TOMOGRAPHIC REFRACTIVE INDEX PROFILE EVALUATION OF NON-SYMMETRICAL GLASS FIBER PREFORMS AND FIBERS THEMSELVES

Abstract

A method for determining the refractive index profile of a preform when the RIP is not substantially symmetrical. (i) The preform is scanned, starting with a first projection angle, and raw data are created representing the object through measured data. (ii) Optionally, the object is rotated and step (i) repeated iteratively until all projection angles have been scanned and all measured data have been created. (iii) The measured data are processed to form a sinogram and, if the optional step (ii) has been completed, the method proceeds to step (v). (iv) The object is rotated and steps (i) and (iii) are repeated iteratively until all projection angles have been scanned. (v) A 2D RIP is calculated. (vi) A line section of interest is selected within the 2D RIP. (vii) A fitting procedure is applied to the line section of interest. (viii) Finally, refractive index steps/gradients and dimensions are determined.

Claims

1. A method for determining a refractive index profile (RIP) of an object, the method comprising: (a) providing the object, the object including an optical object which has a longitudinal axis around which at least one layer extends outwardly wherein the at least one layer lacks a complete rotational symmetry in cross section; (b) scanning the object with a light beam, starting with a first projection angle, and creating raw data representing the object through measured data; (c) optionally rotating the object and repeating step (b) iteratively until all projection angles have been scanned and all measured data have been created; (d) processing the measured data to form a sinogram and, if the optional step (c) has been completed, proceeding to step (f); (e) rotating the object and repeating steps (b) and (d) iteratively until all projection angles have been scanned; (f) calculating a 2D RIP; (g) selecting a line section of interest within the 2D RIP; (h) applying a fitting procedure to the line section of interest; and (i) determining refractive index steps/gradients and dimensions.

2. The method according to claim 1, wherein the step (b) of scanning the object includes directing an entry beam at an entry point into the cylindrical optical object in a direction transverse to the cylinder longitudinal axis, wherein a deflection angle is defined as the angle between an exit beam relative to the entry beam, and y is the distance between the cylinder longitudinal axis and the entry point of the entry beam in a Cartesian coordinate system.

3. The method according to claim 1, wherein the step of calculating the 2D RIP is carried out on the basis of an inverse Radon transformation.

4. The method according to claim 1, wherein the step of processing the measured data to form a sinogram includes calculating a deflection function, calculating a phase-shift diagram, and stacking the phase-shift diagram to form the sinogram.

5. The method according to claim 1, wherein the raw data includes a raw image and the method further comprises the step of improving the quality of the raw image by one or more of using a scanning camera that has an Image ROI feature, gathering several images with different exposures, using a scanning camera with HDR, and removing OVD-diffractions using a filtering algorithm.

6. The method according to claim 1, further comprising the step of improving the quality of the sinogram by one or more of applying a radii-run-out correction and interpolating the sinogram to a fine mesh.

7. The method according to claim 1, wherein the object provided in step (a) is a fiber preform.

8. The method according to claim 7, further comprising the step of using the determined refractive index steps/gradients and dimensions to characterize the fiber preform and adapt a preform manufacturing process.

9. The method according to claim 1, further comprising the step, before the step (f), of determining whether all projection angles have been scanned.

10. The method according to claim 1, further comprising the step, after the step (f), of determining whether significant RIP measurement artifacts are present in the 2D RIP and, if not, determining geometries and refractive index steps and concluding the method.

11. The method according to claim 1, wherein the step (h) of applying the fitting procedure to the selected line section of interest includes extracting start parameters for the fitting procedure from the line section, defining a first fitting region, and determining whether an actual layer is step-like.

12. The method according to claim 11, wherein the fitting procedure further includes, if the actual layer is step-like, generating an analytical deflection curve, transforming the analytical deflection curve to a deflection RIP, and calculating the accuracy of the deflection RIP compared with the initial line section.

13. The method according to claim 1, further comprising, after the step (i), the step (j) of determining whether all line sections of interest within the 2D RIP have been characterized.

14. The method according to claim 13, further comprising repeating steps (g), (h), (i), and (j) iteratively until all line sections of interest within the 2D RIP have been characterized.

15. The method according to claim 14, further comprising calculating a measurement artifact compensated refractive index profile for the object.

Description

BRIEF DESCRIPTION OF THE DRAWING

[0030] The disclosure is best understood from the following detailed description when read in connection with the accompanying drawing. It is emphasized that, according to common practice, the various features of the drawing are not to scale. On the contrary, the dimensions of the various features are arbitrarily expanded or reduced for clarity. The patent or application file contains at least one figure executed in color. Copies of this patent or patent application publication with color figures included in the drawing will be provided by the Office upon request and payment of the necessary fee. Included in the drawing are the following figures:

[0031] FIG. 1 shows the radiation path through an object with a homogeneous refractive index distribution illustrating various geometric relationships schematically;

[0032] FIG. 2 shows the radiation path through an object with a homogeneous refractive index distribution illustrating the measurement artifact in refractive index profiling due to the shaded non-measurable region where no tangential radiation can be detected;

[0033] FIG. 3 illustrates the variables applicable to determining the refractive index profile using computed tomography;

[0034] FIG. 4 is a schematic illustration of a deflection function measurement system;

[0035] FIG. 5A is a graph of an example deflection angle distribution () (in degrees) plotted versus the position along the -axis (in millimeters), after measurement of the deflection function for different orientations for a given projection angle (e.g., =8), during the first phase of an improved method for evaluating non-symmetrical glass fiber preforms;

[0036] FIG. 5B is a graph that highlights the center portion of the graph of FIG. 5A using reduced scales along both the abscissa and ordinate;

[0037] FIG. 6 illustrates the Radon transform which can be used to calculate data during the first phase of the improved method for evaluating non-symmetrical glass fiber preforms;

[0038] FIG. 7 is the sinogram that results from calculating the phase shift and stacking the measured data presented in FIG. 5;

[0039] FIG. 8 shows a diagram of the relative refractive index in the x-y plane (i.e., for a line section of the preform) after applying the inverse of the Radon transform to the sinogram of FIG. 7;

[0040] FIG. 9 is a flow diagram illustrating the steps of a specific embodiment of the improved method for evaluating non-symmetrical glass fiber preforms;

[0041] FIG. 10 is a flow diagram summarizing the steps of a specific embodiment of the fitting procedures which are applied in one step of the improved method illustrated in FIG. 9;

[0042] FIG. 11 shows a graph of the relative refractive index n(r)1.4587 plotted versus the radius r (in millimeters) with a prepared refractive index profile n(r) and a hypothetical refractive index profile n*(r) modeled by evaluation of the profile for a preform with a step profile;

[0043] FIG. 12 is a graph of the relative refractive index n(r)1.4587 plotted versus the radius r (in millimeters) illustrating the generalized fitting procedure;

[0044] FIGS. 13-16 depict an example of applying an embodiment of the method of the present disclosure to a preform having a single elliptical core and, specifically, FIG. 13 is the sinogram created by the method;

[0045] FIG. 14 is a diagram of the relative refractive index n(x, y)1.446 in the x-y plane, illustrating the reconstructed 2D RIP of the preform having a single elliptical core;

