact="compact"><li>multiple interferometers including multiple reference branches and multiple measurement branches having an array of corresponding multiple single core, single mode fibers; and</li><li>a tunable laser to generate light over the range of predetermined wavelengths;</li><li>wherein each fiber of the array of corresponding multiple single core, single mode fibers has a different optical delay.</li></ul></li><li>12. The optical interrogation system in clause 11, wherein the interferometric apparatus further includes: <ul id="ul0004" list-style="none" compact="compact"><li>a collimator arranged to collimate light from the single core and de-collimate light to the single core;</li><li>a microlens array to receive and focus collimated light from the collimator onto the array of corresponding multiple single core, single mode fibers.</li></ul></li><li>13. The optical interrogation system in clause 11, wherein the different optical delays produce multiple coupling coefficients that appear on different ones of the array of corresponding multiple single core, single mode fibers.</li><li>14. The optical interrogation system in clause 13, wherein the single core includes an overlapping grating pattern, each overlapping grating in the overlapping grating pattern being tilted with respect to a longitudinal axis of the single core, and wherein the overlapping grating pattern varies with bend, strain, and twist applied to the sensing fiber.</li><li>15. The optical interrogation system in clause 14, wherein the overlapping grating pattern is associated with a cross-sectional index perturbation for the sensing fiber as a function of distance along the sensing fiber, and the cross-sectional index perturbation is associated with coupling coefficients between back-scattered light propagating modes for the sensing fiber.</li><li>16. The optical interrogation system in clause 15, wherein the interferometric apparatus is configured to measure a phase and an amplitude of the coupling coefficients, and the data processing circuitry is configured to determine differences between the measured phase and amplitude of the coupling coefficients and a predetermined baseline phase and amplitude of the coupling coefficients.</li><li>17. A method in an optical interrogation system having a sensing fiber having a single core, the single core having multiple light propagating modes, comprising: <ul id="ul0005" list-style="none" compact="compact"><li>probing with an interferometric apparatus the single core over a range of predetermined wavelengths and detect measurement interferometric data associated with the multiple light propagating modes of the single core for each predetermined wavelength in the range; and</li><li>processing with a data processor the measurement interferometric data associated with the multiple light propagating modes of the single core to determine a shape-sensing parameter of the sensing fiber.</li></ul></li><li>18. The method in clause 17, further comprising determining a shape of the sensing fiber based on the shape-sensing parameter.</li><li>19. The method in clause 17, further comprising processing the measurement interferometric data associated with the multiple light propagating modes of the single core to determine a plurality of shape-sensing parameters including the shape-sensing parameter, and wherein the plurality of shape-sensing parameters includes strain, bend, and twist parameters.</li><li>20. The method in clause 17, wherein the shape-sensing parameter includes a bend parameter.</li><li>21. The method in clause 17, wherein the shape-sensing parameter includes a twist parameter.</li><li>22. The method in clause 21, or of any of clauses 17 to 20, wherein the single core is shaped to limit a number of the multiple light propagating modes below a predetermined number while providing a predetermined sensitivity to the twist parameter.</li><li>23. The method in clause 17, or of any of clause 18 to 21, wherein the interferometric apparatus includes multiple interferometers including multiple reference branches and multiple measurement branches having an array of corresponding multiple single core, single mode fibers, the method further comprising providing each fiber of the array of corresponding multiple single core, single mode fibers with a different optical delay.</li><li>24. The method in clause 23, further comprising: <ul id="ul0006" list-style="none" compact="compact"><li>generating light over the range of predetermined wavelengths;</li><li>collimating light from the single core and de-collimate light to the single core; and</li><li>receiving and focusing collimated light from the collimator onto the array of corresponding multiple single core, single mode fibers.</li></ul></li><li>25. The method in clause 23, wherein the different optical delays produce multiple coupling coefficients that appear on different ones of the array of corresponding multiple single core, single mode fibers.</li><li>26. The method in clause 25, wherein the single core includes an overlapping grating pattern, each overlapping grating in the overlapping grating pattern being tilted with respect to a longitudinal axis of the single core, and wherein the overlapping grating pattern varies with bend, strain, and twist applied to the sensing fiber.</li><li>27. The method in clause 26, wherein the overlapping grating pattern is associated with a cross-sectional index perturbation for the sensing fiber as a function of distance along the sensing fiber, and the cross-sectional index perturbation is associated with coupling coefficients between back-scattered light propagating modes for the sensing fiber.</li><li>28. The method in clause 27, further comprising: <ul id="ul0007" list-style="none" compact="compact"><li>measuring a phase and an amplitude of the coupling coefficients, and</li><li>determining a difference between the measured coupling coefficient phase and amplitude and a predetermined baseline coupling coefficient phase and amplitude for the sensing fiber.</li></ul></li><li>29. A non-transitory machine-readable medium comprising a plurality of machine-readable instructions which when executed by one or more processors associated with a medical device are adapted to cause the one or more processors to perform the method of any of clauses 17 to 28.</li></ol>