[0046] FIG. 15 is a graph illustrating the line section fitting procedure for data taken along a line section drawn through the core of the diagram of FIG. 14;

[0047] FIG. 16 is a graph of the relative refractive index n(x, y)1.446 against the position r in millimeters, for the preform having a single elliptical core, and compares two different line sections and single versus tomographic measurements;

[0048] FIGS. 17-20 depict an example of applying an embodiment of the method of the present disclosure to a preform having a single core with an octagonal-shaped cladding and, specifically, FIG. 17 is the sinogram created by the method;

[0049] FIG. 18 is a diagram of the relative refractive index n(x, y)1.446 in the x-y plane, illustrating the reconstructed 2D RIP of the preform having a single core with an octagonal-shaped cladding;

[0050] FIG. 19 is a graph illustrating the line section fitting procedure for data taken along a first line section drawn through the core of the diagram of FIG. 18;

[0051] FIG. 20 is a graph illustrating the line section fitting procedure for data taken along a second line section drawn through the core of the diagram of FIG. 18;

[0052] FIGS. 21-24 depict an example of applying an embodiment of the method of the present disclosure to a preform having seven cores and, specifically, FIG. 21 is the sinogram created by the method;

[0053] FIG. 22 is a diagram of the relative refractive index n(x, y)1.4587 in the x-y plane, illustrating the reconstructed 2D RIP of the preform having seven cores;

[0054] FIG. 23 is a graph illustrating the line section fitting procedure for data taken along a first line section drawn through the center core of the diagram of FIG. 22;

[0055] FIG. 24 is a graph illustrating the line section fitting procedure for data taken along a second line section drawn through the center core of the diagram of FIG. 22;

[0056] FIGS. 25-28 depict an example of applying an embodiment of the method of the present disclosure to a preform having six cores and, specifically, FIG. 25 is the sinogram created by the method;

[0057] FIG. 26 is a diagram of the relative refractive index n(x, y)1.4587 in the x-y plane, illustrating the reconstructed 2D RIP of the preform having six cores;

[0058] FIG. 27 is a graph illustrating the line section fitting procedure for data taken along a line section drawn through a first core of the diagram of FIG. 26; and

[0059] FIG. 28 is a graph illustrating the line section fitting procedure for data taken along a line section drawn through another core of the diagram of FIG. 26.

DETAILED DESCRIPTION

[0060] In this specification and in the claims that follow, reference will be made to a number of terms which shall be defined to have the following meanings ascribed to them. Include, includes, including, have, has, having, comprise, comprises, comprising, or like terms mean encompassing but not limited to, that is, inclusive and not exclusive. The term substantially, as used in this document, is a descriptive term that denotes approximation and means considerable in extent or largely but not wholly that which is specified and is intended to avoid a strict numerical boundary to the specified parameter.

[0061] The term about (or approximately) means that amounts, sizes, formulations, parameters, and other quantities and characteristics are not and need not be exact, but may be approximate and/or larger or smaller, as desired, reflecting tolerances, conversion factors, rounding off, measurement error and the like, and other factors known to those of skill in the art. When a value is described to be about or about equal to a certain number, the value is within 10% of the number. For example, a value that is about 10 refers to a value between 9 and 11, inclusive. When the term about is used in describing a value or an end-point of a range, the disclosure should be understood to include the specific value or end-point. Whether or not a numerical value or end-point of a range in the specification recites about, the numerical value or end-point of a range is intended to include two embodiments: one modified by about and one not modified by about. It will be further understood that the end-points of each of the ranges are significant both in relation to the other end-point and independently of the other end-point.

[0062] The term about further references all terms in the range unless otherwise stated. For example, about 1, 2, or 3 is equivalent to about 1, about 2, or about 3, and further comprises from about 1-3, from about 1-2, and from about 2-3. Specific and preferred values disclosed for compositions, components, ingredients, additives, and like aspects, and ranges thereof, are for illustration only; they do not exclude other defined values or other values within defined ranges. The compositions and methods of the disclosure include those having any value or any combination of the values, specific values, more specific values, and preferred values described.

[0063] The indefinite article a or an and its corresponding definite article the as used in this disclosure means at least one, or one or more, unless specified otherwise. Directional terms as used in this disclosurefor example up, down, right, left, front, back, top, bottomare made only with reference to the figures as drawn and the coordinate axis provided with those figures and are not intended to imply absolute orientation.

[0064] The use of computed tomography (CT) to determine the refractive index profile (RIP) of optical fiber preforms is well known. The variables applicable to determining the RIP using CT are defined by the illustration provided in FIG. 3. Depicted is an object with a refractive index distribution n(x, y) surrounded by an index adjustment fluid with a refractive index no such that n.sub.0<n(x, y). The Cartesian coordinates x and y belong to the physical object (e.g., a glass body). When scanning the object, the beam is refracted as soon as it contacts the object and is directed toward the center of the object along the path shown with an arrow. The deflection angle W is defined as the angle between the exit beam and the entry beam.

[0065] During CT, the object is rotated. The mechanism (typically a rotational stage) to hold and rotate the sample is for simplicity not shown in FIG. 4. Tomography requires, however, an accurate way to rotate the sample. Although manual rotation is possible, it is impractical. An electrically driven rotational stage with an encoder is more suitable.

[0066] Rotation of the object through an angle , called the projection angle, leads to the Cartesian coordinate axes x and y. The scanning procedure takes place along the scanning position in the direction of y. For a given and fixed projection angle, such scanning leads to a measured deflection angle (, =fixed). It is necessary to measure the deflection angle distribution (, ) for a large set of projection angles to determine an accurate RIP. Thus, the deflection angle distribution (, ) is measured and the RIP is reconstructed from the deflection angle distribution (where is scanning position and is the measured projection angle).

[0067] The description of the present methods follows with reference to a transparent cylindrical object in the form of a fiber preform. It will be understood by a person skilled in the art, however, that the methods described can be applied generally to any cylindrical object having a refractive index at a given wavelength of radiation, where the corresponding deflection angle distribution can be measured via transverse transmission of the radiation of a wavelength and a target deflection angle distribution function exists that can be expressed as a function which can be fitted to the measured data. Referring now to the drawing, in which like reference numbers refer to like elements throughout the various figures that comprise the drawing, FIG. 4 is a schematic diagram illustrating an example embodiment of a basic deflection angle distribution measurement system 100 that can be used to establish the measured deflection angle distribution function. The deflection angle is calculated by the equation: =arctan(projection on camera/distance to camera).

[0068] The system 100 has a pair of optical axes 15 and a third optical axis (not shown) between a pair of off-axis mirrors 4. One or more laser sources (such as laser diodes 1a, 1b, and 1c) each generate a laser beam (or light beam) and provides or provide the laser beam or beams to one or more beam combiner fibers 2 which delivers or deliver the laser beam or beams to a beam conditioner 3. (In fiber-optic communication, a single-mode optical fiber is an optical fiber designed to carry only a single mode of light: the transverse mode. The standard G.652 defines the most widely used form of single-mode optical fiber.) The one or more beam combiner fibers 2 are either Y-shaped fibers, with two inputs and one output, or X-shaped fibers with cross talk in the spliced region. The beam conditioner 3 is aligned along the first optical axis 15 and alters the properties of the laser beam. The beam emitted by the beam conditioner 3 travels along the first optical axis 15 and engages an off-axis parabolic reflector having the pair of mirrors 4 and an aperture (or pin hole) 5. The first mirror 4 creates parallel beams which pass through the aperture 5; the second mirror 4 focuses those beams along the second optical axis 15.

[0069] The focused beams that reflect from the second mirror 4 of the off-axis parabolic reflector pass through an optical shutter 6 and enter a measuring cell 9. As long as the control voltage to the shutter 6 remains high, the shutter 6 stays open. As soon as the voltage goes low, however, the shutter 6 closes, providing inherent fail-safe operation, i.e., security.

[0070] The measuring cell 9 has opposite sides that are planar and at right-angles to the second optical axis 15. Arranged in the measuring cell 9 is a transparent cylindrical object 11 having a homogeneous or a stepped RIP to be measured, typically in the form of a fiber preform having a core with a higher refractive index surrounded by at least one cladding layer with a lower refractive index. Surrounding the object 11 in the measuring cell 9 is an index adjustment fluid 10. In an example embodiment, the index adjustment fluid 10 is an oil with a refractive index that is close to, but not the same as, the refractive index of the object 11. For simplicity, however, the refractive index of the index adjustment fluid 10 matches the measuring cell 9 so that no additional deflectionsat the entry and exit pointoccur in the embodiment illustrated.

[0071] As an alternative to the widely and commonly used paraffin RI index-matching oils, a glycerol-water-mixture can also be used. The mixing ratio permits relatively easy and accurate adjustment or tuning of the RI to the desired RI level within the range of 1.333 (water RI) and 1.48 (glycerol RI).

[0072] The laser beam enters the measuring cell 9 and is first incident on the object 11 at a first edge of the object 11 and undergoes a first refraction. The first-refracted laser beam then travels through the object 11 and exits the object 11 on the opposite edge, where it undergoes a second refraction and exits the object 11. The deflection angle, identified by the Greek letter W (psi), is defined by the angle of the exit laser beam relative to the incident laser beam. The exit laser beam then passes through a filter (e.g., an infrared long pass filter) 12 and is detected by a photodetector unit. The filter 12 helps to preclude environmental light from adversely impacting measurements. A suitable photodetector unit includes a line scan camera 13 having an optically active sensor 14. (Alternatives to the line scan camera could also be used. Among other suitable cameras are 2D cameras and time delay integration or TDI line scan cameras. The TDI line scan camera is somewhat a hybrid: it is a 2D sensor with, for example, 16,384256 pixels, but its architecture and the sensor extend in one dimension more like a line-scan camera.) The photodetector unit then sends a corresponding detector signal to a controller 16 via data connections 17 for processing.

[0073] The measuring cell 9 is mounted on a linear stage 7 configured both to support the measuring cell 9 and to move the measuring cell 9 in movement directions 8 (e.g., up and down as shown in FIG. 4). By performing measurements of the deflection angle over a range of laser beam heights relative to the central horizontal axis of the measuring cell 9 and the object 11 in the measuring cell 9, the corresponding detector signals received and processed by the controller 16 create the measured deflection angle as described in greater detail below. In other words, the movement of the linear stage 7 allows for the laser beam height to be varied relative to the measuring cell 9 and the object 11 so that the measured deflection function includes a range of radii of the object 11. It is possible, of course, to move the camera relative to the laser beam so that the camera remains aligned with the laser beam as well as vice versa.

[0074] Although in the embodiment illustrated in FIG. 4 the object 11 in the measuring cell 9 is moved (e.g., scanned), in another embodiment other components (such as the laser diodes 1a, 1b, and 1c and the photodetector unit) can all be simultaneously moved (e.g., scanned) relative to the object 11 (which is held stationary) so that the laser beam height can be varied in order to send the laser beam through different parts of the object 11.

[0075] The controller 16 is, for example, a computer that includes a processor unit (e.g., a CPU), a memory unit, and support circuitry all operably interconnected. The processor may be or include any form of a general-purpose computer processor that can be used in an industrial setting. The memory unit includes a computer-readable medium capable of storing instructions (e.g., software) that direct the processor to carry out the methods as described in detail below. The memory unit may be, for example, random-access memory, read-only memory, floppy or hard disk drive, or other form of digital storage. In an example embodiment, the instructions stored in the memory unit are in the form of software that, when executed by the processor, transform the processor into a specific-purpose processor that controls (i.e., directs or causes) the system 100 to carry out one or more of the methods described below. The support circuitry is operably (e.g., electrically) coupled to the processor and may include cache, clock circuits, input/output sub-systems, power supplies, control circuits, and the like.

[0076] The laser diodes 1a, 1b, and 1c; the shutter 6; the linear stage 7; and the line scan camera 13 are each configured to send signals and data to, and receive signals and data from, the controller 16 along a plurality of data connections 17. The data connections 17 may be wired or wireless; any conventional data connections 17 as would be known to an artisan are suitable.

[0077] Many other deflection function measurement systems similar to those described above can be used to obtain the deflection function. Regardless of the system used, a number of methods for determining the RIP of the cylindrical optical object 11 are possible. The deflection angle is calculated by the equation: =arctan(projection on camera/distance to camera).

[0078] As described above, CT is based on the well-known inverse Radon-transformation. This transformation can be written, when applied in the RIP field, as Equation (4)

[00004]$n\left(r,\right)-n_0=\frac{1}{2{}^2}{}_{-/2}^{+/2}{}_{-}^{-}\frac{\left(,\right)}{}.Math.\frac{1}{r\sin \left(-\right)-}dd$

where is the projection angle and is the scanning position (as illustrated in FIG. 3 and discussed above) and is the wavefront phase-shift caused by the object under investigation. (Note that in X-ray CT, (, ) would be an absorption distribution, which leads to a reconstruction of a density distribution.) As always, the result of the Radon transformation is given in polar coordinates (r, ), which are related to the Cartesian coordinates by x=r(cos ) and y=r(sin ). It is common to convert the resulting distribution numerically from n(r, ) to n(x, y). Because the mesh grids of polar and Cartesian coordinates do not match, the reconstructed 2D RIP n(r, ) can be converted to n(x, y) by using interpolations including, but not limited to, nearest neighbor, bilinear, cubic, and spline.

[0079] In contrast to interferometric RI profiling, the phase-shift diagram (, ) is typically not directly measurable. Instead, the deflection function (, ) is accessible and can be converted into the phase-shift diagram (, ) by the approximation given in Equation (5)

[00005]$\left(,\right)={}_0^{}\left({}^{},\right)d{}^{}+\frac{n_0}{12}.Math.{\left[\left(,\right)\right]}^3$ [0080] where no is the reference refractive index at =0.

[0081] As usual for tomography, the measured data typically are pre-processed to improve the quality of the reconstruction. Such pre-processing is true in the RI profiling field, too, where the importance of an accurately defined deflection function origin is well-known. Regardless of how meticulously all mechanical parts and components of the measurement system 100 are manufactured and assembled, it is always beneficial to fine tune the origin numerically. For the evaluation of radii-symmetric profiles different approaches exist to achieve a so-called prepared deflection function, which is the measured deflection function with a fine-tuned origin .sub.prepared(y)=.sub.measured(yy.sub.shift).sub.Offset. The two parameters .sub.Offset and y.sub.shift are very small, but typically not zero, for any particular measurement system 100. Several approaches to define those two parameters for radii-symmetric RIPs are mentioned in U.S. Pat. No. 10,508,973.

[0082] For non-symmetric RIPs, the deflection function for a fixed projection angle can be complex. Such complexity reduces the possibility of achieving a prepared deflection function .sub.prepared(, )=.sub.measured(.sub.shift).sub.Offset. In tomography in general, however, many techniques exist to improve the quality of the reconstructed image. Although most of the research and algorithms to improve the reconstructed image quality focus on X-ray and NMR-CT, some of the methods are applicable in the field of tomographic RIP. For simplicity, just a few examples for improving the quality of the reconstructed image are given in the following disclosure; an artisan skilled in the field can pick and choose additional or alternative methods.

[0083] Defining .sub.shift properly for each measured deflection angle distribution (, ) can be understood as a well-known radii-run-out correction in tomography. A radii-run-out happens when instead of just a rotation along the Z-axis, which varies the projection angle , a trajectory movement occurs. Mathematically, this adds a shift for each projection angle . Radii-run-out affected tomographic reconstructed images typically suffer from smeared edges, indistinct features or transitions, and the like. As a result, a radii-run-out correction can be achieved by minimizing the entropy of the reconstructed image. In this case, .sub.shift of each single projection needs to be varied and the reconstruction calculated. Because this iterative process requires the calculation of many time-consuming reconstructions, it is not very efficient. A preferred alternative approach is explained below.

[0084] In most cases one or even more features within the measured sample can be used for a radii-run-out correction. An artisan skilled in the field can identify a suitable feature by the following criteria. The feature should have a sharp RI step (a RI difference of at least 10.sup.3 over a few micrometers is preferred), so that any edge detection algorithm works well. It is also preferred that the feature defines or even surrounds the region of interest within the reconstructed 2D RIP. This criterion ensures that the edges of the feature are not distorted by the deflection of outer RI variations. In practice, the measurement of each projection angle will be shifted by an individual .sub.shift, so that the feature is aligned concentrically around the origin regardless of how chaotic the inside looks. As illustrated on the sinograms including in the drawing, this means that each column is vertically shifted so that a feature pair is aligned. FIG. 17 illustrates, for example, that a circular feature appears as the straight horizontal lines within the sinogram at 12 mm. A non-circular feature appears as mirrored pairs of sinewaves in the sinogram, where the mirroring axis is =0 mm.

[0085] Besides a proper alignment of the measured deflection function .sub.measured by .sub.shift, the quality of the reconstructed 2D RIP can also be improved by properly defining .sub.offset for each measured projection. Because a measurement system 100 like the one illustrated in FIG. 4 uses a surrounding reference material, which has two beam perpendicular surfaces and a constant refractive index, the resulting phase-shift f through the reference on both opposite sides (=0 and =.sub.ref) must be equal. This constraint can be used to fix .sub.offset for each measured projection. In particular, the deflection function offset .sub.offset is varied, so that the integration spanning the reference material on both sides (setting the upper integration limit p in Equation (5) to the reference position .sub.ref) leads to an equal phase-shift of (=0)=(=.sub.ref). Proper defining of .sub.shift and .sub.Offset allows calculation of a prepared deflection function .sub.prepared(, ) and therefore a prepared phase-shift diagram .sub.prepared(, ).

[0086] Besides the RIP-specific mathematical tools to improve the quality of the reconstruction, more general CT tools also exist. The number of tools is not limited to the following given example about sinogram interpolation. The inverse Radon transformation has an integration over the scanning position as well as over the projection angle . Ideally, both step sizes d and d are very small. A tradeoff exists, however, because a finer mesh size defined by d and d increases the required overall measurement time.

[0087] In addition to decreasing the mesh size by using smaller step sizes d, d, or both d and d, it is possible to interpolate within the sinogram. The interpolation of additional projection angles especially yields excellent results in terms of improving the reconstruction quality. The reason is that the sinogram typically does not change much on the small scale in the direction of d. In contrast, discontinuities typically exist in the direction of d. Starting from one or a few additional, interpolated projection angles between measured projections, the number of projections within the prepared sinogram can reach 10, 20, or even more times the number of measured projections. Many different methods can be applied for the two-dimensional interpolation including, without limitation, nearest neighbor, bilinear, cubic, and spline.

[0088] A frequency domain filter needs to be used to apply the inverse Radon transformation to the prepared phase-shift diagram (, ). Different types of filters are suitable for image reconstruction. Among such filters are a Ram-Lak filter, a Shepp-Logan filter, a Cosine filter, a Hamming window filter, and a Hann window filter. A Ram-Lak filter is also known as a Ramp filter. This filter is sensitive to noise, so it is multiplied by a suitable window to improve the results. A Shepp-Logan filter is obtained by multiplying the Ram-Lak filter by the sinc or sampling function. The Cosine filter is obtained by multiplying the Ram-Lak filter by the cosine function. Similarly, the Hamming and Hann filters are obtained by multiplying the Ram-Lak filter by the Hamming and Hann windows, respectively. The Ram-Lak and Shepp-Logan filters are high-pass filters, which keep the edges information intact. The Cosine, Hamming, and Hann filters are band-pass filters. They are used to smooth the image and remove extra edges from the image.

[0089] In modern CT, plenty of methods exist to improve the quality of the reconstruction, which an artisan skilled in the field can apply here, too. Such improvement always depends, however, on the particular object under investigation: a particular method might be beneficial for a particular object or might just cause additional computational time or might even worsen the result. In the case of large deflections due to large refractive index steps within the object, it can be beneficial to apply back-propagation instead of straight-line back projection. This iterative procedure requires a lot of resources, however, resulting in longer computational time.

[0090] Measurement of the RIP of optical fiber preforms is important for inspection of the quality of optical fiber waveguides and also for specification of additional treatment (e.g., overcoating or etching) of the preform before drawing the fiber. For preforms, non-destructive techniques are particularly useful. The methods disclosed in this document are based on measurement of the deflection of a transversely directed light beam scanned through the preform. The methods are further based on numerical processing of a set of measured deflection functions at many angular projections. The methods disclosed above calculate the RIP still further based upon the assumption that the preform is circular. A method is desired, however, for tomography reconstruction of the RIP for preforms with a highly asymmetric cross section.

[0091] Attempts have been made to address the RI characterization of non-symmetrical fiber preforms. See, for example, K. Toga et al., Microscopic Computer Tomography Measurement of Nonaxisymmetrically Distributed Optical Fiber Refractive Index, Journal of Lightwave Technology, Vol. 6, No. 1, pages 73-79 (1988); and B. Bachim et al., Microinterferometric Optical Phase Tomography for Measuring Small, Asymmetric Refractive-Index Differences in the Profiles of Optical Fibers and Fiber Devices, Applied Optics, Vol. 44, No. 3, pages 316-27 (2005). These two references are incorporated by reference in this document.

[0092] The following disclosure describes the steps of an improved method that provides an accurate RIP for an object that is either not symmetrical or at least not substantially symmetrical. Such preforms may be preforms with a non-intended asymmetry. In addition, the method can be used for multicore preforms, Panda-preforms (used to form a common style of polarization-maintaining fiber, having round and symmetrical stress rods on either side of the core to induce polarization), or preforms for fiber lasers with non-circular interfaces.

[0093] The improved methods for evaluating non-symmetrical glass fiber preforms have two main phases. The first phase involves a tomographic refractive index determination. The second phase involves correction of measurement artifact.

[0094] In the first phase of the method, the deflection function is measured for different orientations (a single scan provides useless information for asymmetrical preforms). The number of measurements taken must balance the accuracy of the data (more measurements are desired to give higher resolution) against the time required to take measurements (fewer measurements are quicker). For example, the system 100 might take about 18 minutes to scan the preform every 2 using one light source, or about 3.6 hours to scan the preform every 0.5 using three light sources (scans in steps of 0.5 means that 720 measurements are made for the full 3600 around the preform). The measurements yield the deflection angle distribution (, ). FIG. 5A is a graph of an example deflection angle distribution (, ) (in degrees) for a simplistic case plotted versus the position along the p-axis (in millimeters) for a given projection angle (0=8). FIG. 5B is a graph that highlights the center portion of the graph of FIG. 5A using reduced scales along both the abscissa and ordinate.

[0095] Once taken, the measurements are subject to tomographic evaluation. Such evaluation includes calculating the phase shift and stacking to form a sinogram. The sinogram provides a convenient way to represent the full set of data acquired during a scan and can be useful to check for systematical errors during data acquisition or to verify that the preprocessing steps (e.g., radii-run out correction) work correctly.

[0096] As illustrated in FIG. 6, the Radon transform is the integral transformation which takes a function defined on the x-y plane to a function R (i.e., the Radon transform) defined on the two-dimensional (, s) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The parameter is the length of a portion of the straight line A-A, s is the distance of the line from the origin, and is the angle that the normal or perpendicular to the line makes with the x-axis. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes (integrating over lines is known as the X-ray transform). It was later generalized to higher-dimensional Euclidean spaces, and more broadly in the context of integral geometry. The inverse Radon transformation is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.

[0097] If a function represents an unknown, then the inverse Radon transformation represents the projection data obtained as the output of a tomographic scan. Hence the inverse of the Radon transformation can be used to reconstruct the original quantity from the projection data, and thus it forms the mathematical underpinning for tomographic reconstruction, also known as iterative reconstruction. The depiction of the inverse Radon transformation data is often called a sinogram because the inverse Radon transformation of an off-center point source is a sinusoid. Consequently, the inverse Radon transformation of a number of scans appears graphically as a number of sine waves with different amplitudes and phases called a sinogram.

[0098] FIG. 7 is the sinogram that results from calculating the phase shift and stacking the measured data presented in FIG. 5A. The phase shift is presented in arbitrary units (abbreviated a.u.). An arbitrary unit is a relative unit of measurement to show the ratio of an amount of a quantity to a predetermined reference measurement. By predetermined is meant determined beforehand, so that the predetermined characteristic must be determined, i.e., chosen or at least known, in advance of some event. FIG. 8 shows a diagram of the relative refractive index in the x-y plane (i.e., for a cross section of the preform) after applying the inverse of the Radon transform. The refractive index is indicated as a relative value based on the refractive index of the surrounding reference glass plate (n.sub.0=1.4587).

[0099] Thus, the refractive index distribution is determined using known techniques (inverse Radon-transformation is preferred). These steps complete the first tomographic phase of the improved method for evaluating non-symmetrical glass fiber preforms. Unfortunately, the result will show the typical measurement artifact of underestimated refractive index steps, distorted profile, rounded edges, and the like.

[0100] The measurement artifact is corrected in the second phase of the method. Correction of the artifact is achieved by applying a method like the 973 method, described above, to a chosen line section of the 2D RIP obtained from the tomographic data (i.e., the starting RIP). A modeled RIP is created after assuming a symmetrical preform based on a few parameters (which will be varied). A distorted RIP is calculated from the modeled RIP by (i) simulating the deflection angle distribution for the modeled RIP, and (ii) calculating the distorted RIP from that simulated distribution. The parameters are varied for the modeled RIP until the distorted RIP is close to the starting RIP. This approach is an approximation. The calculation of the correction is done as if the starting RIP was measured at a preform with rotational symmetry, which is not the case. An important underlying assumption is that the measurement artifact for the actual sample is similar for rotationally symmetrical preforms.

[0101] The starting point for the improved evaluation method is the 2D-RIP based on the tomographic measurement and reconstruction (including all CT-techniques available today for improved imaging). One or more line sections of the regions of interest (cores, side pits, edges, and the like) are selected. According to each of the chosen line sections, an abstract, fictitious, but symmetrical preform is fitted to the chosen line section. For a simplified explanation it is assumed that the abstract, fictitious, symmetrical profile can be fitted directly. The profile cannot be fitted directly, however, because no analytical expression exists for such a RIP (which includes the typical measurement artifact). Therefore, the fitting procedure includes the following steps for each iteration: (i) assuming a symmetrical preform based on a few parameters (which will be varied); (ii) simulating the deflection angle distribution; (iii) reconstructing the symmetrical RIP; and (iv) calculating the deviation of the chosen cross section. By this measuring and evaluation procedure the tomographic 2D RIP reconstruction handles all non-symmetries first (so that no systematical errors occur) and the fitting procedure of the line sections does the restespecially addressing the typical measurement artifact in refractive index profiling.

[0102] FIG. 9 is a flow diagram illustrating the steps of a specific embodiment of the improved method 500 that provides an accurate refractive index profile for an object that does not have a symmetrical cross section or at least a substantially symmetrical cross section. The method 500 determines a radial RIP of the cylindrical optical object 11 having a cylinder longitudinal axis around which at least one layer extends axially asymmetrically. In Step 501 of the method 500, the deflection function is measured beginning with a projection angle (theta) of zero. In Step 502, the object 11 is scanned. It is possible after completion of Step 502 to proceed directly to Step 504 of the method 500.

[0103] Optionally, although not required, a raw image can be created from the scan. In optional Step 503, the quality of the raw image can be improved using one or more of a variety of techniques. For example, the scanning camera has an Image ROI feature that allows the operator to specify the part of the sensor array to be used for image acquisition. ROI is an acronym for region of interest. If an image ROI has been specified, the camera will only transmit pixel data from within that region. On most cameras, this specification significantly increases the maximum frame rate of the camera. The ROI can be brightened. Applying another technique, several images can be gathered with different exposures.

[0104] As another example of a technique that improves the quality of the raw image, a scanning camera with high dynamic range (or HDR) can be used. In a nutshell, HDR helps produce better pixels. The range of luminance levels has been increased via HDR so that enhanced differentiation among whites and blacks is possible. The range of colors has also been increased so that even colors can be more detailed.

[0105] As yet another example of a technique that improves the quality of the raw image, OVD-diffractions can be removed using a filtering algorithm. Outside vapor deposition (OVD) is a process by which glass is deposited in layers to create a porous deposition (glass soot). The porous glass deposition can be cleaned, for example, by flushing with chlorine gas (dehydrogenation) to reduce the OH-percentage. At the same time the porous deposition becomes a glass preform (vitrification). Regardless of the further processing steps like vitrification, the process-related micro layer structures may remain and may affect RI variation on a micro scale, too. These highly frequent RI variations lead to diffractions when scanned with a laser and even a non-coherent light beam.

[0106] Higher diffraction orders reduce the accuracy of the determination of the deflection angle up to really severe cases where the zero-order beam cannot be distinguished by higher orders. Because the high-frequency RI microlayer structures caused by the OVD manufacturing process typically are non-periodic and curved, the diffraction pattern can be quite irregular up to the point where higher diffraction orders can become brighter than those from the zero-order beam. It is well-known to an artisan that only the evaluation of the zero-order beam leads to a useful deflection angle distribution and a plausible RIP reconstruction. Depending on the level of severity of the OVD-diffraction (i.e., the intensity of light within higher diffraction orders), additional pre-processing by way of applying an OVD-filter algorithm may be required.

[0107] After Step 502, which includes scanning the object 11 with a light beam (starting with a first projection angle) and creating raw data representing the object through measured data, the method 500 can proceed along one of two alternative paths. The first alternative path involves rotating the object 11 by a small angle (theta) such as /180 and repeating Step 502 iteratively until all projection angles within the full range (zero to 2) have been scanned and all measured data have been obtained. The method then processes and stacks the measured data to create a phase shift sinogram, and proceeds to Step 508.

[0108] The second alternative path after Step 502 is illustrated in FIG. 9. In Step 504 of the method 500, the deflection function and phase-shift diagram are calculated (as described above). It is also possible to calculate the phase shift diagram directly, without the intermediate step of calculating the deflection function (i.e., the intermediate step is optional). The phase-shift diagram is exported and stacked, also as described above, to form a sinogram in Step 505. The method 500 then asks, in Step 506, whether all projections within the full range (zero to 2) have been measured. If the full range has not yet been measured, then the method 500 implements Step 507. In Step 507, the object 11 is rotated by a small angle (theta) such as /180 and the method 500 returns to Step 502 and scans the object 11. Steps 502, 503 (optional), 504, 505, 506, and 507 are repeated iteratively until all projections within the full range (zero to 2) have been measured and the method 500 can proceed to Step 508.

[0109] In optional Step 508, the quality of the sinogram can be improved using one or more of a variety of techniques. For one example, radii-run-out can be corrected. As another example, the sinogram can be interpolated to a fine mesh.

[0110] In Step 509, the 2D RIP is calculated from the sinogram by applying the inverse Radon transform. This calculation was described above. Step 509 completes the first phase of the method 500.

[0111] Step 510 begins the second phase of the method 500 and asks whether any significant RIP measurement artifacts are present in the 2D RIP. The measurement artifact occurs when a refractive index step transitions from a lower to a higher value (in the direction from outside to inside). This transition is relatively easy to determine for each selected line section. The word significant might specify, for example, the sharpness of the RI step (i.e., the sharper the step, the more significant is the artifact). If the step is a gradient and not sharp, such as for a germanium-doped core with a parabolic shaped RI), the artifact is either not present or can be neglected.

[0112] If the RIP measurement artifacts are not significant, then the method 500 implements Step 520. In Step 520, certain geometries and refractive index steps are determined before the method 500 concludes in Step 580. If significant RIP measurement artifacts are present in the 2D RIP, then the method 500 proceeds to Step 530. In Step 530, a line section of interest (also called a region of interest or ROI) is selected within the 2D RIP. A fitting procedure is then applied to the selected line section in Step 540. More details about the fitting procedures are provided in connection with FIG. 10 below.

[0113] Once the fitting procedure applied in Step 540 meets one or more stop criteria, then the method 500 proceeds to Step 560. A stop criterion might be determined by comparing the result of the fitting procedure against a predetermined accuracy level for the selected line section of the object 11 being evaluated. The predetermined accuracy level may depend upon the particular application. A typical predetermined accuracy is about 90%, a further predetermined accuracy is about 92%, a preferred predetermined accuracy is about 95%, a more preferred predetermined accuracy is about 97%, and a most preferred predetermined accuracy is about 99% or more.

[0114] In Step 560, fitting parameters such as RI-steps, geometries, gradients, and dimensions are exported. The method 500 then asks, in Step 570, whether all regions of interest within the preform have been characterized. If less than all of the regions of interest have been characterized, then the method 500 returns to Step 530 and a new line section of interest is selected within the 2D RIP. Steps 530, 540, and 560 are repeated iteratively until all regions of interest have been characterized and the method 500 can proceed to Step 580 and conclude.

[0115] FIG. 10 is a flow diagram summarizing the steps of a specific embodiment of the fitting procedures applied in Step 540 of the improved method 500. The line section of interest (or ROI) that was selected within the 2D RIP in Step 530 is used as the start of the fitting procedure in Step 541. Start parameters for the fitting procedure are then extracted from the line section in Step 542 and the first fitting region is defined in Step 543. The method 500 then asks, in Step 544, whether the actual layer is step-like. If the answer to the question posed in Step 544 is negative, the method 500 asks in Step 545 whether another inner layer exists. If the answer to the question posed in Step 545 is positive, then the fit region is moved to the next inner layer in Step 546 and the method 500 returns to Step 544. Thus, a loop including Steps 544, 545, and 546 is created until the question posed in Step 544 is answered affirmatively or the question posed in Step 545 is answered negatively.

[0116] If the answer to the question posed in Step 545 is negative, then the method 500 asks, in Step 547, whether the line section is step-like. If the answer to the question posed in Step 547 is positive, then the measurement artifact compensated parameters and line section have been achieved, in Step 548, and the fitting procedure 540 ends. If the answer to the question posed in Step 547 is negative, then the measurement artifact compensated RIP is determined in Step 549 and the measurement artifact compensated line section is achieved in Step 550.

[0117] If the answer to the question posed in Step 544 is positive, then an analytical deflection curve of an axially symmetric RIP is generated that is intended to mimic the measurement artifact as an equivalent axially symmetric RIP in Step 551. The deflection is transformed to a deflection RIP in Step 552. Next, in Step 553, the accuracy of the deflection RIP is calculated in comparison to the initial line section. The accuracy may be measured by the R-squared statistic, which quantifies the predictive accuracy of a statistical model. The statistic shows the proportion of variance in the outcome variable that is explained by the predictions. It is also known as the coefficient of determination, R.sup.2, r.sup.2, and r-squares. R.sup.2 typically has a value in the range of 0 through to 1 (or 100%). A value of 1 indicates that predictions are identical to the observed values; it is not possible to have a value of R.sup.2 of more than 1. A value of 0 indicates that there is no linear relationship between the observed and predicted values, where linear in this context means that it is still possible that there is a non-linear relationship between the observed and predicted values. Finally, a value of 0.5 means that half of the variance in the outcome variable is explained by the model. Sometimes the R.sup.2 is presented as a percentage (e.g., 50%).

[0118] The method 500 then asks, in Step 554, whether certain stop criteria have been fulfilled (e.g., a specific R-squared value was calculated in Step 553). With each iteration of the fitting procedure, the variation of the initial parameters (n.sub.k und r.sub.k) becomes smaller and smaller. Once the variation becomes sufficiently small that it falls below a certain threshold, continuation of the fitting iterations is unnecessary. In practice, it does not make sense to do variations in n.sub.k smaller than, for example, 10.sup.5, because there are other intrinsic errors that limit accuracy. The same observation applies to the layer radii, r.sub.k.

[0119] If the answer to the question posed in Step 554 is negative, then the method 500 proceeds to Step 555 in which the actual fitting parameters n.sub.m and r.sub.m are varied before the method 500 returns to Step 551. Thus, a loop including Steps 551, 552, 553, 554, and 555 is created until the question posed in Step 554 is answered affirmatively. When the answer to the question posed in Step 554 is positive, then the method returns to Step 545.

[0120] The diagram of FIG. 11 shows, for a preform with a simple step profile, a prepared refractive index profile n(r) and a hypothetical refractive index profile n*(r) modeled by evaluation thereof. The refractive index is indicated as a relative value based on the refractive index of the index adjustment fluid (no=1.4587). In the diagram, the relative refractive index n(r)1.4587 is plotted versus the radius r (in millimeters).

[0121] The hypothetical refractive index profile n*(r) already depicts the refractive index profile of the preform to be expected in reality, or it is close to the refractive index profile. The hypothetical refractive index profile is based on the prepared refractive index profile n(r) and the orientation values derived from the profile, which, in turn, include estimation values for the refractive index and the radii from the non-measurable region.

[0122] Using Equation (1), a simulated deflection angle distribution (y) is produced from the hypothetical refractive index profile n*(r) in the next method step. The simulated deflection angle distribution *(y) obtained thereby is thus based on the assumption of a refractive index profile of the preform (namely the hypothetical refractive index profile n*(r)), which, in turn, after correction and evaluation of original measurement values, is derived from a prepared refractive index profile n(r).

[0123] A simulated refractive index profile n(r), plotted with this designation in FIG. 11, is obtained again by transforming the simulated deflection angle distribution (y) by numerical integration of equation (2). The profile has a rounded region 53 between a cladding region 52 and a core region 51. (As mentioned above, a core rod is often surrounded by an inner cladding layer of fluorine- or germanium-doped quartz glass and an outer cladding layer of undoped quartz glass in a preform.) Apart from this rounded region 53, the simulated refractive index n(r) is almost congruent with the prepared refractive index profile n(r). Considering that the assumed refractive index distribution n*(r) considerably differs therefrom, this is remarkable. The similarity is a hint that the assumptions underlying the hypothetical refractive index profile n*(r) are already very close to the real refractive index profile n(r) of the preform. That is, the hypothetical refractive index profile n*(r) in FIG. 11 reflects the real refractive index profile n(r) accurately or at least adequately accurately.

[0124] In practice, an exact match between the simulated refractive index profile n(r) and the prepared refractive index profile n(r) cannot be achieved. It is possible to achieve an adequately and arbitrarily accurate adaptation, however, by iteratively fitting the simulated refractive index profile n(r) to the prepared refractive index profile n(r). An alternative and generalized fitting procedure can be explained with reference to FIG. 12.

[0125] The alternative fitting procedure allows a more accurate evaluation of a RIP that is not fully step-like but has a present measurement artifact due to a sharp RI step from low to high (in the direction from outside to inside). In FIG. 12, certain relative refractive indexes n(r)1.4587 are plotted versus the radius r (in millimeters). The measured data 301 are depicted with an X. The RIP typical measurement artifact appears at r=22 mm, so that the reconstructed measurement differs from the real RIP inside that region. The goal of the fitting procedure is to determine the RI-step and its related artifact error, which can be used to correct the measured data 301 afterward. For that purpose and similar to the previously explained fitting procedure, a simulation helper 302 and the related simulation fit or fitting curve 303 are given. The simulation helper 302 is an assumed RIP, generated with the goal of estimating and covering the RI step at r=22 mm. The assumed RIP of the simulation helper 302 allows generation of a deflection function by using equation (1). Transforming this deflection function by using equation (2) leads to the fitting curve 303.

[0126] For a fixed range (e.g., 21 mm<|r|), the parameters of the simulation helper 302 are varied, so that the fitting curve 303 matches the measured data 301 around the RI step. The region must be small enough so that any inner RI gradients do not affect the fit, but also broad enough that this rounded curve shape can be fitted robustly enough. Depending on the visual feedback, a skilled artisan may tune the chosen region to improve the fitting procedure.

[0127] Adding the difference between the simulation helper 302 and the fitting curve 303 to the measured data 301 leads to the fit result 304. The result is more accurate than the measured RIP. For more complex preforms, this fitting procedure can be repeated as needed on every occurring RI step. In this case, the RIP of the simulation helper 302, the fitting curve 303, and fit result 304 are stitched together accordingly for each of the different regions.

[0128] Even in the particular case where the characterized preform itself does not have such a described RI step but with an RI of undoped fused silica near the outer surface, this method is also extremely useful for the following reason. Setups like apparatus 100 usually have an undoped fused silica measuring cell 9 because this material is homogeneously available and the absolute RI is well-known including its dispersion, temperature-dependency, and other characteristics. It is also state of the art to use the index adjustment fluid 10 to keep the occurring deflection angles small enough so that the apparatus 100 can capture them. If the RI of the index adjustment fluid 10 were above the measuring cell 9 and the RI at the outer region of the glass object 11, the measurement artifact would appear at the boundary of the measuring cell 9 and the index adjustment fluid 10 and would affect the whole inner region, too. The only way to avoid this artifact completely would be if the index adjustment fluid 10 perfectly matched the measuring cell 9 and the outer region of the glass object 11. Besides the practicality issue in general (due to the strong temperature dependency of liquids suitable as the index adjustment fluid 10), this would completely eliminate the boundaries, so that no differentiation between the glass object 9, the index adjustment fluid 10, and the measuring cell 11 would be possible.

EXAMPLES

[0129] The following examples are included to more clearly demonstrate the overall nature of the disclosure. These examples are exemplary, not restrictive, of the disclosure.

[0130] FIGS. 13-16 depict Example 1 for a preform having a single elliptical core. FIG. 13 is the sinogram created by applying the method 500, i.e., rotating the preform, taking measurements, and stacking the measurements left-to-right. Depicted is a single sinusoidal wave shape, reflecting the single core, before application of the Radon transformation. Note the vertical lines reflecting a first line section 310 and a second line section 312. The sinogram shows the magnitude of the deflection angle distribution (, ) in arbitrary units in the - plane.

[0131] FIG. 14 is a diagram of the relative refractive index n(x, y)1.446 in the x-y plane, illustrating the reconstructed 2D RIP of the preform having a single elliptical core. The first line section 310 is taken along the major axis (the longest diameter) of the ellipse between the two vertices of the ellipse. The second line section 312 is taken along the minor axis or shortest diameter of the ellipse.

[0132] Data taken along the second line section 312 drawn through the core of the diagram of FIG. 14 are presented in the graph of FIG. 15, illustrating the line section fitting procedure. The graph plots the relative refractive index n(r)1.446 against the position r in millimeters. FIG. 15 illustrates a main concept of the method 500. The graph of FIG. 15 includes three curves: a curve 320 through the picked, tomographic reconstructed line section data, the simulation fit curve 321, and the fit result (or assumed) curve 322. The assumed RI steps were 0.002 and 0.01, which can be extremely well evaluated with the method 500.

[0133] FIG. 16 highlights the need for a tomographic measurement and the evaluation method therein, using the first line section 310 and the second line section 312. The graph of FIG. 16 plots the relative refractive index n(r)1.446 against the position r in millimeters, for the preform having a single elliptical core, and compares the two different tomographic line sections versus the single related measurements. Curve 330 depicts the single measurement along the first line section 310; curve 332 depicts the single measurement along the second line section 312. Curve 334 depicts the tomography measurement along the first line section 310; curve 336 depicts the tomography measurement along the second line section 312. Thus, the method 500 can be compared to a straight-forward evaluation (a basic inverse Abel transformation). Depending on the orientation during the single measurements, the result can vary greatly. In contrast, the level of the RIP is independent of the chosen line sections. The fit functions and the resulting profiles of the method 500 are omitted to avoid overloading the graph.

[0134] FIGS. 17-20 depict Example 2 for a preform having a single core with an octagonal-shaped cladding which might be considered almost radii-symmetric. The core is made of fluorine-doped quartz glass sold under the registered trademark Fluosil by Heraeus Quarzglas GmbH & Co. KG of Germany. FIG. 17 is the sinogram created by applying the method 500, i.e., rotating the preform, taking measurements, and stacking the measurements left-to-right. Note the vertical lines reflecting a first line section 340 and a second line section 342. The sinogram shows the magnitude of the deflection angle distribution (, ) in arbitrary units in the - plane.

[0135] FIG. 18 is a diagram of the relative refractive index n(x, y)1.446 in the x-y plane, illustrating the reconstructed 2D RIP of the preform having an octagonal-shaped cladding. The first line section 340 is taken along the longer axis through opposing apexes of the octagonal-shaped cladding. The second line section 342 is taken along the shorter axis through opposing flat sections between apexes of the octagonal-shaped cladding.

[0136] Data taken along the line section 342 drawn through the core of the diagram of FIG. 18 are presented in the graph of FIG. 19, illustrating the line section fitting procedure. The graph plots the relative refractive index n(r)1.446 against the position r in millimeters. The graph of FIG. 19 includes four curves: a curve 350 through the measured line section data, the simulation fit curve 351, the fit result (or assumed) curve 352, and the single measurement reconstruction curve 353. The graph shows the merits of the method 500 and highlights the need for the tomographic evaluation upon comparison to the single measurement.

[0137] Data taken along the line section 340 drawn through the core of the diagram of FIG. 18 are presented in the graph of FIG. 20, illustrating the line section fitting procedure. The graph plots the relative refractive index n(r)1.446 against the position r in millimeters. The graph of FIG. 20 includes four curves: a curve 360 through the measured line section data, the simulation fit curve 361, the fit result (or assumed) curve 362, and the single measurement reconstruction curve 363. The graph shows the merits of the method 500 and highlights the need for the tomographic evaluation upon comparison to the single measurement.

[0138] FIGS. 21-24 depict Example 3 for an even more complex preform having seven cores. FIG. 21 is the sinogram created by applying the method 500, i.e., rotating the preform, taking measurements, and stacking the measurements left-to-right. Depicted are six sinusoidal wave shapes, one for each of the non-central cores. Note the vertical lines reflecting a first line section 370 and a second line section 372. The sinogram shows the magnitude of the deflection angle distribution (, ) in arbitrary units in the - plane.

[0139] FIG. 22 is a diagram of the relative refractive index n(x, y)1.4587 in the x-y plane, illustrating the reconstructed 2D RIP of the preform having seven cores. The first line section 370 is a shorter line section taken through the center core. The second line section 372 is a longer line section taken through the center core.

[0140] Data taken along the longer line section 372 drawn through the center core of the diagram of FIG. 22 are presented in the graph of FIG. 23, illustrating the line section fitting procedure. The graph plots the relative refractive index n(r)1.4587 against the position r in millimeters. The graph of FIG. 23 includes three curves: a curve 380 through the measured line section data, the simulation fit curve 381, and the fit result (or assumed) curve 382. The graph shows the merits of the method 500. The Radon transform and its inverse can be applied to transition between the sinogram of FIG. 21 and the graph of FIG. 23.

[0141] Data taken along the shorter line section 370 drawn through the center core of the diagram of FIG. 22 are presented in the graph of FIG. 24, illustrating the line section fitting procedure. The graph plots the relative refractive index n(r)1.4587 against the position r in millimeters. The graph of FIG. 24 includes three curves: a curve 384 through the measured line section data, the simulation fit curve 385, and the fit result (or assumed) curve 386. The graph shows the merits of the method 500.

[0142] FIGS. 25-28 depict Example 4 for a preform having six cores and without any center core. FIG. 25 is the sinogram created by applying the method 500, i.e., rotating the preform, taking measurements, and stacking the measurements left-to-right. Depicted are six sinusoidal wave shapes, one for each of the cores. Note the curved lines reflecting a first line section 390 and a second line section 392. The line sections 390, 392 shown in the sinogram of FIG. 25 appear curved because the line sections 390, 392 within the 2D RIP (see FIG. 26) do not go through the center. The sinogram shows the magnitude of the deflection angle distribution (, ) in arbitrary units in the - plane.

[0143] FIG. 26 is a diagram of the relative refractive index n(x, y)1.4587 in the x-y plane, illustrating the reconstructed 2D RIP of the preform having six cores. The first line section 390 is a shorter line section taken through one of the cores. The second line section 392 is a longer line section taken through another of the cores but including the preform boundaries at around 32 mm.

[0144] Data taken along the longer line section 392 are presented in the graph of FIG. 27, illustrating the line section fitting procedure. The graph plots the relative refractive index n(r)1.4587 against the position r in millimeters. The graph of FIG. 27 includes three curves: a curve 394 through the measured line section data, the simulation fit curve 395, and the fit result (or assumed) curve 396. The graph shows the merits of the method 500.

[0145] Data taken along the shorter line section 390 are presented in the graph of FIG. 28, illustrating the line section fitting procedure. The graph plots the relative refractive index n(r)1.4587 against the position r in millimeters. The graph of FIG. 28 includes three curves: a curve 397 through the measured line section data, the simulation fit curve 398, and the fit result (or assumed) curve 399. The graph shows the merits of the method 500.

[0146] The examples show that the calculated preform parameters, especially the refractive index steps, are much more accurate with the method 500. The method 500 is beneficial for customers of non-symmetrical preforms, including multicore preforms, hollow core fiber preforms, Panda-preforms, or preforms for fiber lasers with non-circular interfaces.

[0147] Although illustrated and described above with reference to certain specific embodiments and examples, the present disclosure is nevertheless not intended to be limited to the details shown. Rather, various modifications may be made in the details within the scope and range of equivalents of the claims and without departing from the spirit of the disclosure. It is expressly intended, for example, that all ranges broadly recited in this document include within their scope all narrower ranges which fall within the broader ranges.